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?


Topic options: 
A. Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.
B. Functions: students learn to define, use, and evaluate functions to model relationships between quantities.
C. Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.
D. Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.
E. Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.
F. Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.
G. Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.
H. Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.
I. Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.
J. Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.
K. Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.


Thought: This problem asks students to reason about ratios of butter amounts to sugar amounts.
Answer: D

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
$$-\frac{1}{4}$$


Topic options: 
A. Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.
B. Functions: students learn to define, use, and evaluate functions to model relationships between quantities.
C. Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.
D. Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.
E. Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.
F. Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.
G. Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.
H. Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.
I. Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.
J. Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.
K. Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.


Thought: This problem teaches students different ways of writing numbers that express the same value. This helps students better understand the complex number system, which includes rational numbers.
Answer: I

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?


Topic options: 
A. Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.
B. Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.
C. Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.
D. Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.
E. Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.
F. Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.
G. Functions: students learn to define, use, and evaluate functions to model relationships between quantities.
H. Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.
I. Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.
J. Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.
K. Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.


Thought: This set of problems asks students to reason with equations, specifically linear equations, such as the one shown in the table in Problem 1.
Answer: F

Now, assign the following problem or activity to one or more relevant topics it teaches.
shot_info: fl_problem_000972

6

fl_problem_001097

7

fl_problem_001392

8
options: Modeling: students learn to choose and use appropriate mathematics and statistics to analyze real-world empirical situations, improve decisions, and report on their conclusions and the reasoning behind them.

Statistics & Probability: students learn to analyze and produce data distributions, and build understanding of univariate and bivariate data. They learn to interpret data, make statistical inferences, justify conclusions, understand rules of probability, and use probability to make decisions.

Number & Operations in Base Ten: students learn to work with the base-ten system and build place value understanding.

Number & Operations - Fractions: students learn to understand fractions as numbers and may work with them using addition, subtraction, multiplication, and division.

Ratios & Proportional Relationships: students learn to recognize, describe, represent, and reason with ratios, rates, proportional relationships, and percent.

Counting & Cardinality: students learn to know number names and the count sequence, count to tell the number of objects, and compare numbers.

Measurement & Data: students learn to work with data and measure attributes such as time, money, length, area, and volume. They may learn to compare measurements to operations and convert between different units of measure.

Functions: students learn to define, use, and evaluate functions to model relationships between quantities.

Operations & Algebra: students learn to solve problems using algebraic thinking and operations such as addition, subtraction, multiplication, and division. They may learn to identify and explain arithmetic patterns, evaluate or manipulate numerical expressions, and reason with equations and inequalities.

Number Systems and Quantity: students learn to understand the complex number system and reason quantitatively.

