**Chapter 11:**
Essential Question: What are some two-dimensional shapes and three-dimensional shapes, and how can you show equal parts of shapes?
Students will recognize and draw shapes with specified attributes, such as a given number of angles or equal faces. They will identify triangles, quadrilaterals, pentagons, hexagons, and cubes. Students will partition rectangles into rows and columns of same-size squares and count the total number. They will also partition circles and rectangles into two, three, or four equal shares, describing the shares using terms like halves, thirds, and fourths.

**Chapter 10:**
Essential Question: How do tally charts, picture graphs, and bar graphs help you solve problems?
Students will draw picture graphs and bar graphs to represent data sets with up to four categories. They will solve simple problems using information presented in bar graphs, including put-together, take-apart, and compare problems.

**Chapter 8:**
Essential Question: What are some methods and tools that can be used to estimate and measure length?
Students will measure and estimate lengths in standard units using tools like rulers, yardsticks, and measuring tapes. They will add and subtract within 100 to solve word problems involving lengths and generate measurement data by measuring objects to the nearest whole unit.

**Chapter 7:**
Essential Question: How do you use the values of coins and bills to find the total value of a group of money, and how do you read times shown on analog and digital clocks?
Students will solve word problems involving dollar bills, quarters, dimes, nickels, and pennies. They will tell and write time from analog and digital clocks to the nearest five minutes using am and pm.

**Chapter 6:**
Essential Question: What are some strategies for adding and subtracting 3-digit numbers?
Students will add and subtract within 1000 using concrete models, drawings, and strategies based on place value and properties of operations. They will relate their strategy to a written method and understand the importance of composing or decomposing tens or hundreds.

**Chapter 5:**
Essential Question: How do you use place value to subtract 2-digit numbers with and without regrouping?
Students will add and subtract within 100 using strategies based on place value and properties of operations. They will explain why addition and subtraction strategies work and use them to solve one and two-step word problems.

**Chapter 4:**
Essential Question: How do you use place value to add 2-digit numbers, and what are some different ways to add 2-digit numbers?
Students will add up to four two-digit numbers using strategies based on place value and properties of operations. They will explain why addition and subtraction strategies work and fluently add and subtract within 100 to solve word problems.

**Chapter 3:**
Essential Question: How can you use patterns and strategies to find sums and differences for basic facts?
Students will add and subtract within 20 using mental strategies and represent and solve problems involving addition and subtraction within 100. They will work with equal groups of objects to gain foundations for multiplication.

**Chapter 2:**
Essential Question: How can you use place value to model, write, and compare 3-digit numbers?
Students will understand that the three digits of a three-digit number represent hundreds, tens, and ones. They will read and write numbers to 1000 using base-ten numerals, number names, and expanded form, and compare two three-digit numbers based on the meanings of the hundreds, tens, and ones digits.

**Chapter 1:**
Essential Question: How do you use place value to find the values of numbers and describe numbers in different ways?
Students will determine whether a group of objects has an odd or even number of members and read and write numbers to 1000 using base-ten numerals, number names, and expanded form. They will count within 1000 and skip count by 5s, 10s, and 100s.


This collection of games and activities is designed to help students improve their understanding of math skills, including:

- Basic arithmetic operations: addition, subtraction, multiplication, and division
- Completing sentences with missing operators
- Mastering math facts for addition, subtraction, multiplication, and division
- Selecting numbers with specific sums, differences, products, or quotients

The games include:
- A racing game where answering math questions correctly helps a penguin win against a bear and a toucan
- A cross-number puzzle where players must place numbers to make each line add up to the same total
- A wagon-jumping game where following instructions helps a player touch a door
- A number-touching game where correct answers increase power and incorrect answers decrease it, with the option to shoot dragons for 10 power units
- A number-creation game where players must find three different ways to make a number and press "go" to win
- A tile-swapping game where players arrange operations to achieve the same result
- A maze game where a mouse must navigate to the end while avoiding a cat

The mixed operations practice games include:
- A mouse game where players move horizontally and vertically, answering easy, moderate, and hard math questions to progress
  - Easy questions: addition or subtraction with numbers from 1 to 100
  - Moderate questions: arithmetic expressions with addition, subtraction, and multiplication using numbers from 1 to 500
  - Hard questions: arithmetic expressions with addition, subtraction, and multiplication using numbers from 1 to 1000
- A Broken Calculator game where players must answer 10 questions using all operations to achieve a target number, with incorrect answers considered wrong if all operations are not used.


Q: What is 5 plus 700?

The answer is 705.

Key examples and related calculations:
- 100 + 700 = 800
- 700 + 700 = 1400
- 7000 + 700 = 7700
- 200 + 500 = 700
- 1500 + 700 = 2200
- 50 + 700 = 750
- 700 + 60 + 8 = 768
- 700 + 4 = 704
- 5% of 700 = 0.05 * 700 = 35

Additional mathematical expressions:
- 700 x $5600 x $532 x $555 x 0.1034 x 0.05
- $8100 + $5.5 + $1.7 = $8,107.20


A rectangular field is to be enclosed by a fence. Two fences parallel to one side of the field divide the field into three rectangular fields. If 2400m of fence are available, find the dimensions that give the maximum area.

Let's denote the length of the field as \(l\) and the width as \(w\). Since there are two dividers, the perimeter is given by \(2l + 4w = 2400\).

We can express \(w\) in terms of \(l\):
\[2l + 4w = 2400\]
\[4w = 2400 - 2l\]
\[w = 600 - 0.5l\]

The area \(A\) of the field is given by \(A = lw\). Substituting \(w\) from the above equation:
\[A = l(600 - 0.5l)\]
\[A = 600l - 0.5l^2\]

To find the maximum area, we can use calculus. Differentiating \(A\) with respect to \(l\):
\[\frac{dA}{dl} = 600 - l\]

Setting \(\frac{dA}{dl} = 0\) to find the maximum:
\[600 - l = 0\]
\[l = 600\]

Since \(\frac{dA}{dl}\) changes sign from positive to negative at \(l = 600\), there is a relative maximum at \(l = 600\).

Substituting \(l = 600\) into the equation for \(w\):
\[w = 600 - 0.5(600)\]
\[w = 600 - 300\]
\[w = 300\]

Therefore, the dimensions that give the maximum area are \(300m \times 600m\).

Alternatively, without using calculus, let the width of the whole rectangle be \(x\) meters and the length be \(y\) meters. Then:
\[4x + 2y = 2400\]
\[2x + y = 1200\]
\[y = 1200 - 2x\]

The area \(A = xy\), so:
\[A = x(1200 - 2x)\]
\[A = -2x^2 + 1200x\]

Completing the square:
\[A = -2(x - 300)^2 + 180000\]

This shows that the maximum area occurs when \(x = 300\), and substituting \(x = 300\) into \(y = 1200 - 2x\) gives \(y = 600\), confirming the dimensions for the maximum area as \(300m \times 600m\).


A standard six-sided fair die is rolled four times. The probability that the product of all four numbers rolled is a perfect square is $\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Find $m+n$.

Notice that, other than the number 5, the remaining numbers 1, 2, 3, 4, 6 are only divisible by 2 and/or 3. We can do some cases on the number of 5's rolled.

Case 1: Four 5's are rolled. This has a probability of $\frac{1}{1296}$.

Case 2: Two 5's are rolled.

Case 3: No 5's are rolled.

To find the number of outcomes for the latter two cases, we will use recursion. Consider a 5-sided die with faces numbered 1, 2, 3, 4, 6. For $n \ge 1$, let $A_n$ equal the number of outcomes after rolling the die $n$ times, with the property that the product is a square. Thus, $A_1 = 2$ as 1 and 4 are the only possibilities.

