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 multiplication to multiply a fraction or whole number by a fraction.

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 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?
problem_activity: Narrative
The purpose of this activity is for students to write expressions for and find the area of shaded regions whose side lengths are a unit fraction and a non-unit fraction. Students look at a series of diagrams with an increasingly large shaded region so they can look for and make use of structure (MP7).
It is important to relate the work here to what students have already learned about the product of a unit fraction and a unit fraction and this is the goal of the synthesis. For example, students have seen that
\(\frac{1}{5} \times \frac{1}{3} = \frac{1}{15}\)
. Since
\(\frac{6}{5}\)
is 6
\(\frac{1}{5}\)
s, the product
\(\frac{6}{5} \times \frac{1}{3}\)
will be 6
\(\frac{1}{15}\)
s or
\(\frac{6}{15}\)
.
MLR8 Discussion Supports.
Prior to solving the problems, invite students to make sense of the situations and take turns sharing their understanding with their partner. Listen for and clarify any questions about the context.
Advances: Listening, Conversing
Launch
Groups of 2
Display image A from the student workbook.
“What is a multiplication expression that represents the shaded region?”(
\(\frac{2}{5} \times \frac{1}{3}\)
,
\(2 \times \frac{1}{15}\)
)
“How does the diagram represent your expression?”(There is
\(\frac{2}{5}\)
of the first row shaded and that row is
\(\frac{1}{3}\)
of the square. There are 2 shaded pieces and each is
\(\frac{1}{15}\)
of the whole square.)
Activity
3–5 minutes: independent work time
3–5 minutes: partner discussion
Monitor for students who:
notice that the product of the numerators represents how many pieces of the square are shaded.
notice that the product of the denominators represents the number of pieces in the whole square.
Student Facing
Write a multiplication expression that represents the shaded region in each diagram.
A
[IMG]
B
[IMG]
C
[IMG]
What patterns do you notice in the multiplication expressions?
Han wrote this equation to represent the area of the shaded region. Explain how the diagram represents the equation.
\(\phantom{2.5cm} \\ \frac {6}{5} \times \frac {1}{3} = \frac {6}{15}\)
[IMG]
Student Response
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Advancing Student Thinking
If students write mathematically correct multiplication expressions that do not represent the product of a unit fraction and a non-unit fraction, write this type of multiplication expression that represents the diagram and ask “How does this expression represent the area of the shaded region in the diagram?”
Activity Synthesis
Ask previously selected students to share the patterns they noticed in the table.
Display the expression from the last problem:
\(\frac {6}{5} \times \frac {1}{3} = \frac {6}{15}\)
.
“How does the diagram show
\(\frac {1}{3}\)
?” (The first row of pieces in a square is
\(\frac{1}{3}\)
of the square.)
“How does the diagram show
\(\frac {6}{5}\)
?” (There are 6 pieces shaded and each one is
\(\frac{1}{5}\)
of the row.)
“How does the diagram show
\(\frac {6}{5} \times \frac {1}{3}\)
?” (There is
\(\frac{6}{5}\)
of a row shaded and that row is
\(\frac{1}{3}\)
of a square unit.)


correct_option_index: 0