few_shots: Example 1:
Problem/activity:
Task
Materials
One 10-frame per pair of students (see example below)
[IMG]
Counters, 10 per pair of students
Decks of number cards 0-5 (with 4 of each number) or two 0-5 dice
Setup
This activity is done in pairs. The students sit side by side with 10-frame and deck of cards or dice. If they have cards, the cards are shuffled and placed face down in front of them.
Action
Player 1 flips a card or rolls the die. They place that many counters on the 10-frame. Player 2 flips a card or rolls the die. They place that many more counters on the 10-frame next to the counters already on the ten frame. The students count the total together.
A recording sheet can be added when the students have been introduced to equations and number sentences. For those students who can handle higher numbers, a deck of 0-10 cards and two 10-frames can be used.
This game can also be played alone.


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


Thought: This task teaches students to understand addition as putting together counters, or adding to already placed counters.
Answer: AB

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:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Mathematical concept/skill options: 
A. Prove and apply trigonometric identities.
B. Model periodic phenomena with trigonometric functions.
C. Interpret functions that arise in applications in terms of the context.
D. Build a function that models a relationship between two quantities.
E. Define, evaluate, and compare functions.
F. Use functions to model relationships between quantities.
G. Understand the concept of a function and use function notation.
H. Analyze functions using different representations.
I. Extend the domain of trigonometric functions using the unit circle.
J. Interpret expressions for functions in terms of the situation they model.
K. Build new functions from existing functions.
L. Construct and compare linear and exponential models and solve problems.


Thought: This problem applies trigonometric identities, including the angle addition formula for cosine.
Answer: A

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: im_task_001477

K

fl_problem_001097

7

fl_problem_001894

HS_0
options: Apply geometric concepts in modeling situations.

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

Prove theorems involving similarity.

Make geometric constructions.

Reason with shapes and their attributes.

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

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

Understand and apply the Pythagorean Theorem.

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

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

Understand and apply theorems about circles.

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

Prove geometric theorems.

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

Apply trigonometry to general triangles.

Understand congruence in terms of rigid motions.

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

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

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

Find arc lengths and areas of sectors of circles.

Define trigonometric ratios and solve problems involving right triangles.

Explain volume formulas and use them to solve problems.

Analyze, compare, create, and compose shapes.

Understand similarity in terms of similarity transformations.

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

Use coordinates to prove simple geometric theorems algebraically.

Experiment with transformations in the plane.
problem_activity: Activity
In this first part of the project, students make sense of the task at hand and determine what they need to know and do to find the most economical shipping box combination. Students should recognize that they will need to:
Find out the measurements of the jewelry boxes and shipping boxes, as well as the costs for mailing a shipping box of each size. Asking students to find out this information themselves will increase the modeling demand (MP4).
Decide on an orientation for the jewelry boxes inside each shipping box and calculate how many jewelry boxes will fit with that particular orientation.
Test out different orientations and how they affect the number of jewelry boxes to be fitted and the cost.
Students may want to think of a strategy for considering different configurations efficiently, rather than testing all of them, which would be time consuming and repetitive. As students make plans to try out different jewelry box orientations and associated calculations, encourage them to work systematically to minimize omissions and errors. Urge students to create drawings or models of the boxes to show the calculations they will need to make.
Consider supporting students by discussing the different orientations of jewelry boxes in a shipping box (Which orientations are possible? How much empty space would result?) and possible ways to use drawings or diagrams to show the arrangements of jewelry boxes.
Launch
Give students 1–2 minutes to read the task statement individually and to ask any clarifying questions. Demonstrate the idea of the task by putting a small box inside a larger box in different orientations. Consider displaying USPS flat-rate boxes, or an image of each of the boxes.
Arrange students in groups of 4. Give students 5 minutes of quiet think time to brainstorm about what information is needed to solve this task and share to it with their group. Give another 5 minutes to plan in groups, followed by time to measure boxes or research box options and dimensions for themselves. Provide access to measuring tools.
USPS flat-rate information:
Small box:
\(5\frac38\)
inches by
\(8\frac58\)
inches by
\(1\frac58\)
inches. Cost:
$
6.80.
Medium box 1:
\(11\)
inches by
\(8\frac12\)
inches by
\( 5\frac12\)
inches. Cost:
$
13.45.
Medium box 2:
\(11\frac78\)
inches by
\(3\frac38\)
inches by
\( 13\frac58\)
inches. Cost:
$
13.45.
Large box:
\(12\)
inches by
\( 12 \)
inches by
\( 5\frac12\)
inches. Cost:
$
18.75.
Reading, Speaking, Representing: MLR6 Three Reads.
Use this routine to orient students to the context of the problem. In the first read, students read the problem with the goal of comprehending the situation (an artist is packing jewelry boxes to ship to a store.). Clarify any unknown language, such as a “flat-rate” box or shipping rates, as needed. For the second read, ask students to identify the quantities and mathematical relationships (number of necklaces ordered, the dimensions of the jewelry box). After the final read, ask students to brainstorm possible strategies they may use to solve the problem.
Design Principle(s): Support sense-making
Student Facing
An artist makes necklaces. She packs each necklace in a small jewelry box that is
\(1\frac34\)
inches by
\(2\frac14\)
inches by
\(\frac34\)
inch.
A department store ordered 270 necklaces. The artist plans to ship the necklaces to the department store using flat-rate shipping boxes from the post office.
Consider the problem: Which of the flat-rate boxes should she use to minimize her shipping cost?
What other information would you need to be able to solve the problem?
Discuss this information with your group. Make a plan for using this information to find the most inexpensive way to ship the jewelry boxes. Once you have agreed on a plan, write down the main steps.
Student Response
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Activity Synthesis
Reconvene as a class before continuing with the next part. Ask each group to share a couple of specific steps they have taken toward answering the question and a couple of steps they plan on taking to move forward. Highlight any ideas students might have about making the problem-solving process more efficient and systematic. If not already mentioned by students, suggest that each group divide up the calculations to be done so each person is responsible for one shipping box.


correct_option_index: 9