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Teaching adjectives to young students can be a fun and engaging experience with the right resources.
This worksheet is designed specifically for class 2 students and includes a variety of activities to help them understand and use adjectives in their writing. With this worksheet, you can watch your students excel in their language skills.
What are adjectives?
Adjectives are words that describe or modify nouns or pronouns. They provide more information about the noun or pronoun, such as its size, color, shape, or personality. For example, in the sentence "The big, red apple," "big" and "red" are adjectives that describe the noun "apple." Adjectives are an important part of language and can help make writing more descriptive and interesting.
Examples of adjectives.
Adjectives can be used to describe a wide range of things, from people and animals to objects and places. Some common examples of adjectives include "happy," "sad," "tall," "short," "round," "square," "blue," "green," "loud," and "quiet." Encourage your class 2 students to come up with their own examples of adjectives and use them in sentences to practice their language skills.
How to use adjectives in sentences.
Adjectives are used to describe nouns or pronouns in a sentence. They provide more information about the noun or pronoun, such as its size, color, shape, or personality. To use adjectives in a sentence, simply place them before the noun or pronoun they are describing. For example, "The big, red apple" or "She is a kind, thoughtful person." Encourage your class 2 students to practice using adjectives in sentences by giving them prompts and asking them to come up with descriptive phrases.
Practice exercises for identifying and using adjectives.
This worksheet is designed to help class 2 students practice identifying and using adjectives in sentences. It includes a variety of exercises, such as matching adjectives to nouns, filling in the blanks with appropriate adjectives, and creating their own sentences using adjectives. By completing these exercises, students will gain a better understanding of how adjectives work and how they can be used to make their writing more descriptive and engaging.
Creative writing prompts using adjectives.
Using adjectives in creative writing can make your stories more vivid and engaging. Here are some prompts to get your class 2 students started:
1. Write a story about a magical forest using at least five descriptive adjectives.
2. Describe your favorite food using as many adjectives as possible.
3. Write a paragraph about a character in a book or movie, using at least three adjectives to describe their personality.
4. Imagine you are on a beach vacation. Write a paragraph describing the scenery using at least five adjectives.
5. Write a story about a superhero with a unique power. Use adjectives to describe their appearance and abilities.
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In conclusion, adjectives worksheet resources are essential for learners looking to develop a strong foundation in English grammar. By working with describing words worksheet materials and practicing with various adjective exercises, learners can improve their understanding of adjective usage and forms. With resources tailored to different grade levels and learning needs, such as adjective worksheet for class 2, learners can build the skills necessary for effective communication and language mastery.
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Equality and other comparisonsEdit
In the last chapter, we used the equals sign to define variables and functions in Haskell as in the following code:
r = 5
That means that the evaluation of the program replaces all occurrences of
5 (within the scope of the definition). Similarly, evaluating the code
f x = x + 3
replaces all occurrences of
f followed by a number (
f's argument) with that number plus three.
Mathematics also uses the equals sign in an important and subtly different way. For instance, consider this simple problem:
Example: Solve the following equation:
Our interest here isn't about representing the value as , or vice-versa. Instead, we read the equation as a proposition that some number gives 5 as result when added to 3. Solving the equation means finding which, if any, values of make that proposition true. In this example, elementary algebra tells us that (i.e. 2 is the number that will make the equation true, giving ).
Comparing values to see if they are equal is also useful in programming. In Haskell, such tests look just like an equation. Since the equals sign is already used for defining things, Haskell uses a double equals sign,
== instead. Enter our proposition above in GHCi:
Prelude> 2 + 3 == 5 True
GHCi returns "True" because is equal to 5. What if we use an equation that is not true?
Prelude> 7 + 3 == 5 False
Nice and coherent. Next, we will use our own functions in these tests. Let's try the function
f we mentioned at the start of the chapter:
Prelude> let f x = x + 3 Prelude> f 2 == 5 True
This works as expected because
f 2 evaluates to
2 + 3.
We can also compare two numerical values to see which one is larger. Haskell provides a number of tests including:
< (less than),
> (greater than),
<= (less than or equal to) and
>= (greater than or equal to). These tests work comparably to
== (equal to). For example, we could use
< alongside the
area function from the previous module to see whether a circle of a certain radius would have an area smaller than some value.
Prelude> let area r = pi * r ^ 2 Prelude> area 5 < 50 False
What is actually going on when GHCi determines whether these arithmetical propositions are true or false? Consider a different but related issue. If we enter an arithmetical expression in GHCi the expression gets evaluated, and the resulting numerical value is displayed on the screen:
Prelude> 2 + 2 4
If we replace the arithmetical expression with an equality comparison, something similar seems to happen:
Prelude> 2 == 2 True
Whereas the "4" returned earlier is a number which represents some kind of count, quantity, etc., "True" is a value that stands for the truth of a proposition. Such values are called truth values, or boolean values. Naturally, only two possible boolean values exist:
Introduction to typesEdit
False are real values, not just an analogy. Boolean values have the same status as numerical values in Haskell, and you can manipulate them in similar ways. One trivial example:
Prelude> True == True True Prelude> True == False False
True is indeed equal to
True is not equal to
False. Now: can you answer whether
2 is equal to
Prelude> 2 == True <interactive>:1:0: No instance for (Num Bool) arising from the literal ‘2’ at <interactive>:1:0 Possible fix: add an instance declaration for (Num Bool) In the first argument of ‘(==)’, namely ‘2’ In the expression: 2 == True In an equation for ‘it’: it = 2 == True
Error! The question just does not make sense. We cannot compare a number with a non-number or a boolean with a non-boolean. Haskell incorporates that notion, and the ugly error message complains about this. Ignoring much of the clutter, the message says that there was a number (
Num) on the left side of the
==, and so some kind of number was expected on the right side; however, a boolean value (
Bool) is not a number, and so the equality test failed.
So, values have types, and these types define limits to what we can or cannot do with the values.
False are values of type
2 is complicated because there are many different types of numbers, so we will defer that explanation until later. Overall, types provide great power because they regulate the behavior of values with rules that make sense, making it easier to write programs that work correctly. We will come back to the topic of types many times as they are very important to Haskell.
An equality test like
2 == 2 is an expression just like
2 + 2; it evaluates to a value in pretty much the same way. The ugly error message we got on the previous example even says so:
In the expression: 2 == True
When we type
2 == 2 in the prompt and GHCi "answers"
True, it is simply evaluating an expression. In fact,
== is itself a function which takes two arguments (which are the left side and the right side of the equality test), but the syntax is notable: Haskell allows two-argument functions to be written as infix operators placed between their arguments. When the function name uses only non-alphanumeric characters, this infix approach is the common use case. If you wish to use such a function in the "standard" way (writing the function name before the arguments, as a prefix operator) the function name must be enclosed in parentheses. So the following expressions are completely equivalent:
Prelude> 4 + 9 == 13 True Prelude> (==) (4 + 9) 13 True
Thus, we see how
(==) works as a function similarly to
areaRect from the previous module. The same considerations apply to the other relational operators we mentioned (
>=) and to the arithmetical operators (
*, etc.) – all are functions that take two arguments and are normally written as infix operators.
In general, we can say that tangible things in Haskell are either values or functions.
Haskell provides three basic functions for further manipulation of truth values as in logic propositions:
(&&)performs the and operation. Given two boolean values, it evaluates to
Trueif both the first and the second are
True, and to
Prelude> (3 < 8) && (False == False) True Prelude> (&&) (6 <= 5) (1 == 1) False
(||)performs the or operation. Given two boolean values, it evaluates to
Trueif at least one of them is
Prelude> (2 + 2 == 5) || (2 > 0) True Prelude> (||) (18 == 17) (9 >= 11) False
notperforms the negation of a boolean value; that is, it converts
Prelude> not (5 * 2 == 10) False
Haskell libraries already include the relational operator function
(/=) for not equal to, but we could easily implement it ourselves as:
x /= y = not (x == y)
Note that we can write operators infix even when defining them. Completely new operators can also be created out of ASCII symbols (which means mostly the common symbols used on a keyboard).
Haskell programs often use boolean operators in convenient and abbreviated syntax. When the same logic is written in alternative styles, we call this syntactic sugar because it sweetens the code from the human perspective. We'll start with guards, a feature that relies on boolean values and allows us to write simple but powerful functions.
Let's implement the absolute value function. The absolute value of a real number is the number with its sign discarded; so if the number is negative (that is, smaller than zero) the sign is inverted; otherwise it remains unchanged. We could write the definition as:
Here, the actual expression to be used for calculating depends on a set of propositions made about . If is true, then we use the first expression, but if is the case, then we use the second expression instead. To express this decision process in Haskell using guards, the implementation could look like this:
Example: The absolute value function.
absolute x | x < 0 = -x | otherwise = x
Remarkably, the above code is about as readable as the corresponding mathematical definition. Let us dissect the components of the definition:
- We start just like a normal function definition, providing a name for the function,
absolute, and saying it will take a single argument, which we will name
- Instead of just following with the
=and the right-hand side of the definition, we enter the two alternatives placed below on separate lines. These alternatives are the guards proper. Note that the whitespace (the indentation of the second and third lines) is not just for aesthetic reasons; it is necessary for the code to be parsed correctly.
- Each of the guards begins with a pipe character,
|. After the pipe, we put an expression which evaluates to a boolean (also called a boolean condition or a predicate), which is followed by the rest of the definition. The function only uses the equals sign and the right-hand side from a line if the predicate evaluates to
otherwisecase is used when none of the preceding predicates evaluate to
True. In this case, if
xis not smaller than zero, it must be greater than or equal to zero, so the final predicate could have just as easily been
x >= 0; but
otherwiseworks just as well.
where and GuardsEdit
where clauses work well along with guards. For instance, consider a function which computes the number of (real) solutions for a quadratic equation, :
numOfRealSolutions a b c | disc > 0 = 2 | disc == 0 = 1 | otherwise = 0 where disc = b^2 - 4*a*c
where definition is within the scope of all of the guards, sparing us from repeating the expression for
- The term is a tribute to the mathematician and philosopher George Boole.
- This function is already provided by Haskell with the name
abs, so in a real-world situation you don't need to provide an implementation yourself.
- We could have joined the lines and written everything in a single line, but it would be less readable.
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Graphing InequalitiesTutorials you should read before this one: Graphing Real Numbers on a Number Line.
A brief explanation
Up until now, we have only had to deal with the equality symbol (=). But what happens when two things are not equal? They can still be described in relationship to one another (compared), and to do that, we use inequality symbols.
Graphing Inequalities Basics
There are five basic inequality symbols: < , , > , and
When graphing an inequality, aside from drawing the number line itself, there are three steps you should follow.
- Read your inequality (correctly!).
- Decide whether to use an open () or closed circle ().
- Finally, figure out which direction you need to shade in.
When I introduced the five basic inequality symbols, notice that I refrained from labeling them. The reason for that is because most students are taught that this symbol (>) automatically means "greater than." This instant labeling of a symbol gets confusing when problems are not written in the normal direction, such as 4 > x. To always graph correctly, you must first read your inequality starting from the variable. The good news is you can still use the old "Alligator Mouth" memory trick. If the mouth is facing your variable, it's greater than. If the tail is facing your variable, it's less than. But only if you always read beginning with the variable! You will understand what I mean after a couple of examples.
Open and Closed Circles
If your symbol has the "equal to option," which is that single line underneath the greater than/less than symbol, then you use a closed circle. The closed circle is only used for the or symbols.
The open circle is used for >, < and
Shading your number line
Less than shades to the Left (they both start with L) because the "lesser" numbers are to the left. Greater than shades to the right, because the further right you go, the "greater" the numbers become.
Sometimes students think that they can always use this memory trick: since inequality symbols look like the arrow ends on a number line, they think they can always shade towards the number line arrow that looks like the inequality symbol. (If you don't know what I'm talking about, good!). DON'T DO THIS. Again, it will get you into trouble for those backwards problems.
|x > -2||We read this inequality as x is "greater than" negative 2. (The mouth is facing the x).|
|Since our inequality does NOT have the equal to option, we use an open circle.|
|Since our inequality reads as "greater than," we shade to the right.|
Always read your inequality starting from the variable.
|4 x||This reads as "x is less than or equal to 4" because this time, our tail is facing our variable.|
|Our inequality symbol does have the equal to option, so we use a closed circle.|
|Less than always shades to the left.|
|-0.3 < x||x is greater than -0.362|
|-0.362 lies between 0 and -1, open circle.|
|Greater than shades to right.|
|This time we have a fraction. Our inequality reads as "x is less than -7/2."|
Remember, convert your fraction to its decimal equivalent = -3.5 (Divide -7 by 2).
-3.5 lies between -3 and -4 on our number line.
|Open circle since our inequality does NOT have the equal to option.|
|Less than shades to left.|
|x -1||This reads as x is not equal to -1.|
|Open circle since -1 is our end number.|
|This time, we shade both directions, because all the numbers that are not equal to -1 are to the left (where the numbers that are less than -1 are located) and to the right (numbers that are greater than -1).|
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After exploring basic probability concepts, students take on the role of game designers to design a fair game for a toy company. They describe the rules for play, explain how probability affects the fairness of the game, and present their game to the toy company’s board of directors trying to persuade them to sell their game.
View how a variety of student-centered assessments are used in the What are the Chances? Unit Plan. These assessments help students and teachers set goals; monitor student progress; provide feedback; assess thinking, processes, performances, products; and reflect on learning throughout the learning cycle.
This unit of study makes use of the Visual Ranking Tool. Examine the Visual Ranking Tool as you plan instruction to learn about the tool and how to use it with your students.
Ask students if they have ever been in a situation where they had bad luck or good luck. Pose the Essential Question: What's fair? Break students into small groups and have them discuss the Essential Question and record their initial responses. Encourage them to talk about why they think life is fair or unfair, as well as what they mean by fair and what luck has to do with fairness. Ask several students to share their responses to the Essential Question and then tell them that they will begin a unit on probability. In this unit they design a fair game and learn how to use probability to determine how to increase their chance of winning. Introduce students to a learning log. The learning log is used to assess student thinking and give them an opportunity to reflect on activities and important questions.
To address the Content Question, What is probability? introduce the idea of probability by discussing the likelihood of events occurring. Encourage students to focus on the language of probability as they use their life experiences to recall events that are certain, impossible, likely, and unlikely to happen. Record these events and introduce students to a probability scale, ranging from zero to one.
Bring in a variety of mathematical-based games. Discuss the rules of each game and brainstorm a list of characteristics that make the game fair. Post these in the classroom to refer back to later in the unit.
1. Overview of activity:
Students use the probability scale to determine how likely an event is to occur. They use prior knowledge to make inferences about the likelihood of an event.
2. Materials needed:
3. Activity procedures:
Stand ten feet away from the trash can and hold the ball of paper in your hand. Ask the students, “What is the likelihood that I will be able to throw the paper into the trash can on my first try?” Focus the discussion on vocabulary such as likely, unlikely, probably, maybe, certain, impossible, and highly unlikely.
Revisit the term “probability” with the class and review its meaning. Probability can be defined as the chance of an event occurring. Ask students to name the instances that they have heard the term used in their everyday lives.
On the board, list the words likely and unlikely. Ask each student if they think it is likely or unlikely you will make the basket and tally their responses. Throw the ball of paper into the trash can. Have a discussion related to the outcome of your throw and if you threw the ball of paper again, Would it result in the same outcome? Does the probability of it going in the trash can increase or decrease each time?
To discuss the Unit Question, How can you measure the likelihood of an event? tell the class that probability can be expressed on a probability scale. Explain to the students that we will examine how you measure likelihood by using this probability scale. Place a long piece of ribbon or string on the floor representing the scale. Ask the students to name a number that would best represent an event that is impossible (0). Choose a student to stand at this position on the scale and hold a card marked, “0 IMPOSSIBLE”. Ask students to name events that are impossible, for example: there will be 12 hours in the day tomorrow, there will be 13 months in the year next year. List student responses on a piece of chart paper.
Now ask students to name a number that would best represent an event that is certain (1), for example: there will be 24 hours in the day tomorrow, there will be 60 minutes in the next hour. Encourage students to name events that are certain and record their responses. Mark 1/2 on the scale and have a student stand halfway between 0 and 1 and hold the card “1/2”. Ask the students to make a prediction about the weather for tomorrow and come up and stand on the position on the walk-on probability scale that best represents the likelihood of their weather prediction coming true. Students need to explain their reasons for standing at a particular spot.
As a check for understanding, have students create their own graphic organizers, making a probability scale and putting events at designated places along the scale. Also, have students reflect on the following Unit Question in their learning logs: How do you measure the likelihood of an event?
Before proceeding with the next activity, click here to set up the What Are the Chances? project in your workspace.
Introduce the Visual Ranking Tool using the demonstration space at Try the Tool. Show students how to rank and compare lists, and how to describe items and explain their relative merit using the comments feature.
The following questions are addressed in the Visual Ranking activity:
Students prioritize and rank the likelihood of certain personal events. The tool activity should spark lively discussions among group members and apply criteria to evaluate the lists.
Have students log in to their Visual Ranking team space. Review with students the prompt for this project: Which of the events are more likely to occur and why? Rank the following events with the one that is most likely to occur on top. As students rank their events remind them to explain their reasoning for each item by using the comments feature of the tool. As students sort their lists, listen to their discussions and ask questions to help groups negotiate, make choices, and express their thinking. Questions such as the following can prompt students to elaborate on their thinking:
After students finish the exercise, have them compare their lists with the lists that were ranked by the other student groups. Direct students to read each other’s comments about the relative merit of each factor. Have students discuss why their lists are alike and different. Suggest that they identify the groups that ranked items most and least like they did. Have similar and dissimilar groups meet to discuss their rankings and rationale behind the order. Encourage groups to revise their thinking based on the things they learn from other groups.
