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.


Standard options: 
A. Represent addition and subtraction with objects, fingers, mental images, drawings (Drawings need not show details, but should show the mathematics in the problem. (This applies wherever drawings are mentioned in the Standards.)), sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.
B. For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.
C. Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).
D. Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
E. Fluently add and subtract within 5.


Thought: This task involves representing addition with objects such as number cards and dice.
Answer: A

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
$$-\frac{1}{4}$$


Standard options: 
A. Solve real-world and mathematical problems involving the four operations with rational numbers. (Computations with rational numbers extend the rules for manipulating fractions to complex fractions.)
B. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
C. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.


Thought: This problem involves understanding fractions and division to divide rational numbers, such as 1, -4, 4, and -1. This problem also involves an understanding of multiplication: if p and q are integers, then -(p/q) = (-p)/q = p/(-q).
Answer: B

Example 3:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Standard options: 
A. Prove the Pythagorean identity sin^(2)(θ) + cos^(2)(θ) = 1 and use it to find sin(θ), cos(θ),or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
B. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.


Thought: An initial step when solving for x in this problem requires the use of the addition formula for cosine on the left side of the equation. Solving this problem does not require use of the Pythagorean identity.
Answer: B

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: im_task_001477

K

fl_problem_001097

7

fl_problem_001894

HS_0
options: Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.

Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.
problem_activity: Optional activity
In this activity, the dots are distributed uniformly in the first square but not in the second square:
[IMG]
However, these dots are drawn so it is not too hard to see that if they were redistributed, each square inch would have 8 dots:
[IMG]
The fact that we have 8 dots per square inch means that if we distributed the dots uniformly throughout the square in the right way, we would see 8 dots in each square inch. This prepares students to be able to interpret the constant of proportionality in the next two activities when working with actual houses per square mile or people per square kilometer. When we speak of “500 houses per square kilometer,” we can think of this as, “If we took all of the houses in the region and spread them out uniformly, then we would see 500 houses in every square kilometer.”
Digital
Launch
Display the image of the two squares with dots. Ask students to describe what is the same and what is different about these squares.
Give students 2–3 minutes of quiet work time, followed by 4–5 minutes of partner work time and whole-class discussion.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, pause to check for understanding after the first 2–3 minutes of work time.
Supports accessibility for: Organization; Attention
Writing, Conversing: MLR5 Co-craft Questions.
Display only the image of the four dot arrays without revealing any of the questions that follow. Give students 1–2 minutes to write a list of possible mathematical questions they could ask about the arrays. Invite students to share their questions with a partner, and then select 2–3 students to share their questions with the whole class. Highlight any questions that refer to how dots are “distributed,” even if students do not use that particular phrase. Finally, reveal the whole problem with text so that students may begin addressing the questions. This helps amplify language related to the distribution of dots.
Design Principle(s): Cultivate conversation; Maximize meta-awareness
Student Facing
The figure shows two arrays, each enclosed by a square that is 2 inches wide.
[IMG]
Let
\(a\)
be the area of the square and
\(d\)
be the number of dots enclosed by the square. For each square, plot a point that represents its values of
\(a\)
and
\(d\)
. Start by choosing an appropriate scale for each axis, using the sliders to adjust the intervals.
Draw lines from
\((0, 0)\)
to each point. For each line, write an equation that represents the proportional relationship.
What is the constant of proportionality for each relationship? What do the constants of proportionality tell us about the dots and squares?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Display the image of the two squares with dots. Ask students to describe what is the same and what is different about these squares.
Give students 2–3 minutes of quiet work time, followed by 4–5 minutes of partner work time and whole-class discussion.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts to support students who benefit from support with organizational skills in problem solving. For example, pause to check for understanding after the first 2–3 minutes of work time.
Supports accessibility for: Organization; Attention
Writing, Conversing: MLR5 Co-craft Questions.
Display only the image of the four dot arrays without revealing any of the questions that follow. Give students 1–2 minutes to write a list of possible mathematical questions they could ask about the arrays. Invite students to share their questions with a partner, and then select 2–3 students to share their questions with the whole class. Highlight any questions that refer to how dots are “distributed,” even if students do not use that particular phrase. Finally, reveal the whole problem with text so that students may begin addressing the questions. This helps amplify language related to the distribution of dots.
Design Principle(s): Cultivate conversation; Maximize meta-awareness
Student Facing
The figure shows two arrays, each enclosed by a square that is 2 inches wide.
[IMG]
Let
\(a\)
be the area of the square and
\(d\)
be the number of dots enclosed by the square. For each square, plot a point that represents its values of
\(a\)
and
\(d\)
.
[IMG]
Draw lines from
\((0, 0)\)
to each point. For each line, write an equation that represents the proportional relationship.
What is the constant of proportionality for each relationship? What do the constants of proportionality tell us about the dots and squares?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Activity Synthesis
The goal of this discussion is for students to make sense of the constant of proportionality in the case where the dots are not uniformly distributed. Invite students to share their interpretations of the constants of proportionality.
Consider asking questions like:
“Why is one of the constants of proportionality larger than the other? How can we see this in the picture of the squares?” (There are more dots in the same area.)
“What are the units of the constants of proportionality?” (The number of dots per square inch).
Students will have more opportunities to think about this when working on the activities that follow.


correct_option_index: 1