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 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_neg_0

7

fl_problem_001894_pos_0

HS_0
standard_description: Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
problem_activity: Optional activity
In this activity, students begin by practicing their understanding of proportions and and probabilities by examining the data set they have available. In the fourth problem, students obtain a sample from the population using tools they choose (MP5)and examine the sample they selected to compare it to the expected proportions and probabilities calculated in the first 3 problems.
The problems are intended for students to use their own data set to answer. Although they are kept in pairs for the entire lesson, this activity should be done individually.
Launch
Keep students in the same groups of 2. Give students 5–7 minutes of quiet work time followed by a whole-class discussion.
If possible, allow students to use their chosen method of random sampling to obtain a sample of 10 for this activity. Have items such as paper clips, scissors, 10-sided polyhedra, and other materials available for student use. The blackline master for the first activity in this lesson contains accurate spinners that could be used to select a random sample.
Action and Expression: Internalize Executive Functions.
Chunk this task into more manageable parts. After students have solved the first 2-3 problems, check-in with either select groups of students or the whole class. Invite students to share the strategies they have used so far as well as any questions they have before continuing.
Supports accessibility for: Organization; Attention
Student Facing
Continue working with the data set your teacher gave you in the previous activity. The data marked with a star all came from students at Springfield Middle School.
When you select the first value for your random sample, what is the probability that it will be a value that came from a student at Springfield Middle School?
What proportion of your entire sample would you expect to be from Springfield Middle School?
If you take a random sample of size 10, how many scores would you expect to be from Springfield Middle School?
Select a random sample of size 10.
Did your random sample have the expected number of scores from Springfield Middle School?
Student Response
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Activity Synthesis
The purpose of this discussion is to connect the ideas of probability and random sampling from the unit.
Consider these questions for discussion:
“How is selecting a sample at random connected to probability?” (A random sample should give each value an equal chance of being chosen. Therefore, each value has a
\(\frac{1}{100}\)
probability of being chosen.)
“How could we simulate the probability of getting at least 2 values in the sample of 10 from Springfield Middle School?” (Since 20% of the values come from Springfield Middle School, we could put 10 blocks in a bag with 2 colored red to represent Springfield Middle School. Draw a block from the bag, and if it is red, it represents a score from Springfield Middle School; replace the block and repeat. Get a sample of 10 and see if the sample has at least 2 red blocks. Repeat this process many times and use the fraction of times there are at least 2 red blocks as an estimate for the probability that a random sample will have at least 2 scores from Springfield Middle School.)
Representing, Speaking: MLR7 Compare and Connect.
Invite students to create a visual display of their random sample of size 10 and response to the question: “Did your random sample have the expected number of scores from Springfield Middle School?” Invite students to investigate each other’s work and compare their responses. Listen for the language students use to describe a random sample and assign a probability of each value being chosen. This will help students connect the ideas of probability and random sampling through discussion.
Design Principle(s): Optimize output (for representation); Cultivate conversation


aligns: no