We have the following equations after considering the possible values of the $n$th roll:
\[A_n = S_{n-1}+B_{n-1}+D_{n-1}+E_{n-1}+2A_{n-1}\]
\[B_n = A_{n-1}+D_{n-1}+F_{n-1}+S_{n-1}+2B_{n-1}\]

Computing $A_2$, $A_3$, $A_4$ gives $A_2 = 10$, $A_3 = 50$, $A_4 = 157$. For Case 3, there are 157 outcomes. For case 2, we multiply by $\binom{4}{2}$ to distribute the two 5's among four rolls. Thus the probability is
\[\frac{1}{1296} + \frac{157}{1296} + \frac{\binom{4}{2} \cdot 12}{1296} = \frac{200}{1296} = \frac{25}{162}\]
meaning our answer is $25 + 162 = 187$.

Alternatively, we can do casework on all the perfect squares from $1^2$ to $36^2$, and how many ways they can be ordered.

Another way to solve this problem is to do casework on the number of 1/4's rolled. Notice that rolling a 1/4 will not affect whether or not the product is a perfect square. This means that in order for the product to be a perfect square, all non 1/4 numbers rolled must come in pairs, with the only exception being the triplet 2,3, 6.

We can also solve this problem using recursion, by defining $A_n$ to be the number of ways to have a product after $n$ rolls where $a$ is odd, and $b$, $c$ are even.

Finally, we have $S_4 = 200$, $\frac{m}{n} = \frac{200}{1296} = \frac{25}{162}$, meaning our answer is $\boxed{187}$.


The implementation of pressure spatial decomposition is based on recognizing the time pattern of flowrate variation pulse sequence in pressure response, constituting a key interpretation procedure for Pressure Pulse-Code Test (PCT). The decomposition algorithm minimizes the correlation functional X between the generating well pressure variation and the pressure trend at the receiving well, after deducting the simulated pressure pulse-code response.

The algorithm is based on two equations:

X[p_{uGG}] = < | p_{G,{\rm tr}}(t) | , |δ p_G(t)| > → min
X[p_{uGR}] = < | p_{R,{\rm tr}}(t) | , |δ p_R(t)| > → min

where p_G(t) and p_R(t) are calculated as:

p_G(t) = p_{G,{\rm tr}}(t) - δ p_G(t) = p_{G,{\rm tr}}(t) - ∫[0,t] p_{uGG}(t-τ) dq_G(τ)
p_R(t) = p_{R,{\rm tr}}(t) - δ p_R(t) = p_{R,{\rm tr}}(t) - ∫[0,t] p_{uGR}(t-τ) dq_G(τ)

The generator's flowrate history q_G(t), generator's wellbore pressure history p_G(t), and receiver's wellbore pressure history p_R(t) are assumed to be known for the whole period of the test. The result of decomposition is a set of unit-rate transient responses, including DTRs p_{uGG}(τ) and CTRs p_{uGR}(τ), which characterize reservoir properties around the generator and between the generator and receiver.

For decomposition to work, the pressure trends at the generator p_{G,{\rm tr}}(t) and receiver p_{R,{\rm tr}}(t) should have minimum correlation with the generating well flowrate variation q_G(t). This means that the pressure trend spectrum should contain minimum overlap with the spectrum of flowrate variation q_G(t) at the generator. The pressure trends may contain low frequencies (created by distant wells) or high frequencies (created by lift instability), which are outside the frequency range of the generating well flow variation q_G(t).

If pressure trends contain components correlated with q_G(t), decomposition is not unique and should not be considered. This may occur when the receiving well or distant wells have been synchronously varying rates with the generating well.


The **slope** of a line is also called the **rate of change**. Given two points on a line, (*x*_{1}, *y*_{1}) and (*x*_{2}, *y*_{2}), the slope is the ratio of the change in *y* to the change in *x*. The slope formula is $m=\genfrac{}{}{0.07ex}{}{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$.

To find the slope, choose any two points. For example, using the points (0, 1) and (1, 2), the slope calculation is:
$\begin{array}{rcl}m& =& \genfrac{}{}{0.07ex}{}{2-1}{1-0}\\ & =& \genfrac{}{}{0.07ex}{}{1}{1}\\ & =& 1\end{array}$

Similarly, using the points (1, 2) and (2, 3), the slope calculation is:
$\begin{array}{rcl}m& =& \genfrac{}{}{0.07ex}{}{3-2}{2-1}\\ & =& \genfrac{}{}{0.07ex}{}{1}{1}\\ & =& 1\end{array}$


Statistics differs from mathematics in fundamental ways, primarily in its purpose of collecting and making sense of data. Context is critical in statistics, distinguishing it from mathematics where context is considered irrelevant detail. As George Cobb and David Moore state, "data are not just numbers, they are numbers with a context." In statistics, context provides meaning, and the interpretation of analysis in context is the ultimate desired outcome.

The importance of communication in statistics is evident in the scoring of free response questions on the AP Statistics Exam, where both statistical knowledge and communication are considered. The Teacher's Guide for AP Statistics (1997) identifies five levels for scoring open-ended statistics questions, with communication being a key aspect. The levels for communication are:

* Highest: clear and concise explanation of what was done, why, and interpretation of results in context
* High: good explanation of what was done and why, with some interpretation of results in context
* Medium: adequate explanation of what was done, but limited interpretation of results in context
* Low: poor explanation of what was done, with little or no interpretation of results in context
* Lowest: no explanation or interpretation of results in context

Sample student responses illustrate the importance of good communication. For example:
* Poor Communication: "Matrix used on calc. X2 = .51686, p= .99996 results in an insignificant p-value. No effective conclusion can be made that there is an association between the active ingredient in the two brands of pills and the pharmacy." (2001 exam, Question 5)
* Weak Communication: "It isn't an outlier because it is within the IQR." (2001 exam, Question 1c)
* Adequate Communication: "10 inches of rainfall is not outstanding at all. The mean is 14.941 with a standard deviation of 6.747. That implies that 67% of the data is between 14.941 ± 6.747. A z score can be used to determine the exact probability." (2001 exam, Question 1c)
* Good Communication: "Since we want at least a pain relief of 50, drug B at 400 milligrams would be better than any of the strengths of drug A because the plot is of averages. For drug A, some patients might have had no pain relief and others 100—so it averaged 50. But drug B at 400 milligrams shows that most of the time the relief would be above 50 therefore bringing the average up to 90." (2000 exam, Question 1c)

To develop good communication skills, AP Statistics teachers can:
* Ask questions that require explanation and interpretation throughout the year
* Not accept "mechanics only" answers as correct on homework, quizzes, or exams
* Not accept "calculator talk" as an explanation of what is being done
* Model good communication for students
* Encourage students to read and write about statistical concepts and processes
* Provide consistent expectations and good feedback

Good statistical communication improves with practice, and frequent opportunities to practice can lead to rewarding results. References include Cobb and Moore's "Mathematics, Statistics, and Teaching" (1997) and the Teacher's Guide for AP Statistics (1997) by Watkins, Roberts, Olsen, and Scheaffer.


I have an array of values and I need to frame an expression such that it chooses the value from the array automatically instead of me assigning a value from the array every time. Let's say I have a dataset that includes 10,000 rows of data. In MATLAB, you can get the first value from an array by using 1 as the index. The syntax is randi([start,end]).

For example, I want to generate a random number between -10 and 10. I type randi([-10,10]) on the command line and get a random number between -10 and 10 as an output.

There are various ways of generating random numbers in MATLAB with different applications. The arrays returned by randi might contain repeated integer values. This behavior is sometimes referred to as sampling with replacement.

One of the most important topics in today’s science and computer simulation is random number generation and Monte Carlo simulation methods. MATLAB uses algorithms to generate pseudorandom and pseudoindependent numbers.

The rng function controls the global stream, which determines how the rand, randi, randn, and randperm functions produce a sequence of random numbers.

For details, see Creating and Controlling a Random Number Stream.

In the simplest scenario for your research, you may need to generate a sequence of uniformly distributed random numbers in MATLAB.

You can try out the differences of MATLAB vs Python with this example:

Matlab also has numerous libraries geared toward scientific and commercial applications including: solving systems of ODEs, signal processing, wavelets, equation solving, linear and nonlinear optimization, neural networks, image processing, interpolation, polynomials, data analysis, Fourier transforms, elementary and special mathematical functions, and digital audio.

