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: Recognize and represent proportional relationships between quantities.

Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2/1/4 miles per hour, equivalently 2 miles per hour.

Use proportional relationships to solve multistep ratio and percent problems. Examples: simple interest, tax, markups and markdowns, gratuities and commissions, fees, percent increase and decrease, percent error.
problem_activity: Activity
This activity gives students an opportunity to put into practice some things they already know about finding percent rates. Additionally, the idea of a fraction of a percent appears for the first time. Encourage students to use any representation they would like to calculate the percentage of appetizers, entrees and desserts. Monitor for students who used various representations and ask them to share during the discussion. The main focus should be on the fractional percentages they encounter in this problem for the first time.
Digital
Launch
Tell students they will be finding some more percentages. Encourage them to use any representation they understand, for example, a double number line or a table. Students in groups of 2. Give students 1–2 minutes of quiet work time, followed by partner then whole-class discussion.
If using the digital activity, students will use an applet to find and check percentages.
Representation: Internalize Comprehension
. Activate or supply background knowledge about diagrams that can be used to represent percent rates such as tables and double number line diagrams.
Supports accessibility for: Memory; Conceptual processing
Writing and Listening: MLR5 Co-Craft Questions.
Before students begin work, display the waiter's situation without revealing the questions. Ask students to write down possible mathematical questions that might be asked about the situation. Invite pairs to compare their questions, and then ask for a few to be shared in a whole-class discussion. Reveal the actual questions about the waiter's situation that students will answer. This will help students make sense of the problem before attempting to solve it.
Design Principle(s): Optimize output for explanation
Student Facing
During one waiter’s shift, he delivered appetizers, entrées, and desserts. What percentage of the dishes were desserts? appetizers? entrées? What do your percentages add up to?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Tell students they will be finding some more percentages. Encourage them to use any representation they understand, for example, a double number line or a table. Students in groups of 2. Give students 1–2 minutes of quiet work time, followed by partner then whole-class discussion.
If using the digital activity, students will use an applet to find and check percentages.
Representation: Internalize Comprehension
. Activate or supply background knowledge about diagrams that can be used to represent percent rates such as tables and double number line diagrams.
Supports accessibility for: Memory; Conceptual processing
Writing and Listening: MLR5 Co-Craft Questions.
Before students begin work, display the waiter's situation without revealing the questions. Ask students to write down possible mathematical questions that might be asked about the situation. Invite pairs to compare their questions, and then ask for a few to be shared in a whole-class discussion. Reveal the actual questions about the waiter's situation that students will answer. This will help students make sense of the problem before attempting to solve it.
Design Principle(s): Optimize output for explanation
Student Facing
During one waiter’s shift, he delivered 13 appetizers, 17 entrées, and 10 desserts.
What percentage of the dishes he delivered were:
desserts?
appetizers?
entrées?
What do your percentages add up to?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
If students round to the nearest percentage, they will get that 33% of the dishes were appetizers and 43% of the dishes were entrées. Along with the 25% desserts, their percentages will sum to 101%. Point out that all of the dishes taken as percentages should sum to 100% and encourage them to be critical of their method and try to figure out where the extra 1% came from.
Activity Synthesis
Select students to share the percentages they calculated for each type of dish the waiter delivered. Depending on the outcome of the warm-up, it may be appropriate once again to display different representations of percentages as rates per 100.
Double number line:
[IMG]
Table:
| number of dishes | percentage |
|------------------|------------|
|        10        |     25     |
|        1         |    2.5     |
|        13        |    32.5    |
|        17        |    42.5    |
Equation: Students may have previously learned to represent relationships like this using an equation in a form
\(y=kx\)
. For example, to find what percent 13 is of 40, they might write
\(13=k \boldcdot 40\)
, and find that
\(k\)
is 0.325 by evaluating
\(13 \div 40\)
. 0.325 is the rate per 1, so 32.5 is the rate per 100.
Students may have never seen a percentage that was not a whole number. Spend a few minutes making sense of this. Ask students:
What do you notice that is different about these percentages from the ones you have looked at before? (Some of these percentages are not whole numbers.)
What do the percentages add up to? (Exactly 100.)
What does 32.5% of 40 mean? (It’s halfway between 32% and 33% of 40.)


correct_option_index: 2