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: Use addition and subtraction within 100 to solve word problems involving lengths that are given in the same units, e.g., by using drawings (such as drawings of rulers) and equations with a symbol for the unknown number to represent the problem.

Represent whole numbers as lengths from 0 on a number line diagram with equally spaced points corresponding to the numbers 0, 1, 2, ..., and represent whole-number sums and differences within 100 on a number line diagram.
problem_activity: Narrative
The purpose of this activity is for students to make sense of different methods they can use to compare three-digit numbers. They analyze the thinking of others and make connections across representations (MP2, MP3). Although students have compared numbers using different representations in prior units, this activity offers them the opportunity to consider using the number line as a tool to compare three-digit numbers. In the synthesis, students will discuss which representation makes it easier to see the comparison. While students could make a case that each of the representations was easier for them, the focus is on Jada’s way, the number line.
MLR8 Discussion Supports.
Display sentence frames to support partner discussion. “_____ and _____ are the same/alike because . . .” and “_____ and _____ are different because . . . .”
Advances: Speaking, Conversing
Launch
Groups of 2
Activity
“Diego, Jada, and Clare were asked to compare 371 and 317. They each represented their thinking differently.”
“Take some time to look over their methods.”
2 minutes: independent work time
“Discuss with your partner how their methods are the same and different.”
4 minutes: partner discussion
“Now try Jada’s way.”
6 minutes: partner work time
Student Facing
Each student compared 371 and 317, but represented their thinking in different ways.
Diego
[IMG]
[IMG]
I see 3 hundreds for each number.
317 only has 1 ten, but 371 has 7 tens.
\(371 > 317\)
Clare
Each has 3 hundreds.
371 has 7 tens, but 317 only has 1 ten.
\(317 < 371\)
Jada
[IMG]
I can see that 371 is farther to the right on my number line, so I know it is greater than 317.
\(371 > 317\)
What is the same and different about these students’ representations?
Discuss with a partner.
Try Jada’s way.
Estimate the location of 483 and 443 on the number line.
Mark each number with a point. Label the point with the number it represents.
[IMG]
Use
\(>\)
,
\(=\)
, or
\(<\)
to compare 483 and 443.
______________________________
Student Response
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Advancing Student Thinking
If students write comparison statements that are not true, ask them to read their statements and consider asking:
“How did you decide which number was greater?”
“How does your statement match the number line?”
Activity Synthesis
“Clare, Diego, and Jada represented their thinking in different ways. Whose method made it easier to see that 371 is greater than 317? Explain.” (Jada’s because you just have to look at which number is farther to the right. You can see 371 is farther from 0. Diego’s because I use diagrams and I can see quickly that the hundreds are the same and there are more tens, but some things were the same.)
As needed, “Jada and Diego wrote
\(371 > 317\)
, but Clare wrote
\(317 < 371\)
. Who do you agree with? Explain.” (They both mean the same thing. 371 is greater than 317, so 317 is less than 371.)


correct_option_index: 1