The binornd function expands scalar inputs to constant arrays with the same dimensions as the other inputs.

For more information, see How to Use the Comma-Separated Lists.

When you create a vector to index into a cell array or structure array, MATLAB returns multiple outputs in a comma-separated list.

I am trying to get a random sample of a matrix which includes NaNs, but I need the indices as well.

This MATLAB function partitions a subset of observations in a random patch extraction datastore into a new datastore.

The same commands will work in MATLAB.

This MATLAB function generates a 1-by-m random variate from the m-dimensional Gaussian mixture distribution gm.

Save the current state of the random number generator and create a 1-by-5 vector of random numbers.

Random Forest for Matlab This toolbox was written for my own education and to give me a chance to explore the models a bit.

In the MATLAB environment, each variable is treated as a matrix or an array irrespective of their data types.

Matlab does not need variables to be declared explicitly.

You can drag and drop an image, send image / svg or a text / plain link to a written file.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

Possible Duplicate: How do I randomly select k points from N points in MATLAB?

This function is useful for selecting a random index of a vector or matrix.

Alternatively, one or more arguments can be scalars.

Since multiple positions may have the same value, the index is very important.

The randperm approach described by @Stewie appears to be the way to go in most cases.

I want to generate a random number between -10 and 10.

This behavior is sometimes referred to as sampling with replacement.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In MATLAB, you can get the first value from an array by using 1 as the index.

For example, I want to generate a random number between -10 and 10.

There are special functions just for vectors; see the vector module for an explanation of how to Use these.

The sequence of numbers produced by randi is determined by the internal settings of the uniform pseudorandom number generator that underlies rand, randi, and randn.

You can control that shared random number generator using rng.

The rng function controls the global stream, which determines how the rand, randi, randn, and randperm functions produce a sequence of random numbers.

For details, see Creating and Controlling a Random Number Stream.

This MATLAB function generates a 1-by-m random variate from the m-dimensional Gaussian mixture distribution gm.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

The same commands will work in MATLAB.

This MATLAB function partitions a subset of observations in a random patch extraction datastore into a new datastore.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In the MATLAB environment, each variable is treated as a matrix or an array irrespective of their data types.

Matlab does not need variables to be declared explicitly.

You can drag and drop an image, send image / svg or a text / plain link to a written file.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

Possible Duplicate: How do I randomly select k points from N points in MATLAB?

This function is useful for selecting a random index of a vector or matrix.

Alternatively, one or more arguments can be scalars.

Since multiple positions may have the same value, the index is very important.

The randperm approach described by @Stewie appears to be the way to go in most cases.

I want to generate a random number between -10 and 10.

This behavior is sometimes referred to as sampling with replacement.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In MATLAB, you can get the first value from an array by using 1 as the index.

For example, I want to generate a random number between -10 and 10.

There are special functions just for vectors; see the vector module for an explanation of how to Use these.

The sequence of numbers produced by randi is determined by the internal settings of the uniform pseudorandom number generator that underlies rand, randi, and randn.

You can control that shared random number generator using rng.

The rng function controls the global stream, which determines how the rand, randi, randn, and randperm functions produce a sequence of random numbers.

For details, see Creating and Controlling a Random Number Stream.

This MATLAB function generates a 1-by-m random variate from the m-dimensional Gaussian mixture distribution gm.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

The same commands will work in MATLAB.

This MATLAB function partitions a subset of observations in a random patch extraction datastore into a new datastore.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In the MATLAB environment, each variable is treated as a matrix or an array irrespective of their data types.

Matlab does not need variables to be declared explicitly.

You can drag and drop an image, send image / svg or a text / plain link to a written file.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

Possible Duplicate: How do I randomly select k points from N points in MATLAB?

This function is useful for selecting a random index of a vector or matrix.

Alternatively, one or more arguments can be scalars.

Since multiple positions may have the same value, the index is very important.

The randperm approach described by @Stewie appears to be the way to go in most cases.

I want to generate a random number between -10 and 10.

This behavior is sometimes referred to as sampling with replacement.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In MATLAB, you can get the first value from an array by using 1 as the index.

For example, I want to generate a random number between -10 and 10.

There are special functions just for vectors; see the vector module for an explanation of how to Use these.

The sequence of numbers produced by randi is determined by the internal settings of the uniform pseudorandom number generator that underlies rand, randi, and randn.

You can control that shared random number generator using rng.

The rng function controls the global stream, which determines how the rand, randi, randn, and randperm functions produce a sequence of random numbers.

For details, see Creating and Controlling a Random Number Stream.

This MATLAB function generates a 1-by-m random variate from the m-dimensional Gaussian mixture distribution gm.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

The same commands will work in MATLAB.

This MATLAB function partitions a subset of observations in a random patch extraction datastore into a new datastore.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In the MATLAB environment, each variable is treated as a matrix or an array irrespective of their data types.

Matlab does not need variables to be declared explicitly.

You can drag and drop an image, send image / svg or a text / plain link to a written file.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

Possible Duplicate: How do I randomly select k points from N points in MATLAB?

This function is useful for selecting a random index of a vector or matrix.

Alternatively, one or more arguments can be scalars.

Since multiple positions may have the same value, the index is very important.

The randperm approach described by @Stewie appears to be the way to go in most cases.

I want to generate a random number between -10 and 10.

This behavior is sometimes referred to as sampling with replacement.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In MATLAB, you can get the first value from an array by using 1 as the index.

For example, I want to generate a random number between -10 and 10.

There are special functions just for vectors; see the vector module for an explanation of how to Use these.

The sequence of numbers produced by randi is determined by the internal settings of the uniform pseudorandom number generator that underlies rand, randi, and randn.

You can control that shared random number generator using rng.

The rng function controls the global stream, which determines how the rand, randi, randn, and randperm functions produce a sequence of random numbers.

For details, see Creating and Controlling a Random Number Stream.

This MATLAB function generates a 1-by-m random variate from the m-dimensional Gaussian mixture distribution gm.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

The same commands will work in MATLAB.

This MATLAB function partitions a subset of observations in a random patch extraction datastore into a new datastore.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In the MATLAB environment, each variable is treated as a matrix or an array irrespective of their data types.

Matlab does not need variables to be declared explicitly.

You can drag and drop an image, send image / svg or a text / plain link to a written file.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

Possible Duplicate: How do I randomly select k points from N points in MATLAB?

This function is useful for selecting a random index of a vector or matrix.

Alternatively, one or more arguments can be scalars.

Since multiple positions may have the same value, the index is very important.

The randperm approach described by @Stewie appears to be the way to go in most cases.

I want to generate a random number between -10 and 10.

This behavior is sometimes referred to as sampling with replacement.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In MATLAB, you can get the first value from an array by using 1 as the index.

For example, I want to generate a random number between -10 and 10.

There are special functions just for vectors; see the vector module for an explanation of how to Use these.

The sequence of numbers produced by randi is determined by the internal settings of the uniform pseudorandom number generator that underlies rand, randi, and randn.

You can control that shared random number generator using rng.

The rng function controls the global stream, which determines how the rand, randi, randn, and randperm functions produce a sequence of random numbers.

For details, see Creating and Controlling a Random Number Stream.

This MATLAB function generates a 1-by-m random variate from the m-dimensional Gaussian mixture distribution gm.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

The same commands will work in MATLAB.

This MATLAB function partitions a subset of observations in a random patch extraction datastore into a new datastore.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In the MATLAB environment, each variable is treated as a matrix or an array irrespective of their data types.

Matlab does not need variables to be declared explicitly.

You can drag and drop an image, send image / svg or a text / plain link to a written file.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

Possible Duplicate: How do I randomly select k points from N points in MATLAB?

This function is useful for selecting a random index of a vector or matrix.

Alternatively, one or more arguments can be scalars.

Since multiple positions may have the same value, the index is very important.

