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: Understand ratio concepts and use ratio reasoning to solve problems.

Analyze proportional relationships and use them to solve real-world and mathematical problems.
problem_activity: Activity
This activity revisits a context seen previously. Students solved problems like this as early as grade 3 without formulating them in terms of ratios or rates (“If 1 cup of rice serves 3 people, how many people can you serve with 12 cups of rice?”). In this activity, they ultimately find an equation for the proportional relationship.
As students find missing values in the table, they should see that they can always multiply the number of food items by the constant of proportionality. When students see this pattern (MP7) and represent the number of people served by
\(x\)
cups of rice (or
\(s\)
spring rolls) as
\(3x\)
(or
\(\frac{1}{2}s\)
), they are expressing regularity in repeated reasoning (MP8).
Only one row in each table is complete. Based on their experience in the previous lesson, students are more likely to multiply the entries in the left-hand column by 3 (or
\(\frac{1}{2}\)
) than to use scale factors, at least for the first and third rows. If they do, they are more likely to see how to complete the last row in each table.
Some students might use unit rates: If 1 cup of rice can serve 3 people, then
\(x\)
cups of rice can serve
\(3x\)
people. So, students have different ways to generate
\(3x\)
as the expression that represents the number of people served by
\(x\)
cups of rice: completing the table for numerical values and continuing the pattern to the last row; or finding the unit rate and using it in the case of
\(x\)
cups.
Monitor students for different approaches as they are working.
Launch
Tell students that this activity revisits a context they worked on in an earlier lesson.
Student Facing
A recipe says that 2 cups of dry rice will serve 6 people. Complete the table as you answer the questions. Be prepared to explain your reasoning.
How many people will 1 cup of rice serve?
How many people will 3 cups of rice serve? 12 cups? 43 cups?
How many people will
\(x\)
cups of rice serve?
| cups of dry rice | number of people |
|------------------|------------------|
|        1         |                  |
|        2         |        6         |
|        3         |                  |
|        12        |                  |
|        43        |                  |
|      \(x\)       |                  |
A recipe says that 6 spring rolls will serve 3 people. Complete the table as you answer the questions. Be prepared to explain your reasoning.
How many people will 1 spring roll serve?
How many people will 10 spring rolls serve? 16 spring rolls? 25 spring rolls?
How many people will
\(n\)
spring rolls serve?
| number of spring rolls | number of people |
|------------------------|------------------|
|           1            |                  |
|           6            |        3         |
|           10           |                  |
|           16           |                  |
|           25           |                  |
|         \(n\)          |                  |
How was completing this table different from the previous table? How was it the same?
Student Response
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Anticipated Misconceptions
If students have trouble encapsulating the relationship with an expression, encourage them to draw diagrams or to verbalize the relationship in words.
Activity Synthesis
Select students who have used the following approaches in the given order:
Recognize that to move from the first column to the second, you multiply by 3.
Say that 3 is the constant of proportionality.
Recognize that the equation
\(2 \boldcdot  {?} = 6\)
can be used to find the constant of proportionality algebraically.
Say that the 3 can be interpreted as the number of people per 1 cup of rice.
If no student sees that these insights are connected to prior work, explicitly connect them with the lessons of the previous two days. At the end of the discussion, suggest to students that we let
\(y\)
represent the number of people who can be served by
\(x\)
cups of rice. Ask students to write an equation that gives the relationship of
\(x\)
and
\(y\)
. (This builds on work from grade 6, but it may be rusty.) Be sure to write the equation
\(y = 3x\)
where all students can see it and help students interpret its meaning in the context: “To find
\(y\)
, the number of people served, we can multiply the number of cups of rice,
\(x\)
, by 3.”
Representation: Internalize Comprehension.
Use color and annotations to illustrate student thinking. As students describe their approaches, use color and annotations to scribe their thinking on a display.
Supports accessibility for: Visual-spatial processing; Conceptual processing
Conversing: MLR7 Compare and Connect.
As students explain their different approaches to completing the tables, ask “What is similar, what is different?” about their methods. Draw students’ attention to the different ways the constant of proportionality was represented across the different strategies. For instance, when discussing first question, ask “Where do you see the 3 in each approach?” These exchanges strengthen students’ mathematical language use and reasoning based on constant of proportionality.
Design Principle(s): Cultivate conversation; Maximize meta-awareness


correct_option_index: 1