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: Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.

Fluently add and subtract multi-digit whole numbers using the standard algorithm.
Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.

Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.
Grade 4 expectations in this domain are limited to whole numbers less than or equal to 1,000,000.
problem_activity: Narrative
In this activity, students use rectangular diagrams and similar reasoning as in earlier activities to represent the multiplication of 2 two-digit numbers. They analyze a progression of diagrams, starting with those that represent multiplication of two-digit and one-digit numbers (18 and 6), a two-digit number and a ten (18 and 10), and then 2 two-digit numbers (18 and 16).
Students may decompose factors in different ways. For example, those who are familiar with multiples of 25 may find it intuitive to decompose
\(25 \times 46\)
as
\(25 \times 40\)
and
\(25 \times 6\)
, rather than decomposing 25 into
\(20 + 5\)
.
MLR8 Discussion Supports.
Pair gestures with verbal directions to clarify the meaning of any unfamiliar terms such as partial product.
Advances: Listening, Representing
Action and Expression: Develop Expression and Communication.
Give students access to base-ten blocks. If students use the blocks for the last question, ask them to also draw a diagram that represents their work with the blocks.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing
Launch
Display the three diagrams in the activity.
“What do you notice? What do you wonder?”
1 minute: quiet think time
1 minute: partner discussion
Activity
“Take a few quiet minutes to work on the first problem. Then, share your thinking with your partner.”
3 minutes: independent work time
3 minutes: partner discussion
Invite students to share their responses: “What multiplication expression can be represented by each diagram?”
“Complete the rest of the activity.”
3 minutes: independent or partner work time
Monitor for students who decompose both factors into tens and ones and those who choose to keep one of the factors intact.
Student Facing
For each diagram, write a multiplication expression that the diagram can represent. Then, find the value of the expression. Use equations to show or explain your reasoning.
[IMG]
[IMG]
[IMG]
How are the diagrams alike? How are they different? Discuss with your partner.
Use a diagram to find each product.
\(13 \times 21\)
\(25 \times 46\)
Student Response
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Activity Synthesis
Select two students to share their diagrams, solutions, and reasoning for the last problem.
“How did you decompose the factors and the diagram?” (I decomposed the 13 into 10 and 3 and the 21 into 20 and 1.)
“What expression could we write to show the partial product represented by each part of the diagram?”
“How do these partial products help us find the value of
\(13 \times 21\)
?” (Adding them gives us the value of
\(13 \times 21\)
:
\(200 + 10 + 3 + 60 = 273\)
)


correct_option_index: 0