The randperm approach described by @Stewie appears to be the way to go in most cases.

I want to generate a random number between -10 and 10.

This behavior is sometimes referred to as sampling with replacement.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In MATLAB, you can get the first value from an array by using 1 as the index.

For example, I want to generate a random number between -10 and 10.

There are special functions just for vectors; see the vector module for an explanation of how to Use these.

The sequence of numbers produced by randi is determined by the internal settings of the uniform pseudorandom number generator that underlies rand, randi, and randn.

You can control that shared random number generator using rng.

The rng function controls the global stream, which determines how the rand, randi, randn, and randperm functions produce a sequence of random numbers.

For details, see Creating and Controlling a Random Number Stream.

This MATLAB function generates a 1-by-m random variate from the m-dimensional Gaussian mixture distribution gm.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

The same commands will work in MATLAB.

This MATLAB function partitions a subset of observations in a random patch extraction datastore into a new datastore.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In the MATLAB environment, each variable is treated as a matrix or an array irrespective of their data types.

Matlab does not need variables to be declared explicitly.

You can drag and drop an image, send image / svg or a text / plain link to a written file.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

Possible Duplicate: How do I randomly select k points from N points in MATLAB?

This function is useful for selecting a random index of a vector or matrix.

Alternatively, one or more arguments can be scalars.

Since multiple positions may have the same value, the index is very important.

The randperm approach described by @Stewie appears to be the way to go in most cases.

I want to generate a random number between -10 and 10.

This behavior is sometimes referred to as sampling with replacement.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In MATLAB, you can get the first value from an array by using 1 as the index.

For example, I want to generate a random number between -10 and 10.

There are special functions just for vectors; see the vector module for an explanation of how to Use these.

The sequence of numbers produced by randi is determined by the internal settings of the uniform pseudorandom number generator that underlies rand, randi, and randn.

You can control that shared random number generator using rng.

The rng function controls the global stream, which determines how the rand, randi, randn, and randperm functions produce a sequence of random numbers.

For details, see Creating and Controlling a Random Number Stream.

This MATLAB function generates a 1-by-m random variate from the m-dimensional Gaussian mixture distribution gm.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

The same commands will work in MATLAB.

This MATLAB function partitions a subset of observations in a random patch extraction datastore into a new datastore.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In the MATLAB environment, each variable is treated as a matrix or an array irrespective of their data types.

Matlab does not need variables to be declared explicitly.

You can drag and drop an image, send image / svg or a text / plain link to a written file.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

Possible Duplicate: How do I randomly select k points from N points in MATLAB?

This function is useful for selecting a random index of a vector or matrix.

Alternatively, one or more arguments can be scalars.

Since multiple positions may have the same value, the index is very important.

The randperm approach described by @Stewie appears to be the way to go in most cases.

I want to generate a random number between -10 and 10.

This behavior is sometimes referred to as sampling with replacement.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In MATLAB, you can get the first value from an array by using 1 as the index.

For example, I want to generate a random number between -10 and 10.

There are special functions just for vectors; see the vector module for an explanation of how to Use these.

The sequence of numbers produced by randi is determined by the internal settings of the uniform pseudorandom number generator that underlies rand, randi, and randn.

You can control that shared random number generator using rng.

The rng function controls the global stream, which determines how the rand, randi, randn, and randperm functions produce a sequence of random numbers.

For details, see Creating and Controlling a Random Number Stream.

This MATLAB function generates a 1-by-m random variate from the m-dimensional Gaussian mixture distribution gm.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

The same commands will work in MATLAB.

This MATLAB function partitions a subset of observations in a random patch extraction datastore into a new datastore.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In the MATLAB environment, each variable is treated as a matrix or an array irrespective of their data types.

Matlab does not need variables to be declared explicitly.

You can drag and drop an image, send image / svg or a text / plain link to a written file.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

Possible Duplicate: How do I randomly select k points from N points in MATLAB?

This function is useful for selecting a random index of a vector or matrix.

Alternatively, one or more arguments can be scalars.

Since multiple positions may have the same value, the index is very important.

The randperm approach described by @Stewie appears to be the way to go in most cases.

I want to generate a random number between -10 and 10.

This behavior is sometimes referred to as sampling with replacement.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In MATLAB, you can get the first value from an array by using 1 as the index.

For example, I want to generate a random number between -10 and 10.

There are special functions just for vectors; see the vector module for an explanation of how to Use these.

The sequence of numbers produced by randi is determined by the internal settings of the uniform pseudorandom number generator that underlies rand, randi, and randn.

You can control that shared random number generator using rng.

The rng function controls the global stream, which determines how the rand, randi, randn, and randperm functions produce a sequence of random numbers.

For details, see Creating and Controlling a Random Number Stream.

This MATLAB function generates a 1-by-m random variate from the m-dimensional Gaussian mixture distribution gm.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

The same commands will work in MATLAB.

This MATLAB function partitions a subset of observations in a random patch extraction datastore into a new datastore.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In the MATLAB environment, each variable is treated as a matrix or an array irrespective of their data types.

Matlab does not need variables to be declared explicitly.

You can drag and drop an image, send image / svg or a text / plain link to a written file.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

Possible Duplicate: How do I randomly select k points from N points in MATLAB?

This function is useful for selecting a random index of a vector or matrix.

Alternatively, one or more arguments can be scalars.

Since multiple positions may have the same value, the index is very important.

The randperm approach described by @Stewie appears to be the way to go in most cases.

I want to generate a random number between -10 and 10.

This behavior is sometimes referred to as sampling with replacement.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In MATLAB, you can get the first value from an array by using 1 as the index.

For example, I want to generate a random number between -10 and 10.

There are special functions just for vectors; see the vector module for an explanation of how to Use these.

The sequence of numbers produced by randi is determined by the internal settings of the uniform pseudorandom number generator that underlies rand, randi, and randn.

You can control that shared random number generator using rng.

The rng function controls the global stream, which determines how the rand, randi, randn, and randperm functions produce a sequence of random numbers.

For details, see Creating and Controlling a Random Number Stream.

This MATLAB function generates a 1-by-m random variate from the m-dimensional Gaussian mixture distribution gm.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

The same commands will work in MATLAB.

This MATLAB function partitions a subset of observations in a random patch extraction datastore into a new datastore.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In the MATLAB environment, each variable is treated as a matrix or an array irrespective of their data types.

Matlab does not need variables to be declared explicitly.

You can drag and drop an image, send image / svg or a text / plain link to a written file.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

Possible Duplicate: How do I randomly select k points from N points in MATLAB?

This function is useful for selecting a random index of a vector or matrix.

Alternatively, one or more arguments can be scalars.

Since multiple positions may have the same value, the index is very important.

The randperm approach described by @Stewie appears to be the way to go in most cases.

I want to generate a random number between -10 and 10.

This behavior is sometimes referred to as sampling with replacement.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In MATLAB, you can get the first value from an array by using 1 as the index.

For example, I want to generate a random number between -10 and 10.

There are special functions just for vectors; see the vector module for an explanation of how to Use these.

The sequence of numbers produced by randi is determined by the internal settings of the uniform pseudorandom number generator that underlies rand, randi, and randn.

You can control that shared random number generator using rng.

The rng function controls the global stream, which determines how the rand, randi, randn, and randperm functions produce a sequence of random numbers.

For details, see Creating and Controlling a Random Number Stream.

This MATLAB function generates a 1-by-m random variate from the m-dimensional Gaussian mixture distribution gm.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

The same commands will work in MATLAB.

This MATLAB function partitions a subset of observations in a random patch extraction datastore into a new datastore.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In the MATLAB environment, each variable is treated as a matrix or an array irrespective of their data types.

Matlab does not need variables to be declared explicitly.

You can drag and drop an image, send image / svg or a text / plain link to a written file.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

Possible Duplicate: How do I randomly select k points from N points in MATLAB?

This function is useful for selecting a random index of a vector or matrix.

Alternatively, one or more arguments can be scalars.

