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.


Concept/skill: 
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.

Answer: 
Yes. This task involves representing addition with objects such as number cards and dice.

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
$$-\frac{1}{4}$$


Concept/skill: 
Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

Answer: 
Yes. 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).

Example 3:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Concept/skill: 
Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.

Answer: 
Yes. 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.

Now, does the problem or activity shown below enable students to learn the full intent of the following concept or skill?

shot_info: im_task_001477_pos_0

K

fl_problem_001097_pos_0

7

fl_problem_001894_pos_0

HS_0
standard_description: Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.
problem_activity: Activity
In this activity students continue to build fluency operating with signed numbers as they match different expressions that have the same value. Students look for and use the relationship between inverse operations (MP7).
As students work, identify groups that make connections between the operations, for example they notice that subtracting a number is the same as adding its opposite, or that dividing is the same as multiplying by the multiplicative inverse.
Launch
Arrange students in groups of 2. Distribute pre-cut slips from the blackline master.
Student Facing
Your teacher will give you a set of cards. Group them into pairs of expressions that have the same value.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
If students struggle to find matches, encourage them to think of the operations in different ways. You might ask:
How else can you think of ____ (subtraction, addition, multiplication, division)?
Activity Synthesis
Select students to share their strategies. Highlight stategies that compared the structure of the expressions instead of just the final answers.
Discuss:
Subtracting a number is equivalent to adding the additive inverse.
A number and its additive inverse have oppostie signs (but the same magnitude).
Dividing by a number is equivalent to multiplying by the multiplicative inverse.
A number and its multiplicative inverse have the same sign (but different magnitudes).
If desired, have students order their pairs of equivalent expressions from least to greatest to review ordering rational numbers.
Engagement: Develop Effort and Persistence.
Break the class into small discussion groups and then invite a representative from each group to report back to the whole class.
Supports accessibility for: Attention; Social-emotional skills
Speaking, Representing: MLR7 Compare and Connect.
Use this routine to support whole-class discussion. Ask students to consider what is the same and what is different about the structure of each expression. Draw students’ attention to the association between quantities (e.g., adding the additive inverse is subtracting; multiplying by the multiplicative inverse is dividing). These exchanges strengthen students’ mathematical language use and reasoning to describe when comparing inverse-operation relationships involving rational numbers.
Design Principle(s): Support sense-making; Maximize meta-awareness


aligns: no