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: Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?

Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. (Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade.)

Interpret multiplication as scaling (resizing), by:
- Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.
- Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.
problem_activity: Narrative
The purpose of this activity is for students to observe and use the structure of diagrams to find areas of shaded regions with non-unit fraction side lengths. Students build on what they learned in the previous activity, solidifying their understanding of why the numerator of a product of two fractions is the product of the numerators and the denominator of a product of fractions is the product of the denominators.
Action and Expression: Internalize Executive Functions.
Invite students to verbalize their strategy for writing multiplication expressions to represent the area of the shaded rectangle in each figure before they begin. Students can speak quietly to themselves, or share with a partner.
Supports accessibility for: Organization, Conceptual Processing, Language
Launch
Groups of 2
Activity
“Start working on completing the table independently. After a couple minutes, you'll work with your partner to complete the table and answer the rest of the questions.”
1–2 minutes: independent work time
5-8 minutes: partner work time
Monitor for students who:
notice the area of the shaded regions is always twentieths
write the expression
\(\frac{6}{5}\times\frac{4}{5}\)
to represent the shaded region of the last diagram in the table
explain that the expression
\(\frac{6\times4}{5\times4}\)
represents the shaded part of the last diagram in the table because
\(6\times4\)
represents the number of pieces that are shaded and
\(4 \times 5\)
represents the number of those pieces in the unit square
Student Facing
Complete the table.
|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                diagram                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                | multiplication                       expression | shaded area                       (square units) |
|-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|-------------------------------------------------|--------------------------------------------------|
| A                |https://cms-im.s3.amazonaws.com/ui5g959sm5gv0dm8y5muaeuevuxb?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-147492.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-147492.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T165553Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=da65b42438b700b09e62286557fa329bb057a4e3c43b71d5a338b3b21f3a4bad|  |https://cms-im.s3.amazonaws.com/ui5g959sm5gv0dm8y5muaeuevuxb?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-147492.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-147492.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T165553Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=da65b42438b700b09e62286557fa329bb057a4e3c43b71d5a338b3b21f3a4bad| image:: https://cms-im.s3.amazonaws.com/ui5g959sm5gv0dm8y5muaeuevuxb?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-147492.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-147492.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T165553Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=da65b42438b700b09e62286557fa329bb057a4e3c43b71d5a338b3b21f3a4bad |                                                 |                                                  |
| B                |https://cms-im.s3.amazonaws.com/dx240g4b4kkym1gkgs7khx0yw7ri?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-149071.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-149071.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T165553Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=216cb8ca62660faf0bab04b1fb04fd987604857004a334e4036d514267493e92|  |https://cms-im.s3.amazonaws.com/dx240g4b4kkym1gkgs7khx0yw7ri?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-149071.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-149071.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T165553Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=216cb8ca62660faf0bab04b1fb04fd987604857004a334e4036d514267493e92| image:: https://cms-im.s3.amazonaws.com/dx240g4b4kkym1gkgs7khx0yw7ri?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-149071.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-149071.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T165553Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=216cb8ca62660faf0bab04b1fb04fd987604857004a334e4036d514267493e92 |                                                 |                                                  |
|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                diagram                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                | multiplication                       expression | shaded area                       (square units) |
|-------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|-------------------------------------------------|--------------------------------------------------|
| C                |https://cms-im.s3.amazonaws.com/9mw9803l8i0vjwwj588r8xkqowrt?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-149072.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-149072.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T165553Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=80f7eec3ac67d9991858cc362c8526df4c2ac740536b8a7383a5ba2034426a7e|  |https://cms-im.s3.amazonaws.com/9mw9803l8i0vjwwj588r8xkqowrt?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-149072.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-149072.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T165553Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=80f7eec3ac67d9991858cc362c8526df4c2ac740536b8a7383a5ba2034426a7e| image:: https://cms-im.s3.amazonaws.com/9mw9803l8i0vjwwj588r8xkqowrt?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-149072.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-149072.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T165553Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=80f7eec3ac67d9991858cc362c8526df4c2ac740536b8a7383a5ba2034426a7e |                                                 |                                                  |
| D                |https://cms-im.s3.amazonaws.com/wy0o9qune26s9u7htzi5ilsfmts4?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-149073.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-149073.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T165553Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=57e693708d062f1a9de3113218a2c369f9dbe92ca8b9a751cd2729b2d6cbb65d|  |https://cms-im.s3.amazonaws.com/wy0o9qune26s9u7htzi5ilsfmts4?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-149073.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-149073.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T165553Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=57e693708d062f1a9de3113218a2c369f9dbe92ca8b9a751cd2729b2d6cbb65d| image:: https://cms-im.s3.amazonaws.com/wy0o9qune26s9u7htzi5ilsfmts4?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-149073.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-149073.svg&response-content-type=image%2Fsvg%2Bxml&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T165553Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=57e693708d062f1a9de3113218a2c369f9dbe92ca8b9a751cd2729b2d6cbb65d |                                                 |                                                  |
What patterns do you notice in the table?
Explain or show how the expression
\(\frac{6\times4}{5\times4}\)
represents the last diagram in the table.
Student Response
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Advancing Student Thinking
If students do not explain that the product of the numerators represents the number of pieces in the shaded region and the product of the denominators represents the number of pieces in the whole, consider asking: “What is the same and what is different about diagrams A and B?”
Activity Synthesis
Ask previously selected students to share their reasoning.
“How do the expressions in the table represent the number of pieces shaded in and the size of the pieces shaded in?” (If we multiply the numerators, we get the number of pieces that are shaded in. If we multiply the denominators, we get the size of the pieces.)
Refer to diagrams and draw on each diagram to show how the multiplication of the numerators and denominators represents the number of shaded pieces and the size of the shaded pieces (that is, the number of those pieces in the whole).


correct_option_index: 1