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.


Your response, in json format:
{
  "explanation": "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.


Your response, in json format:
{
  "explanation": "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.


Your response, in json format:
{
  "explanation": "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: Understand that rewriting an expression in different forms in a problem context can shed light on the problem and how the quantities in it are related. For example, a + 0.05a = 1.05a means that “increase by 5%” is the same as “multiply by 1.05.”

Apply properties of operations as strategies to add, subtract, factor, and expand linear expressions with rational coefficients.
problem_activity: Activity
This activity offers four word problems. Depending on time constraints, you may have all students complete all four problems or assign a different problem to each group. The problems increase in difficulty. It is suggested that students create a visual display of one of the problems and do a gallery walk or presentation, but if time is short you may choose to just have students work in their workbooks or devices.
Launch
Keep students in the same groups. Either instruct students to complete all four problems or assign one problem to each group. If opting to have students do presentations or a gallery walk, distribute tools for making a visual display.
Give students 5–6 minutes quiet work time and partner discussion followed by a whole-class discussion or gallery walk.
Representation: Internalize Comprehension.
Represent the same information through different modalities by using diagrams, equations, or other drawings to depict the situation. If students are unsure where to begin, suggest that they draw a diagram to help organize the information provided. During the Synthesis, annotate drawings to illustrate connections between representations.
Supports accessibility for: Conceptual processing; Visual-spatial processing
Reading, Representing: MLR6 Three Reads.
Use this routine to support reading comprehension of the first problem without solving it for students. For the first read, read the situation to students, without revealing the final question. Ask students "What is this question about?" (A store is having a sale. Diego is buying shoes with a coupon). In the second read, ask students to name the important quantities (e.g., discount of 20%, coupon for
$
3 off of the regular price, Diego pays
$
18.40), and then create a diagram to represent the relationships among these quantities. After the third read, ask students to brainstorm possible strategies to answer the question, “What was the original price before the sale and without the coupon?” This will help students connect the language in the word problem and reasoning needed to solve percent problems.
Design Principle(s): Support sense-making; Maximize meta-awareness
Student Facing
A store is having a sale where all shoes are discounted by 20%. Diego has a coupon for
$
3 off of the regular price for one pair of shoes. The store first applies the coupon and then takes 20% off of the reduced price. If Diego pays
$
18.40 for a pair of shoes, what was their original price before the sale and without the coupon?
Before the sale, the store had 100 pairs of flip flops in stock. After selling some, they notice that
\(\frac35\)
of the flip flops they have left are blue. If the store has 39 pairs of blue flip flops, how many pairs of flip flops (any color) have they sold?
When the store had sold
\(\frac29\)
of the boots that were on display, they brought out another 34 pairs from the stock room. If that gave them 174 pairs of boots out, how many pairs were on display originally?
On the morning of the sale, the store donated 50 pairs of shoes to a homeless shelter. Then they sold 64% of their remaining inventory during the sale. If the store had 288 pairs after the donation and the sale, how many pairs of shoes did they have at the start?
Student Response
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Student Facing
Are you ready for more?
A coffee shop offers a special: 33% extra free or 33% off the regular price. Which offer is a better deal? Explain your reasoning.
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
If students created a visual display and you opt to conduct a gallery walk, ask students to post their solutions. Distribute sticky notes and ask students to read others’ solutions, using the sticky notes to leave questions or comments. Give students a moment to review any questions or comments left on their display.
Invite any students who chose to draw a diagram to share; have the class agree or disagree with their diagrams and suggest any revisions. Next, invite students who did not try to draw a diagram to share strategies. Ask students about any difficulties they had creating the expressions and equations. Highlight equivalent expressions that represent the same quantity and different strategies for solving equations.


correct_option_index: 0