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: Apply geometric concepts in modeling situations.

Identify and describe shapes (squares, circles, triangles, rectangles, hexagons, cubes, cones, cylinders, and spheres).

Draw, construct, and describe geometrical figures and describe the relationships between them.

Reason with shapes and their attributes.

Understand and apply theorems about circles.

Draw and identify lines and angles, and classify shapes by properties of their lines and angles.

Understand congruence in terms of rigid motions.

Understand similarity in terms of similarity transformations.

Translate between the geometric description and the equation for a conic section.

Analyze, compare, create, and compose shapes.

Solve real-life and mathematical problems involving angle measure, area, surface area, and volume.

Apply trigonometry to general triangles.

Understand and apply the Pythagorean Theorem.

Find arc lengths and areas of sectors of circles.

Visualize relationships between two-dimensional and three-dimensional objects.

Solve real-world and mathematical problems involving area, surface area, and volume.

Use coordinates to prove simple geometric theorems algebraically.

Classify two-dimensional figures into categories based on their properties.

Graph points on the coordinate plane to solve real-world and mathematical problems.

Solve real-world and mathematical problems involving volume of cylinders, cones, and spheres.

Explain volume formulas and use them to solve problems.

Define trigonometric ratios and solve problems involving right triangles.

Experiment with transformations in the plane.

Understand congruence and similarity using physical models, transparencies, or geometry software.

Prove theorems involving similarity.

Make geometric constructions.

Prove geometric theorems.
problem_activity: Activity
In this activity, students apply what they have learned previously about surface area and volume to different situations (MP4). Students have to consider whether they are finding the surface area or volume before answering each question. In addition, students apply proportional reasoning to find the cost of the vinyl that is needed. This is an opportunity for students to revisit this prior understanding in a geometry context.
As students work on the task, monitor for students who are using different methods to decompose or compose the base of the object to calculate the area.
Launch
Arrange students in groups of 2. Make sure students are familiar with the terms “foam” and “vinyl.” For example, it may help to explain that many binders are made out of cardboard covered with vinyl. In the diagram, all measurements have been rounded to the nearest inch.
Give students 3–5 minutes of quiet work time followed by time to share their answers with a partner. Follow with a whole-class discussion.
Representation: Internalize Comprehension.
Activate or supply background knowledge by reviewing an image or video of a foam play structure. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing
Student Facing
At a daycare, Kiran sees children climbing on this foam play structure.
[IMG]
Kiran is thinking about building a structure like this for his younger cousins to play on.
The entire structure is made out of soft foam so the children don’t hurt themselves. How much foam would Kiran need to build this play structure?
The entire structure is covered with vinyl so it is easy to wipe clean. How much vinyl would Kiran need to build this play structure?
The foam costs 0.8¢ per in
3
. Here is a table that lists the costs for different amounts of vinyl. What is the total cost for all the foam and vinyl needed to build this play structure?
| vinyl (in                             2                            ) | cost ($) |
|----------------------------------------------------------------------|----------|
|                                  75                                  |   0.45   |
|                                 125                                  |   0.75   |
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?
When he examines the play structure more closely, Kiran realizes it is really two separate pieces that are next to each other.
[IMG]
How does this affect the amount of foam in the play structure?
How does this affect the amount of vinyl covering the play structure?
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
For the first question, students may try to figure out the total mass of foam needed instead of the total volume. Point out that they are not given any information about how much the foam weighs and prompt them to look for a different way of answering the question “how much foam?”
Some students might see the 0.8¢ for the unit price and confuse that with
$
0.80. Remind students of their work with fractional percentages in a previous unit and that 0.8¢ must be less than 1¢.
Activity Synthesis
Select previously identified students to share how they calculated the area of the base.
Consider asking some of the following questions:
“Are there other ways to calculate the area of the base?”
“How did you know if you had to calculate surface area or volume for this problem?”
“If Kiran buys a big block of foam that is 36 inches wide, 20 inches deep, and 10 inches tall and cuts it into this shape, what shapes would he be cutting off?” (A rectangular prism for the step on the left side and a triangular prism for the slide part on the right.)
“How much more would the big block of foam cost than your calculations?” (This method wastes a volume of
\(36 \boldcdot 20 \boldcdot 10 - 4,\!960 = 2,\!240\)
cubic inches of foam. This would be an extra
$
17.92 since
\(2,\!240 \boldcdot 0.008 = 17.92\)
.)
If Kiran decides not to cover the bottom of the structure with vinyl, how much would he save?” (This reduces the area of vinyl needed by another
\(36\boldcdot 20 = 720\)
square inches. This would save
$
4.32 since
\(720 \boldcdot 0.006 = 4.32\)
.)
Conversing, Listening: MLR7 Compare and Connect.
Use this routine when students share their strategies for calculating the volume of the play structure. Ask students to consider what is the same and what is different about each approach. In this discussion, listen for and amplify comments that refer to the way the figure was decomposed. Draw students’ attention to the various ways area and perimeter of the base were found and how these were represented in each strategy. These exchanges can strengthen students’ mathematical language use and reasoning based on volume and surface area.
Design Principle(s): Maximize meta-awareness


correct_option_index: 10