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?


Concept/skill: 
Recognize and represent proportional relationships between quantities.

Answer: 
No. Usually problems that teach recognizing and representing proportional relationships between quantities involve deciding whether two quantities are in a proportional relationship or identifying the constant of proportionality, and then representing proportional relationships by equations or explaining proportional relationship using coordinate graphs.

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
$$-\frac{1}{4}$$


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

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:
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?


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.

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, does the problem or activity shown below enable students to learn the full intent of the following concept or skill?

shot_info: fl_problem_000972_neg_0

6

fl_problem_001097_pos_0

7

fl_problem_001392_neg_0

8
standard_description: Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.
problem_activity: Activity
Students practice using precise wording (MP6) to describe the positive or negative association between two variables given scatter plots of data.
Launch
Display the scatterplot for all to see. Remind students that we investigated the relationship between car weight and fuel efficiency earlier.
[IMG]
Ask students how they would describe the relationship between weight and fuel efficiency. After 30 seconds of quiet think time, select 1–2 students to share their responses. (The scatter plot shows that for these cars, as the weight of a car increases, its fuel efficiency decreases.)
Give students 3–5 minutes to construct similar sentences to describe scatter plots of three other data sets followed by a whole-class discussion.
Student Facing
For each scatter plot, decide if there is an association between the two variables, and describe the situation using one of these sentences:
For these data, as ________________ increases, ________________ tends to increase.
For these data, as ________________ increases, ________________ tends to decrease.
For these data, ________________ and ________________ do not appear to be related.
[IMG]
[IMG]
[IMG]
Student Response
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Anticipated Misconceptions
Students may assume they need to use each sentence exactly one time. Let them know it is acceptable to use a sentence more than once and that it is acceptable to not use a sentence.
Activity Synthesis
The purpose of the discussion is to talk about trends in data based on the representations in scatter plots.
Consider asking some of the following questions:
“Is it surprising that the price decreases as mileage increases? Why doesn’t mileage predict price perfectly?” (There are other factors, like whether the car has any damage or has any extra features.)
“Is it surprising that heavier diamonds cost more? Why doesn’t weight predict price perfectly?” (There are other factors, like the cut, the clarity, etc.)
“Is it surprising that energy consumption goes up as the temperature increases?” (No, because of air conditioning.)
For the last scatter plot, highlight the outliers by asking:
“What might cause more energy consumption on a cool day?” (Laundry, using power tools, etc.)
“What might cause less energy consumption on a hot day?” (Being gone from home, etc.)
Students may notice that the association between high temperature and energy consumed is more variable than the other situations. There is still a positive association or positive trend, but we would describe the association as “weaker.”
Speaking: MLR8 Discussion Supports.
Use this routine to amplify mathematical uses of language to communicate about the relationship between the two quantities represented in each scatter plot. As students describe the trends, press for details by requesting that students challenge an idea, elaborate on an idea, or give an example of their process. Revoice student ideas to model mathematical language use in order to clarify, apply appropriate language, and involve more students.
Design Principle(s): Support sense-making; Optimize output (for explanation)


aligns: yes