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.


Your response:
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.


Your response:
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.


Your response:
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: Add and subtract within 20.

Represent and analyze quantitative relationships between dependent and independent variables.

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

Solve systems of equations.

Multiply and divide within 100.

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

Create equations that describe numbers or relationships

Understand the relationship between zeros and factors of polynomials.

Work with addition and subtraction equations.

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

Represent and solve problems involving multiplication and division.

Use the four operations with whole numbers to solve problems.

Represent and solve problems involving addition and subtraction.

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

Use polynomial identities to solve problems.

Perform arithmetic operations on polynomials.

Apply and extend previous understandings of arithmetic to algebraic expressions.

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

Analyze patterns and relationships.

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

Work with radicals and integer exponents.

Write expressions in equivalent forms to solve problems.

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

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

Gain familiarity with factors and multiples.

Reason about and solve one-variable equations and inequalities.

Represent and solve equations and inequalities graphically.

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

Rewrite rational expressions.

Use properties of operations to generate equivalent expressions.

Interpret the structure of expressions.

Generate and analyze patterns.

Write and interpret numerical expressions.

Solve equations and inequalities in one variable.
problem_activity: Activity
The goal of this activity is to write an expression that models the volume of a box as a function of the side length
\(x\)
of the square cutouts. Students then graph the function, use the graph to estimate the value of
\(x\)
that will produce the box with the largest volume, and establish a reasonable domain for the function by considering the context.
During the discussion, students interpret their expression in context (MP2). Of particular interest, values of
\(x\)
greater than 5.5 inches result in positive volumes even though cutouts of this size do not make sense when constructing a box from an 8.5 inch by 11 inch sheet of paper. There is no need for students to rewrite the expression for
\(V(x)\)
at this time in standard form. In later lessons, students will practice rewriting equations for polynomials in different forms.
Launch
Arrange students in groups of 2. Tell students there are many possible answers for the first question. After quiet work time, ask students to compare their response to their partner’s and decide which plan they like best. Select 2–3 groups to share their plans. Provide access to devices that can run Desmos or other graphing technology.
Representation: Internalize Comprehension.
Use annotations to highlight connections between representations in a problem. For example, prior to developing an equation for volume, have students label the edges of their box with the calculations they used to determine the length, width, and height.
Supports accessibility for: Visual-spatial processing
Student Facing
[IMG]
The volume
\(V(x)\)
in cubic inches of the open-top box is a function of the side length
\(x\)
in inches of the square cutouts. Make a plan to figure out how to construct the box with the largest volume.
Pause here so your teacher can review your plan.
Write an expression for
\(V(x)\)
.
Use graphing technology to create a graph representing
\(V(x)\)
. Approximate the value of
\(x\)
that would allow you to construct an open-top box with the largest volume possible from one piece of paper.
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?
The surface area
\(A(x)\)
in square inches of the open-top box is also a function of the side length
\(x\)
in inches of the square cutouts.
Find one expression for
\(A(x)\)
by summing the area of the five faces of our open-top box.
Find another expression for
\(A(x)\)
by subtracting the area of the cutouts from the area of the paper.
Show algebraically that these two expressions are equivalent.
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 forget that the width and length of the paper are based on removing length
\(x\)
twice due to cutting out squares at each corner. Encourage these students to revisit their work from the previous activity and think about how they can calculate length and width without measuring directly.
If students have trouble finding a general expression for the volume of the box, encourage them to revisit their results from the previous activity and write the volumes of the boxes without simplifying their answers (e.g., 6.5 in
\(\times\)
9 in
\(\times\)
1 in instead of 58.5 in
3
) in order to see the pattern more easily.
Activity Synthesis
The purpose of this discussion is for students to think critically about the equation they’ve written to model the volume of the box. Here are some questions for discussion:
“What is a value of
\(x\)
that wouldn’t make sense?” (
\(x=6\)
would not make sense, because the sheet of paper is only 8.5 inches long on one side, so cutting out 6 inch by 6 inch squares from each corner is not possible.)
“What is the smallest value that would work for
\(x\)
? The largest?” (The smallest value is some number greater than 0 inches that you could still use as a side length to cut out the squares and fold up the box sides to make a very short box. The largest value is some number less than half of 8.5 inches that you could still use as a side length to cut out the squares and fold up the box sides to make a very tall box.)
“What is the range of
\(x\)
values where the volume is increasing? Decreasing?” (The volume increases as the cutouts get larger, until
\(x\)
is about 1.6 inches. Then the volume decreases until
\(x\)
is 4.25 inches, which creates a box with a volume of 0. The volume also starts increasing again when
\(x\)
is about 4.9 inches, but these values of
\(x\)
don’t make sense because we can’t make cutouts that large.)
If no students bring up the word
domain
during the discussion, remind students of the meaning of the word and select students to recommend a reasonable domain for the function
\(V\)
.
Conclude the discussion by telling students that
\(V\)
is a type of
polynomial
function and that they will learn about other situations polynomial functions can model along with what graphs of polynomial functions can look like in future lessons.
Conversing: MLR2 Collect and Display.
During the discussion, listen for and collect the language students use to interpret the equation they’ve written to model the volume of the box. Call students’ attention to language such as “range of values,” “groups of parentheses,” or other key parts of the equation representing the length, width, or height. Write the students’ words and phrases on a visual display and update it throughout the remainder of the lesson. Use this opportunity to define a “polynomial function.” Remind students to borrow language from the display as needed. This will help students use mathematical language to interpret the equation for the volume of the box.
Design Principle(s): Maximize meta-awareness


correct_option_index: 6