The Visual Ranking Tool space below represents one team’s ranking on this project. The view you see is functional. You can roll over the white icon to see the group’s comments and click the compare button to see how different groups ranked the items.
Project Name: What are the Chances? (Click here to set up this project in your workspace)
Question: Which of the events are more likely to occur and why?
Explore an interactive demo.
Using a projector system and networked computer display the lists and discuss general themes that appear. Ask students to consider: Is any factor consistently in the top of the ranking? At the bottom of the ranking? How is where the factors are located (at the top or bottom) related to their degree of likelihood?
Students have gained some understanding of how likely or unlikely an event is to occur. To gauge prior knowledge, have students respond in their learning log to the questions, How can you be sure you’ve placed the factors in the correct order? and, How do you think you measure the likelihood of an event? Review the entries and differentiate instruction based on student responses.
Tell students that today’s lesson will focus on the exploration of the Content Question: How do you predict probable outcomes? In this activity students will be making inferences to predict outcomes and drawing conclusions about possible results. Create a spinner like the example below (you can use card stock, a brad (brass tack) and a paper clip) and ask them to name all of the possible outcomes (green, red, blue). Have students predict what color the spinner is most likely to land on and justify their responses.
Put students into groups of three or four and give each group a “secret spinner” (that you have created) in an envelope. Tell the students these are NOT to be shared with other groups, as they are top secret. Give each student a copy of the Secret Spinner handout.
Examples of secret spinners:
After predicting the results of their upcoming experiment, instruct the students to spin the spinner 30 times and keep track of their results on a frequency table.
Each group then shares the results of their probability experiment with the rest of the class. Based on the data presented, the class will predict what they think the group’s secret spinner looked like. The group then reveals their secret spinner for the class to see. Discuss the results.
Ask students to conduct a think-pair-share to address the following question: How do you predict probable outcomes on such things as spinners?
Collect the Secret Spinner handouts. Have students respond in their learning logs to the following questions; How do you predict probable outcomes? and How can understanding probable outcomes help you change your luck? Review the entries to assess students' acquisition of key concepts and modify instruction as necessary.
To reinforce inferring skills and allow students to experience another probability lesson, conduct the following activity. Introduce the activity by displaying a large square number cube that has the numbers one to six. Ask students to name all of the possible outcomes that they could roll using the number cube (1,2,3,4,5,6). Ask students to determine if one number has a better chance than another when rolled and to explain their reasoning. Tell students that they will continue with the exploration of the Content Question, “How do you predict probable outcomes?” by examining dice in today’s lesson.
Use the computer software program at What are your Chances?*
This program simulates the rolling of a number cube. Display the software program (see Internet resources) so that students can look at 1,000 rolls of a number cube and analyze the results.
Then ask what the possible sums are if the two dice are rolled. To tap prior knowledge, present the following scenario:
Mario and Amanda are playing a dice game. Each time the dice are rolled, they find the sum of the dots. Mario gets a point every time a 10 is rolled. Amanda gets a point every time an 8 is rolled. Mario thinks he will win because he predicts a 10 will occur most often. Amanda disagrees and thinks she will win because she thinks an 8 will occur most often.
Ask students to write in their learning log whether they agree with Mario or Amanda or if they think some other sum will occur most often. Ask them to give their reasoning for their prediction.
Then have students work in groups to investigate the chances for rolling a particular sum. Have each person in the group create a number line for the possible sums (2,3,4,5,6,7,8,9,10,11,12) and place “x’s” each time the sum is rolled. Have students roll the dice 15 times. Create a classroom frequency distribution graph (a number line with the “x’s” to represent how many times each sum occurred). Ask students to compare their own group data to the whole class data. Ask students if the chances are the same for all of the sums, and if not, which ones are more likely to occur and which ones are least likely to occur.
Introduce students to the idea that a table can be a useful tool in showing the possible outcomes (mathematically) of the sums of two dice. After modeling how to fill in the table, have students complete it:
Ask students the following questions as you circulate through the room making observations and taking notes:
Introduce the probability game, Is this Game Fair?
Ivan and Rhonda found some chips with odd markings and decided to make up a game using them. They played the game a few times, but Rhonda said it wasn’t fair. Play their game and then decide if you think it is fair (each player has an equal chance of winning).
1 chip with an A side and a B side
1 chip with an A side and a C side
1 chip with a B side and a C side
Flip all 3 chips at once.
Ivan gets a point if there is a match.
Rhonda gets a point if there is no match.
Break students into pairs and have them play the game and tally the points in a T-chart. Ask each pair to share their tallies with the whole class and record them on a large sheet of paper. The students will decide that the game is unfair after seeing the class results. Then combine the pairs of students into groups of four or five and ask them to make up a fair game using these three chips so that Ivan and Rhonda would agree that they each have an equal chance of winning. Have the groups share their revised games with the whole class.
Now that students have had additional experiences with probability and gained new knowledge, have them revisit these questions in their learning log:
Students apply what they have learned as they take on the role of game designers to create a new game for children ages eight through ten. Create an environment that fosters cooperation and decision making, by inviting local business owners to share in the product development process and by having students give feedback to one another. Providing opportunities for students to receive input from others will allow them to investigate alternatives they may not have considered themselves. Each team of designers creates a game using spinners, number cubes, or chips, and
Hand out and discuss the project rubric and the student checklist. Check for student understanding of project expectations and make sure students are using the checklist to guide the creation of the project.
Once students have designed and tested their games they need to create a multimedia presentation that will explain their game to the audience at Game Night. In the presentation students address the Curriculum-Framing Questions, How does probability affect fairness?, How can you measure the likelihood of an event?, and What's fair?
To help students with the planning and implementing of their game idea and multimedia presentation remind them to use the student checklist to monitor their progress and the project rubric to assess their work. Check for student understanding of project expectations and make sure students are using the checklist to guide the creation of the project. To help students become self-directed learners, pose the following questions to guide their work:
Invite parents, school faculty, and the toy and business representatives to attend a Game Night to recognize student work and learning. Students present their slideshows to the participants and then have time to play the games. Ask guests to give students feedback about their game.
Return to the Essential Question: What's fair? Ask students to think about how they responded to the question at the beginning of the unit. Have them write their thoughts about fairness, chance, and probability in their learning logs. Encourage them to write about what they have learned about these things over the course of the unit and to provide as much detail and examples as possible. Have students complete the self-reflection about their work on the project.
A teacher contributed this idea for a classroom project. A team of educators expanded the plan into the example you see here.
Grade Level: 3-5
Topics: Probability, Statistics
Higher-Order Thinking Skills: Implementation, Prediction
Key Learnings: Degrees of Likelihood; Predicting Skills; Understanding Probability; Determining Fairness
Time Needed: Seven 45-minute lessons
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http://www.intel.com/content/www/us/en/education/k12/thinking-tools/visual-ranking/examples/unit-plans/what-chances.html
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<urn:uuid:42464385-9d2e-472a-a946-f486df439c8a>
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Teaching the order of operations is a fundamental skill that students must learn in mathematics. Without an understanding of the order of operations, students will struggle with more advanced concepts in mathematics when they move through the grades.
Teaching the Order of Operations Rule (PEMDAS or GEMDAS)
The order of operations, also known as the PEMDAS rule, is a set of rules that determine the order in which mathematical operations should be performed. The PEDMAS rule stands for Parentheses, Exponents, Division, Multiplication, Addition, and Subtraction.
Using Real-Life Examples for Teaching the Order of Operations
When teaching the order of operations, it is important to start with the basics. Begin by explaining what the PEDMAS rule stands for and why it is important. Use real-life examples to demonstrate how the rule is used in everyday situations. One of the best real life situations for order of operations is budgeting.
I like to explain to students when planning a budget, you need to be able to perform mathematical operations in the correct order. For example, if you have a total income of $1,000 and want to save 20% of it, you would first multiply $1,000 by 0.20 to find the amount you need to save: $1,000 x 0.20 = $200. You would then subtract $200 from $1,000 to find the amount you have left for expenses: $1,000 – $200 = $800.
Examples of Mathematical Expressions
Once students have a basic understanding of the PEDMAS rule, introduce them to examples of mathematical expressions that require the rule to be applied. For example, consider the expression 3 + 5 x 2. Starting with simple problems will make it easier for them to go back if they make an error, and then progress to more difficult expressions.
Using a Variety of Examples and Practice Problems
It is important to use a variety of examples when teaching the order of operations. Use simple examples at first and gradually increase the complexity of the expressions. Encourage students to use mental math whenever possible, as this will help them to develop their problem-solving skills. In addition to using examples, it is also important to provide students with practice problems.
Games and Activities
One helpful way to teach the order of operations is to use games or activities. For example, create a game in which students have to solve a series of expressions using the PEDMAS rule. Alternatively, use a group activity in which students work together to solve a complex expression, with each student responsible for one or two operations. Click here to see all of the no prep resources I have for the order of operations.
Emphasizing the Importance of Following the Rule
When teaching the order of operations, it is important to emphasize the importance of following the rule. Remind students that the order of operations is not optional and that failing to follow the rule can lead to incorrect answers.
Encourage students to take their time when solving expressions and to double-check their work to make sure they followed the steps properly. This will relate back to real-life examples because if they calculate something wrong, especially with finances it could be a mistake that causes them to be short on money.
Assessing Students’ Understanding
Finally, it is important to assess students’ understanding of the PEDMAS rule. Use quizzes, tests, or other assessment methods to evaluate their knowledge and provide feedback on their progress. Encourage students to ask questions if they are unsure of anything, and be prepared to provide additional support or guidance as needed. This can be a difficult skill for them because it’s different than what they’re most likely used to, so it will require some reteaching with some students.
Teaching the order of operations is a critical skill that students must learn in mathematics. By using a variety of examples, practice problems, and games, students can develop a strong understanding of the PEDMAS rule and its importance in mathematics. By emphasizing the importance of following the rule and providing feedback on students’ work, teachers can help their students to develop strong problem-solving skills and achieve success in mathematics.
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https://thatonecheerfulclassroom.com/7-best-tips-for-teaching-the-order-of-operations/
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<urn:uuid:40354063-240f-4663-93fd-94cc9ad985f9>
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Back to: COMPUTER SCIENCE SS2
Welcome to Class !!
We are eager to have you join us !!
In today’s Computer Science class, We will continue learning about programming in BASIC. We hope you enjoy the class!
There are cases where the precedence rule may cause a problem. For example:
The BASIC expression LET Y = A/B+ C would produce undesired results because A would be divided by B and the result added to C. The solution to this problem is in the use of parentheses. If we let parentheses override the order of precedence (but maintain the order of precedence within the parentheses), the result will be satisfactory. Now let’s examine the previous example using parentheses.
With the parentheses, B is added to C and the sum of this operation is divided into A, giving the correct result.
Sometimes more than one set of parentheses may be needed to tell the BASIC language in what order to execute the arithmetic operations.
Parentheses inside parentheses
The BASIC expression to accomplish this would be: Let A+B/2^2
For this expression to give us the correct results, A and B must be added first, then the sum divided by C, and finally that result is squared. When parentheses within parentheses are used, the innermost parentheses will be evaluated first. The addition has lower precedence than either division or exponentiation; therefore, A + B must be in the inner parentheses. The division has lower precedence than exponentiation, so (A+B)/C also must be enclosed in parentheses, ((A+B)/C), to ensure it is performed next.
The important thing to remember is the parentheses may be used to override the normal order of precedence.
The parentheses rule says:
- Computations inside parentheses are performed first.
- If there are parentheses inside parentheses, the operations inside the inner pair are performed first.
Simple problems can be solved with BASIC by using only two or three instructions. These are the END, PRINT and LET statements. To use these effectively, you must know how they work and what rules must be followed in using them.
The END statement, which must be the last statement in every BASIC program, has two functions. It indicates to the compiler that there are no more BASIC statements for it to translate and it terminates execution of the program.
The PRINT statement is used to instruct the computer to output something either on the terminal or the printer. The standard print line in BASIC is divided into print zones or fields of 16 spaces each.
The two punctuation marks used in PRINT statements are the comma and semicolon. A comma used as a separator in a PRINT statement causes standard spacing and a semicolon causes packed spacing.
Any Information enclosed in quotation marks in a PRINT statement will be printed exactly as it appears in the program.
The LET statement can be used to assign a constant value to a variable name, a variable to a variable name, or the results of an expression to a variable name. The equal sign in a LET statement does not indicate algebraic equality, rather it means to be assigned the value of. The value assigned by a LET statement is stored in the computer’s memory; therefore, it can be referenced by its variable name.
Both the PRINT and LET statements may contain expressions with arithmetic operations. These arithmetic operations must be specified by the appropriate operation symbol. Should you forget to include the symbol, the computer will not insert it for you but will give you an error message.
Constants and variables are used to refer to numeric values or character strings. A constant is a whole or decimal number or character string whose value does not change. A variable name is an arbitrary name you select and you and the computer used to refer to a value stored in the computer’s memory. This value may vary during the execution of the program but can contain only one value at a time.
- Define a ‘function’?
- What does the BASIC acronym represent in Programming?
- State the parenthesis rule.
We have come to the end of this class. We do hope you enjoyed the class?
Should you have any further question, feel free to ask in the comment section below and trust us to respond as soon as possible.
In our next class, we will be learning about Computer Maintainance. We are very much eager to meet you there.Your Opinion Matters! Quickly tell us how to improve your Learning Experience HERE
Pass WAEC, JAMB, NECO, BECE In One Sitting CLICK HERE!
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https://classnotes.ng/lesson/introduction-to-basic-programming-parenthesis-computer-science-ss2/
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<urn:uuid:01c529da-3fb6-4871-97ad-74c1d4cab3bc>
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This powerful KS2 grammar resources pack provides everything you need to teach a series of five lessons on adjectives, culminating in an extended writing task where children can use their grammatical understanding in context.
This primary resource pack includes:
- Adjectives challenge sheet
- Noun cards
- Writing plan
- Uplevelling writing worksheet
- Teacher’s notes
What is an adjective?
An adjective is a word naming an attribute of a noun. Or, more plainly, it is a word that describes a person or thing. Adjectives are words that modify other words to make language more interesting or specific.
The National Curriculum guidance offers the following advice:
“The surest way to identify adjectives is by the ways they can be used:
- before a noun, to make the noun’s meaning more specific (ie to modify the noun), or
- after the verb be, as its complement.
Adjectives cannot be modified by other adjectives. This distinguishes them from nouns, which can be. Adjectives are sometimes called ‘describing words’ because they pick out single characteristics such as size or colour. This is often true, but it doesn’t help to distinguish adjectives from other word classes because verbs, nouns and adverbs can do the same thing.”
National Curriculum English programmes of study links
Pupils should be taught to use the grammar for year 2 in English Appendix 2, and to use and understand the grammatical terminology in English Appendix 2 in discussing their writing.
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https://www.plazoom.com/resource/grammar-burst-y2-adjectives/
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Practice with the Circle Formula:
Once students have derived the formula for a circle using the Pythagorean Theorem, teachers can help students work through the first two examples of page two of the lesson. The first example asks students to write the equation of a circle given a point and a radius, while the second question asks students to find the formula given two points. This will require students to use a prior skill of calculating length using the distance formula. Teachers should give students time to talk through with a partner how to solve this question before telling them to use the distance formula, a graphical representations can help for students to visualize this.
We will then ask students to graph a circle when given the formula. Example 3 can be tricky for students since there are no h and k term for students.
Finishing Class notes:
Page 4 of notes asks students to find the area and circumference of a circle when given the formula of the circle. There is a review activity which digs deeper into the idea and differences behind perimeter/circumference and area. This review may not be necessary for students who have a strong understanding of these topics, and could be kept in the notes for classes who could use a reinforcement of these topics.
The last part of the lesson includes a host of practice questions for students to apply their knowledge. If teachers have a chance, they can review questions #8 and #9 with students since question #8 asks students to write the equation of a circle, and the other ask students to graph a circle when given the equation of a circle.
The exit ticket for this lesson asks students to determine the radius and center for a given circle, and also to write the equation of a circle when given the center and radius length.
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https://betterlesson.com/lesson/442617/finding-the-circle-formula?from=mtp_lesson
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<urn:uuid:6c258bce-0cae-471b-8290-ad2256738081>
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Was That Fair?
The teacher poses several questions to spark conversation and critical thinking about the meaning of fairness. Students work together to create a definition of fairness.
The learner will:
- respond in discussion to a scenario about a fairness issue.
- give a personal non-verbal response to several questions.
- discuss and come to consensus on a definition for fairness.
Talk to friends and family about the meaning and examples of fairness.
Give the following scenario: At a high school, a girl named Stephanie wants to take shop class because she would like to build things with wood and tools. The counselor smiled when Stephanie asked to sign up for shop and said, "You don't want to take that class; you'll be the only girl in the class. Let's get you into an art class." The school does not have a rule that prohibits girls from taking the class.
Discuss the scenario: Was the counselor being fair? Why or why not? Why do you think the counselor discouraged Stephanie from taking the class?What do you think Stephanie should do? Why do you think this decision could be difficult for Stephanie? Can you think of a time when rules were fair but the practice wasn't fair? (Example: In 1870, the 15th Amendment granted men of color the right to vote, but many were still kept from voting.)
