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: 
Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.

Answer: 
No. Though this task involves addition, it does not involve any word problems.

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_neg_0

K

fl_problem_001097_neg_0

7

fl_problem_001894_pos_0

HS_0
standard_description: Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.
problem_activity: Activity
Students build their sense for the sizes of angles measured in radians by shading sectors with given central angle measures.
Launch
Arrange students in groups of 2. Distribute 1 set of pre-cut slips to each group.
Student Facing
Your teacher will give you a set of cards with angle measures on them. Place the cards upside down in a pile. Choose 1 student to go first. This student should draw a card, then on either circle shade a sector of the circle whose central angle is the measure on the card that was drawn.
Take turns repeating these steps. If you are shading in a circle that already has a shaded sector, choose a spot next to the already-shaded sectors—don’t leave any gaps. You might have to draw additional lines to break the sectors into smaller pieces.
Continue until an angle is drawn that won’t fit in any of the sectors that are still blank.
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[IMG]
When you’re finished, answer these questions about each circle:
What is the central angle measure for the remaining unshaded sector?
What is the central angle measure for the block of shaded sectors?
Student Response
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Anticipated Misconceptions
If students struggle to figure out the size of the angles on the cards they draw, ask them what radian measure 180 degrees is equivalent to (
\(\pi\)
radians). Then, ask them to find the measures of the central angles of each sector on the 2 circles (
\(\frac{\pi}{4}\)
and
\(\frac{\pi}{6}\)
).
Activity Synthesis
The goal is to gather strategies for understanding the sizes of angles measured in radians. Here are some questions for discussion:
“For groups that drew a card with
\(\frac{\pi}{8}\)
radians, how did you figure out what size sector to shade?” (Sample response: Each sector in the first circle had a central angle measuring
\(\frac{\pi}{4}\)
radians, so
\(\frac{\pi}{8}\)
radians is half that.)
“What is the sum of the angle measures for the shaded and unshaded sector blocks, and why?” (The angle measures add to
\(2\pi\)
, because together they make up the whole circle.)
“How could you shade a sector with central angle measure
\(\frac{\pi}{24}\)
radians in each of the circles?” (This would be one-sixth of a sector in the first circle, and one-quarter of a sector in the second circle.)


aligns: yes