few_shots: 
Example 1:
Concept/skill: 
Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

Problem/activity:
You’re mixing ingredients for cookies. The recipe says to combine 6 tablespoons, or ¾ stick, of butter with 1 cup of sugar. You accidentally mix in a whole stick of butter (8 tablespoons) with the cup of sugar. How can you fix this?


Answer: 
Yes. This problem solves a real-world problem involving mixing ingredients for cookies. A student doing this problem would need to reason about equivalent ratios of butter amounts to sugar amounts.

Example 2:
Concept/skill: 
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

Problem/activity:
For each pair of numbers, determine if the values are equivalent. Explain your reasoning for each one.
a.
$$-0.25$$
and
$$\frac{1}{-4}$$
b.
$$-\left ( \frac{1}{4} \right )$$
and
$$\frac{-1}{-4}$$
c.
$$\frac{-1}{4}$$
and
$$-\frac{1}{4}$$


Answer: 
Yes. This problem involves understanding fractions and division to divide rational numbers, such as 1, -4, 4, and -1. This problem also involves an understanding of multiplication: if p and q are integers, then -(p/q) = (-p)/q = p/(-q).

Example 3:
Concept/skill: 
Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

Problem/activity:
Problem 1
The table below contains single equations in a single variable, single equations in two variables, and pairs of simultaneous equations in two variables. Check the appropriate box to show whether there is no solution, exactly 1 solution, or if there is an infinite number of solutions.
|                                                          | **           No solution          ** | **           Exactly 1 solution          ** | **           Infinite number of solutions          ** |
|----------------------------------------------------------|--------------------------------------|---------------------------------------------|-------------------------------------------------------|
|                       $${9=4x+7}$$                       |                                      |                                             |                                                       |
|                       $${y=3y+5}$$                       |                                      |                                             |                                                       |
|                    $${y={x\over2}}$$                     |                                      |                                             |                                                       |
|                     $${3x+6y-9=0}$$                      |                                      |                                             |                                                       |
|    $${x=y}$$                        $${1.25x=1.25y}$$    |                                      |                                             |                                                       |
| $${9.9=6x+8y}$$                        $${9x=2.5y-8.8}$$ |                                      |                                             |                                                       |
| $${11x-2y=1.5}$$                        $${11x-2y=2.5}$$ |                                      |                                             |                                                       |
Problem 2
Kim has a small container and a large container as shown.
[IMG]
It takes 16 of the small containers to fill the large container. Three small containers leave 1.95 gallons of space in the big container.
What is the size of each of the two containers?


Answer: 
No. Completing neither of these problems requires replacing one equation by the sum of that equation and a multiple of the other equation.

Now, determine whether the following problem or activity can enable students to understand the concepts or skills in the provided description.