Ask the students to respond to some questions about fairness by giving a thumbs-up if they answer "yes," thumbs-down if they answer "no," and palm flat for "I don't know." After each question, discuss student responses. (Note: some of these are fair, but will create discussion about variations of these that are not fair.) Questions:
- Is it fair to give one student a B and another student an A?
- Is it fair to ask one member of a family to do a chore and not another member of the family?
- Is it fair to let boys and not girls play basketball on a team?
- Is it fair to hire a man for a job that a woman also applied for?
- Is it fair to tell a black man he must give up his seat for a white woman on a crowded bus?
- Is it fair to tell a bus passenger to give up his seat for a person with a cane?
- Is it fair for a teacher to ignore a tardy when someone is late for class?
- Is it fair for an employer to pay a woman more than a man for the same job?
- Is it fair to force someone to work for no pay if they don't get caught?
- Is it possible for unequal treatment to be fair?
- Is it possible for equal treatment to be unfair?
Have the students write down what it means to be fair and then share their ideas with the class. Discuss and develop a class definition of fairness. Save the definition for the next lesson.
Strand PHIL.I Definitions of Philanthropy
Standard DP 02. Roles of Government, Business, and Philanthropy
Benchmark MS.2 Give examples of needs not met by the government, business, or family sectors.
Strand PHIL.II Philanthropy and Civil Society
Standard PCS 02. Diverse Cultures
Benchmark MS.5 Discuss examples of groups denied their rights in history.
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https://www.learningtogive.org/units/character-education-fairness-grade-7/was-fair
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<urn:uuid:0ed115a7-3059-4f09-804b-fd627c420109>
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It's not important for National Curriculum Computing at KS3 that you understand how to create a program to draw a circle, but, if you're interested, here are two techniques explained. GCSE Maths now includes trigonometry and Pythagoras in both tiers, so these techniques should be suitable for students of GCSE Computer Science.
GCSE Maths students will have been introduced to trigonometry - sines and cosines - for their Maths exam. If you don't know what sines and cosines are, or you would like a reminder of what they are, check out the trigonometry page.
This method of drawing a circle works by varying the angle n and using trigonometry to work out where the point (x,y) would be for that angle. As you could hopefully see from the animation, for a circle of radius 1, x would be cos(n) and y would be sin(n). For a larger circle, you just need to multiply by the radius. The program draws a circle of radius 400, so the x-coordinate is 400 x cos(n), and the y-coordinate is 400 x sin(n).
There is, however, one slight complication. Unlike your calculator, most programming languages (and applications such as Excel) measure angles in radians, rather than degrees. The program therefore uses the RAD() function to convert the angle from degrees to radians before calculating the sine or cosine.
You might find the Maths for this method more straightforward - and so does the computer! Square roots are quicker for the computer to calculate than sines and cosines, which explains why this method is quicker.
Pythagoras works with right-angled triangles. Imagine a triangle with one corner in the centre of the circle, and another on the circumference. The hypotenuse (the longest side) will always be equal in length to the radius.
This method of drawing the circle varies the x-position. From the x-position, we know the width of the triangle, and we know the length of the hypotenuse, so we can use Pythagoras to work out the height of the triangle to give us the y-position.
If we assume that (0, 0) is in the middle of the circle, then the value of x is the width of the triangle - it might be negative, but as we're going to square it, it doesn't matter. If the hypotenuse is 400, then y2 = 4002 - x2, or y is the square root of (160000 - x2).
More recently I've discovered that you can use randomness to plot points on the circumference of a circle - the method is described on Guilherme Kunigami's blog, Circles and Randomness and is included my my file of BBC BASIC programs (see my blog In Praise of Slowness).
This method has the benefit of only using arithmetic - no sines, cosines, or square roots - but the downside is that the points on the circumference aren't joined, so you need more of them to give the appearance of a full circle.
I've created a version of the random circle using Python with Turtle, but you could even do it in any programming language that allows you to plot points, including Scratch.
Random numbers can be used for all sorts of things, including using the Monte Carlo Method for calculating pi.
If you would like to see how these techniques work in practice, watch the video on programming efficiency on the AdvancedICT YouTube channel.
If you would like to demonstrate this techniques to your students yourself using a BBC Model B emulator (or even a real one - there are still plenty around) then have a look at my blog, In Praise of Slowness.
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http://ccgi.virnuls.plus.com/mathematics/circles.html
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Order of Operations
Before your students use parentheses in math, they need to be clear about the order of operations without parentheses. Start by reviewing the rules for order of operations, and then show students how parentheses can affect that order.
Materials: Overhead projector or front board
Prerequisite Skills and Concepts: Students should have a working knowledge of order of operations for addition, subtraction, multiplication, and division. Students should also have mastered the basic facts for all four operations.
- Ask: What operation do I perform first in the expression 5 7 + 3? Why?
Write the expression on the front board or overhead projector. If students don't remember the rules for order of operations, remind them that multiplication and division come before addition and subtraction.
- Ask: What is the value of this expression?
Walk students through evaluating the expression. 5 7 = 35, so the expression becomes 35 + 3, which equals 38.
- Ask: What happens if I switch the addition and multiplication symbols? What value would I get?
Rewrite the expression as 5 + 7 3, and work through the evaluation. 7 3 = 21, so the expression becomes 5 + 21, which equals 26.
- Ask: Did we get different answers when we changed the operations?
This result will probably not surprise your students too much. They most likely know that performing different operations on the same numbers will give different answers.
- Ask: What if I wanted to keep the multiplication and addition symbols in the same place but get a different answer? How do you think I could do that?
Discuss the question for a short time, then write 5 (7 + 3) on the board.
- Point to the parentheses.
- Say: We call these symbols parentheses. If there are parentheses in an expression, do whatever is inside the parentheses first.
- Ask: What is inside the parentheses in the expression 5 (7 + 3)?
Make sure that students can identify the parts of the expression before continuing.
- Say: Now, let's find the value of the expression with parentheses. (The value is 5 10, or 50.)
Is that the same value we got before?
Help students notice that the answer isn't the same as either the original expression or the expression with the operation symbols switched.
Give students a few more examples, showing an expression with and without parentheses. Have student volunteers evaluate the expressions and compare the answers.
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http://www.eduplace.com/math/mathsteps/4/a/4.orderop.ideas.html
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<urn:uuid:0c6d431d-e6d6-4f8b-93fb-2ef69e4c536a>
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Students will need the Geometric Solids Tool for this lesson.
As students are exploring the various geometric solids, begin a class discussion which includes the following points.
Students should note the following characteristics:
- Each solid has flat sides called faces.
- Each solid has edges to connect the faces.
- Each solid has corners that connect the edges. (Note that the activity sheet refers to corners as "vertices", so you may wish to familiarize students with this vocabulary.)
If you have these shapes in your classroom, have students find the shapes that match those in this computer activity.
This lesson is designed to help students focus on the properties
of each shape. The tools allow them to color the faces, edges, or
corners (vertices) easily by holding the shift key while clicking the
mouse on the paint palette and then on a face. (The edges are always
white and the corners black.)
Distribute the Exploring Geometric Solids activity sheet to the students.
Teacher Note: For today's lesson, students will only
complete the table on the activity sheet. Save the three questions
below the table for tomorrow's lesson.
As students complete the activity sheet, guide them as needed.
For example, when asking "How many sides does each face have?", guide
the student to click the given face, and color the sides. When
determining the number of faces, it may be helpful to color the faces
and count as they color. (It's interesting to observe students doing
this: some color all faces the same color, while others color the faces
After determining the number of faces, students are asked to
count the number of edges and vertices (corners) in the solid. Before
actually counting the number of edges, students may wish to "guess" the
For example, working with the dodecahedron, a student may say
each face has five sides — a pentagon. She counts 12 faces. In guessing
the number of edges, she may estimate 60 and give as her reason,
"5 sides × 12 faces = 60 edges." Counting the edges with the result
of 30 provided an opportunity to have her take a second look at the
dodecahedron and find out why. (Each edge is shared by two faces.)
Taking advantage of opportunities such as these enriches student understanding.
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http://illuminations.nctm.org/Lesson.aspx?id=1486
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We begin our study of angles by learning what they are, how to name them, and ways in which angles can be classified. These concepts are explained below.
An angle is formed when two rays meet at a common endpoint, or vertex. The two sides of the angle are the rays, and the point that unites them is called the vertex.
The vertices are shown in red in the diagram above
Angles can be named in various ways. One way is to use the ? symbol accompanied by three letters. The first and third letters indicate points on the two rays. The letter in the middle is the vertex. Note that the first and third letters are interchangeable because they both measure the same angle. Another way to label an angle is by just using the ? symbol accompanied by the vertex point alone. However, this method only works when there is only one angle at the vertex point. If more than one angle is formed at a vertex point, we need to specify which angle we are talking about by naming it in a different way. Finally, the last way to label an angle is by using the ? symbol accompanied by the letter or number shown between the angle. The different ways of labeling an angle are shown below.
The angle above can be called ?ABC, ?CBA, ?B, or ??
Angles can be measured in degrees or radians. For the time being, we will strictly talk about angles in terms of their degree measure. The symbol for degrees is °. Angles can measure from 0° up to 360°. Angles with no measure are called zero angles, while angles of 360° are full rotations. For our study of geometry, we will primarily focus on three important classifications of angles: acute, obtuse, and right.
A right angle is an angle whose measure is exactly 90°. An easy way to determine whether an angle is a right angle is by considering whether a small square could fit perfectly in the corner of the intersection of the two lines that form the angle. While you would need a protractor to give a more precise measurement, this can give you an approximation of whether or not an angle is close to 90°.
An acute angle is an angle whose measure is less than 90°. For these kinds of angles, a square could not fit perfectly at the intersection of the two lines that form them.
Obtuse angles have measures greater than 90° but less than 180°.
If an angle's measure is 180°, it is called a straight angle. Straight angles are just lines with three points on them.
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http://www.wyzant.com/help/math/geometry/lines_and_angles/introduction_to_angles
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<urn:uuid:fe35a9fb-acbc-41d2-9502-f60b78919deb>
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This starter activity allows pupils to reflect on their own experiences of being treated unfairly to help them understand the concept of racism.
In this activity, pupils will think about and discuss the impacts of racism. Use the 'How does racism make people feel?' activity sheets to support pupils and facilitate classroom discussion.
There are two versions to choose from.
Pupils will learn about allyship and how they can be a good ally and challenge racist behaviour.
As an extension activity, set this task for pupils to design their own poster to show why allies are important and write a class pledge.
This could be set as a home learning activity.
It is recommended to refer to the teacher guidance document before teaching the lesson. This document includes key information and tips for delivering the session. Key topic vocabulary can be found at the end of the document.
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https://plprimarystars.com/resources/nrfr-racism-inclusion
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<urn:uuid:570cbe83-b226-4796-9e7e-0f486da74036>
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Introduction to Angles
We begin our study of angles by learning what they are, how to name them, and ways in which angles can be classified. These concepts are explained below.
An angle is formed when two rays meet at a common endpoint, or vertex. The two sides of the angle are the rays, and the point that unites them is called the vertex.
The vertices are shown in red in the diagram above
Angles can be named in various ways. One way is to use the ? symbol accompanied by three letters. The first and third letters indicate points on the two rays. The letter in the middle is the vertex. Note that the first and third letters are interchangeable because they both measure the same angle. Another way to label an angle is by just using the ? symbol accompanied by the vertex point alone. However, this method only works when there is only one angle at the vertex point. If more than one angle is formed at a vertex point, we need to specify which angle we are talking about by naming it in a different way. Finally, the last way to label an angle is by using the ? symbol accompanied by the letter or number shown between the angle. The different ways of labeling an angle are shown below.
The angle above can be called ?ABC, ?CBA, ?B, or ??
Classifications of Angles
Angles can be measured in degrees or radians. For the time being, we will strictly talk about angles in terms of their degree measure. The symbol for degrees is °. Angles can measure from 0° up to 360°. Angles with no measure are called zero angles, while angles of 360° are full rotations. For our study of geometry, we will primarily focus on three important classifications of angles: acute, obtuse, and right.
A right angle is an angle whose measure is exactly 90°. An easy way to determine whether an angle is a right angle is by considering whether a small square could fit perfectly in the corner of the intersection of the two lines that form the angle. While you would need a protractor to give a more precise measurement, this can give you an approximation of whether or not an angle is close to 90°.
An acute angle is an angle whose measure is less than 90°. For these kinds of angles, a square could not fit perfectly at the intersection of the two lines that form them.
Obtuse angles have measures greater than 90° but less than 180°.
If an angle's measure is 180°, it is called a straight angle. Straight angles are just lines with three points on them.
| 5.1875
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http://www.wyzant.com/resources/lessons/math/geometry/lines_and_angles/introduction_to_angles
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<urn:uuid:88f58e65-9745-4d38-a8a6-6b31ebcb4737>
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In this lesson we will introduce comparison operators and Boolean operators in Python. We must first of all know that comparison operators are widely used in programming languages and are very easy concepts to learn.
Comparison operators in Python
Comparison operators are used mostly in conditional statements (if else), which we will discuss later. We specify that the comparison operators can return only two values: True or False.
So let’s take an example, assuming that the variable a is equal to 5 and b is equal to 6.
== equal – Ex: a == b returns False
! = different – Ex: a! = b returns True
> greater – Ex: a> b returns False
< lesser – Ex: a <b returns True
Operator >= greater than or equal – Ex: a> = b returns False
<= less than or equal – Ex: a <= b returns True
Try running these examples in interactive mode.
Example of using comparison operators in Python
Now let’s do other examples of use. So let’s go back to interactive mode and type:
>>>name = ‘Alan’ # I assign the string Alan to name
>>>name == ‘Alan’ #I compare the variable name with the string Alan
>>>name == ‘Tom’ #I compare the variable name with the string Tom
As we can see, clearly having assigned the string to name Alan the following comparison gives the value True while the comparison with the string Tom gives me False.
Let’s see some examples of the use of comparison operators in Python:
name = 'Alan' name_2 = 'Tom' print(name == name_2) print(name != name_2) number = 5 number_2 = 10 print(number == number_2) print(number != number_2) print(number >= number_2) print(number <= number_2) print(number > number_2) print(number < number_2)
In this example we first took two strings as input, then we compared them to see if they are equal or not.
In the second part we took 2 numbers as input and compared them with the comparison operators, just studied.
In this lesson we have introduced comparison operators in Python, we will see later how to apply them together with conditional operators.
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https://www.codingcreativo.it/en/comparison-operators-in-python/
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In the context of programming, a function is a named sequence of statements that performs a computation. When you define a function, you specify the name and the sequence of statements. Later, you can “call” the function by name.
def is a keyword that indicates that this is a function definition. For example:
>>> def tax(bill): ... """Adds 8% tax to a restaurant bill.""" ... bill *= 1.08 ... print "With tax: %f" % bill ... return bill ... >>> meal_cost = 100 >>> meal_with_tax = tax(meal_cost) With tax: 108.000000 >>>
*=means add that percentage.
tip *=1.6 is the same as adding tip value a 60% :
tip +tip +0.6.
The function components are:
The header, which includes the def keyword, the name of the function, and any parameters the function requires.
An optional comment that explains what the function does.
The body, which describes the procedures the function carries out. The body is indented, just like for conditional statements.
If you look the example above, after
def we find the function argument, between brackets.The empty parentheses after the name indicate that this function doesn’t take any
arguments. Then after the colon and idented to the rigth we find the sentences to run when we call the function.
After defining a function, it must be called to be implemented.
Define a function that returns the square of a number, and call it with the argument 10.
>>> def square(n): ... """Returns the square of a number.""" ... squared = n**2 ... print "%d squared is %d." % (n, squared) ... return squared ... ... # Call the square function on line 9! Make sure to ... # include the number 10 between the parentheses. ... >>> square(10) 10 squared is 100. 100
Make the same and kind of function as in Practice 1, but using two arguments:base, exponent.
>>> def power(base, exponent): # Add your parameters here! ... result = base**exponent ... print "%d to the power of %d is %d." % (base, exponent, result) ... >>> power(37,4) # Add your arguments here! 37 to the power of 4 is 1874161.
A function can call another function, for example:
one_good_turn (which adds 1 to the number it takes in as an argument) and
deserves_another (which adds 2).
Change the body of
deserves_another so that it always adds 2 to the output of
>>> def one_good_turn(n): ... return n + 1 ... >>> def deserves_another(n): ... return one_good_turn(n) + 2 ... >>> deserves_another(7) 10 >>>
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https://erlerobotics.gitbooks.io/erle-robotics-learning-python-gitbook-free/content/functions/function_basics.html
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We are learning to: * Describe characters in a story (e.g., their traits, motivations, or feelings) Students will be successful when they: * Use key details (e.g., a character's thoughts, words, or actions) to describe the character.
What is Characterisation?
Characterisation is the way an author or an actor describes or shows what a character is like. It helps to make the characters seem believable or life-like to the reader or audience.
5 Methods of Characterisation
Physical descriptions: This is where the physical appearance and features of the character is described. This could be their height, hair colour, eye colour, clothes, etc. If your students are unsure where to begin, why don't they use their own physical description to build their characters?
