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: Make geometric constructions.

Explain volume formulas and use them to solve problems.

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

Understand similarity in terms of similarity transformations.

Prove geometric theorems.

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

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

Apply geometric concepts in modeling situations.

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

Prove theorems involving similarity.

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

Understand and apply theorems about circles.

Define trigonometric ratios and solve problems involving right triangles.

Analyze, compare, create, and compose shapes.

Understand congruence in terms of rigid motions.

Apply trigonometry to general triangles.

Understand and apply the Pythagorean Theorem.

Reason with shapes and their attributes.

Find arc lengths and areas of sectors of circles.

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

Experiment with transformations in the plane.

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

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

Use coordinates to prove simple geometric theorems algebraically.

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

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.
problem_activity: Narrative
The purpose of this activity is for students to partition rectangles to create rows and columns of equal-size squares. In the launch, students build an array with tiles and then represent it on a rectangle with tick marks as guidance. They will partition rectangles without tick marks in the next activity.
MLR2 Collect and Display.
Synthesis: Direct attention to words collected and displayed from the previous lessons. Include the word “partition.” Invite students to borrow language from the display as needed, and update it throughout the lesson.
Advances: Conversing, Reading
Required Materials
Materials to Gather
Inch tiles
Rulers
Launch
Groups of 2
Give students rulers and inch tiles.
Activity
“Use 8 tiles to make a rectangle. Your tiles should be in 2 rows.”
1 minute: independent work time
“Now, draw lines in the rectangle to show the squares. It should have the same number of equal-size squares as the rectangle you made out of tiles. You may use a ruler if it helps you.”
1 minute: independent work time
“You will draw lines to partition 2 more rectangles into equal-size squares. You can also use the tiles to build it first, if that helps.”
10 minutes: independent work time
Student Facing
Build a rectangle with 8 tiles arranged in 2 rows. Use a ruler to partition the rectangle to match the rectangle you made.
[IMG]
Use a ruler to partition the rectangle using the tick marks as a guide.
[IMG]
How many rows of equal-size squares did you make?
How many columns did you make?
Write 2 equations to represent the total number of equal-size squares.
Use a ruler to partition the rectangle using the tick marks as a guide.
[IMG]
How many rows of equal-size squares did you make?
How many columns did you make?
Write 2 equations to represent the total number of equal-size squares.
Student Response
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Advancing Student Thinking
If students create equations that match the total number of squares, but do not represent the sum of squares in each row or the sum of squares in each column, consider asking:
“How does your equation match the array?”
“Can you write an equation with equal addends that shows the sum of each row or column?”
Activity Synthesis
Invite 1
–
2 students to share their rectangles and equations for their rectangle with 3 rows and 5 columns. (
\(3 + 3 + 3 + 3 + 3 = 15\)
or
\(5 + 5 + 5 = 15\)
)
Draw a rectangle and partition to show:
[IMG]
“How is this partition different from what we did in this activity?” (The squares aren’t the same size and the last column has 4 not 3.)


correct_option_index: 17