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: 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
Previously, students found percentages of 100 cents. In this activity, they reason about percentages of 1 dollar.
One important question to think about here is how students know or decide how the numbers on the double number line diagram should be aligned. Students build on their extensive work on equivalent ratios and double number lines to make sense of percentages and “per 100” reasoning.
Launch
Recap that in the previous activity students found percentages of 100 cents. Tell students they will now find percentages of 1 dollar. Draw their attention to the fact that, on the double number line, the 1 dollar and 100% are lined up vertically to reflect this.
Keep students in the same groups. Give students 2–3 minutes of quiet think time, and then ask them to share their responses with their partner. Display and read aloud the following questions. Ask partners to use them to guide their discussion.
How did each of you arrive at your answers for the first two questions?
Where do your answers fall on the double number line diagram? How do you know?
Are your answers the same for the third question? If they are not, can they both be correct? If they are, can you think of another answer that would also be correct?
Student Facing
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A
$
1 coin is worth 100% of the value of a dollar. Here is a double number line that shows this.
[IMG]
The coins in Jada’s pocket are worth 75% of a dollar. How much are they worth (in dollars)?
The coins in Diego’s pocket are worth 150% of a dollar. How much are they worth (in dollars)?
Elena has 3 quarters and 5 dimes. What percentage of a dollar does she have?
Student Response
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Anticipated Misconceptions
Based on previous work with labeling number lines less than 1, students may label the tick marks with fractions instead of the decimal value of the coins. This may not be helpful for answering the first two questions, but provides an opportunity to discuss alternative ways to label the number line given the context of the problem. Consider prompting them to write fractional values as cents or to rewrite the cents as dollar values.
Activity Synthesis
Select students who used the provided double number line to share their reasoning. This is an opportunity to refresh students’ number line reasoning. Some students may see the four equally spaced tick marks from 0 to 1 and conclude that each is worth 0.25, or
\(\frac14\)
. Others may fill in the 0.50 first, as it is half of 1, then the 0.25 for half of 0.50, and then use additive thinking to fill in the other tick mark values along the top.
Some students may reason in terms of equivalent ratios and say, for example, that since 100 divided by 4 is 25, then
\(1\div4=0.25\)
must be 25% of 1. They would then assign the 0.25 value to the first tick mark and use additive thinking to conclude that 0.75 is 75% of a dollar. Ask students who used such an approach to present last to emphasize that the familiar ratio thinking applies to percentage problems as well, even though the % symbol may be unfamiliar. If no students took this approach, illustrate it to make this point.
Speaking, Listening: MLR7 Compare and Connect.
As students prepare a visual display of how they made sense of the problem, look for students who labeled the tick marks on the double number line with fractions or cents instead of dollar values. This may result in answers such as
\(¾\)
, or 75 cents is 75% of a dollar rather than 0.75 is 75% of a dollar. Although 75 cents is 75% of a dollar, the number line should be labeled with the decimal value of the coins in dollars. As students investigate each other's work, ask students to share what worked or did not work well in the way they labeled the double number line. Is there a particular advantage to using decimals instead of fractions to label the double number line? Emphasize that although there are several ways to label the double number line given the context of the problem, certain methods are more helpful for answering the question. This will foster students’ meta-awareness and support constructive conversations as they compare the various ways to label a double number line given a context.
Design Principles(s): Cultivate conversation; Maximize meta-awareness


aligns: no