Actions: Characters can be motivated and built through what they do rather than what they look like. Try adding dramatic events and actions to teach your reader a little more about the character. How a character reacts to an action or event, can tell us a lot. What the character does tells us a lot about him/her, as well as how the character behaves and his or her attitude. Is the character a good person or a bad person? Is the character helpful to others or selfish?
Inner thoughts: Use inner thoughts as a way to describe and build a detailed character. A character's inner thoughts include the thoughts and feelings that they may not share with the other characters in a story.
Reactions: How does the character make the other characters feel? Do they feel scared, happy, or confused? This helps the reader have a better understanding of all the characters.
Speech: Finally, speech is all about what their characters say rather than what they do. It can also refer to the tone and way they say. For example, the character might speak in a shy, quiet manner or in a nervous manner. The character might speak intelligently or in a rude manner.
When designing characters, they might think about questions such as:
What is the character's name?
How does the character behave? (Do they behave differently alone than when around people?)
How old are they?
Where are they from?
Direct vs Indirect Characterisation
When we describe them directly, we simply explain what they're feeling to the audience.
She was feeling nervous.
When we describe them indirectly, we give the reader an impression of what they're feeling through descriptions of their appearance and their actions.
She bit down on her lip, her thoughts racing through her mind. Her hands shook as they held the interview notes.
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https://www.mrsrussellcreaneyps.com/characterisation.html
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What you will learn: Kindergartners start with numbers, representing and comparing them. We start with taking help from pictures and objects. We progress towards written numerals, modelling simple addition and subtraction, counting, combining, and segregating sets. Get introduced to the base ten system. Compose numbers from 11-19 in tens and ones. We introduce ourselves to basic geometric shapes like circles, squares, triangles etc. We know about solid shapes like cyclinders, cubes, and spheres. Learn to name 2-d and 3-d shapes. Compose simple shapes to form larger shapes.
Here is the list of all the common core standards for this grade. There are some sample worksheets on the page. Please subscribe to access the whole content in its best form. All of our worksheets are free for non-commercial and personal use. Click on any link to view, print, or download the worksheets.
|K.CC.A Know number names and the count sequence.|
Count to 100 by ones and by tens.
Count forward beginning from a given number within the known sequence (instead of having to begin at 1).
Write numbers from 0 to 20. Represent a number of objects with a written numeral 0-20 (with 0 representing a count of no objects).
|K.CC.B Count to tell the number of objects.|
Understand the relationship between numbers and quantities; connect counting to cardinality. K.CC.B.4.A When counting objects, say the number names in the standard order, pairing each object with one and only one number name and each number name with one and only one object. K.CC.B.4.B Understand that the last number name said tells the number of objects counted. The number of objects is the same regardless of their arrangement or the order in which they were counted. K.CC.B.4.C Understand that each successive number name refers to a quantity that is one larger.
Count to answer "how many?" questions about as many as 20 things arranged in a line, a rectangular array, or a circle, or as many as 10 things in a scattered configuration; given a number from 1-20, count out that many objects.
|K.CC.C Compare numbers.|
Identify whether the number of objects in one group is greater than, less than, or equal to the number of objects in another group, e.g., by using matching and counting strategies.
Compare two numbers between 1 and 10 presented as written numerals.
|K.OA.A Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.|
Represent addition and subtraction with objects, fingers, mental images, drawings, sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.
Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).
For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.
Fluently add and subtract within 5.
|K.NBT.A Work with numbers 11-19 to gain foundations for place value.|
Compose and decompose numbers from 11 to 19 into ten ones and some further ones, e.g., by using objects or drawings, and record each composition or decomposition by a drawing or equation (e.g., 18 = 10 + 8); understand that these numbers are composed of ten ones and one, two, three, four, five, six, seven, eight, or nine ones.
|K.MD.A Describe and compare measurable attributes.|
Describe measurable attributes of objects, such as length or weight. Describe several measurable attributes of a single object.
Directly compare two objects with a measurable attribute in common, to see which object has "more of"/"less of" the attribute, and describe the difference. For example, directly compare the heights of two children and describe one child as taller/shorter.
|K.MD.B Classify objects and count the number of objects in each category.|
Classify objects into given categories; count the numbers of objects in each category and sort the categories by count.
|K.G.A Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).|
Describe objects in the environment using names of shapes, and describe the relative positions of these objects using terms such as above, below, beside, in front of, behind, and next to.
Correctly name shapes regardless of their orientations or overall size.
Identify shapes as two-dimensional (lying in a plane, "flat") or three-dimensional ("solid").
|K.G.B Analyze, compare, create, and compose shapes.|
Analyze and compare two- and three-dimensional shapes, in different sizes and orientations, using informal language to describe their similarities, differences, parts (e.g., number of sides and vertices/"corners") and other attributes (e.g., having sides of equal length).
Model shapes in the world by building shapes from components (e.g., sticks and clay balls) and drawing shapes.
Compose simple shapes to form larger shapes.
Children will count the beads on the abacus and learn how many more than ten makes that number.
Core Standard: K.CC.B.4
This a single - digit subtraction word problems worksheet.
Core Standard: K.OA.A.2
Color the diamonds blue, and color the ovals red.
Core Standard: K.G.B.4
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https://www.biglearners.com/common-core/worksheets/kindergarten/math
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To learn about probability and fairness, students participate in several chance activities and examine a few games for fairness. Student groups become game designers who are asked to design a fair game for a toy company describing the rules for play and explaining mathematically why the game is fair. Finally, groups present their game to a fictional toy company’s board of directors convincing them to sell their game.
View how a variety of student-centered assessments are used in the Fair Games Unit Plan. These assessments help students and teachers set goals; monitor student progress; provide feedback; assess thinking, processes, performances, products; and reflect on learning throughout the instructional cycle.
Set the Stage
Ask students if they have ever been in a situation that was not fair. Pose the Essential Question, Is life fair? Break students into small groups and have them discuss the Essential Question and record their initial responses. Encourage them to talk about why they think life is fair or unfair, as well as what they mean by fair and what determines fairness. Ask several students to share their responses to the Essential Question and then tell them that they will begin a unit to learn how to use mathematics to determine the fairness of games.
Introduce a math journal to students. This journal will be used to record answers to questions, prompts, and problems.
What are the Chances? Activity
This activity addresses the Content Questions, What is probability? and How do you measure the likelihood of an event?
Overview of activity:
Introduce the idea of probability by discussing the likelihood of events occurring. Encourage students to focus on the language of probability as they use their life experiences to recall events that are certain, impossible, likely, and unlikely to happen. Record these events and introduce students to a probability scale, ranging from zero to one. This activity is intended to get students involved in talking about probability.
The following are a series of activities that are meant to lead the students to understanding “chance”.
What are the Chances? Mystery Pasta Activity
In preparation for this lesson, fill three bags with the following proportions of shell and elbow pasta shapes. Write the three populations on the chalkboard:
Bag 1: 8 shells, 16 elbow
Bag 2: 16 shells, 8 elbow
Bag 3: 4 shells, 20 elbow
Is it a Fair Game? Rock, Paper, Scissors
Officially known as Rock Paper Scissors or RPS, this game is also known in parts of the world as Jenken, Jan Ken Pon, Roshambo, Shnik Shnak Shnuk, Ching Chong Chow, Farggling, Scissors Paper Stone. Divide the class into pairs (player A and player B) and have them play the game 15 times. Use chart paper or an overhead projector to record the results of player A in red and player B in a different color (How many A players won game 1,2,3,etc? How many B players won? How many ties?) Compare the results. Ask the class, "Is this game fair?" (explain that this means equal chance of winning for all players) Ask students to explain why they think it is fair. Try to elicit from students that it is fair because each student has an equally likely or equal chance of winning (50% or 1/2). Introduce students to a tree diagram as a visual tool for keeping track of the possible outcomes of this game: This is known as a probability tree. To address the Content Question, What is the difference between experimental and theoretical probability? compare this mathematical model with what happened when students played the game (theoretical vs. experimental probability).
Player A wins 3/9 or 1/3
Player B wins 3/9 or 1/3
Tie 3/9 or 1/3
Ask the students to play the game now with three players using the following rules:
Ask students to consider the following questions, Is this game fair? Why or why not? What determines fairness? Ask students to construct a probability tree in their math journals to determine the possible outcomes (There will be 27 outcomes—three more branches off of each of the above nine possibilities. It is not fair because player C has more chances of winning than players A and B)
Remind students of the Essential Question they discussed at the beginning of the unit, Is life fair? Does fairness in life relate to fairness in games and if so, how? And if not, why not?
Rolling Dice: What are the Chances? Activity
Introduce the activity by discussing the possible outcomes that can be obtained when a die is rolled. Students should be able to identify that the possible outcomes are the numbers from 1 to 6. Then ask, What are the possible sums if the two dice are rolled? Have students work in groups to investigate the chances for rolling a particular sum. Have each person in the group create a number line for the possible sums (2,3,4,5,6,7,8,9,10,11,12) and place “x’s” each time the sum is rolled. Have students roll the dice 15 times. Create a classroom frequency distribution graph (a number line with the “x’s” to represent how many times each sum occurred). Ask students to compare their own group data to the whole class data. Ask students, Are all sums equally likely to occur? If not, which ones are more likely to occur and which ones are least likely to occur?
Introduce students to the idea that a table can be a useful tool in showing the possible outcomes (mathematically) of the sums of two dice. After getting students started on the table in their math journals, have them complete it:
Ask students the following questions:
Which sum is most likely to occur on the next roll of dice? Least likely? Why?
How many total possible outcomes? (36)
How many times does each sum appear in the table?
What does this tell us? (The probability of that sum occurring; for example: 9 appears 4 times, so there is 4/36 or 1/9 probability of rolling a 9). Have them record responses in their math journals.
Looking at the Competition
In the following activity, students create a fair game based upon what they’ve learned in the previous activities and game. Bring what they’ve learned together by providing several games that use probability and chance. Allow students to play the games while recording why or why not they think the game is fair. Once student groups have played at least two different games, have the whole group discuss and list the common reasons the games were fair and how chance was involved.
In math journals, have students reflect on what they’ve learned from playing the games and brainstorm ideas for designing their new game. Offer a list of questions for to students to think about.
Putting it all Together
Have students share with their group in round-robin fashion what they brainstormed in their journals the previous day. Students then apply what they have learned as they take on the roles of game designers responding to an advertisement of a toy company that wants to create and sell a new game for children ages 11-13. Create an environment that fosters creative thinking by having students give and receive peer feedback and invite local business owners to share in the process of creating a product to sell. Each team of designers needs to create a game using number cubes, cards, or pasta to advance play, describe the rules for play, and explain why the game is fair using probability and graphical organizers (tables, lists, tree diagrams). Have students refer back to their math journals to connect what they’ve learned to create a new game. Have them create a multimedia presentation of their game to present to the fictional board of directors (parents, school faculty, local toy and business representatives) and address the Curriculum-Framing Questions. Show the sample presentation to students as an example and give students opportunities to ask questions and get any clarification needed. Hand out the project rubric and presentation checklist to discuss project expectations. Have students use the checklist to guide the creation of the slideshow presentation. Check for student understanding and guide students in using the rubric and checklist to create quality work.
To help students with the planning and implementing of their game idea, encourage students to use the following guiding questions to promote metacognition skills:
Model a think-aloud beforehand, so students are aware of strategies to use while exploring these questions in-depth. While students are using these metacognitive guiding questions, take anecdotal notes to document students’ thinking processes.
Invite parents, school faculty, and local toy and business representatives to attend a Game Night to recognize student work and learning. Students present their slideshows to the participants and then have time to play the games. Guests are invited to give students feedback about their game.
Return to the Essential Question, Is life fair? Ask students to think about how they responded to the question at the beginning of the unit. Have them write their thoughts in their journals about fairness, chance, and probability. Encourage them to write about what they have learned about these things over the course of the unit and to provide as much detail and examples as possible. As a final assessment, students fill out the self-reflection to reflect on what they’ve learned.
English Language Learner
A teacher participated in the Intel® Teach Program, which resulted in this idea for a classroom project. A team of teachers expanded the plan into the example you see here.
This unit is aligned to Common Core State Standards for Math.
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http://www.intel.com/content/www/us/en/education/k12/project-design/unit-plans/fairgames.html
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You can compare strings in Python using the equality (
==) and comparison (
>=) operators. There are no special methods to compare two strings. In this article, you’ll learn how each of the operators work when comparing strings.
Python string comparison compares the characters in both strings one by one. When different characters are found, then their Unicode code point values are compared. The character with the lower Unicode value is considered to be smaller.
Declare the string variable:
fruit1 = 'Apple'
The following table shows the results of comparing identical strings (
Apple) using different operators.
|Not equal to||
|Less than or equal to||
|Greater than or equal to||
Both the strings are exactly the same. In other words, they’re equal. The equality operator and the other equal to operators return
If you compare strings of different values, then you get the exact opposite output.
If you compare strings that contain the same substring, such as
ApplePie, then the longer string is considered larger.
This example code takes and compares input from the user. Then the program uses the results of the comparison to print additional information about the alphabetical order of the input strings. In this case, the program assumes that the smaller string comes before the larger string.
fruit1 = input('Enter the name of the first fruit:\n') fruit2 = input('Enter the name of the second fruit:\n') if fruit1 < fruit2: print(fruit1 + " comes before " + fruit2 + " in the dictionary.") elif fruit1 > fruit2: print(fruit1 + " comes after " + fruit2 + " in the dictionary.") else: print(fruit1 + " and " + fruit2 + " are the same.")
Here’s an example of the potential output when you enter different values:
OutputEnter the name of first fruit: Apple Enter the name of second fruit: Banana Apple comes before Banana in the dictionary.
Here’s an example of the potential output when you enter identical strings:
OutputEnter the name of first fruit: Orange Enter the name of second fruit: Orange Orange and Orange are the same.
Note: For this example to work, the user needs to enter either only upper case or only lower case for the first letter of both input strings. For example, if the user enters the strings
Banana, then the output will be
apple comes after Banana in the dictionary, which is incorrect.
This discrepancy occurs because the Unicode code point values of uppercase letters are always smaller than the Unicode code point values of lowercase letters: the value of
a is 97 and the value of
B is 66. You can test this yourself by using the
ord() function to print the Unicode code point value of the characters.
In this article you learned how to compare strings in Python using the equality (
==) and comparison (
>=) operators. Continue your learning about Python strings.
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what if I want to get the difference in term of percentage.For instance , Apple and apple instead of getting false can I get a percentage of similarity like 93%
You missed one thing, if it’s ‘applebanana’ and ‘appleorange’ then ‘appleorange’ is greater than ‘applebanana’. Hopefully, this helps.
when comparing strings, is only unicode of first letter considered or addition of unicodes of all the letters is considered?
print(‘Apple’ < ‘ApplePie’) does not return True because of the length. print(‘2’ < ‘11’) will return False.
- Ammar S Salman
your day of love may bring the gratitude of others for life.
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Basic String Operations
Strings are bits of text. They can be defined as anything between quotes:
astring = "Hello world!"
As you can see, the first thing you learned was printing a simple sentence. This sentence was stored by Python as a string. However, instead of immediately printing strings out, we will explore the various things you can do to them.
That prints out 12, because "Hello world!" is 12 characters long, including punctuation and spaces.
That prints out 4, because the location of the first occurrence of the letter "o" is 4 characters away from the first character. Notice how there are actually two o's in the phrase - this method only recognizes the first.
But why didn't it print out 5? Isn't "o" the fifth character in the string? To make things more simple, Python (and most other programming languages) start things at 0 instead of 1. So the index of "o" is 4.
For those of you using silly fonts, that is a lowercase L, not a number one. This counts the number of l's in the string. Therefore, it should print 3.
This prints a slice of the string, starting at index 3, and ending at index 6. But why 6 and not 7? Again, most programming languages do this - it makes doing math inside those brackets easier.
If you just have one number in the brackets, it will give you the single character at that index. If you leave out the first number but keep the colon, it will give you a slice from the start to the number you left in. If you leave out the second number, if will give you a slice from the first number to the end.
You can even put negative numbers inside the brackets. They are an easy way of starting at the end of the string instead of the beginning. This way, -3 means "3rd character from the end".
print astring.upper() print astring.lower()
These make a new string with all letters converted to uppercase and lowercase, respectively.
print astring.startswith("Hello") print astring.endswith("asdfasdfasdf")
This is used to determine whether the string starts with something or ends with something, respectively. The first one will print True, as the string starts with "Hello". The second one will print False, as the string certainly does not end with "asdfasdfasdf".
afewwords = astring.split(" ")
This splits the string into a bunch of strings grouped together in a list. Since this example splits at a space, the first item in the list will be "Hello", and the second will be "world!".
Try to fix the code to print out the correct information by changing the string.
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http://www.learnpython.org/Basic_String_Operations
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On this series of worksheets, we will focus on the type of verb that describe an action. The worksheets begin with identification activities we will then transition to helping you use these terms in your own writing. There are two types of action verbs that are used. A transitive verb form are words that affect a specific object. A good example of this is the word washes. You can only be washing one thing at a time, so this is a direct object. Intransitive verbs do not apply to a specific person or thing, instead they focus on what the subject of the action is doing. A few common intransitive words would include yell, play, and go.
An action verb is a word that shows what someone or something is doing. Underline all of them within the series of sentences that you are seeing.
Find the word that is making it all happen. You can circle them or show them in anyway that you choose to.
If the bolded noun in each sentence is an action verb, label its role. If it does not serve this role, circle "no."