shot_info: fl_problem_000972_pos_0

6

fl_problem_001097_pos_0

7

fl_problem_001392_neg_0

8
standard_description: Use the mean and standard deviation of a data set to fit it to a normal distribution and to estimate population percentages. Recognize that there are data sets for which such a procedure is not appropriate. Use calculators, spreadsheets, and tables to estimate areas under the normal curve.
problem_activity: Activity
The purpose of this activity is to let students investigate what happens to the standard deviation using different data sets. The goal is for students to make conjectures about what standard deviation measures and how relative size of the standard deviation can be estimated from the shape of the distribution. In particular, students should recognize that adding or subtracting the same value from each value in the data set will change the mean by the same amount, but the standard deviation remains unchanged. Multiplying or dividing each value in the data set by the same value scales both the mean and standard deviation by the same value.
Monitor for students who
Check values without much thought until the mean or standard deviation is correct.
Use a starting set of values, then modify a few of them up or down to match the statistics given.
Use symmetry to get the mean correct.
Adjust the data by multiplying all of the values by a number to change the spread of the distribution to get close to the standard deviation.
Identify students who create data sets with different types of distributions for “10 different numbers that have a mean of 1” and “10 different numbers with a standard deviation as close to 2.5 as you can get in one minute.”
Launch
Demonstrate how to find standard deviation and mean using the technology available. In Algebra 1, students are dealing with the entire population, not sampling, so the population standard deviation is used.
Open the Spreadsheet & Statistics tool from Math Tools, or navigate to
https://www.geogebra.org/classic/spreadsheet
. Enter the values in column A. Select all of column A, and then choose the button that looks like a histogram and “One Variable Analysis.” Click the button that says
\(\Sigma x\)
. The population standard deviation is labeled
\(\sigma\)
. Additionally, the spreadsheet command =SD(A1:A10) will compute the population standard deviation for data in cells A1 through A10.
Arrange students in groups of 2. Give students time to work through the first two questions, followed by a whole-class discussion.
Conversing: MLR 5 Co-Craft Questions.
Display only the first line of this task (“Use technology to find the mean and the standard deviation for the data in the dot plots”) and the dot plots, and ask pairs of students to write possible mathematical questions about the distributions. Then, invite pairs to share their questions with the class. This helps students produce the language of mathematical questions and talk about the relationships between the two quantities in this task (mean and standard deviation).
Design Principle(s): Cultivate conversation
Student Facing
Use technology to find the mean and the standard deviation for the data in the dot plots.
What do you notice about the mean and standard deviation you and your partner found for the three dot plots?
Invent some data that fits the conditions. Be prepared to share your data set and reasoning for choice of values.
Partner 1
​Partner 2
Dot plots:
[IMG]
[IMG]
[IMG]
Dot plots:
[IMG]
[IMG]
[IMG]
​​​​​​
Conditions:
10 numbers with a standard deviation equal to the standard deviation of your first dot plot with a mean of 6.
10 numbers with a standard deviation three times greater than the data in the first row.
10 different numbers with a standard deviation as close to 2 as you can get in 1 minute.
Conditions:
10 numbers with a standard deviation equal to the standard deviation of your first dot plot with a mean of 12.
10 numbers with a standard deviation four times greater than the data in the first row.
10 different numbers with a standard deviation as close to 2 as you can get in 1 minute.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
The purpose of this discussion is to understand that the standard deviation is a measure of variability related to the mean of the data set. The discussion also provides an opportunity for students to discuss what they notice and wonder about the mean and standard deviation.
Select previously identified students in the order listed in the narrative to share their distributions and methods for finding values that worked.
For at least one student’s data set that had a standard deviation close to 2.5, ask them how they might adapt the data so that it has a mean of 1. (Shift the data up or down so that it has a mean of 1.)
Here are some questions for discussion.
“What do you think that standard deviation measures? Why do you think that?” (I think that it measures variability because it behaved like MAD. When all the values were the same it was zero.)
“Why is the standard deviation the same for {1,2,3,4,5} and {-2,-1,0,1,2}?” (For each data set: 1) the values to the left of the mean are a distance of 2 and 1 from the mean and the values to the right of the mean are a distance 2 and 1 from the mean, and 2) the middle value is 0 away from the mean. This results in the standard deviation being the same.)
“Why is the standard deviation different for {-4, -2, 0, 2, 4} and {-4, -3, -2, -1, 0}?” (The values for the first data set are twice the distance from the mean as the values in the second data set. That makes the standard deviation for the first set greater than the standard deviation of the second data set.)
“When is the standard deviation equal to zero?” (It is zero when all the values are the same as each other or when there is no variability.)
“Was your mean the same as your partner’s mean in the fifth match?” (Answers vary. Sample response: My mean was much different from my partners. Mine was 5 and theirs was 10.)
“How did using technology help or hinder your mathematical thinking about standard deviation and mean?” (It really helped me because I did not get bogged down with the calculations and I could look for patterns in the data, the data displays, and the statistics. In particular, having to come up with my own data and then see what happened to the statistics was really helpful.)
Action and Expression: Internalize Executive Functions.
Provide students with a graphic organizer with standard deviation at the center, connecting to it all the related concepts mentioned during the discussion.
Supports accessibility for: Language; Organization


aligns: no