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 to multiply a fraction or whole number by a fraction.

Answer: 
No. The expressions shown in this problem do not involve multiplying fractions with other fractions, nor whole numbers with fractions.

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 following problem or activity enable students to learn the full intent of the following concept or skill?

shot_info: im_task_001477_pos_0

K

fl_problem_001097_neg_0

7

fl_problem_001894_pos_0

HS_0
standard_description: Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠0, and use rate language in the context of a ratio relationship. For example, “This recipe has a ratio of 3 cups of flour to 4 cups of sugar, so there is 3/4 cup of flour for each cup of sugar.” “We paid $75 for 15 hamburgers, which is a rate of $5 per hamburger.” (Expectations for unit rates in this grade are limited to non-complex fractions.)
problem_activity: Activity
This activity is intended to build confidence and facility in writing an equation for a proportional relationship. Students build on their understanding that measurement conversions can be represented by proportional relationships, which they studied in an earlier lesson and will revisit in future lessons. Students are expected to use methods developed earlier: organize data in a table, write and solve an equation to determine the constant of proportionality, and use repeated reasoning (MP8) to arrive at an equation. In addition, they are expected to identify the relationship between the constant of proportionality when going from centimeters to meters and vice versa (reciprocals). Since students have already explored a similar relationship (centimeters and millimeters) in a previous lesson, this activity may go very quickly.
As students work, look for students writing an equation like
\(100k = 1\)
(for table 2) as a step to finding the constant of proportionality. Encourage them to say how they would solve the equation. Ask students to say why using this equation makes sense in the scenario.
Launch
Introduce the task: “In a previous lesson, you examined the relationship between millimeters and centimeters. Today we will examine the relationship between centimeters and meters.”
Student Facing
There are 100 centimeters (cm) in every meter (m).
| length (m) | length (cm) |
|------------|-------------|
|     1      |     100     |
|    0.94    |             |
|    1.67    |             |
|   57.24    |             |
|   \(x\)    |             |
| length (cm) | length (m) |
|-------------|------------|
|     100     |     1      |
|     250     |            |
|    78.2     |            |
|    123.9    |            |
|    \(y\)    |            |
Complete each of the tables.
For each table, find the constant of proportionality.
What is the relationship between these constants of proportionality?
For each table, write an equation for the proportional relationship. Let
\(x\)
represent a length measured in meters and
\(y\)
represent the same length measured in centimeters.
Student Response
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Student Facing
Are you ready for more?
How many cubic centimeters are there in a cubic meter?
How do you convert cubic centimeters to cubic meters?
How do you convert the other way?
Student Response
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for free access to Extension Student Response.
Activity Synthesis
Invite students to share their answers. These questions can guide the discussion:
"How can we find an equation for each table?"
"Where does the constant of proportionality occur in the table and equation?"
"What is the relationship between the two constants of proportionality? How can you use the equations to see why this should be true?"
The equations can help students see why the constants of proportionality are reciprocals:
\(\displaystyle y = 100x\)
\(\displaystyle \left(\frac{1}{100}\right)y = \frac{1}{100} \boldcdot (100x)\)
\(\displaystyle \left(\frac{1}{100}\right)y = x\)
\(\displaystyle x = \left(\frac{1}{100}\right)y\)
This line of reasoning illustrated above should be accessible to students, because it builds on grade 6 work with expressions and equations.
Ask students to interpret the meaning of the equations in the context: "What do the equations tell us about the conversion from meters to centimeters and back?"
Given the length in meters, to find the length in centimeters, multiply the number of centimeters by 100.
Given the length in centimeters, to find the the length in meters, multiply the number of meters by
\(\frac{1}{100}\)
.
Engagement: Develop Effort and Persistence.
Encourage and support opportunities for peer interactions. Prior to the whole-class discussion, invite students to share their work with a partner. Display sentence frames to support student conversation such as: “First, I _____ because….,” “I noticed _____ so I….,” “Why did you . . .?,” and “I agree/disagree because….”
Supports accessibility for: Language; Social-emotional skills
Speaking: MLR8 Discussion supports.
Provide students with sentence frames such as “I found the constant of proportionality by _____” to help explain their reasoning. Revoice student ideas to model mathematical language use by restating a response as a question in order to clarify, and apply appropriate language.
Design Principle(s): Maximize linguistic and cognitive awareness


aligns: no