Geometry: students learn to classify geometric figures by their properties, and understand and compare the relationships between them. Students develop and use formulas to compute lengths, areas, and volumes, and they use transformations to generate new shapes and compare existing ones. Geometry can be studied with and without coordinates.
problem_activity: Activity
This activity builds on the previous one as students plot a different relationship, comparing the length of each square diagonal to its area. Students use the length measurements from the previous activity and use them to calculate the area of the squares. Unlike in the previous activity, this time both the table and the graph show that the relationship between diagonal length and area is
not
close to being proportional. As a result, the measurement error does not play a role in this activity. Rather, the focus is on identifying that a relationship is not proportional: this is an example of MP7 because students will identify that there is no constant of proportionality in the table (not even allowing for measurement error) and will correspondingly notice that the plotted points are not close to lying on a line.
Launch
Keep students in the same groups. Tell them to calculate the area of the same 3 squares they measured in the previous activity. Give students 4 minutes of group work time. Display this blank grid and have students plots points for their measurements.
[IMG]
Give students 2–3 more minutes of group work time followed by whole-class discussion.
Representation: Internalize Comprehension.
Activate or supply background knowledge about calculating the area of a square. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing
Student Facing
In the table, record the length of the diagonal for each of your assigned squares from the previous activity. Next, calculate the area of each of your squares.
|          | diagonal (cm) | area (cm                       2                      ) |
|----------|---------------|---------------------------------------------------------|
| square A |               |                                                         |
| square B |               |                                                         |
| square C |               |                                                         |
| square D |               |                                                         |
| square E |               |                                                         |
| square F |               |                                                         |
| square G |               |                                                         |
| square H |               |                                                         |
| square I |               |                                                         |
Pause here so your teacher can review your work. Be prepared to share your values with the class.
Examine the class graph of these values. What do you notice?
How is the relationship between the diagonal and area of a square the same as the relationship between the diagonal and perimeter of a square from the previous activity? How is it different?
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?
Here is a rough map of a neighborhood.
[IMG]
There are 4 mail routes during the week.
On Monday, the mail truck follows the route A-B-E-F-G-H-A, which is 14 miles long.
On Tuesday, the mail truck follows the route B-C-D-E-F-G-B, which is 22 miles long.
On Wednesday, the truck follows the route A-B-C-D-E-F-G-H-A, which is 24 miles long.
On Thursday, the mail truck follows the route B-E-F-G-B.
How long is the route on Thursdays?
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
Some students may struggle to calculate the area of the squares, or may use the length of the diagonal as if it were the side length. Prompt them with questions like “How can you calculate the area of a rectangle?” “What is the length and width of your square?”
Some students may measure the side length of each square again, instead of dividing the perimeter from the previous activity by 4. This strategy is allowable, although you can also prompt them to consider if they no longer had access to a ruler, is there a way they could use the information they already recorded to find this measurement.
Some students may struggle to organize the information. Prompt them to add a column to the table in the previous activity to record the side length of the squares.
Activity Synthesis
The goal of this discussion is for students to recognize that the relationship between the length of the diagonal and the area of squares of different sizes is not a proportional relationship.
Invite students to share their observations about the graph. Important points include:
As the length of the diagonal increases, the area of the square increases.
The points do not look like they lie on a line and definitely do not lie on a line that goes through
\((0,0)\)
.
The area of the square looks to grow at a faster rate for larger diagonals than for the short diagonals.
To reinforce that the relationship is not proportional, instruct students to calculate the quotient of the area divided by the length of the diagonal for their square. Consider displaying a table of their measurements and adding a third column to record their quotients. Students should realize that the differences between these quotients are too large to be accounted for by just measurement error.
| diagonal (cm) | area (cm                   2                  ) | \(\text{area} \div \text{diagonal}\) |
|---------------|-------------------------------------------------|--------------------------------------|
|      8.2      |                      33.6                       |                 4.1                  |
|      4.2      |                        9                        |                 2.1                  |
|      6.8      |                       23                        |                 3.4                  |
|      2.2      |                       2.4                       |                 1.1                  |
|     12.6      |                      77.4                       |                 6.1                  |
If time permits, consider using this picture to reinforce that length scales linearly while area does not:
[IMG]
“If the side length of the small square is
\(s\)
, what is the side length of the large square?” (
\(2s\)
)
“If the diagonal of the small square is
\(d\)
, what is the diagonal of the large square?” (
\(2d\)
)
“If the area of the small square is
\(A\)
, what is the area of the large square?” (
\(4A\)
)
“Is the relationship between the diagonal and the area proportional?” (No, when the diagonal doubles, the area quadruples; if the relationship were proportional, both the diagonal and the area would double.)
Speaking, Listening: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. For each observation that is shared, ask students to restate what they heard using precise mathematical language. Consider providing students time to restate what they heard to a partner before selecting one or two students to share with the class. Ask the original speaker(s) if their peer was accurately able to restate their thinking. Call students' attention to any words or phrases that helped to clarify the original statement. This provides more students with an opportunity to produce language as they interpret the reasoning of others.
Design Principle(s): Support sense-making


correct_option_index: 4