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: 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.

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.)

Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.

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.
problem_activity: Narrative
The purpose of this activity is for students to match situations to expressions and then find the value of the expressions (MP2). Students see expressions that show both quotients of a whole number by a fraction and quotients of a fraction by a whole number. They need to think carefully about the situations to make sure the expression they choose matches the situation (MP2).
Launch
Groups of 2
Display the image from student book:
[IMG]
“What do you notice? What do you wonder?” (They look like seeds. How many are in the bowls? Why are there 2 bowls?)
“The small bowl is filled with
\(\frac {1}{4}\)
cup of kernels. That is one serving of popcorn kernels. What does ‘one serving’ mean?” (It is the amount you are supposed to eat at one time.)
“About how many servings are in the big bowl?” (Answers vary. Sample responses range from 10 to 15.)
Activity
1–2 minutes: independent think time
5–8 minutes: partner work time
Monitor for students who:
draw diagrams to represent the situations
describe the relationship between the dividend and divisor using language such as “groups of”
use the value of the expression to describe why the expression matches the situation
Student Facing
Match each problem to an expression that represents the problem. Some expressions will not have a match. Be prepared to explain your reasoning.
One serving of popcorn is
\(\frac {1}{4}\)
cup of kernels. There are 3 cups of kernels in the bowl. How many servings are in the bowl?
[IMG]
One serving of orange juice is
\(\frac {1}{4}\)
liter. The container of juice holds 2 liters. How many servings are in the container?
One serving of granola is
\(\frac {1}{2}\)
cup. The bag of granola holds 5 cups. How many servings are in the bag?
\(\frac {1}{4} \div 3 \)
\(\frac {1}{2} \div 5\)
\(3 \div \frac {1}{4}\)
\(\frac {1}{4} \div 2\)
\(5 \div \frac {1}{2}\)
\(2 \div \frac {1}{4}\)
Find the value of each expression.
Student Response
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Advancing Student Thinking
If students do not correctly match the situations to the expressions, refer to each situation and ask, “How does the situation represent division?”
Activity Synthesis
Ask previously selected students to share their solutions.
“How did you decide which expression matched which situation?” (I thought about the number of things being divided and whether it was a fraction or a whole number. I thought about the size of the pieces.)
Display:
\(3 \div \frac {1}{4}= 12\)
“Describe how this equation represents the situation in the first problem.” (There are 3 cups of kernels in the bowl and a serving is
\(\frac {1}{4}\)
a cup, so there are 12 servings in the bowl.)
Display:
\(\frac {1}{4} \div 3 = \frac {1}{12}\)
“How do you know that this equation does not match the situation?” (The answer is a fraction. That doesn’t make sense because there is more than 1 serving in 3 cups.)


correct_option_index: 1