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: Generate two numerical patterns using two given rules. Identify apparent relationships between corresponding terms. Form ordered pairs consisting of corresponding terms from the two patterns, and graph the ordered pairs on a coordinate plane. For example, given the rule “Add 3” and the starting number 0, and given the rule “Add 6” and the starting number 0, generate terms in the resulting sequences, and observe that the terms in one sequence are twice the corresponding terms in the other sequence. Explain informally why this is so.
problem_activity: Narrative
The purpose of this activity is for students to practice interpreting relationships between patterns generated from two different rules. Students may need to generate patterns beyond the boxes provided. Encourage them to continue the pattern as needed. Students may describe the patterns and relationships in different, but accurate ways. Encourage them to notice as many relationships as they can and describe the relationships in whatever way makes sense to them. This is the first time students see two patterns where the numbers in one pattern are not multiples of the numbers in the other pattern.
This activity uses
MLR7 Compare and Connect
. Advances: Representing, Conversing.
Launch
Groups of 2
Activity
5 minutes: independent work time
2 minutes: partner discussion
MLR7 Compare and Connect
“Create a visual display that shows your thinking about the relationships between each set of patterns. Include details such as notes, diagrams, and drawings to help others understand your thinking.”
2–5 minutes: independent or group work
3–5 minutes: gallery walk
Student Facing
Complete the patterns for each set of rules.
What are some relationships between the patterns for each set of rules? Be prepared to explain your thinking.
Set A
Rule 1: Start at 0. Keep adding 3.
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Rule 2: Start at 0. Keep adding 6.
[IMG]
Set B
Rule 1: Start at 4. Keep adding 3.
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Rule 2: Start at 9. Keep adding 6.
[IMG]
Set C
Rule 1: Start at 0. Keep adding 5.
[IMG]
Rule 2: Start at 0. Keep adding 3.
[IMG]
Student Response
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Activity Synthesis
“What is the same and what is different in the way we represented the relationships between the patterns?” (Some of us used numbers and symbols, some of us wrote sentences.)
30 seconds: quiet think time
1 minute: partner discussion
“What is the same and what is different about Set A and Set B?” (They both add the same amount, but one set starts at 0. Set B has a lot more odd numbers in it. In set A, each number in rule 2 is double the number in rule 1, but in set B, each number in rule 2 is 1 more than double the number in rule 1.)
“For which pair of rules was it most challenging to notice and describe relationships?” (Set B because there was no multiple that worked to get from one pattern to the other. Set C because there weren’t doubles or halves.)


correct_option_index: 0