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?


Mathematical concept/skill options: 
A. Analyze proportional relationships and use them to solve real-world and mathematical problems.
B. Understand ratio concepts and use ratio reasoning to solve problems.


Thought: This problem focuses on the ratio between two ingredients. Students do not have to analyze proportional relationships in-depth, which is reserved for a higher grade level, but rather use ratios to find missing values.
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
$$-\frac{1}{4}$$


Mathematical concept/skill options: 
A. Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
B. Reason quantitatively and use units to solve problems.
C. Apply and extend previous understandings of numbers to the system of rational numbers.
D. Represent complex numbers and their operations on the complex plane.
E. Perform arithmetic operations with complex numbers.
F. Compute fluently with multi-digit numbers and find common factors and multiples.
G. Represent and model with vector quantities.
H. Perform operations on vectors.
I. Use properties of rational and irrational numbers.
J. Use complex numbers in polynomial identities and equations.
K. Extend the properties of exponents to rational exponents.
L. Know that there are numbers that are not rational, and approximate them by rational numbers.
M. Perform operations on matrices and use matrices in applications.
N. Apply and extend previous understandings of multiplication and division to divide fractions by fractions.


Thought: This problem involves understanding fractions, multiplication, and division.
Answer: A

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?


Mathematical concept/skill options: 
A. Write expressions in equivalent forms to solve problems.
B. Understand properties of multiplication and the relationship between multiplication and division.
C. Interpret the structure of expressions.
D. Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
E. Use polynomial identities to solve problems.
F. Understand the relationship between zeros and factors of polynomials.
G. Write and interpret numerical expressions.
H. Work with equal groups of objects to gain foundations for multiplication.
I. Perform arithmetic operations on polynomials.
J. Create equations that describe numbers or relationships
K. Generate and analyze patterns.
L. Analyze and solve linear equations and pairs of simultaneous linear equations.
M. Understand solving equations as a process of reasoning and explain the reasoning.
N. Apply and extend previous understandings of arithmetic to algebraic expressions.
O. Rewrite rational expressions.
P. Represent and solve problems involving multiplication and division.
Q. Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
R. Reason about and solve one-variable equations and inequalities.
S. Understand and apply properties of operations and the relationship between addition and subtraction.
T. Understand the connections between proportional relationships, lines, and linear equations.
U. Multiply and divide within 100.
V. Gain familiarity with factors and multiples.
W. Use the four operations with whole numbers to solve problems.
X. Work with addition and subtraction equations.
Y. Represent and solve problems involving addition and subtraction.
Z. Solve problems involving the four operations, and identify and explain patterns in arithmetic.
AA. Solve equations and inequalities in one variable.
AB. Solve systems of equations.
AC. Represent and analyze quantitative relationships between dependent and independent variables.
AD. Work with radicals and integer exponents.
AE. Represent and solve equations and inequalities graphically.
AF. Analyze patterns and relationships.
AG. Use properties of operations to generate equivalent expressions.
AH. Add and subtract within 20.


Thought: This set of problems involves solving linear equations and pairs of linear equations.
Answer: L

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: fl_problem_000972

6

fl_problem_001097

7

fl_problem_001392

8
options: Rewrite rational expressions.

Perform arithmetic operations on polynomials.

Write and interpret numerical expressions.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Work with addition and subtraction equations.

Work with equal groups of objects to gain foundations for multiplication.

Understand the relationship between zeros and factors of polynomials.

Use the four operations with whole numbers to solve problems.

Represent and solve problems involving multiplication and division.

Use polynomial identities to solve problems.

Create equations that describe numbers or relationships

Understand and apply properties of operations and the relationship between addition and subtraction.

Represent and analyze quantitative relationships between dependent and independent variables.

Solve systems of equations.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Generate and analyze patterns.

Write expressions in equivalent forms to solve problems.

Work with radicals and integer exponents.

Represent and solve equations and inequalities graphically.

Analyze and solve linear equations and pairs of simultaneous linear equations.

Represent and solve problems involving addition and subtraction.

Use properties of operations to generate equivalent expressions.

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Interpret the structure of expressions.

Understand properties of multiplication and the relationship between multiplication and division.

Add and subtract within 20.

Solve equations and inequalities in one variable.

Analyze patterns and relationships.

Gain familiarity with factors and multiples.

Multiply and divide within 100.

