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: 
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.

Answer: 
Yes. 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.

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: 
Analyze and solve pairs of simultaneous linear equations.

Answer: 
Yes. 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.

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

shot_info: fl_problem_000972_pos_0

6

fl_problem_001097_pos_0

7

fl_problem_001392_pos_0

8
standard_description: Prove the Laws of Sines and Cosines and use them to solve problems.
problem_activity: Narrative
In this activity, students solve addition problems using money based on the Pattern Block Puzzles they sketched in the last activity.
The launch of the activity is an opportunity for students to reason which of the two designs will cost more before they complete any computations.
MLR7 Compare and Connect.
Synthesis: After all strategies have been presented, lead a discussion comparing, contrasting, and connecting the different approaches. Ask, “Did anyone solve the problem the same way, but would explain it differently?”, “What did the representations have in common?”, “How were they different?”
Advances: Representing, Conversing
Launch
Groups of 2
“At the Pattern Block Store, each pattern block shape has a different price.”
Display the price list.
“Which of the two puzzles that you sketched do you think will cost more to make? Puzzle 1 or 2?”
30 seconds: quiet think time
1 minute: partner discussion
Share responses.
Activity
“Use the price list to figure out how much your two puzzles would cost.”
5 minutes: independent work time
5 minutes: partner discussion
As students work, look for different addition strategies. Monitor for students who:
add like quantities first
add bigger or smaller coins first
add some coins mentally
Student Facing
|                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                        shape                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              cost                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                              |
|--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|--------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------|
| |https://cms-im.s3.amazonaws.com/cv5e6biplridvvjo2ed3aj5h1l0e?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-150143.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-150143.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=20240208T164143Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=0347509318fc91e6302bb64febc22d05e7fffde820a42d8a1c974d802d04e917|  |https://cms-im.s3.amazonaws.com/cv5e6biplridvvjo2ed3aj5h1l0e?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-150143.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-150143.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=20240208T164143Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=0347509318fc91e6302bb64febc22d05e7fffde820a42d8a1c974d802d04e917| image:: https://cms-im.s3.amazonaws.com/cv5e6biplridvvjo2ed3aj5h1l0e?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-150143.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-150143.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=20240208T164143Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=0347509318fc91e6302bb64febc22d05e7fffde820a42d8a1c974d802d04e917 | |https://cms-im.s3.amazonaws.com/mbgbaf4nkk0vsawoqjlubrf00b4z?response-content-disposition=inline%3B%20filename%3D%22quarter-back.png%22%3B%20filename%2A%3DUTF-8%27%27quarter-back.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T164143Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=5ac6aff10bd46b5365f46ffeab054cd6ff7d700dfc9c660e3d011fd4c9676951|  |https://cms-im.s3.amazonaws.com/mbgbaf4nkk0vsawoqjlubrf00b4z?response-content-disposition=inline%3B%20filename%3D%22quarter-back.png%22%3B%20filename%2A%3DUTF-8%27%27quarter-back.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T164143Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=5ac6aff10bd46b5365f46ffeab054cd6ff7d700dfc9c660e3d011fd4c9676951| image:: https://cms-im.s3.amazonaws.com/mbgbaf4nkk0vsawoqjlubrf00b4z?response-content-disposition=inline%3B%20filename%3D%22quarter-back.png%22%3B%20filename%2A%3DUTF-8%27%27quarter-back.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T164143Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=5ac6aff10bd46b5365f46ffeab054cd6ff7d700dfc9c660e3d011fd4c9676951 |
| |https://cms-im.s3.amazonaws.com/jbxq4b7lr973twd7cju57q0oulmj?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-150144.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-150144.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=20240208T164143Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=cc2008c7b66cc9ba4e3feaf6d1cf65f2728db5a857421988d5a8147858504207|  |https://cms-im.s3.amazonaws.com/jbxq4b7lr973twd7cju57q0oulmj?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-150144.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-150144.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=20240208T164143Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=cc2008c7b66cc9ba4e3feaf6d1cf65f2728db5a857421988d5a8147858504207| image:: https://cms-im.s3.amazonaws.com/jbxq4b7lr973twd7cju57q0oulmj?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-150144.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-150144.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=20240208T164143Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=cc2008c7b66cc9ba4e3feaf6d1cf65f2728db5a857421988d5a8147858504207 |       |https://cms-im.s3.amazonaws.com/cc7y30885x6mfqcvcubex3157eqw?response-content-disposition=inline%3B%20filename%3D%22dime-front.png%22%3B%20filename%2A%3DUTF-8%27%27dime-front.