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

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fl_problem_001894

HS_0
options: Draw, construct, and describe geometrical figures and describe the relationships between them.

Use coordinates to prove simple geometric theorems algebraically.

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

Find arc lengths and areas of sectors of circles.

Prove theorems involving similarity.

Make geometric constructions.

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

Explain volume formulas and use them to solve problems.

Analyze, compare, create, and compose shapes.

Understand and apply the Pythagorean Theorem.

Apply trigonometry to general triangles.

Define trigonometric ratios and solve problems involving right triangles.

Apply geometric concepts in modeling situations.

Reason with shapes and their attributes.

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

Understand and apply theorems about circles.

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

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

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

Understand similarity in terms of similarity transformations.

Prove geometric theorems.

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

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

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

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

Understand congruence in terms of rigid motions.

Experiment with transformations in the plane.
problem_activity: Activity
In the previous unit, students studied scale drawings of real-world objects. In grade 8, they will study dilations and similarity. The purpose of this activity is to use students’ recent study of scale drawings as a transition to the study of proportional relationships. Initially, students may describe the difference between Moons A, B, C, and D in qualitative terms, e.g., “D is more squished than the others.” They may also use the term “scaled copies,” which appeared in the work of the previous unit, but struggle to identify measurements to use in these figures that consist only of curved sides. It is important to ask students to articulate what they mean by “squished” in quantitative terms, for example, by talking about the height relative to the width and helping students to define “height” and “width” of a moon in some appropriate way. Once students have that, they can note that the height of the enclosing rectangle is always one and a half times its width for Moons A, B, and C, but not D, or they might note that the distance tip to tip is three times the width of the widest part of the moon for Moons A, B, and C, but not so for D.
[IMG]
As students explore these transformations, ask questions with the goal of having students articulate that for two images to look like scaled copies of each other, the ratios of the side lengths need to be the same.
In the third question, students are asked to represent the situation with tables and double number line diagrams, providing students with an opportunity to recall these representations from their work in grade 6.
Digital
Launch
Give students 3 minutes of quiet think time and tell them to pause after the first two questions. After the quiet think time, ask students to discuss their answers with a partner to describe how Moon D is different. Use MLR 2 (Collect and Display) as students share. Record the explanations that students are using to describe the moons. Ensure that students see some ways to measure lengths associated with the moons, then complete the last question.
If using the digital activity, give students 3 minutes of quiet think time and tell them to pause after the first two questions. After the quiet think time, students can discuss answers with a partner. Based on student conversations, you may want to have a whole-group discussion to ensure they see a way to measure the lengths associated with the moons before they attempt to answer the last question.
Action and Expression: Internalize Executive Functions.
To support development of organizational skills, check in with students within the first 2–3 minutes of work time. Check to make sure students have a method for comparing the measurements of each moon. For example, create a rectangle around each moon and compare the width-height ratios.
Supports accessibility for: Memory; Organization
Student Facing
Here are four different crescent moon shapes.
What do Moons A, B, and C all have in common that Moon D doesn’t?
Use numbers to describe how Moons A, B, and C are different from Moon D.
Use a table or a double number line to show how Moons A, B, and C are different from Moon D.
Student Response
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for free access to Student Response.
Student Facing
Are you ready for more?
Can you make one moon cover another by changing its size? What does that tell you about its dimensions?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Print
Launch
Give students 3 minutes of quiet think time and tell them to pause after the first two questions. After the quiet think time, ask students to discuss their answers with a partner to describe how Moon D is different. Use MLR 2 (Collect and Display) as students share. Record the explanations that students are using to describe the moons. Ensure that students see some ways to measure lengths associated with the moons, then complete the last question.
If using the digital activity, give students 3 minutes of quiet think time and tell them to pause after the first two questions. After the quiet think time, students can discuss answers with a partner. Based on student conversations, you may want to have a whole-group discussion to ensure they see a way to measure the lengths associated with the moons before they attempt to answer the last question.
Action and Expression: Internalize Executive Functions.
To support development of organizational skills, check in with students within the first 2–3 minutes of work time. Check to make sure students have a method for comparing the measurements of each moon. For example, create a rectangle around each moon and compare the width-height ratios.
Supports accessibility for: Memory; Organization
Student Facing
Here are four different crescent moon shapes.
[IMG]
What do Moons A, B, and C all have in common that Moon D doesn’t?
Use numbers to describe how Moons A, B, and C are different from Moon D.
Use a table or a double number line to show how Moons A, B, and C are different from Moon D.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
For question 2, students might attempt to find the area of each moon by counting individual square units. Suggest that they create a rectangle around each moon instead and compare the width-height ratios.
For question 3, if students are not sure how to set up these representations, providing a template may be helpful.
Activity Synthesis
As students suggest ways to characterize the difference between Moons A, B, and C and Moon D, ask questions that help them clarify and make their statements more precise. For example,
“What does it mean to be ‘smashed down?’ What measurements might you make to show that this is true?”
“Is there anything else about what A, B, and C have in common that you can identify?”
“What things might we measure about these moons to be able to talk about what makes them different in a more precise way?”
“How can you represent the ratios of the measurements you are comparing using a table and a double number line diagram?”
If some students are still struggling with the tables or diagrams, ask students who were successful to share their representations with the class.
Speaking: MLR8 Discussion Supports.
To aid students in producing statements about comparisons of proportional relationships. Provide sentence frames for students to use when they are comparing and contrasting such as: “All _____ have _____ except _____,” “What makes _____ different from the others is _____.” Ask students to further add to the statement to clarify the comparisons if they used descriptor words such as “wider, narrower, etc.” Improved statements should include mathematical language to describe the measurements (such as length, width, ratio, etc.). This will help students practice and develop language for comparisons.
Design Principle(s): Support sense-making


correct_option_index: 0