Since multiple positions may have the same value, the index is very important.

The randperm approach described by @Stewie appears to be the way to go in most cases.

I want to generate a random number between -10 and 10.

This behavior is sometimes referred to as sampling with replacement.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In MATLAB, you can get the first value from an array by using 1 as the index.

For example, I want to generate a random number between -10 and 10.

There are special functions just for vectors; see the vector module for an explanation of how to Use these.

The sequence of numbers produced by randi is determined by the internal settings of the uniform pseudorandom number generator that underlies rand, randi, and randn.

You can control that shared random number generator using rng.

The rng function controls the global stream, which determines how the rand, randi, randn, and randperm functions produce a sequence of random numbers.

For details, see Creating and Controlling a Random Number Stream.

This MATLAB function generates a 1-by-m random variate from the m-dimensional Gaussian mixture distribution gm.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

The same commands will work in MATLAB.

This MATLAB function partitions a subset of observations in a random patch extraction datastore into a new datastore.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In the MATLAB environment, each variable is treated as a matrix or an array irrespective of their data types.

Matlab does not need variables to be declared explicitly.

You can drag and drop an image, send image / svg or a text / plain link to a written file.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

Possible Duplicate: How do I randomly select k points from N points in MATLAB?

This function is useful for selecting a random index of a vector or matrix.

Alternatively, one or more arguments can be scalars.

Since multiple positions may have the same value, the index is very important.

The randperm approach described by @Stewie appears to be the way to go in most cases.

I want to generate a random number between -10 and 10.

This behavior is sometimes referred to as sampling with replacement.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In MATLAB, you can get the first value from an array by using 1 as the index.

For example, I want to generate a random number between -10 and 10.

There are special functions just for vectors; see the vector module for an explanation of how to Use these.

The sequence of numbers produced by randi is determined by the internal settings of the uniform pseudorandom number generator that underlies rand, randi, and randn.

You can control that shared random number generator using rng.

The rng function controls the global stream, which determines how the rand, randi, randn, and randperm functions produce a sequence of random numbers.

For details, see Creating and Controlling a Random Number Stream.

This MATLAB function generates a 1-by-m random variate from the m-dimensional Gaussian mixture distribution gm.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

The same commands will work in MATLAB.

This MATLAB function partitions a subset of observations in a random patch extraction datastore into a new datastore.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In the MATLAB environment, each variable is treated as a matrix or an array irrespective of their data types.

Matlab does not need variables to be declared explicitly.

You can drag and drop an image, send image / svg or a text / plain link to a written file.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

Possible Duplicate: How do I randomly select k points from N points in MATLAB?

This function is useful for selecting a random index of a vector or matrix.

Alternatively, one or more arguments can be scalars.

Since multiple positions may have the same value, the index is very important.

The randperm approach described by @Stewie appears to be the way to go in most cases.

I want to generate a random number between -10 and 10.

This behavior is sometimes referred to as sampling with replacement.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In MATLAB, you can get the first value from an array by using 1 as the index.

For example, I want to generate a random number between -10 and 10.

There are special functions just for vectors; see the vector module for an explanation of how to Use these.

The sequence of numbers produced by randi is determined by the internal settings of the uniform pseudorandom number generator that underlies rand, randi, and randn.

You can control that shared random number generator using rng.

The rng function controls the global stream, which determines how the rand, randi, randn, and randperm functions produce a sequence of random numbers.

For details, see Creating and Controlling a Random Number Stream.

This MATLAB function generates a 1-by-m random variate from the m-dimensional Gaussian mixture distribution gm.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

The same commands will work in MATLAB.

This MATLAB function partitions a subset of observations in a random patch extraction datastore into a new datastore.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In the MATLAB environment, each variable is treated as a matrix or an array irrespective of their data types.

Matlab does not need variables to be declared explicitly.

You can drag and drop an image, send image / svg or a text / plain link to a written file.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

Possible Duplicate: How do I randomly select k points from N points in MATLAB?

This function is useful for selecting a random index of a vector or matrix.

Alternatively, one or more arguments can be scalars.

Since multiple positions may have the same value, the index is very important.

The randperm approach described by @Stewie appears to be the way to go in most cases.

I want to generate a random number between -10 and 10.

This behavior is sometimes referred to as sampling with replacement.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In MATLAB, you can get the first value from an array by using 1 as the index.

For example, I want to generate a random number between -10 and 10.

There are special functions just for vectors; see the vector module for an explanation of how to Use these.

The sequence of numbers produced by randi is determined by the internal settings of the uniform pseudorandom number generator that underlies rand, randi, and randn.

You can control that shared random number generator using rng.

The rng function controls the global stream, which determines how the rand, randi, randn, and randperm functions produce a sequence of random numbers.

For details, see Creating and Controlling a Random Number Stream.

This MATLAB function generates a 1-by-m random variate from the m-dimensional Gaussian mixture distribution gm.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

The same commands will work in MATLAB.

This MATLAB function partitions a subset of observations in a random patch extraction datastore into a new datastore.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In the MATLAB environment, each variable is treated as a matrix or an array irrespective of their data types.

Matlab does not need variables to be declared explicitly.

You can drag and drop an image, send image / svg or a text / plain link to a written file.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

Possible Duplicate: How do I randomly select k points from N points in MATLAB?

This function is useful for selecting a random index of a vector or matrix.

Alternatively, one or more arguments can be scalars.

Since multiple positions may have the same value, the index is very important.

The randperm approach described by @Stewie appears to be the way to go in most cases.

I want to generate a random number between -10 and 10.

This behavior is sometimes referred to as sampling with replacement.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In MATLAB, you can get the first value from an array by using 1 as the index.

For example, I want to generate a random number between -10 and 10.

There are special functions just for vectors; see the vector module for an explanation of how to Use these.

The sequence of numbers produced by randi is determined by the internal settings of the uniform pseudorandom number generator that underlies rand, randi, and randn.

You can control that shared random number generator using rng.

The rng function controls the global stream, which determines how the rand, randi, randn, and randperm functions produce a sequence of random numbers.

For details, see Creating and Controlling a Random Number Stream.

This MATLAB function generates a 1-by-m random variate from the m-dimensional Gaussian mixture distribution gm.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

The same commands will work in MATLAB.

This MATLAB function partitions a subset of observations in a random patch extraction datastore into a new datastore.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In the MATLAB environment, each variable is treated as a matrix or an array irrespective of their data types.

Matlab does not need variables to be declared explicitly.

You can drag and drop an image, send image / svg or a text / plain link to a written file.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

Possible Duplicate: How do I randomly select k points from N points in MATLAB?

This function is useful for selecting a random index of a vector or matrix.

Alternatively, one or more arguments can be scalars.

Since multiple positions may have the same value, the index is very important.

The randperm approach described by @Stewie appears to be the way to go in most cases.

I want to generate a random number between -10 and 10.

This behavior is sometimes referred to as sampling with replacement.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In MATLAB, you can get the first value from an array by using 1 as the index.

For example, I want to generate a random number between -10 and 10.

There are special functions just for vectors; see the vector module for an explanation of how to Use these.

The sequence of numbers produced by randi is determined by the internal settings of the uniform pseudorandom number generator that underlies rand, randi, and randn.

You can control that shared random number generator using rng.

The rng function controls the global stream, which determines how the rand, randi, randn, and randperm functions produce a sequence of random numbers.

For details, see Creating and Controlling a Random Number Stream.

This MATLAB function generates a 1-by-m random variate from the m-dimensional Gaussian mixture distribution gm.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

The same commands will work in MATLAB.

This MATLAB function partitions a subset of observations in a random patch extraction datastore into a new datastore.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In the MATLAB environment, each variable is treated as a matrix or an array irrespective of their data types.

Matlab does not need variables to be declared explicitly.

You can drag and drop an image, send image / svg or a text / plain link to a written file.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

Possible Duplicate: How do I randomly select k points from N points in MATLAB?

This function is useful for selecting a random index of a vector or matrix.