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T164143Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=ed89067d01fbdb1013e28bfcff88ddb6c25812308117e2bbcefd3ce44f699d7a|  |https://cms-im.s3.amazonaws.com/cc7y30885x6mfqcvcubex3157eqw?response-content-disposition=inline%3B%20filename%3D%22dime-front.png%22%3B%20filename%2A%3DUTF-8%27%27dime-front.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T164143Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=ed89067d01fbdb1013e28bfcff88ddb6c25812308117e2bbcefd3ce44f699d7a| image:: https://cms-im.s3.amazonaws.com/cc7y30885x6mfqcvcubex3157eqw?response-content-disposition=inline%3B%20filename%3D%22dime-front.png%22%3B%20filename%2A%3DUTF-8%27%27dime-front.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T164143Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=ed89067d01fbdb1013e28bfcff88ddb6c25812308117e2bbcefd3ce44f699d7a       |
| |https://cms-im.s3.amazonaws.com/eifk4j1l1h1tmht1t4etxeq8tveu?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-150145.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-150145.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=20240208T164143Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=58c5ac52a6d4f25d2d6c3c613126807263a2a66bf9e4aef477a79db9cedd3c1b|  |https://cms-im.s3.amazonaws.com/eifk4j1l1h1tmht1t4etxeq8tveu?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-150145.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-150145.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=20240208T164143Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=58c5ac52a6d4f25d2d6c3c613126807263a2a66bf9e4aef477a79db9cedd3c1b| image:: https://cms-im.s3.amazonaws.com/eifk4j1l1h1tmht1t4etxeq8tveu?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-150145.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-150145.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=20240208T164143Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=58c5ac52a6d4f25d2d6c3c613126807263a2a66bf9e4aef477a79db9cedd3c1b | |https://cms-im.s3.amazonaws.com/eafp6vkvl2h847xnayhv4z1hsgpk?response-content-disposition=inline%3B%20filename%3D%22nickel-front.png%22%3B%20filename%2A%3DUTF-8%27%27nickel-front.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T164143Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=364dd5652c695c8ea37c72709081c93a7d637b3606704f096422eac5b76b6266|  |https://cms-im.s3.amazonaws.com/eafp6vkvl2h847xnayhv4z1hsgpk?response-content-disposition=inline%3B%20filename%3D%22nickel-front.png%22%3B%20filename%2A%3DUTF-8%27%27nickel-front.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T164143Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=364dd5652c695c8ea37c72709081c93a7d637b3606704f096422eac5b76b6266| image:: https://cms-im.s3.amazonaws.com/eafp6vkvl2h847xnayhv4z1hsgpk?response-content-disposition=inline%3B%20filename%3D%22nickel-front.png%22%3B%20filename%2A%3DUTF-8%27%27nickel-front.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T164143Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=364dd5652c695c8ea37c72709081c93a7d637b3606704f096422eac5b76b6266 |
| |https://cms-im.s3.amazonaws.com/nwbexfpbl484uucueevlb0pvhxm8?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-150146.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-150146.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=20240208T164143Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=f63c01510af539188608375c3095abe0b87f05e2c40893f0cba48aec0cb42d1c|  |https://cms-im.s3.amazonaws.com/nwbexfpbl484uucueevlb0pvhxm8?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-150146.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-150146.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=20240208T164143Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=f63c01510af539188608375c3095abe0b87f05e2c40893f0cba48aec0cb42d1c| image:: https://cms-im.s3.amazonaws.com/nwbexfpbl484uucueevlb0pvhxm8?response-content-disposition=attachment%3B%20filename%3D%22tikz-file-150146.svg%22%3B%20filename%2A%3DUTF-8%27%27tikz-file-150146.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=20240208T164143Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=f63c01510af539188608375c3095abe0b87f05e2c40893f0cba48aec0cb42d1c |       |https://cms-im.s3.amazonaws.com/s4d93o4nmnqcgb18bzjs965plev5?response-content-disposition=inline%3B%20filename%3D%22penny-back.png%22%3B%20filename%2A%3DUTF-8%27%27penny-back.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T164143Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=b1fa77470c010d1273eed7fa1b9b1afb99d84f2212108c0c6bfbdf297fa2308e|  |https://cms-im.s3.amazonaws.com/s4d93o4nmnqcgb18bzjs965plev5?response-content-disposition=inline%3B%20filename%3D%22penny-back.png%22%3B%20filename%2A%3DUTF-8%27%27penny-back.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T164143Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=b1fa77470c010d1273eed7fa1b9b1afb99d84f2212108c0c6bfbdf297fa2308e| image:: https://cms-im.s3.amazonaws.com/s4d93o4nmnqcgb18bzjs965plev5?response-content-disposition=inline%3B%20filename%3D%22penny-back.png%22%3B%20filename%2A%3DUTF-8%27%27penny-back.png&response-content-type=image%2Fpng&X-Amz-Algorithm=AWS4-HMAC-SHA256&X-Amz-Credential=AKIAXQCCIHWF3XOEFOW4%2F20240208%2Fus-east-1%2Fs3%2Faws4_request&X-Amz-Date=20240208T164143Z&X-Amz-Expires=604800&X-Amz-SignedHeaders=host&X-Amz-Signature=b1fa77470c010d1273eed7fa1b9b1afb99d84f2212108c0c6bfbdf297fa2308e       |
How much would the two puzzles you sketched cost at the Pattern Block Store? Show or explain your reasoning.
Puzzle 1:
Puzzle 2:
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
Invite previously selected students to share.
“Is it possible to redesign the puzzle to save more money?” (Yes, I can switch a blue rhombus with two triangles.)


aligns: no