Mrs. Willoughby watches us on Saturday nights. The word in this case is "watches". This worksheet floats a slightly higher level of vocabulary.
Write what is actually taking place on the line that is provided for you? You will write the term of interest found in each sentence.
Find the word that is doing something about it. These are quick and sweet. A nice way to do a mass review of this skill.
The robot stared at me. The robot waved at me. How many did you see in the past two sentences?
Verbs express either action or a state of being. Another way to think of it is that a verb can express either a physical activity or a mental activity.
We have you work with much longer sentences in this worksheet. This allows use to take this skill on to the next level.
Some of these sentences have multiple actions to identify. Just take your time and identify all of them as you see.
Some words will present as thoughts that are not tangible. A good way to approach these parts of speech is to spot movement of somekind.
Read each sentence. Does the underlined word refer to an action or to a state/condition? Write your answer on the line.
See how well you can perform this skill, now that you have had some practice at it.
What are Action Verbs?
There are many different words in the English language that are used to describe what someone or something is doing. These are critical words to any sentence because they tell us what is going on. Without the presence of these words, the sentences do not make sense or offer much information to the reader. It can be things as simple as walking or complex as writing a fractional algorithm to solve a world health crisis. Any time a single word is used to describe what a subject is doing; we refer to that word as an action verb. In addition to these words being able to describe a physical action they can also be used to describe mental things and things we cannot see taking place such as thinking or feeling. These types of words come in many different tenses and should be examined in depth to best know which form of the word is the most fitting.
You cannot express a complete thought without using a verb. They are an essential part of a complete sentence.
These parts of speech are added to a sentence to describe an action that can be done by a person, an animal, a thing, or even nature.
For example, a bartender pouring a drink, a dog barking, a kite flying, or the wind blowing are all action verbs. You will often use them to answer this type of question: Can I DO that?
Consider this example:
Sophia updated her resume before leaving the office.
Are 'resume' or 'before' things you can do? Can you 'office'? What about 'updated' and 'leaving'? Can you update? Can you leave? Of course, you can! So updated and leaving are both action verbs in this sentence.
Examples of Action Verbs
The crowd exploded with cheers when the Queen appeared on the palace's balcony and waved.
In this sentence, exploded, appeared, and waved are all examples of action verbs.
The football team stared gloomily out of the window as the rain ruined their match.
Here, the words stared and ruined serve this purpose in an impactful manner.
Gregory wanted a horse for his birthday, but his parents could only afford horse-riding lessons.
In this sentence, wanted and afford are examples of action verbs.
The marathon runners forced themselves to move faster as soon as they caught sight of the finish line.
Here, forced, move, and caught are all serving this role that describe the runners' actions.
Examples of Impactful Action Verbs
If you lead or initiate a project, you can use the following action verbs to describe your action:
If you develop or build the said project from the ground up, consider the following action verbs:
If you want to highlight cost-effectiveness, pick any of these action verbs:
To show a boost in a company's sales or revenue, try these:
To show improvement or change, consider these action verbs:
You can show team management and leadership skills through:
Want to find appropriate words for researching, analyzing, or fact-finding? Try these:
If you've spoken, lobbied, or communicated effectively, show off your wordplay with:
Proud to achieve your targets? Win any awards? Hit the goals? How about these:
Addressing customers' concerns and fulfilling their needs can be described by these:
If you raised funds, discovered resources, or brought in a new partner, try swapping responsible for, with:
If you fulfill a company's requirements, implement a certain change, or regulate a process, try using anyone of these action verbs:
Action verbs help identify any action performed by the subject of a sentence. If you want your sentence to sound complete, you need to use these specific parts of speech.
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https://www.easyteacherworksheets.com/langarts/2/actionverbs.html
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These exercises and activities are for students to use independently of the teacher to practice number properties. Some of these activities would be suitable for homework. Others require follow-up during teaching sessions.
There are a lot of misconceptions around the use of the equals sign. Some students seem to think that it means “work out the answer”. Consequently, no equals signs have been provided in exercises where simple additions are required. Students with such an understanding may also think that 4 + 6 = 10 is a correct way to write the sentence, but not 10 = 4 + 6. (Hence number 25 in exercise 1). Such a question may be posed as part of practice, then discussed at a later teaching session, or could be included as part of a lesson.
It has been noted that in recent years there has not been the same emphasis on developing an understanding of inequalities in primary mathematics. This is not intended, but may mean that some students come through without the understandings they have had in the past. Consequently some teaching may need to be provided before students understand the signs and the concepts required by this exercise.
Students don't really know a fact until they can recognise and use it in all three formats. Exercise 3 not only provides practice in recognising the facts in these other formats, but also introduces students to lower level algebra. Mental computation (or recall of known facts) of such simple equations is most sensible method of solution.
Students should have been working with shapes as unknowns for quite some time before reaching secondary school, so should be conversant with what is expected in using a shape in a sentence/equation. In this example, however, the meaning of the unknown has changed. Firstly there are two different shapes – which traditionally would mean that they represent different numbers (though there is the special case where they are the same.) Students may need to discuss this before attempting the problem). However, the unknowns do not represent a single number in this context. This too may to be introduced – that there could be lots of possibilities for such an equation (though is likely to arise naturally if you ask them all to think of two numbers that add up to ten.
A single addition fact should be able to be turned into related subtraction facts and simple subtractions should be able to be solved using knowledge of basic addition facts. However, for many students, subtraction understanding lags behind addition understanding. Making the link between addition and subtraction is thus essential teaching at this level.
One issue with providing word problems in an exercise alongside simple number problems is that some students learn not to read the words, and simply to pull out the numbers “and do the same to them”. To address this problem, this exercise includes a variety of formats of problem. In fact, number one requires a subtraction with the numbers 6 and 4 – rather than an addition, while others include change unknown format – so could equally be an addition or a subtraction that relies on their compatible number knowledge. This exercise thus provides a good basis for a teaching session around “what words tell us that we should be adding the numbers…” In this teaching session, students could be encouraged to develop a list of words commonly used to indicate that the operations of addition and subtraction are to be used.
Students learn a lot of mathematics (things that are not necessarily directly taught – or intended to be taught) by identifying patterns. Often, the better we are at identifying patterns, the better we are at mathematics. These exercises look to harness patterning to help students realise that knowing these facts to 10 mean that they can also answer a whole load of other problems. Both exercises require follow-up discussion – and additional practice built around consolidating these discoveries. For example, students could make a poster showing how to use their facts to 10 to answer other problems. They should also do some practice work in using this new skill – in all 3 formats, start, change and result unknown.
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In Topic A, students learn the notation related to roots (8.EE.A.2). The definition for irrational numbers relies on students’ understanding of rational numbers, that is, students know that rational numbers are points on a number line (6.NS.C.6) and that every quotient of integers (with a non-zero divisor) is a rational number (7.NS.A.2). Then irrational numbers are numbers that can be placed in their approximate positions on a number line and not expressed as a quotient of integers. Though the term “irrational” is not introduced until Topic B, students learn that irrational numbers exist and are different from rational numbers. Students learn to find positive square roots and cube roots of expressions and know that there is only one such number (8.EE.A.2). Topic A includes some extension work on simplifying perfect square factors of radicals in preparation for Algebra I.
Grade 8 Mathematics Module 7, Topic A, Overview
Resources may contain links to sites external to the EngageNY.org website. These sites may not be within the jurisdiction of NYSED and in such cases NYSED is not responsible for its content.
Common Core Learning Standards
|8.NS.1||Know that numbers that are not rational are called irrational. Understand informally that every...|
|8.NS.2||Use rational approximations of irrational numbers to compare the size of irrational numbers, locate...|
|8.EE.2||Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3...|
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Observe the differences in the graphs when <, ≤, =, >, and ≥ are used.
- Students will graph the solution to simple inequalities in one variable and describe the solution using correct vocabulary and symbols.
- Students will use appropriate tools strategically (CCSS Mathematical Practices).
- Students will look for and express regularity in repeated reasoning (CCSS Mathematical Practices).
- Boundary point
- Open (non-inclusive) intervals
- Closed (inclusive) intervals
About the Lesson
This lesson involves observing the differences in the graphs when <, ≤, =, >, and ≥ are used. Students will make conjectures as to when to shade to the left, right, or not at all, as well as to whether the boundary point is shaded (included). As a result, students will:
- Understand how to graph the solution to an inequality in one variable on the number line.
- Describe the solution of a linear inequality in one variable, given the graph, using correct vocabulary and symbols.
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What do we mean by 'fair'?
Concept stretching with questions
SOMETIMES YOU MAY want your pupils to focus on a particular concept in a structured way through reflecting on a set of questions you provide. You might also use these kinds of questions as a model for the pupils’ own ongoing questioning about concepts. For example, here is a series of questions following on from an initial question about friendship: ‘Should we always agree with our friends? If we disagree with our friends, should tell them we disagree? Are there times when we shouldn’t tell our friends we disagree with them? Are there times when it is essential that we should tell our friends when we disagree with them?’
The key document will explain how to use the opinions presented in this section.
Please note: sometimes the links we create in our materials become obsolete or don’t work for a variety of reasons. If you find a link doesn’t work, please let us know, citing the name of the resource. Send a message to: firstname.lastname@example.org
Questions to ask young children (3 7)
* How many shapes can you see right now?
* Do you have a favourite shape? Why?
* Is every shape like the one youre describing, your favourite?
* Does everything have a shape?
* What shape is a teddy bear?
* What shape are you?
* What shape are your dreams?
* Do sounds have shapes?
* When you look at a circle, how do you know its a circle?
Exploring the similarities and differences between reality, perception, virtual reality and fiction.
Six classroom activities with which to explore the concepts of democracy.
Sample dialogue, questions and activities to explore the concepts of proof and evidence.
Example dialogue, questions and activities exploring the concept of zero.
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This week in week 11 we started Graphing Inequalities. The majority of this is related to our graphing from last year in grade 10 math.
Linear graphs are known as straight line graphs with the equation of y=mx +b.
m is the slope ( ) b is the y intercept
For example: on a graph looks like:
Now when graphing inequalities you have to find a sign that will have a true statement after you have chosen a point on the graph to substitute x and y.
Note that > and < are broken lines on the graph and and are solid lines like the above graph ( because it is equal to so it includes the line)
- (greater than zero)
- (greater or equal to zero)
- (less than zero)
- (less than or equal to zero)
Now to graph the inequality you have to choose a point on the graph that makes the expression true for example:
to make things easy use (0,0) as the point
- 0 < 6
This statement is true because 0 is smaller than 6. This means that the side that has the coordinate of (0,0) will be shaded in. This will also have a broken line because it is not equal to.
Now when we change the sign to greater to (>) it will flip because 0 would not be greater than 6
- 0 > 6
- FALSE STATEMENT
If the sign was or the line would be solid.
You can also graph a parabola
Step one: Graph the parabola by putting it into standard form.
Graph it from here.
Step two: chose a point on the graph and make a true statement.
- TRUE STATEMENT, 0 is smaller than two
The side with (0,0) as a coordinate will be shaded in and the line will be solid because it is equal to.
Now if the sign was change the inside of the parabola would be shaded in.
This is how to graph inequalities with linear equations as well as quadratic equations.
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-For example, in 2*3 =6 the factors would be 2 and 3
(Note: more examples can be included by the teacher if needed)
- Pull up the factor tree game on the interactive board or on the projector. Play the game as a class with volunteers going to the board after you have modeled an example first.
- Example: “Let's look at the number 24. What two numbers multiplied together can give us 24? It can be any two sets that give me 24 such as 2*12, 3*8, 6*4 (explain to students that if you do 1*24 it would lead you back to the same place of still finding factors of 24).” Next type in any two factors (let’s use 6*4 for this example) and ask students, “What factors make 6 and what factors make 4?” At this point, the game will note 6 and 4 as squares because they are composite and can be broken down by additional factors. Explain to students, “Since 6 is composite, it can be broken down by two factors such as 2 and 3 because 2*3=6. 4 can be broken down into the factors 2 and 2 because 2*2 =4.” Then ask “Can the 2’s and the 3 be broken down by anything else except one (1*3=3)? (Students should reply no if they reply yes then ask what other multiplication facts can make these numbers to clear up confusion).” “These numbers are prime because they have no other factors other than one and itself so they will be represented in circles”
- This game notes prime numbers circled and composite numbers in a square shape. This is helpful so students can differentiate between the two numbers as well as see that prime numbers do not break down.
- After a few examples, make sure to explain how the answer can be checked. The game mentions not only the word “correct” for the answers but displays why the answer is correct with the prime factors of the number written out, for example:
-Prime factorization of: 20
-This can be checked by multiplying 5*2*2 = 20
(Note: We multiply the prime numbers out which makes the original number)
(Note: Walk around the class and check the work of students as they figure out the answer)
- Students will be creating a visual representation of a number by showing the factors of each number and writing it correctly on a poster board.
- Students will choose a composite number up to 100. The teacher will show on the projector a list of composite numbers students can choose from.
(Note: Gifted/advanced students can use higher numbers)
- The teacher will note what number the students choose and will make sure all students have different numbers.
(Note: allow students to think of a different number if students have the same number)
Encourage students to use higher numbers instead of just one digit numerals.
The teacher will show a completed copy of the number 100. Make sure to explain that all composite factors should be represented one way (in the teacher example they are all in pink squares) and prime factors represented in a different way (in the teacher example they are in orange stars).
The teacher will have popsicle sticks available that students can use for their factor tree model. Students are free to use markers, construction paper, glue, or any materials they need for their poster.
(Note: the teacher will need to have materials ready and out so students can use)
(Note: give students an appropriate deadline for this assignment)
- Class discussion: Have a few students (those that have finished their poster) share their poster with the class.
- Ask students:
- What their prime and composite numbers are and how they know this.
- How to check if their prime factorization is correct.
(Note: Have the rest of the students share their poster after the deadline as the closure to this activity)
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In a previous lesson we alluded to "literal text" in programming.
By this we meant text in a program that is always taken on face value; verbatim as
it stands. Taken literally by the computer with no further processing. Text such as this in programming
is often referred to as a string. Note the key element here is what the computer outputs in response to
print(2+2) and print("2+2"). The former yields 4, the latter yields 2+2. Do you know why?
A string is programming is a "string" (or sequence, series, succession) of characters, tacked together,
to form a word or a sentence. A string is usually enclosed in double or single quotes. The double (or single)
quotes are special, as they tell the computer to not evaluate or interpret what's inside
of the quotes in any way...just take it as it stands. This is the nature of a string.
This lesson will show you what it means to "add" two string together, often called "concatenation."
The concatenation operator is two dots put together like this: ...
string1 .. string2
Move the mouse over a dotted box for more information.
The result of the .. operation on string1 and string2 is the concatenation of the two strings.
Now you try. Fill in some strings into the a= and b= lines.
Type your code here:
See your results here:
This code will run, but the output will be nonsense. Fill in some text in between the
double quotes in the a= and b= lines. The c = a .. b line will do the
concatenation. What do you think the output will be given your assignments
of a and b?
Share your code
Show a friend, family member, or teacher what you've done!
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For more grammar guides, click here.
How to Teach Single and Plural Nouns
In English, we often alter the form of nouns to show whether we are referring to one or more than one instance of the noun. For example, one cat refers to a single instance of a cat, whereas cats refers to more than one. While English marks the difference between one and two or more of an item, other languages may not mark this difference, or have a more complex system. Russian, for instance, marks nouns differently for one, two to four, or five or more of the same noun.
When to Teach
The idea of plural nouns is typically taught near the start of beginner level courses and may be reinforced in an elementary course. Once the regular rules have been learnt, a teacher may just point out irregular plural forms as new words are introduced at higher levels.
Ideally, before introducing plural nouns, the singular versions will have been taught. Also useful are some numbers so that students will be able to use these in combination with the plural forms.
As mentioned above, the key meaning of a plural noun is to distinguish that the speaker is referring to more than one of an item.
Countable nouns typically have a singular and a plural form. Some nouns that are usually uncountable can also have countable usages, in which case they tend to follow the regular rules for producing their plural form.
The singular form is typically the citation form that is found in the dictionary. For example:
The regular rule for forming plurals in English is to add -s after the base form. For example, dog becomes dogs.
Some words however require -es. Any noun ending in -ch, -x, -sh, -ss, -s or -z will require an -es (watches, foxes, bushes, classes, buses, quizzes). Notice also that quiz requires a second -z.
Nouns ending with -o either require -s or -es. The simple way to tell is whether it has a vowel before the -o or a consonant. A vowel would mean that it only requires -s (stereos) whereas a consonant would require -es (tomatoes).
Words ending in a -y will also change depending on whether the -y is preceded by a vowel or consonant. If a vowel, only an -s is needed (days). If a consonant, the -y is replaced with an -i and -es is added to the end (babies).
Nouns ending with an -f or -fe also have more complicated rules. The -f will be changed to a -v and -es or -s will be added so that in both cases the ending becomes -ves (knives, wolves).
Some nouns that have come into the language from Latin or Greek retain their own rules for forming the plural form (fungi, stadia, data)
Some plurals in English do not follow such nice rules. These are irregular nouns and include such examples as children, mice, and sheep. In some cases, these irregular nouns do not change. Many of these are very common words and therefore are a source of frustration to new students of English.
Some uncountable nouns can have countable uses. In this case they tend to take an -s (times, sugars, coffees).
Finally, there exist some nouns in English that only appear in a plural form (scissors, mathematics, trousers).
