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: Add up to four two-digit numbers using strategies based on place value and properties of operations.

Explain why addition and subtraction strategies work, using place value and the properties of operations. (Explanations may be supported by drawings or objects.)

Mentally add 10 or 100 to a given number 100–900, and mentally subtract 10 or 100 from a given number 100–900.

Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Add and subtract within 1000, using concrete models or drawings and strategies based on place value, properties of operations, and/or the relationship between addition and subtraction; relate the strategy to a written method. Understand that in adding or subtracting three-digit numbers, one adds or subtracts hundreds and hundreds, tens and tens, ones and ones; and sometimes it is necessary to compose or decompose tens or hundreds.
problem_activity: Narrative
The purpose of this activity is for students to interpret and compare representations that show decomposing a ten to subtract by place. One student shows decomposing a ten by crossing off a ten and drawing 10 ones. The other representation shows a student who begins their drawing with a ten decomposed into 10 ones. Students compare and make connections between the representations and a set of equations that also shows how to find the value of the difference (MP3).
This activity uses
MLR2 Collect and Display
. Advances: conversing, reading, writing
Representation: Internalize Comprehension.
Invite students to identify which details were most useful from each strategy to solve the problem. Display the sentence frame, “The next time I have a subtraction task, I will look for . . . .“ Encourage decomposition, drawing pictures or using manipulatives as the main strategies.
Supports accessibility for: Attention, Memory
Required Materials
Materials to Gather
Base-ten blocks
Launch
Groups of 2
Give students access to base-ten blocks.
Activity
“Diego and Elena drew base-ten diagrams to find the value of
\(82-9\)
.”
“Think about what is the same or different on your own. Then, discuss your ideas with your partner.”
1 minute: quiet think time
2
–
3 minutes: partner discussion
MLR2 Collect and Display
Circulate, listen for, and collect the language students use to describe the diagrams and how the ten is decomposed. Listen for: break apart a ten, decompose, tens, need more ones.
Record students’ words and phrases on a visual display and update it throughout the lesson.
Share responses. Consider asking:
“Did Diego and Elena find the same value for
\(82-9\)
? How do you know?”
“What is the difference between their diagrams?”
“Tyler found the value by using equations. Diego says Tyler’s equations match his diagram. Elena says the equations match her diagram. Who do you agree with?”
2 minutes: independent work time
4 minutes: partner discussion
Monitor for students who agree with Diego, Elena, or both and can explain their reasoning with connections to the diagrams.
Student Facing
Diego and Elena drew base-ten diagrams to find the value of
\(82-9\)
.
Diego
[IMG]
Elena
[IMG]
Compare Deigo’s work to Elena’s.
What is the same?
What is different?
Tyler used equations to show his thinking.
\(82 - 9\)
\(82 = 70 + 12\)
\(12 - 9 = 3\)
\(70 + 3 = 73\)
Diego says Tyler’s work matches his diagram.
Elena says Tyler’s work matches her diagram.
Who do you agree with? Explain.
Student Response
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Advancing Student Thinking
Some students may only recognize 82 as 8 tens and 2 ones. Consider asking:
“What was the value of Elena’s blocks before she started subtracting? Explain.”
“How could you make 82 with base-ten blocks if you only had 7 tens?” Allow students to share how they could represent 82 with 7 tens and 12 ones.
Activity Synthesis
Invite previously identified students to share.
“Diego, Elena, and Tyler saw they needed more ones before they could subtract ones from ones. They showed decomposing a ten in different ways.”
Display the list of words and phrases recorded during the activity.
“Here are some of the words we used to describe subtracting by place.”
“Are there any other words or phrases that are important to include on our display?”
As students share responses, update the display, by adding (or replacing) language, diagrams, or annotations.
Remind students to borrow language from the display as needed.


correct_option_index: 3