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: Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions. (Students able to multiply fractions in general can develop strategies to divide fractions in general, by reasoning about the relationship between multiplication and division. But division of a fraction by a fraction is not a requirement at this grade.)

Interpret multiplication as scaling (resizing), by:
- Explaining why multiplying a given number by a fraction greater than 1 results in a product greater than the given number (recognizing multiplication by whole numbers greater than 1 as a familiar case); explaining why multiplying a given number by a fraction less than 1 results in a product smaller than the given number; and relating the principle of fraction equivalence a/b = (n×a)/(n×b) to the effect of multiplying a/b by 1.
- Comparing the size of a product to the size of one factor on the basis of the size of the other factor, without performing the indicated multiplication.

Interpret a fraction as division of the numerator by the denominator (a/b = a ÷ b). Solve word problems involving division of whole numbers leading to answers in the form of fractions or mixed numbers, e.g., by using visual fraction models or equations to represent the problem. For example, interpret 3/4 as the result of dividing 3 by 4, noting that 3/4 multiplied by 4 equals 3, and that when 3 wholes are shared equally among 4 people each person has a share of size 3/4. If 9 people want to share a 50-pound sack of rice equally by weight, how many pounds of rice should each person get? Between what two whole numbers does your answer lie?

Solve real world problems involving multiplication of fractions and mixed numbers, e.g., by using visual fraction models or equations to represent the problem.

Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
problem_activity: Narrative
The purpose of this activity is for students to practice multiplying fractions. The structure of the activity is identical to the previous one except that the goal is to have the smallest product. Monitor for students who identify the common structure with the previous game and place the larger numbers in the denominator and the smaller ones in the numerator (MP8).
Engagement: Internalize Self-Regulation.
Synthesis: Provide students an opportunity to self-assess and reflect on their own progress. For example, ask students to compare their calculated products to their partner’s. Encourage students to include how each factor impacted the product in their comparisons.
Supports accessibility for: Conceptual Understanding, Language
Required Materials
Materials to Gather
Paper clips
Required Preparation
Each group of 2 needs a paper clip.
Launch
Groups of 2
“We are going to play another round of Fraction Multiplication Compare, but this time the person with the smallest product is the winner.”
“Will you use the same strategy that you used when trying to make the greatest product?" (No because there the goal was to get the biggest product.)
1 minute: quiet think time
1 minute: partner discussion
Give each group a paper clip.
“Play Fraction Multiplication Compare with your partner.”
Activity
10–15 minutes: partner work time
Student Facing
[IMG]
Use the directions to play Fraction Multiplication Compare with your partner.
Spin the spinner.
Write the number you spun in one of the four blank boxes.
Player two spins and writes the number on their game board.
Continue taking turns until all four blank boxes are filled.
Multiply your fractions.
The player with the smallest product wins.
Play again.
Round 1
\(\frac{\boxed{\phantom{111\frac{1}{1}}}}{\boxed{\phantom{111\frac{1}{1}}}}\, \times\, \frac{\boxed{\phantom{111\frac{1}{1}}}}{\boxed{\phantom{111\frac{1}{1}}}}\, =\, \underline{\hspace{1cm}}\)
Round 2
\(\frac{\boxed{\phantom{111\frac{1}{1}}}}{\boxed{\phantom{111\frac{1}{1}}}}\, \times \, \frac{\boxed{\phantom{111\frac{1}{1}}}}{\boxed{\phantom{111\frac{1}{1}}}}\, = \,\underline{\hspace{1cm}}\)
What strategy did you use to choose where to write the numbers?
Student Response
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Activity Synthesis
“How was this game the same as the earlier version of Fraction Multiplication Compare?” (I knew where to put big numbers like 8 and 9 and small numbers like 1 and 2. It was hard to know where to put the middle numbers like 4 or 5.)
“How did your strategy change when trying to make the smallest product?” (It was the opposite of the first game. I tried to put the biggest numbers in the denominator and the smallest numbers in the numerator).


correct_option_index: 4