When applying the regular rules for making plural nouns, there are generally three sounds that students will need to make /z/, /s/ or /ɪz/.
The /z/ sound occurs after a voiced consonant, while the /s/ phoneme will occur after an unvoiced consonant.
The /ɪz/ sound adds an additional syllable to the word. It appears where a noun ends -ch, -x, -sh, -ss, -s or -z. For example, in buses, but not in clothes.
Problems for Students
Students often find the treatment of plurals in English confusing. In particular:
- Whether a noun is countable or uncountable may not be the same as in their L1;
- Irregular plurals cause particular difficulty for students in remembering their forms and which relates to singular or plural;
- Adding -s to irregular plural forms (childrens);
- Nouns that have come from other languages may lose their foreign language endings (focuses vs foci) or take on a singular form even though they are originally plural (a criteria).
- Ensuring the verb following a single or plural noun is correctly conjugated.
With young learners, animals are a common theme for introducing or practising singular and plural forms. Many irregular plural forms occur with animals such as mice, geese, sheep and fish. Not to mention that parts of animals would also likely lead to including feet and teeth.
Another common context for plurals with young learners is body parts as students will learn to say that they have two eyes, ten fingers but only one nose.
Food is another strong theme to introduce singular and plurals. However, food also uncovers one of the common problems that students have. Typically, many foods are uncountable such as bread, cheese or coffee. Each of these can be used countably though, with a supermarket boasting of having a wide range of breads and cheeses or somebody ordering three coffees in a cafe.
The most effective way of presenting plural nouns is likely to be using visuals such as pictures or realia. As plurals are often taught on beginner courses, students should really know the singular form of the nouns for the objects selected. Typical items could therefore be classroom objects (pens, tables, students), fruit (apples, bananas, oranges), animals (dogs, cats, rabbits) or body parts (eyes, ears, legs).
During the presentation stage, you are likely to need to show students one of the item, and then two or more of the item.
Classroom/School (Photo) Hunt
Make a list of items that are in the classroom or in the school. The students could go around the school to see how many of different items they can find and write this number. They should then be able to report back with this number.
Alternatively, you could give students different lists and tell them to take pictures of the items. One student may have chair, for example, while another has chairs. Students need to pay attention to the plural forms to ensure their pictures are correct and they finish fastest.
Another variation on this would be provide instructions for a photograph to be taken that must include a number of objects, some of which are in the singular and others in the plural. Students should first find the items and then take a photograph that includes them all.
For this you need to prepare some pictures which have either a singular item on them or several of that item. Students then sort them into singular or plural.
Another sorting activity would be to sort the pictures according to their plural ending (-s, -es, -ves, -ies, or irregular).
To help students hear the plural forms you could use bingo. For this, you would need to either prepare bingo cards or have students write 5 nouns from a list (of about 12 – 6 x singular, 6 x corresponding plurals). You call the words one by one in any order (note the words you have said). The first student to hear all of their words and tick them on their bingo card or list wins. Students can then take over calling out the words in their group.
Alien Guess Who
For this game, you need to create a number of pictures of aliens with differing numbers of facial features. For example, one may have three noses and six eyes, while another has four mouths and eight ears. A student then chooses one of the aliens and the other students have to ask questions to work out which of the aliens they are thinking of.
Songs are a great source of language. Because listening is enjoyable (subject to taste), students often don’t mind listening to songs repeatedly. Therefore they come to know the lyrics, which if containing clear examples of a language point, provide memorable examples for students.
Virtually every song contains plural nouns, or at least singular nouns. One easy to generate activity could be to take the lyrics to a song that students like, replace the nouns with a choice between their singular and plural forms and give it to the students. Have students try to guess if it is plural or singular first from any contextual clues and then have them listen to check.
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Suffixes are added to the ends of words in order to change their root form. Sometimes nouns will become adjectives, sometimes verbs will become nouns, etc. We also use suffixes to create the plural forms of words, and to make sure word tenses remain consistent throughout a sentence. End all of your words the right way with these guys and some more tender loving care. These activity sheets will provide short sentences and prompts in order to help your students learn how to use suffixes correctly. Answer sheets have been provided. If you work through all of the worksheets, down below, you will start to see a pattern that is true to the English language.
A suffix comes at the end of a word. When the suffix "-er" is added to the end of a verb, it means one of the following.
"-er" is added to the end of a verb, it means one of the following: - Someone who does something - Something that does something - Someone who provides or is involved with something
Add the correct word part, -ful or -less to each term below to create a word that matches the meaning given.
You will choose from several different choices to create a new term by adding a suffix of your choice.
You can use suffixes to create new words. Choose one of the chunk that are provided to create a word that has the meaning provided. Write the term that you have created on the line.
When you add certain suffixes to nouns and verbs, you can create adjectives. You may have to change, drop, or add letters before adding the suffix to some words.
Add one of the four suffixes above to each word to create an adjective. Write the new term on the line.
You will be given a word and you will apply new parts to it and the end result will be a brand new term.
You will add suffixes to a given base term to create an entire new word. In some cases you will create a synonym in other case, it could be quite the opposite.
This is a series of additional tasks that track with the previous worksheet, but they do push it in a different direction.
This is going to be a familiar series of tasks to help you practice this skill in a few different ways.
Add suffixes to each base word. Write the new term on the first line. On the second line, write the part of speech that this new term represents.
Locate the suffix in each word. Write the suffix on the first line. What part of speech is it? Write the part of speech on the second line.
See how many words you can make that contain the suffixes you identified above. Write them on the lines.
Cut out the cards. Shuffle them, and place them face-down on a hard surface. On each turn, a player draws a card and says a word that contains the suffix on the card. Then the player keeps the card. Words cannot be repeated.
If the player is unable to think of a word, the card is returned face-down and put back into play.
Play continues until all cards have been drawn and a word has been said for each suffix.
When we add an ending (-s, -ed or -ing) to the end of a word, the spelling changes but the meaning does not. When we add a 's', we make the word a plural, meaning more than one.
Copy the sentences below, changing the underlined word to the correct usage by adding -ed or -ing.
What Are Suffixes?
Broadcast media has radically changed how people learn new things. Learning new languages has always been associated with different forms of media, and rightly so. We can all agree that our trusty vintage television helped us learn much more about English than Miss Nancy from first grade. People can quickly learn how to speak different languages but understanding the different complexities and rules of grammar lie in a whole other ballpark.
So today, we'll be talking about one of the foundations of the English language, suffixes.
To understand what suffixes are, we need to define an important term called the “affix.” You might recall the word from your school days, but what is it?
Affixes are words, letters, or groups of letters that can be added to a root word to form a different word or to change the root word’s meaning. There are two main kinds of affixes: prefixes and suffixes. Consider the root word form and the word conforming. Conforming is made by adding the prefix con- before the root word and the suffix -ing after it.
Most commonly they are used to broaden our use of language and spice up the things we say, read, and write. They are often used to display a part of speech of a word. They also elude to the verb tense of the words and can tell us if a word is singular or plural. A good number of suffixes have multiple meanings and we need to see where they sit within the context of the sentences to determine their meaning. These word parts also have a way of changing the spelling of words. Some words would not be words without the presence of a suffix. An academic example is the word unforgettable. We cannot unforget something, but something can be unforgettable for certain.
Therefore, we can define a suffix as a letter, or group of letters, that is added at the end of a word to form new words or change the root word’s meaning and function. They can be as small as a single letter or as big as a four-letter word. However, there’s a stark difference between words that form suffixes and prefixes.
Words that form prefixes can sometimes make sense independently, like the prefix per- in the word perform. However, words that form suffixes don’t typically mean anything on their own, like the suffix -ing in the word suffering.
Take a look at the following letters that are used to form suffixes:
|-s||runs, girls, picks|
|-ly||bravely, gravely, markedly|
|-ment||enjoyment, movement, achievement|
Importance and Function
Suffixes take a base or root word and change its meaning or grammatical function. Therefore, suffixes hold extreme significance in English as they enrich the vocabulary. Using just one root word, you can form several associated new words by adding a simple suffix at the end. An easy example is the word art; take the suffix -ist, and you get artist, a person skilled in art!
Similarly, suffixes are also used to form comparative adjectives. For instance, add the suffixes -er and -est to the adjective brave, and you get the comparative braver and the superlative bravest. Let’s explore some examples of suffixes in detail.
Examples of Suffixes
There are different suffixes, just as there are different kinds of words. Suffixes can modify nouns, verbs, and adjectives, or they can also modify them to change their forms. For example, the word read can be changed into the noun reader by adding the suffix -er. Here’s a list of some commonly used suffixes:
In Noun Form
|-age||a particular condition||marriage, bondage|
|-ee||performer of an action||referee, interviewee|
In Verb Form
|-ate||do something||exterminate, congratulate|
|-ize||to become||metamorphosize, humanize|
In Adjective Form
|-ous||characterized by an attribute||joyous, pious, religious|
|-istic||to characterize by a trait||pessimistic, optimistic|
Learning about suffixes helps us expand our vocabulary. They’re easy to explain and can also aid children in faster learning. A little kid who only knows the word pay can use different suffixes to form the words payer, payment, paying, payable, and payee all in one go!
The English language isn’t that challenging to understand once you grasp the important concepts. Educationists can help young individuals advance their language prowess using easy-to-understand concepts like the use of affixes. We hope you now know what suffixes are and will use them to help others!
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To introduce the notation for recursive patterns, begin by asking
students to find patterns and predict the next number in the following
- 5, 8, 11, 14, …
- 4, 6, 8, 10, 12, …
- 10, 20, 30, 40, …
Give students some time to work through these examples and
ask them to write down how they got their answers. Have students share
their answers with the class. Students may have different answers, so
allow students who have different methods to explain how they got their
answers. Correct any misconceptions.
Discuss students' responses with the following:
- The next number in the first pattern is 17 (add 3 to the previous number).
17 = 14 + 3
Ask them how to get the next number: 20 = 17 + 3
Students should be able to explain that to get the next number, you add 3 to the previous number.
In mathematical language this is An = An – 1 + 3 and is called a recursive rule. Recursion is the root of recursive.
|Recursion is defining the next (subsequent) term in a pattern (sequence) by using the term(s) that came before it.|
The terms subsequent and sequence are formal
language that can be substituted into the definition if a formal
definition is needed. If students are not used to these formal words,
it may be helpful to define math terms using their vocabulary.
- The next number in the second pattern is 14 (add 2 to the previous number): 14 = 12 + 2
Ask them how to get the next number: 16 = 14 + 2
Tell them that in order to get the next number, you keep adding 2 to the previous term. In mathematical language this is An = An – 1 + 2
- In the third pattern, the next number is 50 (add 10 to the term before).
Ask students to write this pattern using the new recursion notation.
Check that students know what the subscripts refer to.
At your discretion, you may choose to talk about the first term as the
“0th” term or as the “1st” term. Note that students easily grasp that
the first term is the “1st” term or n = 1, but may have trouble with n = 0 as the 1st term.
There is a rule that can be applied to all 3 of the patterns. Wait for students to think about this. If necessary, suggest the next number in any pattern is double the term before, minus the term before that.
Ask students to try it out for each one. Check that they see how this works.
- From pattern 1: 2 × 14 – 11 = 17
- From pattern 2: 2 × 12 – 10 = 14
- From pattern 3: 2 × 40 – 30 = 50
Challenge the class find the recursive rule of An = 2An – 1 – An – 2 and explain that n – 2 refers to the term that is 2 before.
To check for understanding, ask them to write a rule for 3, 7, 11, 15, ….
Ask students to find the next term and to write 2 recursive rules. An = An – 1 + 4 and An = 2An – 1 – An – 2 will both work. If they need more examples, use either or both of the following:
- 6, 11, 16, 21, …
- 22, 25, 28, 31, …
Before moving on, make sure to explain recursion again, and make sure students have a working definition written down.
Finding Recursive Patterns Using Trains
Explain to students that they will use two lengths of cars to form trains:
Show students the trains below and explain that even though they use the same cars, they are two different trains.
| || || |
|a 4-train made from 2 cars of length 1 and 1 car of length 2 || ||a 4-train train made from 1 car of length 1, 1 car of length 2, and another car of length 1 |
Explain that the train below is a train of length 5 made from 1 car of length 1 and 2 cars of length 2.
Give students an opportunity to ask questions, and allow them to
hold the trains if it helps them to explain their thinking. Emphasizing
the difference between train length and car length at
the beginning helps students to talk about the different types of
trains. It gives them a common language. To assess their understanding,
hold up a train and ask students to describe it to you. Repeat this
until all students understand. Then, have students work in groups
of 2–4 to build the trains. This will decrease the number of train
combinations you will have to check.
Distribute the following: train sets; Counting the Trains
activity sheet; grid paper; and colored pencils, markers, or crayons.
Before the lesson, prepare set of Cuisenaire train cards (or paper
strips if you are using them as substitutes). Make sure that each set
has 50 cars of length 1, 20 cars of length 2, 15 cars of length 3,
10 cars of length 4, and 5 cars of length 5.
Students who can’t see certain colors could put a number in each
car. Explain that studnets should use the grid paper and the colors to
record their trains. Remind students that they must build and record
their trains. It might help to ask, "If I were to take your cars away,
would you still be able to tell me what ALL the combinations you built
were by reading what is on your paper?" Suggest that they use black to
represent the white trains to avoid problems with recording white
Make sure that students have built and recorded all of the
different trains they need to complete Question 1 on the activity
sheet. When students say they have them all, be sure to ask how they
know they have them all. If they are missing trains, you can tell them
that they are missing some. You might need to point out another train
that has the same cars or point to another train of a different length
that looks similar. When all else fails, give them the pieces they need
and let them build the train(s) they are missing.
If some students finish building all the trains sooner than others,
encourage them to move on to the subsequent questions. Students who
complete the activity sheet before others are done can always do the Extension activities listed below.
Lead a whole group discussion, allowing students to present how they
figured out the number of trains of length 5 and including the
Students should be able to see that the trains of length 5 are made
from adding the trains of length 3 and length 4, and see that A5 = A4 + A3. Ask them to explain how they made trains of length 6 and to show you the recursive rule (A6 = A5 + A4).
At this point, ask students how to find the trains of length n. They should be able to tell you that you take the train before (n – 1) and the train before that (n – 2), which leads to the rule An = An – 1 + An – 2.
Ask students to present or explain their table and scatter plot. Have
students discuss any difficulties they ran into in making the table.
You may want to point out that train length should be on the x‑axis
since it is the independent variable (save this discussion if this is
not part of your curriculum). When discussing the scatter plot, ask
students if they think the graph is linear. Have them explain their
reasoning. Depending on the students’ exposure to exponential
functions, they may be able to explain why the function is exponential.
The common ratio for this function is approximately (1+√5)/2
(which is about 1.618) and is directly related to Fibonacci numbers. A
discussion of finding a regression line and a common ratio may be
appropriate, depending on students’ prior experience with regression
and exponential functions. You may want to ask students to calculate an
exponential regression for the data in Questions 4 and 5 on their
Directions for calculating a regression line on the TI-83/TI-84 calculator are available on the Using the TI-83 or TI-84 for Regression sheet. The calculator gives y = 0.685 · 1.632x as a regression line with an r‑squared
value of 0.998, which indicates that it is a very good fit. If students
use more data points, the regression comes closer and closer to
To help get at why this is, use the Golden Ratio lesson in the Extensions section. The Fibonacci Rabbits
activity sheet from that lesson can also be used as an assessment.
However, More Trains, the next lesson in this unit, involves another
pattern that looks exponential and will lead students to a better
understanding of lines of best fit.
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Our Regular and Irregular Verbs lesson plan teaches students about both regular and irregular verbs, including what they are and how they are used. During this lesson, students are asked to work collaboratively with their classmates to use a specific list of verbs in a conversation with each other, demonstrating that they understand how to use them. Students are also asked to fill in different tenses of specific verbs (for example, they need to give the past participle form of the verb “fill”).
At the end of the lesson, students will be able to form and use regular and irregular verbs.
Common Core State Standards: CCSS.ELA-LITERACY.L.3.1.D
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The Basics of Conditional Expressions: Exploring the Concept
A conditional expression, also known as a conditional statement or a ternary operator, is a concept widely used in programming languages to make decisions based on conditions. It allows programmers to write code that performs different actions depending on whether a certain condition is true or false. By incorporating conditional expressions into their code, developers can create more flexible and dynamic programs.
The syntax of a conditional expression typically consists of three components: the condition, the expression to be executed if the condition is true, and the expression to be executed if the condition is false. In most programming languages, including Python, the syntax follows the format "condition ? expression_if_true : expression_if_false". This concise syntax makes conditional expressions a powerful tool for writing efficient and readable code. By understanding the concept and syntax of conditional expressions, developers can unlock the full potential of their programming language and build more sophisticated applications.
Syntax Breakdown: How to Write a Ternary Operator in Python
To write a ternary operator in Python, the syntax follows a specific structure. It consists of three elements: the condition, the value to return if the condition is true, and the value to return if the condition is false. The overall syntax looks like this:
value_if_true if condition else value_if_false
Let's break down the components further. The "condition" is any expression that evaluates to either True or False. It can be a simple comparison, such as x > y, or a more complex expression involving logical operators, such as x > y and z < w. The "value_if_true" is the result to return if the condition is true, and "value_if_false" is the result to return if the condition is false. These values can be any valid Python expression, from a single variable to a complex calculation.