Alternatively, one or more arguments can be scalars.

Since multiple positions may have the same value, the index is very important.

The randperm approach described by @Stewie appears to be the way to go in most cases.

I want to generate a random number between -10 and 10.

This behavior is sometimes referred to as sampling with replacement.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In MATLAB, you can get the first value from an array by using 1 as the index.

For example, I want to generate a random number between -10 and 10.

There are special functions just for vectors; see the vector module for an explanation of how to Use these.

The sequence of numbers produced by randi is determined by the internal settings of the uniform pseudorandom number generator that underlies rand, randi, and randn.

You can control that shared random number generator using rng.

The rng function controls the global stream, which determines how the rand, randi, randn, and randperm functions produce a sequence of random numbers.

For details, see Creating and Controlling a Random Number Stream.

This MATLAB function generates a 1-by-m random variate from the m-dimensional Gaussian mixture distribution gm.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

The same commands will work in MATLAB.

This MATLAB function partitions a subset of observations in a random patch extraction datastore into a new datastore.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In the MATLAB environment, each variable is treated as a matrix or an array irrespective of their data types.

Matlab does not need variables to be declared explicitly.

You can drag and drop an image, send image / svg or a text / plain link to a written file.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

Possible Duplicate: How do I randomly select k points from N points in MATLAB?

This function is useful for selecting a random index of a vector or matrix.

Alternatively, one or more arguments can be scalars.

Since multiple positions may have the same value, the index is very important.

The randperm approach described by @Stewie appears to be the way to go in most cases.

I want to generate a random number between -10 and 10.

This behavior is sometimes referred to as sampling with replacement.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In MATLAB, you can get the first value from an array by using 1 as the index.

For example, I want to generate a random number between -10 and 10.

There are special functions just for vectors; see the vector module for an explanation of how to Use these.

The sequence of numbers produced by randi is determined by the internal settings of the uniform pseudorandom number generator that underlies rand, randi, and randn.

You can control that shared random number generator using rng.

The rng function controls the global stream, which determines how the rand, randi, randn, and randperm functions produce a sequence of random numbers.

For details, see Creating and Controlling a Random Number Stream.

This MATLAB function generates a 1-by-m random variate from the m-dimensional Gaussian mixture distribution gm.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

The same commands will work in MATLAB.

This MATLAB function partitions a subset of observations in a random patch extraction datastore into a new datastore.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In the MATLAB environment, each variable is treated as a matrix or an array irrespective of their data types.

Matlab does not need variables to be declared explicitly.

You can drag and drop an image, send image / svg or a text / plain link to a written file.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

Possible Duplicate: How do I randomly select k points from N points in MATLAB?

This function is useful for selecting a random index of a vector or matrix.

Alternatively, one or more arguments can be scalars.

Since multiple positions may have the same value, the index is very important.

The randperm approach described by @Stewie appears to be the way to go in most cases.

I want to generate a random number between -10 and 10.

This behavior is sometimes referred to as sampling with replacement.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In MATLAB, you can get the first value from an array by using 1 as the index.

For example, I want to generate a random number between -10 and 10.

There are special functions just for vectors; see the vector module for an explanation of how to Use these.

The sequence of numbers produced by randi is determined by the internal settings of the uniform pseudorandom number generator that underlies rand, randi, and randn.

You can control that shared random number generator using rng.

The rng function controls the global stream, which determines how the rand, randi, randn, and randperm functions produce a sequence of random numbers.

For details, see Creating and Controlling a Random Number Stream.

This MATLAB function generates a 1-by-m random variate from the m-dimensional Gaussian mixture distribution gm.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

The same commands will work in MATLAB.

This MATLAB function partitions a subset of observations in a random patch extraction datastore into a new datastore.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In the MATLAB environment, each variable is treated as a matrix or an array irrespective of their data types.

Matlab does not need variables to be declared explicitly.

You can drag and drop an image, send image / svg or a text / plain link to a written file.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

Possible Duplicate: How do I randomly select k points from N points in MATLAB?

This function is useful for selecting a random index of a vector or matrix.

Alternatively, one or more arguments can be scalars.

Since multiple positions may have the same value, the index is very important.

The randperm approach described by @Stewie appears to be the way to go in most cases.

I want to generate a random number between -10 and 10.

This behavior is sometimes referred to as sampling with replacement.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In MATLAB, you can get the first value from an array by using 1 as the index.

For example, I want to generate a random number between -10 and 10.

There are special functions just for vectors; see the vector module for an explanation of how to Use these.

The sequence of numbers produced by randi is determined by the internal settings of the uniform pseudorandom number generator that underlies rand, randi, and randn.

You can control that shared random number generator using rng.

The rng function controls the global stream, which determines how the rand, randi, randn, and randperm functions produce a sequence of random numbers.

For details, see Creating and Controlling a Random Number Stream.

This MATLAB function generates a 1-by-m random variate from the m-dimensional Gaussian mixture distribution gm.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

The same commands will work in MATLAB.

This MATLAB function partitions a subset of observations in a random patch extraction datastore into a new datastore.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In the MATLAB environment, each variable is treated as a matrix or an array irrespective of their data types.

Matlab does not need variables to be declared explicitly.

You can drag and drop an image, send image / svg or a text / plain link to a written file.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

Possible Duplicate: How do I randomly select k points from N points in MATLAB?

This function is useful for selecting a random index of a vector or matrix.

Alternatively, one or more arguments can be scalars.

Since multiple positions may have the same value, the index is very important.

The randperm approach described by @Stewie appears to be the way to go in most cases.

I want to generate a random number between -10 and 10.

This behavior is sometimes referred to as sampling with replacement.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In MATLAB, you can get the first value from an array by using 1 as the index.

For example, I want to generate a random number between -10 and 10.

There are special functions just for vectors; see the vector module for an explanation of how to Use these.

The sequence of numbers produced by randi is determined by the internal settings of the uniform pseudorandom number generator that underlies rand, randi, and randn.

You can control that shared random number generator using rng.

The rng function controls the global stream, which determines how the rand, randi, randn, and randperm functions produce a sequence of random numbers.

For details, see Creating and Controlling a Random Number Stream.

This MATLAB function generates a 1-by-m random variate from the m-dimensional Gaussian mixture distribution gm.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

The same commands will work in MATLAB.

This MATLAB function partitions a subset of observations in a random patch extraction datastore into a new datastore.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In the MATLAB environment, each variable is treated as a matrix or an array irrespective of their data types.

Matlab does not need variables to be declared explicitly.

You can drag and drop an image, send image / svg or a text / plain link to a written file.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

Possible Duplicate: How do I randomly select k points from N points in MATLAB?

This function is useful for selecting a random index of a vector or matrix.

Alternatively, one or more arguments can be scalars.

Since multiple positions may have the same value, the index is very important.

The randperm approach described by @Stewie appears to be the way to go in most cases.

I want to generate a random number between -10 and 10.

This behavior is sometimes referred to as sampling with replacement.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In MATLAB, you can get the first value from an array by using 1 as the index.

For example, I want to generate a random number between -10 and 10.

There are special functions just for vectors; see the vector module for an explanation of how to Use these.

The sequence of numbers produced by randi is determined by the internal settings of the uniform pseudorandom number generator that underlies rand, randi, and randn.

You can control that shared random number generator using rng.

The rng function controls the global stream, which determines how the rand, randi, randn, and randperm functions produce a sequence of random numbers.

For details, see Creating and Controlling a Random Number Stream.

This MATLAB function generates a 1-by-m random variate from the m-dimensional Gaussian mixture distribution gm.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

The same commands will work in MATLAB.

This MATLAB function partitions a subset of observations in a random patch extraction datastore into a new datastore.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In the MATLAB environment, each variable is treated as a matrix or an array irrespective of their data types.

Matlab does not need variables to be declared explicitly.

You can drag and drop an image, send image / svg or a text / plain link to a written file.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

Possible Duplicate: How do I randomly select k points from N points in MATLAB?

