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.


Standard options: 
A. 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.
B. For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.
C. Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = 2 + 3 and 5 = 4 + 1).
D. Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.
E. Fluently add and subtract within 5.


Thought: This task involves representing addition with objects such as number cards and dice.
Answer: A

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
$$-\frac{1}{4}$$


Standard options: 
A. Solve real-world and mathematical problems involving the four operations with rational numbers. (Computations with rational numbers extend the rules for manipulating fractions to complex fractions.)
B. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.
C. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram.


Thought: 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).
Answer: B

Example 3:
Problem/activity:
Find the general solution for:
$${\mathrm{cos}\left({\pi\over2}+x\right)=\mathrm{tan}{\pi\over4}}$$


Standard options: 
A. Prove the Pythagorean identity sin^(2)(θ) + cos^(2)(θ) = 1 and use it to find sin(θ), cos(θ),or tan(θ) given sin(θ), cos(θ), or tan(θ) and the quadrant of the angle.
B. Prove the addition and subtraction formulas for sine, cosine, and tangent and use them to solve problems.


Thought: 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. Solving this problem does not require use of the Pythagorean identity.
Answer: B

Now, assign the following problem or activity to one or more relevant standards it aligns with.
shot_info: im_task_001477

K

fl_problem_001097

7

fl_problem_001894

HS_0
options: Use facts about supplementary, complementary, vertical, and adjacent angles in a multi-step problem to write and solve simple equations for an unknown angle in a figure.

Know the formulas for the area and circumference of a circle and use them to solve problems; give an informal derivation of the relationship between the circumference and area of a circle.

Solve real-world and mathematical problems involving area, volume and surface area of two- and three-dimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms.
problem_activity: Activity
The purpose of this activity is for students to find the areas of regions involving different-sized circles and compare the strategies used. The first question introduces subtraction as a strategy to find the area around the outside of a circle. The second question introduces division to find the area of fractions of a circle.
Monitor for students who use different strategies for finding the area, including:
Calculating 30.96 square units for Figure A,
\(4 \boldcdot 7.74\)
for Figure B, and
\(9 \boldcdot 3.44\)
for Figure C
Realizing that all 3 figures end up being
\(144 - 113.04\)
so their areas had to be equal
Noticing that Figure B is composed of 4 scaled copies of Figure A, each with a scale factor of
\(\frac12\)
and therefore
\(\frac14\)
as much area. Figure C is composed of 9 scaled copies of Figure A, each with a scale factor of
\(\frac13\)
and therefore
\(\frac19\)
as much area. (If a student does not make this realization, it is
not
necessary for the teacher to bring it up during the discussion.)
Calculating 8.28 square units for Figure D and 8.71 square units for Figure E
Realizing that Figures D has 2 fewer squares and 2 more quarter-circles (which are smaller than the squares) so it must have a smaller area
The task affords an opportunity for students to engage in MP6. Some students may leave the area expressions in terms of
\(\pi\)
, but others may use an approximation for
\(\pi\)
. It is very important, particularly for the first problem, that they use the
same
approximation for
\(\pi\)
in all 3 figures. Otherwise they will end up with numerically different answers.
Launch
Arrange students in groups of 2. Display the image in the first question and ask students to make a prediction before calculating. Give 30 seconds of quiet think time before sharing with their partner. Give students quiet work time followed by a whole-class discussion.
Student Facing
Each square has a side length of 12 units. Compare the areas of the shaded regions in the 3 figures. Which figure has the largest shaded region? Explain or show your reasoning.
[IMG]
Each square in Figures D and E has a side length of 1 unit. Compare the area of the two figures. Which figure has more area? How much more? Explain or show your reasoning.
[IMG]
Student Response
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Student Facing
Are you ready for more?
Which figure has a longer perimeter, Figure D or Figure E? How much longer?
Student Response
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for free access to Extension Student Response.
Anticipated Misconceptions
In the first question, students may not know how to find the radius of the circles. Suggest having them cut off the shaded regions and rearrange them to show that the length of each side fits half way across the circle (marking the radius).
In the second question, students might benefit from cutting and rearranging the figures. Some students might assume, based on previous activities, that the areas of both figures are equal. However, Figure D has more pieces that are parts of a circle, and Figure E has more units that are a full square. Ask students whether the fourth of the circle has the same area as the square.
Activity Synthesis
There are two main goals for this discussion: for students to notice ways to be more efficient when comparing the areas of the regions and to be introduced to expressing answers in terms of pi.
Display Figures A, B, and C for all to see. Ask selected students to share their reasoning. Sequence the strategies from most calculations to most efficient.
If there were selected students who determined the areas were equal before calculating, ask them to share how they could tell. If there were no selected students, ask the class how we could determine that the area of the shaded regions in Figures A, B, and C were equal before calculating the answer of 30.96.
Next, focus the discussion on leaving answers in terms of
\(\pi\)
for each figure. Explain to students that in Figure A, the radius of the circle is 6, so the area of the circular region is
\(\pi \boldcdot 6^2\)
. Instead of multiplying by an approximation of
\(\pi\)
, we can express this answer as
\(36\pi\)
. This is called answering
in terms of
\(\pi\)
. Consider writing “
\(36\pi\)
” inside the large circle of Figure A.
Discuss:
In terms of
\(\pi\)
, what is the area of one of the circular regions in Figure B? (
\(9\pi\)
)
What is the combined area of all four circles in Figure B? (
\(4 \boldcdot 9\pi\)
, or
\(36\pi\)
)
What is the area of one of the circular regions in Figure C? (
\(4\pi\)
)
What is the combined area of all nine circles in Figure C? (
\(9 \boldcdot 4\pi\)
, or
\(36\pi\)
)
Consider writing “
\(9\pi\)
” and “
\(4\pi\)
” inside some of the circles in Figures B and C. Explain that the area of the shaded region for each of these figures is
\(144 - 36\pi\)
.
[IMG]
Discuss how students' strategies differed between the first problem (about Figures A, B, and C) and the second problem (about Figures D and E) and why.
Ask students to express the area of Figures D and E in terms of
\(\pi\)
. Record and display their answers of
\(2 + 2\pi\)
and
\(4 + 1.5\pi\)
for all to see. Ask students to discuss how they can tell Figure E’s area is larger than Figure D’s area when they are both written in terms of
\(\pi\)
.
Writing, Speaking, Listening: MLR1 Stronger and Clearer Each Time.
After students have determined which figure has the largest shaded region, ask students to show their work and provide a brief explanation of their reasoning. Ask each student to meet with 2–3 other partners for feedback. Provide students with prompts for feedback that will help them strengthen their ideas and clarify their language (e.g., “How did you find the radius of each circle in the figure?”, “Why did you . . . ?”, etc.). Students can borrow ideas and language from each partner to refine and clarify their original explanation. This will help students refine their own explanation and learn about different strategies to find area.
Design Principles(s): Optimize output (for explanation); Maximize meta-awareness


correct_option_index: 1