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 to multiply a fraction or whole number by a fraction.

Answer: 
No. The expressions shown in this problem do not involve multiplying fractions with other fractions, nor whole numbers with fractions.

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.
{"table": [{"_attributes": {"align": "center", "class": ["table", "table-bordered"], "style": "table-layout:fixed; width:70%"}, "tbody": [{"tr": [{"td": [{"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "No solution"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "Exactly 1 solution"}]}, {"_attributes": {"style": "text-align:center"}, "strong": [{"_value": "Infinite number of solutions"}]}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9=4x+7}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${y=3y+5}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${y={x\\over2}}$$"}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${3x+6y-9=0}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${x=y}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${1.25x=1.25y}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9.9=6x+8y}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${9x=2.5y-8.8}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}]}, {"td": [{"p": [{"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${11x-2y=1.5}$$"}]}, {"_attributes": {"style": "text-align:center"}, "span": [{"_attributes": {"class": ["math-tex"]}, "_value": "$${11x-2y=2.5}$$"}]}]}, {"_attributes": {"style": "text-align:center"}}, {"_attributes": {"style": "text-align:center"}}, {}]}]}]}]}
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_neg_0

7

fl_problem_001392_neg_0

8
standard_description: 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.”
problem_activity: Activity
In this task, students calculate and interpret both
\(\frac{a}{b}\)
and
\(\frac{b}{a}\)
from a ratio
\(a:b\)
presented in a context. They work with less-familiar units. The term
unit rate
is introduced so that students have a general name for a “how many per 1” quantity.
In the first half of the task, students practice computing unit rates from ratios. In the second half they practice selecting the better unit rate to use (
\(\frac{a}{b}\)
or
\(\frac{b}{a}\)
) based on the question posed.
As students work on the second half of the task, identify 1–2 students per question to share their choice of unit rate and how it was used to answer the question.
Launch
Recap that in the previous activity the ratio of 15 cups water for every 6 cups oats can be expressed as two rates “per 1”. These rates are 0.4 cups of oats per cup of water or 2.5 cups of water per cup of oats. Emphasize that, in a table, each of these rates reflects a value paired with a “1” in a row, and that both can be useful depending on the problem at hand. Tell students that we call 0.4 and 2.5 “unit rates” and that a unit rate means “the amount of one quantity for 1 of another quantity.”
Arrange students in groups of 2. Tell students that they will now solve some problems using unit rates. Give students 3–4 minutes to complete the first half of the task (the first three problems). Ask them to share their responses with their partner and come to an agreement before moving on to the second half. Clarify that “oz” is an abbreviation for “ounce.”
Representation: Develop Language and Symbols.
Create a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of these terms. Include the following term and maintain the display for reference throughout the unit: unit rate.
Supports accessibility for: Conceptual processing; Language
Writing, Listening: MLR6 Three Reads.
Use this routine to set students up to comprehend each situation, without solving the questions for them. In the first read, students read the situation with the goal of comprehending the situation (e.g., a cheesecake recipe contains cream cheese and sugar, Mai’s family drinks milk regularly, Tyler has raffle tickets). In the second read, ask students to identify the important quantities by asking them what can be counted or measured (e.g., 12 oz of cream cheese with 15 oz of sugar, 10 gallons of milk every 6 weeks, $16 for 4 raffle tickets). In the third read, ask students to brainstorm possible mathematical solution strategies to answer the question. This helps students connect the ratio language used in the problem to identify a strategy for finding a unit rate.
Design Principle(s): Support sense-making
Student Facing
For each situation, find the
unit rates
.
A cheesecake recipe says, “Mix 12 oz of cream cheese with 15 oz of sugar.”
How many ounces of cream cheese are there for every ounce of sugar?
How many ounces of sugar is that for every ounce of cream cheese?
Mai’s family drinks a total of 10 gallons of milk every 6 weeks.
How many gallons of milk does the family drink per week?
How many weeks does it take the family to consume 1 gallon of milk?
Tyler paid
$
16 for 4 raffle tickets.
What is the price per ticket?
How many tickets is that per dollar?
For each problem, decide which unit rate from the previous situations you prefer to use. Next, solve the problem, and show your thinking.
If Lin wants to make extra cheesecake filling, how much cream cheese will she need to mix with 35 ounces of sugar?
How many weeks will it take Mai’s family to finish 3 gallons of milk?
How much would all 1,000 raffle tickets cost?
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?
Write a “deal” on tickets for Tyler’s raffle that sounds good, but is actually a little worse than just buying tickets at the normal price.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
If students are not sure how to use the unit rates they found for each situation to answer the second half of the task, remind them of how the oatmeal problem was solved. Suggest that this problem is similar because they can scale up from a unit rate to answer the questions.
Activity Synthesis
Invite previously identified students to share their work on the second half (the last three questions) of the task.
Though the task prompts students to think in terms of unit rate, some students may still reason in ways that feel safer. For example, to find out how much cream cheese Lin would mix with 35 oz of sugar, they may double the 12 oz of cream cheese to 15 oz of sugar ratio to obtain 24 oz of cream cheese for 30 oz of sugar, and then add 4 oz more of cream cheese for the additional 5 oz of sugar. Such lines of reasoning show depth of understanding and should be celebrated. Guide students to also see, however, that some problems (such as the milk problem) can be more efficiently solved using unit rates.
For the ticket problem, students may comment that
\(\frac14\)
of a ticket costing a dollar does not make sense, since it is not possible to purchase
\(\frac14\)
of a ticket. Take this opportunity to applaud the student(s) for reasoning about the interpretation of the number in the context, which is an example of engaging in MP2. If students do not raise this concern, ask: “How can
\(\frac14\)
of a ticket costing a dollar make sense?” Students may argue that the quantity, on its own, does not make sense. Challenge them to figure out how the rate could be used in the context of the problem. For example, ask, “If I had
$
80, how many tickets could I buy? What if I had
$
75? Can the ‘
\(\frac14\)
of a ticket per dollar’ rate help answer these questions?”


aligns: no