This function is useful for selecting a random index of a vector or matrix.

Alternatively, one or more arguments can be scalars.

Since multiple positions may have the same value, the index is very important.

The randperm approach described by @Stewie appears to be the way to go in most cases.

I want to generate a random number between -10 and 10.

This behavior is sometimes referred to as sampling with replacement.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In MATLAB, you can get the first value from an array by using 1 as the index.

For example, I want to generate a random number between -10 and 10.

There are special functions just for vectors; see the vector module for an explanation of how to Use these.

The sequence of numbers produced by randi is determined by the internal settings of the uniform pseudorandom number generator that underlies rand, randi, and randn.

You can control that shared random number generator using rng.

The rng function controls the global stream, which determines how the rand, randi, randn, and randperm functions produce a sequence of random numbers.

For details, see Creating and Controlling a Random Number Stream.

This MATLAB function generates a 1-by-m random variate from the m-dimensional Gaussian mixture distribution gm.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

The same commands will work in MATLAB.

This MATLAB function partitions a subset of observations in a random patch extraction datastore into a new datastore.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In the MATLAB environment, each variable is treated as a matrix or an array irrespective of their data types.

Matlab does not need variables to be declared explicitly.

You can drag and drop an image, send image / svg or a text / plain link to a written file.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

Possible Duplicate: How do I randomly select k points from N points in MATLAB?

This function is useful for selecting a random index of a vector or matrix.

Alternatively, one or more arguments can be scalars.

Since multiple positions may have the same value, the index is very important.

The randperm approach described by @Stewie appears to be the way to go in most cases.

I want to generate a random number between -10 and 10.

This behavior is sometimes referred to as sampling with replacement.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In MATLAB, you can get the first value from an array by using 1 as the index.

For example, I want to generate a random number between -10 and 10.

There are special functions just for vectors; see the vector module for an explanation of how to Use these.

The sequence of numbers produced by randi is determined by the internal settings of the uniform pseudorandom number generator that underlies rand, randi, and randn.

You can control that shared random number generator using rng.

The rng function controls the global stream, which determines how the rand, randi, randn, and randperm functions produce a sequence of random numbers.

For details, see Creating and Controlling a Random Number Stream.

This MATLAB function generates a 1-by-m random variate from the m-dimensional Gaussian mixture distribution gm.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

The same commands will work in MATLAB.

This MATLAB function partitions a subset of observations in a random patch extraction datastore into a new datastore.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In the MATLAB environment, each variable is treated as a matrix or an array irrespective of their data types.

Matlab does not need variables to be declared explicitly.

You can drag and drop an image, send image / svg or a text / plain link to a written file.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

Possible Duplicate: How do I randomly select k points from N points in MATLAB?

This function is useful for selecting a random index of a vector or matrix.

Alternatively, one or more arguments can be scalars.

Since multiple positions may have the same value, the index is very important.

The randperm approach described by @Stewie appears to be the way to go in most cases.

I want to generate a random number between -10 and 10.

This behavior is sometimes referred to as sampling with replacement.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In MATLAB, you can get the first value from an array by using 1 as the index.

For example, I want to generate a random number between -10 and 10.

There are special functions just for vectors; see the vector module for an explanation of how to Use these.

The sequence of numbers produced by randi is determined by the internal settings of the uniform pseudorandom number generator that underlies rand, randi, and randn.

You can control that shared random number generator using rng.

The rng function controls the global stream, which determines how the rand, randi, randn, and randperm functions produce a sequence of random numbers.

For details, see Creating and Controlling a Random Number Stream.

This MATLAB function generates a 1-by-m random variate from the m-dimensional Gaussian mixture distribution gm.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

The same commands will work in MATLAB.

This MATLAB function partitions a subset of observations in a random patch extraction datastore into a new datastore.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In the MATLAB environment, each variable is treated as a matrix or an array irrespective of their data types.

Matlab does not need variables to be declared explicitly.

You can drag and drop an image, send image / svg or a text / plain link to a written file.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

Possible Duplicate: How do I randomly select k points from N points in MATLAB?

This function is useful for selecting a random index of a vector or matrix.

Alternatively, one or more arguments can be scalars.

Since multiple positions may have the same value, the index is very important.

The randperm approach described by @Stewie appears to be the way to go in most cases.

I want to generate a random number between -10 and 10.

This behavior is sometimes referred to as sampling with replacement.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In MATLAB, you can get the first value from an array by using 1 as the index.

For example, I want to generate a random number between -10 and 10.

There are special functions just for vectors; see the vector module for an explanation of how to Use these.

The sequence of numbers produced by randi is determined by the internal settings of the uniform pseudorandom number generator that underlies rand, randi, and randn.

You can control that shared random number generator using rng.

The rng function controls the global stream, which determines how the rand, randi, randn, and randperm functions produce a sequence of random numbers.

For details, see Creating and Controlling a Random Number Stream.

This MATLAB function generates a 1-by-m random variate from the m-dimensional Gaussian mixture distribution gm.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

The same commands will work in MATLAB.

This MATLAB function partitions a subset of observations in a random patch extraction datastore into a new datastore.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In the MATLAB environment, each variable is treated as a matrix or an array irrespective of their data types.

Matlab does not need variables to be declared explicitly.

You can drag and drop an image, send image / svg or a text / plain link to a written file.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

Possible Duplicate: How do I randomly select k points from N points in MATLAB?

This function is useful for selecting a random index of a vector or matrix.

Alternatively, one or more arguments can be scalars.

Since multiple positions may have the same value, the index is very important.

The randperm approach described by @Stewie appears to be the way to go in most cases.

I want to generate a random number between -10 and 10.

This behavior is sometimes referred to as sampling with replacement.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In MATLAB, you can get the first value from an array by using 1 as the index.

For example, I want to generate a random number between -10 and 10.

There are special functions just for vectors; see the vector module for an explanation of how to Use these.

The sequence of numbers produced by randi is determined by the internal settings of the uniform pseudorandom number generator that underlies rand, randi, and randn.

You can control that shared random number generator using rng.

The rng function controls the global stream, which determines how the rand, randi, randn, and randperm functions produce a sequence of random numbers.

For details, see Creating and Controlling a Random Number Stream.

This MATLAB function generates a 1-by-m random variate from the m-dimensional Gaussian mixture distribution gm.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

The same commands will work in MATLAB.

This MATLAB function partitions a subset of observations in a random patch extraction datastore into a new datastore.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In the MATLAB environment, each variable is treated as a matrix or an array irrespective of their data types.

Matlab does not need variables to be declared explicitly.

You can drag and drop an image, send image / svg or a text / plain link to a written file.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

Possible Duplicate: How do I randomly select k points from N points in MATLAB?

This function is useful for selecting a random index of a vector or matrix.

Alternatively, one or more arguments can be scalars.

Since multiple positions may have the same value, the index is very important.

The randperm approach described by @Stewie appears to be the way to go in most cases.

I want to generate a random number between -10 and 10.

This behavior is sometimes referred to as sampling with replacement.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In MATLAB, you can get the first value from an array by using 1 as the index.

For example, I want to generate a random number between -10 and 10.

There are special functions just for vectors; see the vector module for an explanation of how to Use these.

The sequence of numbers produced by randi is determined by the internal settings of the uniform pseudorandom number generator that underlies rand, randi, and randn.

You can control that shared random number generator using rng.

The rng function controls the global stream, which determines how the rand, randi, randn, and randperm functions produce a sequence of random numbers.

For details, see Creating and Controlling a Random Number Stream.

This MATLAB function generates a 1-by-m random variate from the m-dimensional Gaussian mixture distribution gm.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

The same commands will work in MATLAB.

This MATLAB function partitions a subset of observations in a random patch extraction datastore into a new datastore.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number in the interval (0,1).

In the MATLAB environment, each variable is treated as a matrix or an array irrespective of their data types.

Matlab does not need variables to be declared explicitly.

You can drag and drop an image, send image / svg or a text / plain link to a written file.

