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

Apply trigonometry to general triangles.

Analyze, compare, create, and compose shapes.

Define trigonometric ratios and solve problems involving right triangles.

Understand congruence in terms of rigid motions.

Use coordinates to prove simple geometric theorems algebraically.

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

Reason with shapes and their attributes.

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

Explain volume formulas and use them to solve problems.

Prove theorems involving similarity.

Understand and apply the Pythagorean Theorem.

Find arc lengths and areas of sectors of circles.

Understand and apply theorems about circles.

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

Make geometric constructions.

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

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

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

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

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

Understand similarity in terms of similarity transformations.

Apply geometric concepts in modeling situations.

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

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

Prove geometric theorems.

Experiment with transformations in the plane.
problem_activity: Activity
In this activity, students think about a hemisphere fitting inside the smallest possible box (or rectangular prism). Students reason that the smallest box in which a hemisphere can fit has a square base with edge length that is the same as the diameter of the hemisphere, and its height will be the radius of the hemisphere. Further, if we calculate the volume of the box, we have an upper bound for the volume of the hemisphere. This activity prepares students for the next, where they will calculate the upper and lower bounds for the volume of a hemisphere by considering cylinders and cones that fit outside or inside the hemisphere.
Launch
Ask students if they are familiar with the word
sphere
, and if they can think of examples of spheres. Some examples they might come up with are: ping pong balls, soap bubbles, baseballs, volleyballs, a globe. If students don’t come up with many examples, offer your own, or perhaps display an Internet image search for “sphere.” If you brought in any examples of physical spheres, display these or allow students to hold them. Explain to students that the radius of a sphere is the distance from the center of the sphere (the point in the exact middle) to any point on the sphere. If you brought in physical examples of spheres, ask students to rank them from smallest diameter to largest diameter.
Ask students if they are familiar with the word hemisphere, and if they can think of examples of hemispheres. Some examples are hemispherical (or ‘half’) balance boards, half of Earth, a dome or planetarium, and convex security mirrors. Display an example, illustration, or diagram of a hemisphere for all to see, pointing out how the radius is the distance from the center of the flat side of the sphere to any point on the curved surface of the sphere.
Tell students that in this activity, they will think about building a box (or rectangular prism) around a hemisphere. Arrange students in groups of 2. Ask students to quietly work through the first question and then share their reasoning with their partner. Pause for a whole-class discussion, and invite students to share their responses. The purpose of this problem is to see how the measurements of a hemisphere determine the dimensions of the prism. Ensure everyone understands the box must have edge lengths 6, 6, and 3 inches, and that the hemisphere must have a volume that is less than
\(108\)
cubic inches (since
\(6^2\boldcdot 3=108\)
). Discuss how much less students
think
the volume of the sphere is and their reasoning, then have students move onto the second problem with their partner.
Representation: Develop Language and Symbols.
Create a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of: sphere and hemisphere.
Supports accessibility for: Memory; Language
Reading, Speaking, Listening: MLR6 Three Reads.
Use this routine to support comprehension of the situation before looking at the questions. In the first read, students read the problem with the goal of comprehending the situation (e.g., Mai has a paperweight that is a dome. She wants to design a box for it.). If needed, discuss the meaning of unfamiliar terms at this time. Use the second read to identify the important quantities by asking students what can be counted or measured (e.g., the radius of the dome, the area of the base of the dome, the side lengths of the box). In the third read, ask students to brainstorm possible strategies to answer the questions. This helps students connect the language in the word problem and the reasoning needed to solve the problem keeping the intended level of cognitive demand in the task.
Design Principle(s): Support sense-making​
Student Facing
[IMG]
Mai has a dome paperweight that she can use as a magnifier. The paperweight is shaped like a hemisphere made of solid glass, so she wants to design a box to keep it in so it won't get broken. Her paperweight has a radius of 3 cm.
What should the dimensions of the inside of box be so the box is as small as possible?
What is the volume of the box?
What is a reasonable estimate for the volume of the paperweight?
Tyler has a different box with side lengths that are twice as long as the sides of Mai's box. Tyler's box is just large enough to hold a different glass paperweight.
What is the volume of the new box?
What is a reasonable estimate for the volume of this glass paperweight?
How many times bigger do you think the volume of the paperweight in this box is than the volume of Mai's paperweight? Explain your thinking.
Student Response
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Anticipated Misconceptions
Students may believe that since this box is not a cube, but a rectangular prism, the volume will not increase by 8 when the side lengths are doubled. Since volume of a rectangular prism is the product of length, width, and height, the side lengths do not need to be equal; if
every
side length is doubled, then the volume will be increased by
\(2 \boldcdot 2 \boldcdot 2 = 8\)
.
Activity Synthesis
Invite students to share their answers to the second question. The purpose of this discussion is for students to see that since a rectangular prism’s volume gets larger by a factor of
\(2^3\)
when edge lengths are doubled then it would make sense for a hemisphere’s volume to do the same. Ensure that students understand the volume calculated for the box holding the hemisphere is greater than the actual volume of the hemisphere because of the space left around the paperweight in the box and that this will be the upper bound of the volume of a sphere.
Tell students that in the next activity they are going to investigate a better way to estimate the volume of a hemisphere. If time allows, ask students to suggest shapes they are already familiar with that they could use to find the volume of a hemisphere.


correct_option_index: 6