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: 
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

Answer: 
No. This problem does not directly teach how side ratios in right triangles lead to definitions of trigonometric ratios. Instead, it involves solving a trigonometric function.

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_neg_0

HS_0
standard_description: Explain and use the relationship between the sine and cosine of complementary angles.
problem_activity: Activity
In this activity students build off the specific calculations from the previous lesson to generalize the perimeter of a polygon inscribed in a circle of radius 1. The relatively unstructured presentation of this activity is purposeful. Students work with their groups to determine what information they need, how to calculate in the specific cases, and how they can express those repeated procedures in a generalized formula (MP8). Monitor for groups who have generalized any part of the process.
Digital
Launch
Encourage students to refer to the examples from the previous lesson as they work to generalize.
Action and Expression: Internalize Executive Functions.
Chunk this task into manageable parts for students who benefit from support with organizational skills in problem solving. Check in with students after the first 2–3 minutes of work time. Look for students who are struggling to begin and review examples from the previous lesson. Record their thinking on a display and keep the work visible as students continue to work.
Supports accessibility for: Organization; Attention
Student Facing
The applet shows a regular
\(n\)
-sided polygon inscribed in a circle.
Come up with a general formula for the perimeter of the polygon in terms of
\(n\)
.  Explain or show your reasoning.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Encourage students to refer to the examples from the previous lesson as they work to generalize.
Action and Expression: Internalize Executive Functions.
Chunk this task into manageable parts for students who benefit from support with organizational skills in problem solving. Check in with students after the first 2–3 minutes of work time. Look for students who are struggling to begin and review examples from the previous lesson. Record their thinking on a display and keep the work visible as students continue to work.
Supports accessibility for: Organization; Attention
Student Facing
Here is one part of a regular
\(n\)
-sided polygon inscribed in a circle of radius 1.
Come up with a general formula for the perimeter of the polygon in terms of
\(n\)
.  Explain or show your reasoning.
[IMG]
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 are struggling invite them to go back to the problems from the previous lesson to generalize the process. (Draw in the altitude. Find the measure of the central angle. Find the length of the opposite leg.) Suggest students generalize each step before trying to write a single formula.
Activity Synthesis
Invite previously selected groups to share one step at a time:
generalize the angle measure
generalize the segment length
extend to the whole perimeter
Invite another student to summarize by explaining where each piece of
\(P=2n \boldcdot \sin \left( \frac{360}{2n} \right)\)
appears in the diagram.
Representing, Conversing: MLR7 Compare and Connect.
Use this routine to prepare students for the whole-class discussion. At the appropriate time, invite students to create a visual display showing their general formula for the perimeter of the polygon in terms of
\(n\)
and their reasoning. Allow students time to quietly circulate and analyze the formulas and reasoning in at least 2 other displays in the room. Give students quiet think time to consider what is the same and what is different. Next, ask students to find a partner to discuss what they noticed. Listen for and amplify observations that highlight what information each group used, any calculations they completed, and how they expressed those procedures in a generalized formula.
Design Principle(s): Optimize output (for generalization); Cultivate conversation


aligns: yes