few_shots: Example 1:
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?


Standard options: 
A. Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.”
B. 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.
C. Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.” (Expectations for unit rates in this grade are limited to non-complex fractions.)


Thought: 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.
Answer: B

Example 2:
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
$$-\frac{1}{4}$$


Standard options: 
A. Solve real-world and mathematical problems involving the four operations with rational numbers. (Computations with rational numbers extend the rules for manipulating fractions to complex fractions.)
B. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
C. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.


Thought: 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).
Answer: B

Example 3:
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?


Standard options: 
A. Solve linear equations in one variable.
B. Analyze and solve pairs of simultaneous linear equations.


Thought: The last three rows of the table in the first problem involves solving pairs of simultaneous linear equations. The second word problem can also be represented as two simultaneous linear equations.
Answer: B

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: fl_problem_000972

6

fl_problem_001097

7

fl_problem_001392

8
options: Informally assess the degree of visual overlap of two numerical data distributions with similar variabilities, measuring the difference between the centers by expressing it as a multiple of a measure of variability. For example, the mean height of players on the basketball team is 10 cm greater than the mean height of players on the soccer team, about twice the variability (mean absolute deviation) on either team; on a dot plot, the separation between the two distributions of heights is noticeable.

Use measures of center and measures of variability for numerical data from random samples to draw informal comparative inferences about two populations. For example, decide whether the words in a chapter of a seventh-grade science book are generally longer than the words in a chapter of a fourth-grade science book.
problem_activity: Activity
Following the review from the previous activity, students are asked to use these calculations to compare two groups more formally. Students are shown one quantifiable method of determining whether the two groups are relatively close or relatively very different in the discussion following the activity involving describing the difference of the measures of center as a multiple of the variability (MP2). The important idea for students to grasp from this activity is that the measures of center and measures of variability of the groups work together to give an idea of how similar or different the groups are.
As students work to compare heights of the two families, monitor for students who:
Create dot plots of the data and look at the overlap visually.
Compute measures of center to compare the groups numerically.
Also compute measures of variability to compare the measures of center for each group accounting for the spread of values.
Launch
Keep students in groups of 2.
Introduce students to the idea that both the measures of center as well as the measures of variability are important when comparing data sets by asking students about these contexts.
“Two groups of adults have mean weights that are different by 10 pounds. Are the two groups very different in weight?”
“Two groups of 8 year olds have weights that are different by 10 pounds. Are the two groups very different in weight?”
“Two groups of birds have weights that are different by 3.2 ounces (0.2 pounds). Are the two groups very different in weight?”
Without knowing more about the exact numbers, it seems that the adults might not be that different. A typical adult might weigh 180 pounds and, depending on diet and activity, even one person’s weight might go up or down by 10 pounds in a few weeks. It might be hard to tell these two groups apart even with the 10 pound difference in means.
On the other hand, 8 year olds typically weigh about 50 pounds, so a 10 pound difference can mean a lot more. A group with an average weight of 50 pounds is 25% heavier than a group with an average weight of 40 pounds.
For the birds, it might be hard to say how different the groups are unless we have more information. If the two groups contain the same type of bird (e.g., parrots), we might expect the weights to be fairly consistent, so a 3.2 ounce difference could be a lot. If the two groups each contain a variety of types (from hummingbirds up to emus), the 3.2 ounce difference might not be very much.
Based on these examples, it helps to know not only the means (or medians), but also the variability of the groups. We saw this in the previous activity with the overlap in the dot plots.
For the work in this activity, tell students to try to give some values to help back up their comparison of the two groups.
Allow students 1 minute of quiet think time to examine ways of approaching the problem, followed by partner work time and a whole-class discussion.
Representation: Internalize Comprehension.
Activate or supply background knowledge of measures of centers. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing
Conversing: MLR2 Collect and Display.
Use this routine to capture the language students use as they compare the heights of these two families. Circulate and listen to student talk during small-group and whole-class discussion. Record the words and phrases students use on a display for all to see. Invite students to borrow from—or add more language to—the display throughout the remainder of the lesson. This will help students read and use mathematical language during their paired and whole-group discussions.
Design Principle(s): Optimize output (for comparison); Maximize meta-awareness
Student Facing
Compare the heights of these two families. Explain or show your reasoning.
The heights (in inches) of Noah’s family members: 28, 39, 41, 52, 63, 66, 71
The heights (in inches) of Jada’s family members: 49, 60, 68, 70, 71, 73, 77
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
If Jada's family adopts newborn twins who are each 18 inches tall, does this change your thinking? Explain your reasoning.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
The purpose of this discussion is for students to begin to be more formal in their comparison of data from different groups. In particular, a general rule is established that will be used in this unit.
Select students to share their approaches in the sequence outlined in the Activity Narrative.
The heights of Noah’s and Jada’s families overlap more than the heights of the gymnastics and volleyball teams. The difference in means for the two families is about the same as between the two teams.
The difference between the mean heights of the volleyball and gymnastics teams is
\(78.17 - 62.75\)
, or 15.42 inches.
The difference between the mean heights of Jada’s and Noah’s families is
\(66.86 - 51.43\)
, or 15.43 inches. The variability in heights for the families is greater than the variability in heights for the teams.
Explain: The difference between the means is not enough information to know whether or not the data sets are very different. One way to express the amount of overlap is to divide the difference in means by the (larger) mean absolute deviation.
Demonstrate for students how to do this calculation for the volleyball and gymnastics teams:
The difference in means is more than 6 times the measure of variability, because
\(15.42 \div 2.46 \approx 6.3\)
.
Leave the calculation for the two teams displayed. Ask students to do the calculation for Jada’s and Noah’s families.
The difference in means is a little more than 1 time the measure of variability, because
\(15.43 \div 13.22 \approx 1.2\)
.
As a general rule, we will consider it a large difference between the data sets if the difference in means is
more than twice
the mean absolute deviation. If the mean absolute deviation is different for each group, use the larger one for this calculation.
For students who ask why twice the MAD is used rather than some other value, defer the question for later in the unit. A later lesson titled Do They Carry More? will address this question in more detail.


correct_option_index: 0