When writing a ternary operator, it is important to remember that the condition must be enclosed in parentheses. This ensures that the condition is evaluated as a single expression. Moreover, it is worth noting that the ternary operator in Python is read from left to right, similar to a normal sentence. This allows for a more intuitive understanding of the code's logic. By following these syntax guidelines, you can effectively write ternary operators in Python for concise and efficient decision-making code.
Evaluating Conditions: Working with Boolean Expressions
Boolean expressions are an essential part of programming as they allow us to evaluate conditions and make decisions within our code. In Python, boolean expressions are built using comparison operators such as "==", ">", "<", etc., that compare two values and return either True or False. These expressions can be used in various scenarios, such as conditional statements, loops, and logical operations.
When evaluating conditions using boolean expressions, it's important to understand the concept of truthiness and falsiness. In Python, certain values are considered "truthy" and will evaluate to True, while others are considered "falsy" and will evaluate to False. For example, any non-zero number, non-empty strings, non-empty lists, and non-empty dictionaries are considered truthy. On the other hand, the numbers 0 and 0.0, an empty string, an empty list, an empty dictionary, and the value None are considered falsy. Understanding these concepts is crucial when working with boolean expressions to ensure accurate condition evaluation.
Utilizing the Ternary Operator for Simpler Decision Making
The ternary operator is a powerful tool in Python that allows for simpler decision making within code. By leveraging this operator, developers can write more concise and efficient conditional expressions, reducing the need for lengthy if-else statements. The syntax of the ternary operator is straightforward, consisting of three parts: the condition, the expression to be executed if the condition is true, and the expression to be executed if the condition is false.
One of the advantages of utilizing the ternary operator is its ability to streamline code and improve readability. Instead of writing multiple lines of if-else statements, developers can condense their decision-making logic into a single line. For example, instead of writing:
if x > 0: result = 'Positive' else: result = 'Non-positive'
The ternary operator allows us to write:
result = 'Positive' if x > 0 else 'Non-positive'
This not only reduces the number of lines and improves code readability, but also makes the intention of the code more explicit. The ternary operator is particularly useful when the conditions are simple and the outcome is a single expression. However, it's important to use it judiciously and consider readability when dealing with complex conditions or multiple expressions. By leveraging the ternary operator effectively, developers can enhance code efficiency and simplify decision making in Python.
Importance of Ternary Operator: Streamlining Code Efficiency
The ternary operator in Python is a powerful tool for streamlining code efficiency. By providing a concise way to make decisions within a single line, it allows for more compact and clear code. This can greatly enhance the readability and maintainability of a program, especially when dealing with simple conditions where a full if-else statement would be excessive.
In addition to improving code readability, the ternary operator also helps in improving code efficiency. By avoiding the need for explicit branching with if-else statements, the ternary operator reduces the number of lines and unnecessary computations, resulting in faster and more efficient code execution. This is particularly important when working with large datasets or time-sensitive operations, where even small gains in efficiency can have a significant impact. Overall, the ternary operator is a valuable tool in Python for optimizing code and improving overall performance.
Handling Complex Conditions: Nesting Ternary Operators
In order to handle more complex conditions, programmers often employ the technique of nesting ternary operators within one another. This allows for multiple conditions to be evaluated and decisions to be made based on the outcome of those evaluations. By nesting ternary operators, developers can effectively create more intricate decision-making processes within their code.
When nesting ternary operators, it is important to keep the code readable and maintainable. It is recommended to use parentheses to clearly define the order of evaluation and to separate different levels of nesting. By doing so, it becomes easier for future programmers to understand the logic behind the nested ternary operators and to make any necessary changes or modifications. However, it is important to note that nesting ternary operators can quickly become convoluted and decrease code readability, so it should be used judiciously.
Best Practices: When to Use and When to Avoid Ternary Operators
Before diving into the best practices for using ternary operators, it's important to understand that these practices are not definitive rules set in stone. Instead, they serve as guidelines to help you make informed decisions based on the specific context and requirements of your code.
One of the primary considerations when deciding whether to use a ternary operator is code readability. Ternary operators can be concise and offer a compact way to express simple conditions. However, if the condition becomes too complex or involves multiple operations, it may be more appropriate to use an if-else statement. Opting for an if-else statement in such cases can enhance the readability and maintainability of your code, making it easier for both you and other developers to understand and debug in the future.
Another aspect to consider is the potential impact on code maintainability. While ternary operators can be a powerful tool, they have the potential to make code harder to understand, especially if used excessively or inappropriately. Overusing ternary operators can lead to code that is difficult to follow and modify, which can make it prone to errors or make it harder to maintain in the long run. It's important to strike a balance and evaluate the trade-offs between conciseness and readability in each specific code scenario.
Ternary Operator vs. If-Else Statement: A Comparison
Ternary operators and if-else statements are both powerful tools in Python for making decisions based on certain conditions. However, there are some key differences between the two that can affect the way you write and structure your code.
One major difference is the syntax. Ternary operators typically have a more concise and compact syntax compared to if-else statements. They reduce the number of lines needed to achieve the same result, making the code appear cleaner and more streamlined. On the other hand, if-else statements offer more flexibility and can handle more complex conditions with multiple elif branches.
Another difference is the readability and understandability of the code. Ternary operators are known for their brevity, which can be a double-edged sword. While they can make simple decisions very readable, they can become difficult to understand when nested or used for longer expressions. If-else statements, on the other hand, have a more explicit structure, making it easier to follow the flow of the code and understand the logic behind the decision-making process.
In conclusion, the choice between ternary operators and if-else statements ultimately depends on the specific requirements of your code and your personal coding style. Ternary operators offer a concise and clean syntax for simple decisions, while if-else statements provide more flexibility and readability for complex conditions. It is important to consider the context, maintainability, and understandability of your code when deciding which approach to use.
Examples and Use Cases of Ternary Operators in Real-World Python Code
Ternary operators can be incredibly useful in real-world Python code, providing a concise and efficient way to make decisions. One common use case is in assigning values based on conditions. For example, let's say we have a variable called "temperature" that represents the current temperature. We can use a ternary operator to determine whether it is hot or cold, and assign an appropriate string value to another variable called "weather_condition". The code would look like this:
weather_condition = "hot" if temperature > 25 else "cold"
In this example, if the temperature is greater than 25, the value of "weather_condition" will be set to "hot"; otherwise, it will be set to "cold". This is a simple and readable way to assign values based on conditions, making the code more efficient and easier to understand.
Tips and Tricks: Enhancing Readability and Understanding of Ternary Operators
Enhancing the readability and understanding of ternary operators can greatly improve the overall quality of your code. One useful tip is to use parentheses to clearly separate the different components of the expression. This can help to avoid any confusion or ambiguity, especially when dealing with complex conditions or nested ternary operators. By using parentheses, you can clearly indicate the grouping of different parts of the expression, making it easier for others (and even yourself) to comprehend the logic behind it.
Another trick to enhance readability is to avoid relying too heavily on nested ternary operators. While they can be useful in certain situations, they can quickly become difficult to follow if nested too deeply. Instead, consider breaking down complex conditions into multiple steps using if-else statements. This not only improves readability but also allows for easier debugging and maintenance in the long run. Remember, the goal is to write code that is not only efficient but also clear and understandable by others who may need to work with it in the future.
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(Python Tutorial – 017)
In this tutorial, you will learn to create a recursive function (a function that calls itself).
What is recursion?
Recursion is the process of defining something in terms of itself.
A physical world example would be to place two parallel mirrors facing each other. Any object in between them would be reflected recursively.
Python Recursive Function
In Python, we know that a function can call other functions. It is even possible for the function to call itself. These types of construct are termed as recursive functions.
The following image shows the working of a recursive function called
Following is an example of a recursive function to find the factorial of an integer.
Factorial of a number is the product of all the integers from 1 to that number. For example, the factorial of 6 (denoted as 6!) is 1*2*3*4*5*6 = 720.
Example of a recursive function
def factorial(x): """This is a recursive function to find the factorial of an integer""" if x == 1: return 1 else: return (x * factorial(x-1)) num = 3 print("The factorial of", num, "is", factorial(num))
The factorial of 3 is 6
In the above example,
factorial() is a recursive function as it calls itself.
When we call this function with a positive integer, it will recursively call itself by decreasing the number.
Each function multiplies the number with the factorial of the number below it until it is equal to one. This recursive call can be explained in the following steps.
factorial(3) # 1st call with 3 3 * factorial(2) # 2nd call with 2 3 * 2 * factorial(1) # 3rd call with 1 3 * 2 * 1 # return from 3rd call as number=1 3 * 2 # return from 2nd call 6 # return from 1st call
Let’s look at an image that shows a step-by-step process of what is going on:
Our recursion ends when the number reduces to 1. This is called the base condition.
Every recursive function must have a base condition that stops the recursion or else the function calls itself infinitely.
The Python interpreter limits the depths of recursion to help avoid infinite recursions, resulting in stack overflows.
By default, the maximum depth of recursion is 1000. If the limit is crossed, it results in
RecursionError. Let’s look at one such condition.
def recursor(): recursor() recursor()
Traceback (most recent call last): File "<string>", line 3, in <module> File "<string>", line 2, in a File "<string>", line 2, in a File "<string>", line 2, in a [Previous line repeated 996 more times] RecursionError: maximum recursion depth exceeded
Advantages of Recursion
- Recursive functions make the code look clean and elegant.
- A complex task can be broken down into simpler sub-problems using recursion.
- Sequence generation is easier with recursion than using some nested iteration.
Disadvantages of Recursion
- Sometimes the logic behind recursion is hard to follow through.
- Recursive calls are expensive (inefficient) as they take up a lot of memory and time.
- Recursive functions are hard to debug.
Disclaimer: The information and code presented within this recipe/tutorial is only for educational and coaching purposes for beginners and developers. Anyone can practice and apply the recipe/tutorial presented here, but the reader is taking full responsibility for his/her actions. The author (content curator) of this recipe (code / program) has made every effort to ensure the accuracy of the information was correct at time of publication. The author (content curator) does not assume and hereby disclaims any liability to any party for any loss, damage, or disruption caused by errors or omissions, whether such errors or omissions result from accident, negligence, or any other cause. The information presented here could also be found in public knowledge domains.
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Learn Java by Example: Java Program to Find Factorial of a Number Using Recursion
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In these Year 2 verbs in the present tense worksheets, your pupils explore the dynamic world of verbs by circling the word written in the present tense in pairs of words. Following this, children encounter a list of words and a set of sentences with missing parts, prompting them to choose the correct word to ensure grammatical correctness in terms of present tense. To cap it off, your class is presented with sentences, and their task is to underline the verb in each one.
This comprehensive worksheet elevates the understanding of verbs to the next level, helping learners realise that verbs can be wielded in the present tense to vividly convey “something happening now”.
Our Year 2 grammar worksheets are aligned with the KS1 primary national curriculum and can be used with your ideas for primary learning activities, differentiation, homework and lesson plans.
You may also like our Year 2 verbs in the past tense worksheets if you like this.
Explore our full collection of Year 2 English worksheets.
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Lesson Plans and Worksheets
Browse by Subject
Common Noun Teacher Resources
Find Common Noun educational ideas and activities
In this proper nouns and common nouns usage worksheet, students identify nouns in sentences, label common and proper nouns in sentences, fill in blanks with common or proper nouns, and choose multiple choice answers that are names of proper nouns. Students write forty-one answers.
In this recognizing proper nouns and common nouns instructional activity, students identify proper and common nouns in sentences by labeling underlined words in sentences, correcting capitalization mistakes, writing proper nouns for given categories, and reviewing and assessing knowledge. Students write forty-six answers.
Fifth graders review what nouns, pronouns and verbs are though grammar games. For this grammar lesson, 5th graders identify proper and common nouns, singular and plural nouns and verb agreement with nouns. Students make an original slide show that explains each of these topics.
Reinforce nouns in your eighth grade classroom with an introductory conventions lesson plan. After reviewing the definitions for singular, plural, and collective nouns, kids view a series of slides to determine which nouns belong in which category. Turn the lesson plan into a grammar game for review.
Fourth graders complete a writing assignment that focuses on using quotation marks and common and proper nouns within the context of writing a story. They listen to the book "The Snowbelly Family of Chillyville Inn," discuss when to use quotation marks and proper and common nouns, and write a Christmas story.
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In this lesson, we will focus on indirect questions in the English language.
A question is an interrogative expression often used to find out or confirmation information. There are a lot of different types of questions in English such as object, subject, duration, tag, negative, indirect, open, closed, etc… .
An indirect question is a grammatical structure in which two clauses are joined together to form a question. Indirect questions are used to express extreme politeness when asking someone who you do not know an interrogative question.
A direct question is considered to be very direct and informal. They are normal questions that we can ask friends, family members, and people who we know well.
An indirect question is considered to be more formal and polite. We normally ask people who we do not know or have a relationship with an indirect question.
There are several key clauses, which are used to ask indirect questions:
Do you know … ? Can you tell me … ? Could you tell me … ?
I would like to know …
There are three important grammatical changes between direct and indirect questions.
1. Word Order
When we start using an indirect question form, the word order in the second clause is the same as a positive statement, not a question.
|Direct:||What time is it?|
|Indirect:||Could you tell me (what) time it is?|
|Incorrect:||Could you tell me what time is it?|
2. Auxiliary Verbs (did, do, does)
If the direct question uses the auxiliary verb ‘do’ (i.e. does, did, do), it is left out of the indirect question.
|Direct:||Where does she live?|
|Indirect:||Do you know where she lives?|
|Incorrect:||Do you know where does she live?|
|Direct:||How many replacement parts do you need?|
|Indirect:|| I would like to know (how many) replacement parts |
|Incorrect:||I would like to know how many replacement parts do you need.|
3. ‘Yes’ or ‘No” responses
If the direct question can be answered with ‘Yes’ or ‘No’, the indirect question needs ‘if’ or‘whether.’
|Direct:||Does the manager attend the monthly meetings?|
|Indirect:||Do you know (whether) the manager attends the monthly meetings?|
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Comparing and Ordering Fractions, Mixed Numbers, and Decimals
Decimal numbers are another way to write fractions or mixed numbers. You can calculate with decimals as you do with whole numbers because they follow the same place-value rules as whole numbers. A digit in any place has a value 10 times the value of the digit one place to the right.
The 1 in the hundreds place has a value of 100ten times the value of the 1 in the tens place, which has a value of 10. The 1 in the ones place has a value of 1ten times the value of the 1 in the tenths place, which has a value of 0.1, or
Students may have trouble reading a decimal number. The biggest confusion seems to come with the use of the word and. It's best to reserve this word to denote the decimal point, trying not to use it when reading a whole number.
Read 125 as "one hundred twenty-five."
Read 100.25 as "one hundred and twenty-five hundredths."
Read 0.12 as "twelve hundredths."
It's also common to hear the decimal point read as point, followed by a listing of the digits in order: zero point one two. Try not to do this when you're teaching students about decimals and fractions as it undermines your efforts to establish the relationship between the two forms.
A good model for decimal numbers is a square grid. Use
This grid represents one whole (1) or 100 of 100 equal parts,
This grid represents 1 hundredth (0.01). As a fraction, 1 part of 100 equal parts is
This grid represents 10 hundredths or 1 tenth
These grids represent 1 and 25 hundredths, 1.25, and
Students will soon see that they don't need to depend on the grids to compare decimal numbers because they can compare and order decimals just as they compare and order whole numbers: Line up the digits. Begin in the greatest place. Find the place where the digits are different. Compare. Just as 8 ones is greater than 6 ones, 8 tenths is greater than 6 tenths. The only time this method might lead to confusion is when the decimal numbers, like 0.15 and 0.6, don't have the same number of digits. If you try to align the digits on the right, you might think that 0.15 is greater than 0.6. It's extremely important for students to understand that when they line up whole numbers for comparison or computation, they're lining up the ones places. When they compare decimals, if they always line up the decimal points, they won't get confused. A place-value chart can help them keep things straight. Students who still have difficulty can use zeros as placeholders.
As with whole numbers, start comparing the digits in the greatest place. Both numbers have the same number of ones, so compare tenths. Since 6 tenths is greater than 1 tenth,
This model is especially useful when comparing decimals and a mix of fractions with different denominators. As you move from left to right on a number line, numberswhether they're fractions, decimals, or mixed numbersincrease or become greater. As you move from right to left, numbers decrease or become less.
Another method is to use a place-value chart. For example, order these numbers from least to greatest: 1.40,
First, change the fractions to decimals. Write the decimals in hundredths. To compare, begin in the greatest place. Compare the ones. They are the same. Compare the tenths: 5 tenths are greater than 4 tenths, 4 tenths are greater than 2 tenths, so the final order from least to greatest is
Because decimal numbers follow similar rules as whole numbers, you add and subtract decimal numbers like whole numbers. To be sure the place values line up, first line up the decimal points. You may wish to write zeros as placeholders if needed. Then add or subtract as you would with whole numbers. Write the decimal point in the answer. Estimate to check addition; add to check subtraction.
Rounding and estimating with decimal numbers follow similar procedures to rounding and estimating with whole numbers. When rounding, find the place you want to round to. Look at the digit to the right. Round as you do with whole numbers. Rounding to the nearest whole number:
15.5 rounds to 16
Rounding to the nearest tenth:
1.55 rounds to 1.6
To estimate sums and differences with decimal numbers, have students first round each decimal to the nearest whole number and then add the rounded numbers.
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Common Core Standards: Math
1. Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions.