Random number stream, specified as the MATLAB default random number stream or RandStream.

Example: s = RandStream('mlfg6331_64') creates a random number stream that uses the multiplicative lagged Fibonacci generator algorithm.

Possible Duplicate: How do I randomly select k points from N points in MATLAB?

This function is useful for selecting a random index of a vector or matrix.

Alternatively, one or more arguments can be scalars.

Since multiple positions may have the same value, the index is very important.

The randperm approach described by @Stewie appears to be the way to go in most cases.

I want to generate a random number between -10 and 10.

This behavior is sometimes referred to as sampling with replacement.

The same commands will work in MATLAB.

This MATLAB function returns a single uniformly distributed random number


## Overview

Students analyze the data they have collected to answer their question for the unit project. They will also complete a short Self Check. Students are given class time to work on their projects, using the time to analyze their data, finding the different measures and/or graphing their data.

## Key Concepts

Students will look at all of the tools that they have to analyze data, including:
- Graphic representations: line plots, box plots, and histograms
- Measures of center and spread: mean, median, mode, range, and the five-number summary

## Goals and Learning Objectives

- Complete the project, or progress far enough to complete it outside of class.
- Review measures of center and spread and the three types of graphs explored in the unit.
- Check knowledge of box plots and measures of center and spread.

## Project Rubric

Have students look at the rubric and talk with a partner to describe it as completely as possible in their own words. Students with disabilities may have a more challenging time identifying areas of improvement to target in their projects. English language learners may need additional support understanding the rubric.

## Organize and Analyze Project Data

Students will work in their groups to make decisions about their project. Explain that they need to be clear about what is left to do on their project by the end of Work Time. Check with students as they work. Ensure students understand the measures of center (mean and median) and how to calculate them.

## Mathematics

As students work, ask them which type of graph (or graphs) would be most useful for their data sets and why. Provide interventions for students who have difficulty getting started, choosing the best measure of what is typical, or selecting an inappropriate plot or bin width.

## Interventions

- For students having trouble getting started: try making a line plot of the data to see what it looks like.
- For students having trouble choosing the best measure of what is typical: consider calculating the MAD, looking at the line plot, and checking for outliers.
- For students choosing an inappropriate plot or bin width: ask questions about the data set, such as what type of plot would show big gaps or a value much greater than all the others.

## Work Time

Organize and analyze the data collected for the project. Consider what measures can be calculated to help draw conclusions and what graph(s) will best represent the data. Keep the rubric in mind as you work on the project.

## Prepare a Presentation

Work on preparing the project presentation, including the data set, at least one graph, a measure of center and/or spread, a summary of what the graph and measures tell about the data, and a conclusion about what is typical for the set of data.

## Challenge Problem

Suppose you had collected twice as much data. Do you think having twice the amount of data would change your conclusion, or reinforce it? Which of your measures would change? How would your graph(s) change?

## Make Connections

Have students share their thoughts about the project and discuss possible additions to the rubric. Consider what specific mathematical representations should be used in the unit project, what an appropriate level of effort is, how to know if the conclusions are correct, and what to evaluate that is specific to a statistics unit.

## Performance Task

Look at the rubric again and consider what should be added to the column "Specific to This Project" or the row at the bottom. Discuss these questions with a partner and write down any ideas. Be prepared to discuss your ideas as a class.

## Measures of Center and Spread

Complete the Self Check by creating a possible data set so that the mean of the data set is 15 inches. Show your work and explain how you created the data set.

## Reflect on Your Work

Write a reflection about the ideas discussed in class today, including what your group accomplished and your next steps. Share your reflections with your group.


Given a triangle ABC, its sides are divided internally at A′, B′, C′, in extreme and mean ratio. Let M be the area of triangle ABC. The division of the sides is such that BA′ = x, where x² = a · (a − x), yielding x = (−a ± a√5)/2.

The area of triangle AB′C′ is calculated as (1/8) * (area of triangle ABC) * (√5 - 2) for one of the smaller triangles, and similarly for triangles BC′A′ and CA′B′.

The sum of the areas of these three smaller triangles is 3 * (√5 - 2) * M. Consequently, the area of triangle A′B′C′ is (7 - 3√5) * M.


**Unit Overview:** Students apply properties of number systems to equations involving various functions, including polynomial, rational, radical, and absolute value functions. They sketch graphs and identify zeros of polynomials, recognizing patterns to identify polynomial identities and geometric series.

**Enduring Understanding:**
- Equations can be presented in multiple equivalent forms through manipulation, extending rational expression understandings to equation functions.
- Zeros of polynomials can be determined using various methods.
- Graphs of polynomials can be sketched by identifying key elements.

**Essential Questions:**
1. What is the purpose of graphing functions?
2. How can extraneous solutions be interpreted?
3. Why are the zeros of a function important?
4. How can identifying patterns be beneficial?


**Laws of Arithmetic**

The Commutative Law for Addition states that the order of addition does not change the result: a + b = b + a. For example, 5 + 7 = 7 + 5 = 12. The Commutative Law for Multiplication states that the order of multiplication does not change the result: a x b = b x a. For example, 3 x 6 = 6 x 3 = 18.

The Associative Law for Addition states that the order in which we add numbers does not change the result: (a + b) + c = a + (b + c). For example, (4 + 6) + 5 = 4 + (6 + 5) = 15. The Associative Law for Multiplication states that the order in which we multiply numbers does not change the result: (a x b) x c = a x (b x c). For example, (2 x 3) x 4 = 2 x (3 x 4) = 24.

The Distributive Law states that multiplication distributes over addition: a x (b + c) = (a x b) + (a x c). For example, 3 x (2 + 5) = (3 x 2) + (3 x 5) = 21.

**Inverse and Identity Elements**

The additive inverse of a number a is -a, such that a + (-a) = 0. For example, 5 + (-5) = 0. The multiplicative inverse of a number a is 1/a, such that a x (1/a) = 1. For example, 5 x (1/5) = 1.

The identity element for addition is 0, such that a + 0 = a. For example, 5 + 0 = 5. The identity element for multiplication is 1, such that a x 1 = a. For example, 5 x 1 = 5.

**Distance Formula**

The distance between two points (x1, y1) and (x2, y2) on a Cartesian graph is given by the formula: z^2 = (x2 - x1)^2 + (y2 - y1)^2, where z is the distance between the two points. Taking the square root of both sides, we get: z = sqrt((x2 - x1)^2 + (y2 - y1)^2).

**Exponent Rules**

When multiplying two numbers with the same base, we add the exponents: x^m x x^n = x^(m + n). For example, 2^3 x 2^4 = 2^(3 + 4) = 2^7.

When dividing two numbers with the same base, we subtract the exponents: x^m / x^n = x^(m - n). For example, 2^7 / 2^4 = 2^(7 - 4) = 2^3.

When raising a power to another power, we multiply the exponents: (x^m)^n = x^(m x n). For example, (x^5)^3 = x^(5 x 3) = x^15.

When taking the nth root of a number, we raise it to the power of 1/n: (x^m)^(1/n) = x^(m/n). For example, (x^15)^(1/3) = x^(15/3) = x^5.

**Solving Equations**

To solve the equation 4x - 9 = 7 + 9x + 19, we first combine like terms: 4x - 9 = 26 + 9x. Then, we add 9 to both sides to get: 4x = 35 + 9x. Next, we subtract 9x from both sides to get: -5x = 35. Finally, we multiply both sides by -1/5 to solve for x: x = 7.

**Factoring Polynomials**

To factor the polynomial x^5 - 2x^4 - 15x^3, we can factor out the common factor of x^3: x^3(x^2 - 2x - 15). Factoring the quadratic expression, we get: x^3(x - 5)(x + 3).

To simplify the rational expression (x^4)(x + 4)(x + 3) / (x^3)(x + 3)(x - 5), we can cancel out the common factors: (x^4)(x + 4) / (x^3)(x - 5). Simplifying further, we get: x(x + 4) / (x - 5).