Understand the connections between proportional relationships, lines, and linear equations.

Understand solving equations as a process of reasoning and explain the reasoning.

Reason about and solve one-variable equations and inequalities.
problem_activity: Activity
In this task, students are presented with a situation that leads to a linear relationship that is not proportional because there is a non-zero starting amount. By trying to answer the question, “How many cups are needed to get to a height of 50 cm?” the students explore the rate of change, which is the increase per cup after the first cup. The rate of change can be seen in the graph as the slope. Students use representations and ideas from previous lessons on proportional relationships. They use tables and graphs to represent the situation and make deductions by generalizing from repeated reasoning (MP8), arguing that each additional cup increases the height of the stack by the same amount. As students are working, suggest that students make a graph or a table if they are stuck or if they have trouble explaining their reasoning. Students should be prepared to share their strategies with the class.
Students can attack this problem using any method. If they get stuck, here are some things it would be useful to figure out:
How much height does each cup add?
Is the first cup different from the others?
Create a graph or table to help you reason about this problem.
As students work, identify students who use different strategies, i.e. graphs, tables, equations.
This activity and the next were written using a particular type of cup. Photos are included of all measurements needed, so these activities can be used without any additional preparation. However, if desired, the lesson could be modified so that students are measuring stacks of actual cups. The numbers then may have to be adjusted since the cups come in different sizes. Teachers who want to use real cups should measure the height of two stacks ahead of time, find the rate of change and make sure that it is approximately constant. Note that rounding error will likely play a role unless the number of cups in the stacks is chosen carefully, so some flexibility in answers may be necessary if students measure actual stacks of cups.
Launch
Before students look at the picture with all of the given information, display just this part of the picture.
[IMG]
Ask students how many cups they see, and how tall the stack is. (There are 6 cups, and the stack is 15 cm tall.) Then, ask students to quietly make a prediction about the height of 12 cups and give you a silent signal when they have a response. When everyone has come up with an answer, ask a few students to share their responses and write these on the board. It is likely that some students will say 30 centimeters, and some might have an argument for why the height of 12 cups must be less than 30 centimeters. This question will be resolved as soon as students look at the photo in the task. Tell students that their job is to figure out how many cups would be needed in order to stack them to a height of 50 centimeters.
Arrange students in groups of 2–4. Give access to rulers and graph paper. Allow individual think time of 3 minutes before students work together in groups.
Representation: Internalize Comprehension.
Begin the activity with concrete or familiar contexts. Set up a display of foam cups next to a ruler, as depicted. Encourage students to consider the rate of change and whether or not there is a proportional relationship.
Supports accessibility for: Conceptual processing; Memory
Conversing, Representing, Writing: MLR2 Collect and Display.
As students discuss, listen for and collect the language students use to describe whether they think the relationship is proportional. Write the students’ words and phrases on a visual display and update it throughout the remainder of the lesson. Remind students to borrow language from the display as needed. This will help students read and use mathematical language during their paired and whole-group discussions.
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness
Student Facing
We have two stacks of styrofoam cups.
One stack has 6 cups, and its height is 15 cm.
The other stack has 12 cups, and its height is 23 cm.
How many cups are needed for a stack with a height of 50 cm?
[IMG]
Student Response
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Anticipated Misconceptions
Students compute
\(\frac{15}{6}=2.5\)
or
\(\frac{23}{12}=1.9\)
and use this as the increase per cup. Measurement is approximate; students may be looking for exact, nice numbers. Make sure numbers agree approximately between students, but reassure them that they are close enough.
Activity Synthesis
Poll the class about the number of cups they came up with to reach a total height of 50 cm.
If the class used real cups, stack enough of them to get to a height of 50 cm, and let students check if they were correct. Otherwise display this photo:
[IMG]
Ask students who used different strategies to share their reasoning with the class.
Ask students if they think the relationship is proportional. If there is a disagreement, ask one person per opinion to share their reasons. Encourage them to come to the conclusion that it is not proportional since doubling the number of cups from 6 to 12 did not double the height. Students can see that when stacking the cups only the rim adds to the height. The first cup adds more height since there are additionally 7 cm from the bottom of the cup to the bottom of the rim.
Tell students that even though the relationship is not proportional, this relationship has things in common with proportional relationships, and that they will explore this in the next activity.


correct_option_index: 31