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.)


Your response:
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.


Your response:
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.


Your response:
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: Fluently multiply multi-digit whole numbers using the standard algorithm.

Find whole-number quotients of whole numbers with up to four-digit dividends and two-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Add, subtract, multiply, and divide decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method and explain the reasoning used.
problem_activity: Narrative
The purpose of this activity is for students to estimate whole number products in the context of volume. In the next activity students will calculate the smallest and largest volumes within the range recommended for each type of bird. The estimates here may or may not fall within the range, depending on the numbers students pick. When making reasoned estimates, there is always some tension between accuracy and using the most friendly numbers. During the synthesis, students explain the different strategies they use to make reasonable estimates with calculations that they can perform as simply as possible, often mentally (MP3).
Students may need a quick reminder of how to find the volume of a rectangular prism. If needed, remind students that the volume of a rectangular prism is the product of the length, width, and height, or alternatively, the product of the area of a base and the height for that base.
Launch
Display table from task.
“What do you notice and wonder about the table?” (There are different kinds of birds listed. There are side lengths for the floor but the height is a range. What do the numbers mean? Do smaller birds have smaller houses?)
1 minute: quiet think time
Share and record student responses.
If needed, display images of different kinds of birds.
Activity
2 minutes: quiet think time
5 minutes: partner work time
Monitor for students who consider friendly numbers to use for the height and also for estimating the product of the area of the floor and the height.
Student Facing
Different types of birds use different types of houses. The table gives you the recommended side lengths for birdhouses of various species.
|     type of bird      | side lengths of floor | height | volume estimate |
|-----------------------|-----------------------|--------|-----------------|
|       chickadee       |                       |        |                 |
|       wood duck       |                       |        |                 |
|       barn owl        |                       |        |                 |
| red-headed woodpecker |                       |        |                 |
|       bluebird        |                       |        |                 |
|        swallow        |                       |        |                 |
Estimate a possible volume for each birdhouse. Be prepared to explain your reasoning.
Student Response
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Activity Synthesis
“How did you estimate the volume of a house for a wood duck?”
I know
\(10 \times 18\)
is 180 and I multiplied by 10 since that just adds a zero.
I also used
\(10 \times 18 = 180\)
but multiplied by 20 since that is between 10 and 24. I got
\(3,600\)
cubic inches.
“How did you estimate the volume of a house for a red-headed woodpecker?”
I found that the area of the floor is 36 square inches. I multiplied this by 10 to get 360 and by 20 to get 720 and then picked a number in between, 500 cubic inches.
I found the area of the floor was 36 square inches. I rounded that to 40 since it is a nicer number and then found
\(40 \times 12\)
which is 480.
“Which numbers are the friendliest for estimating products? Why?” (10 is the friendliest because I can use place value. Multiples of 10, like 20, are also friendly as I can multiply by 10 and then double.)


correct_option_index: 0