We've finally gotten to the coolest part of statistics. You know we're right, but your students might not. Time to convince them.
Students should be able to predict the outcome of an event and express it numerically. That means that if you have ten chocolates of different flavors and you pick one at random, there will be a 1 in 10 chance or a probability of 0.1 that you'll get the one chocolate you actually like. (So it's not Murphy's Law. It's just statistics.)
But life isn't like a box of chocolates (or is it?), and numbers aren't always that pretty. In that case, students should know to use the equation P(X = a) = nCa • pa • qn – a, where P is our probability, n is our total number of trials, X is our event, a is the number of successes of our event, and q is the probability of failure. Also, remember that
(As a side note, those exclamation points are factorials, not our attempt at making things more exciting.)
So why is this the coolest part of statistics? The pretty pictures, of course. Using "graphical displays" just means making charts that represent the probability of our outcome occurring a out of n times, where a goes from 0 to n. That's called a probability distribution, and we usually use histograms (cousin of the bar graph) for these purposes.
For instance, let's take a fair six-sided die. If we roll it 12 times, what is the probability that we'll never get the number 1? Or that all 12 rolls will result in a 1? The probability distribution will tell us all that and then some.
The best way to teach this concept to your students is through as many examples as possible. First, try to see if they can logically understand where the peak of P(X) should occur, and then have them actually calculate and graph it out. With enough practice, they'll be ready to tackle any box of chocolates they come across.
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http://www.shmoop.com/common-core-standards/ccss-hs-s-md-1.html
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7.11. Format operator¶
The format operator,
% allows us to
construct strings, replacing parts of the strings with the data stored
in variables. When applied to integers,
% is the modulus
operator. But when the first operand is a string,
% is the
The first operand is the format string, which contains one or more format sequences that specify how the second operand is formatted. The result is a string.
For example, the format sequence
%d means that the second operand
should be formatted as an integer (“d” stands for “decimal”):
>>>camels = 42 >>>print('%d' % camels) 42
The result is the string “42”, which is not to be confused with the integer value 42.
A format sequence can appear anywhere in the string, so you can embed a value in a sentence:
If there is more than one format sequence in the string, the second argument has to be a tuple [A tuple is a sequence of comma-separated values inside a pair of parentheses. We will cover tuples in Chapter 10]. Each format sequence is matched with an element of the tuple, in order.
The following example uses
%d to format an integer,
%g to format
a floating-point number (don’t ask why), and
%s to format a string:
The number of elements in the tuple must match the number of format sequences in the string. The types of the elements also must match the format sequences:
>>> '%d %d %d' % (1, 2) TypeError: not enough arguments for format string >>> '%d' % 'dollars' TypeError: %d format: a number is required, not str
In the first example, there aren’t enough elements; in the second, the element is the wrong type.
The format operator is powerful, but it can be difficult to use. You can read more about it at
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o Order of operations
o Review the order of operations in the context of algebra
o Learn how to manipulate expressions in a way that maintains an equality
o Understand why substitution can be used in algebra
An important skill in algebra is the ability to perform mathematical manipulations of expressions and equalities. When we deal with simple numbers, such as 2 + 9 = 11, there is seldom a need to make changes to simplify or expand the expressions (2 + 9 and 11 are equivalent expressions). If we deal with an algebraic expression, such as 2 + x = 9, then we must extend our ability to perform arithmetic operations to deal with variables as well as numbers.
Order of Operations
It is helpful to briefly review the order of operations in the context of algebra. When performing any series of operations expressed in mathematical form, perform them in the following order:
1. Any operations in parentheses
Within a set of parentheses, there may be a series of operations-in this case, simply start back at the first step for the expression in the parentheses.
Practice Problem: Evaluate the following expression using x = 4: 3x2 - (4x + 1).
Solution: Use the rules for order of operations. First, evaluate the expression in parentheses, using x = 4:
4x + 1 = 4(4) + 1 = 16 + 1 = 17
Next, evaluate the exponent in the first term:
x2 = (4)2 = 16
Let's now take a look at the full expression:
3(16) – 17
Now, evaluate the product of 3 and 16, then subtract 17.
3 16 – 17 = 48 – 17 = 31
The solution is then 31.
First, we need to recognize the importance of the "equal sign" (=) in math: if two expressions are equal, then they are indeed equal. Although this fact may seem obvious, it has several important implications for algebra. Let's take the following equality as an example for our discussion:
Remember, both sides of this equality are equal to one another. As a result, if we perform the same operation, whatever it is, on both sides of the equation, the resulting expressions will still be equal. For instance, if we have two bananas on the left and two bananas on the right, then the numbers of bananas on each side are equal. Thus, if you take one banana away from the left and also take one away from the right, then you still have the same number on both sides. Indeed, if you add x bananas to the left and you add x bananas to the right, you still have the same number on each side, even if you don't know what x is.
So, let's consider our algebraic expression example above. Since both sides are equal, we can do whatever we want to the expressions and still have an equality as long as we perform the same operations, in the same sequence, on both sides. (Note that the sequence is important--the best way to approach this process is to perform one and only one operation on both sides before performing a different operation.) What if we decide to add 2 to both sides?
The resulting expressions are still equal, because we performed the same operation on both sides. We can even simplify the result as follows (using associativity and commutativity).
These expressions are all still equal. Although the expressions have changed, the equality has not. Note carefully that in this case we have modified the equality so that the constant number value only appears on the right side. Let's do the same for multiples of x by subtracting 3x from both sides of the equality. We will then apply our rules of real numbers.
So, we can perform operations on both sides of the equal sign regardless of whether they involve just numbers or numbers and variables, and still maintain the equality. Note that the final result above has only three terms total: this is a simpler equality than the one we started with (which had five terms).
We can also perform multiplication and division operations in the same manner. For instance, consider the following equality.
Since these expressions are in the form of fractions, they can be difficult to work with. We can simplify them, however, by applying what we know about equalities. Let's first multiply both sides by 5x; the equality will still hold. Again, we'll use our rules for real numbers: associativity, identity, and distributivity.
So, multiplication and division (note that 5x divided by 5x is just 1) work the same way for algebraic expressions containing numbers as they do for simple numbers. As you become more familiar with these and other algebraic operations, you will become more able to perform them without needing to recognize the specific rules (for instance, associativity) that justify them.
Although it may or may not seem obvious, there is one additional point that we need to consider regarding equalities. Let's say a particular equality contains a variable x. The values of x that satisfy this equality also satisfy other equalities that are derived from this equality using (most) legitimate algebraic operations. (There are some limitations to this statement, but they can typically be addressed case by case.) Let's consider an example: 3 + x = 5. Obviously, the value x = 2 satisfies this equality, since 3 + 2 = 5. What if we perform some algebraic operation, however? Let's add 2 to both sides, then multiply both sides by 5.
3 + x + 2 = 5 + 2
x + 5 = 7
5(x + 5) = 5(7)
5x + 25 = 35
We know that 10 + 25 = 35, so we know that 5x must be equal to 10. Thus, x = 2 once more. We can note from this example that the correct value (or values) for x does not change if the equality is manipulated in a legitimate manner.
Practice Problem: Justify the labeled steps in the following sequence of algebraic manipulations:
Solution: To solve this problem, we must recall the basic rules discussed in the previous chapter and apply them to algebra. In the case of A, two terms are reversed, so the justification is commutativity. For B, the factor t is distributed as a factor to each term in parentheses, so this step is justified by distributivity. In C, the product of t and is 1, which is the property of (multiplicative) inverses. The step corresponding to D can be justified by associativity, and the step corresponding to E can be justified by (additive) inverses.
As noted above, we can make changes to an equality and still maintain the equality. Likewise, we can also substitute one variable or variable expression for another variable or variable expression. For instance, let's say we have the following equality.
Now, let's say that we want to change x into a different form (for whatever reason). Let's say we want x to be called z instead--so, x = z. Then,
is the same as the previous expression. On the other hand, we might say that x = 3y, so that
Again, this expression maintains the equality. If a particular value of y satisfies this equality, then the corresponding x value (using the relationship x = 3y) also satisfies the preceding expression.
Practice Problem: Substitute p = 3n + 4 into the expression p + 3 – .
Solution: The problem statement gives us a new variable expression that we can substitute into the overall expression as follows. Make the substitution for each instance of p. (Note, however, that there is nothing fundamentally wrong with making the substitution for only one instance of p; this leaves us with a new expression containing two variables instead of one. Because the variables are related, however, it is simplest to have the final expression in terms of just one variable.)
p + 3 – = (3n + 4) + 3 –
We can simplify this result somewhat using our rules of manipulation:
(3n + 4) + 3 – = 3n + (4 + 3) – Associativity
3n + (4 + 3) – = 3n + 7 –
This is one potential expression that results from the substitution described in the problem.
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Our Introduction to Syllables lesson plan teaches students all about syllables, including what they are and how to determine the number of syllables in a word. During this lesson, students are asked to work with a partner to practice looking at a list of words, reading them aloud, determining the number of syllables, and then sorting them into different categories based on the number of syllables. Students are also asked to read a short story and circle all of the single syllable words, demonstrating their understanding of the lesson material.
At the end of the lesson, students will be able to define syllable and identify the number of syllables in grade-appropriate words.
Common Core State Standards: CCSS.ELA-LITERACY.RF.1.3.D
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https://learnbright.org/lessons/reading/introduction-to-syllables/
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What Are Equivalent Fractions? Explained For Primary School
Equivalent fractions come up a lot in KS2 maths and some children, parents, and even teachers at primary school can be a little unsure as to what they are and how to find them. This article aims to make things a little clearer.
- What are equivalent fractions?
- To understand equivalent fractions, make sure you know the basics of fractions
- Examples of equivalent fractions
- How to work out equivalent fractions
- When do children learn about equivalent fractions in primary school?
- How do equivalent fractions relate to other areas of maths?
- Equivalent fractions questions
- Equivalent fractions resources
What are equivalent fractions?
Equivalent fractions are two or more fractions that are all equal even though they different numerators and denominators. For example the fraction 1/2 is equivalent to (or the same as) 25/50 or 500/1000.
Each time the fraction in its simplest form is ‘one half’.
Remember, a fraction is a part of a whole: the denominator (bottom number) represents how many equal parts the whole is split into; the numerator (top number) represents the amount of those parts.
To understand equivalent fractions, make sure you know the basics of fractions
If the concept of equivalent fractions already sounds a bit confusing and you’re not yet clear on what the difference is between whole numbers, denominators of a fraction and different numerators you may want to loop back to our fractions for kids article.
This breaks down the first fraction steps that Key Stage 1 and Key Stage 2 children must take at school, together with clear examples of how to find the value of a fraction using concrete resources, maths manipulatives, pictorial representations and number lines; the difference between unit fractions and non unit fractions; all the way up to proper and improper fractions.
It’s been written as a guide for children and parents to work through together in clear digestible chunks.
Equivalent Fractions: Understanding and Comparing Fractions Worksheets
Download these FREE understanding and comparing fractions worksheets for Year 3 pupils, intended to help pupils independently practise what they've been learning.
Examples of equivalent fractions
Here are some examples of equivalent fractions using a bar model and showing the ‘parts’ each numerator is referring to out of the ‘whole’ ie the denominator.
4/6 = four out of six parts, also shown as a :
Although 8/12 may look like a different fraction, it is actually equivalent to 4/6 because eight out of 12 parts is the same as four out of six parts, as shown below:
2/3, or two out of three, is another fraction equivalent to both 4/6 and 8/12.
The three fractions 2/3, 4/6 and 8/12 are shown below respectively in a fraction wall to demonstrate their equivalence.
How to work out equivalent fractions
To work out equivalent fractions, both the numerator and denominator of a fraction must be multiplied by the same number. What this means is in fact you’re multiplying by 1, and we know that multiplying by 1 doesn’t change the original number so the fraction will be equivalent.
For example you can multiply by 2/2 or 6/6 and you’re still multiplying by 1.
Equivalent fractions to 3/5
3/5 x 2/2 = 6/10
3/5 x 3/3 = 9/15
3/5 x 4/4 = 12/20
So, 3/5 = 6/10 = 9/15 = 12/20.
Another way to find equivalent fractions is to divide both the numerator and the denominator of the fraction by the same number – this is called simplifying fractions, because both the numerator and denominator digits will get smaller.
For example, to simplify the fraction 9/12, find a number that both the numerator and denominator can be divided by (also known as a ‘common factor’), such as 3.
9/12 ÷ 3/3 = 3/4, so 9/12 and 3/4 are equivalent fractions, with 3/4 being the fraction in its simplest form.
When do children learn about equivalent fractions in primary school?
Equivalent fractions KS2
The concept of equivalent fractions isn’t introduced until Year 3, where children recognise and show, using diagrams, equivalent fractions with small denominators.
In Year 4, they will recognise and show, using diagrams, families of common equivalent fractions. The National Curriculum’s non-statutory guidance also advises that pupils use factors and multiples to recognise equivalent fractions and simplify where appropriate (for example, 6/9 = 2/3 or ¼ = 2/8).
In Year 5, pupils are taught to identify, name and write equivalent fractions of a given fraction, represented visually, including tenths and hundredths.
In Year 6, they will begin learning how to add fractions and subtracting fractions with different denominators and mixed numbers, using the concept of equivalent fractions. Non-statutory guidance for Year 6 suggests that common factors can be related to finding equivalent fractions and that children practise fraction questions for calculations with simple fractions… including listing equivalent fractions to identify fractions with common denominators.
How do equivalent fractions relate to other areas of maths?
Children will need to have a strong knowledge of equivalent fractions to be able to convert between fractions, decimals and percentages. Knowledge of times tables, the lowest common multiple and highest common factor are also important for equivalent fractions
Wondering about how to explain other key maths vocabulary to your children? Check out our Primary Maths Dictionary, or try these other terms related to equivalent fractions:
- What Is A Unit Fraction?
- What Is BODMAS (and BIDMAS)?
- Properties of shape
- What are 2D shapes?
- What are 3D shapes?
Equivalent fractions questions
1. Write the missing values: 3/4 = 9/? = ?/24
(Answer: 12, 18)
2. Circle the two fractions that have the same value:
(Answer: ½ and 5/10)
3. Tick two shapes that have ¾ shaded.
(Answer: top left (6/8) and bottom right (12/16) as both = 3/4)
4. Shade ¼ of this shape.
(Answer: Any 9 triangles shaded)
5. Ahmed says, ‘One-third of this shape is shaded.’ Is he correct? Explain how you know.
(Answer: Yes – it would be 2/6 (imagine the middle square split into halves too) which = 1/3)
In maths, ‘equivalent’ means that two (or more) values, quantities etc. are the same.
Equivalent fractions are fractions that may look different but are actually represent the same quantity. 2/3 and 6/9 are examples of equivalent fractions.
Equivalent fractions can be explained as fractions that have different numerators and denominators but represent the same value.
- 10 Fun, Simple Fraction Games For KS1 & KS2
- How to Simplify Fractions: A Primary School Guide
- How To Divide Fractions: Step By Step Guide
Equivalent fractions resources
- Year 3 Equivalent Fractions Worksheet
- Year 6 Equivalent Fractions, Decimals and Percentages Worksheet
- Tarsia Puzzle Equivalent Fractions and Decimals (Year 5)
- Printable Maths Resources Fraction Walls
Online 1-to-1 maths lessons trusted by schools and teachers
Every week Third Space Learning’s maths specialist tutors support thousands of primary school children with weekly online 1-to-1 lessons and maths interventions. Since 2013 we’ve helped over 150,000 children become more confident, able mathematicians. Learn more or request a personalised quote to speak to us about your needs and how we can help.
Primary school tuition targeted to the needs of each child and closely following the National Curriculum.
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A great tip to follow when learning new forms of punctuation is to
always read sentences aloud, if you are not sure how it should be punctuated.
Punctuation marks are put in place in writing to demonstrate to the reader how a passage was constructed and how it should be read. A comma tells us to take a slight pause. The exclamation mark tells us that the writer had strong feelings coming off a sentence. The period tells us that a thought has ended. A question mark, well, tells us we are looking for a response. When we want to connect two sentences or thoughts, we use a semi-colon. Students have a great deal of difficulty with that symbol's usage. These worksheets will help students master the proper punctuation of various sentences.
Colons - These symbols are used to add a layer of separation between two independent clauses. In most cases the second clause provides an explanation for the first.
Comma Use Within Sentences - When used in this fashion the comma helps the reader gauge what in the sentence is the most important.
Ellipsis Omission - These symbols help writers save space in their work and help point out important things.
End Punctuation - Students will learn what should be placed at the end of a sentence and why.
Exclamation Marks - When you what to express a heightened level of excitement, these are your go to.
Italics and Underlining - We use this type of formatting to make our words or phrases stick out. We show you how to proper use them.
Periods - We show you how to properly use this symbol to end off complete thoughts.
Question Marks - When you are looking for answers, this is your go to symbol of choice.
Quotation Marks - When you want to compile that exact words that someone has vocalized and written.
The Rules - These worksheets focus on the general rules that govern the use of these marks and symbols.
Semicolons - We use these to join to clauses or parts of a sentence when they are given equal emphasis.
Using Apostrophes - They have many different uses including displaying possession and plurals.
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Top 1 million highest-educational-quality documents from HuggingFaceFW/fineweb-edu, filtered to score >= 4.0 (scale 0-5).
Quick Use
from datasets import load_dataset
ds = load_dataset("blythet/fineweb-edu-top1m", split="train")
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