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.


Your response:
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.


Your response:
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.


Your response:
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: Write a function that describes a relationship between two quantities.

Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.
problem_activity: Optional activity
The purpose of this task is for students to write a recursive definition for a sequence that represents a mathematical context and to create other representations of the sequence.
Monitor for groups who create their recursive definitions in different ways to share during the whole-class discussion. For example, some students may first create a table showing step number and the associated values while others may draw additional steps.
Allow students to use graph paper to sketch their graphs if needed. Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Display the image for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.
If not brought up by students, ask “How would you describe the total number of small squares in Step 3 compared to the total number of small squares in Step 2?” (There are 9, or
\(3^2\)
, more squares in Step 4 than in Step 3.)
Arrange students in groups of 2. Encourage them to check in with their partner frequently as they work through the task.
Action and Expression: Internalize Executive Functions.
Provide students with grid or graph paper to organize their work with representations of geometric and arithmetic sequences.
Supports accessibility for: Language; Organization
Student Facing
Here is a pattern where the number of small squares increases with each new step.
[IMG]
Write a recursive definition for the total number of small squares
\(S(n)\)
in Step
\(n\)
.
Sketch a graph of
\(S\)
that shows Steps 1 to 7.
Is this sequence geometric, arithmetic, or neither? Be prepared to explain how you know.
Student Response
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for free access to Student Response.
Student Facing
Are you ready for more?
Start with a circle. If you make 1 cut, you have 2 pieces. If you make 2 cuts, you can have a maximum of 4 pieces. If you make 3 cuts, you can have a
maximum
of 7 pieces.
Draw a picture to show how 3 cuts can give 7 pieces.
Find the maximum number of pieces you can get from 4 cuts.
From 5 cuts.
Can you find a function that gives the maximum number of pieces from
\(n\)
cuts?
Student Response
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click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
Students may not be sure where to begin with the graph since no axes are provided in the task statement. Encourage these students to first figure out what values they need to plot before drawing, scaling, and labeling their axes.
Activity Synthesis
The goal of this discussion is for students to share how they reasoned about a recursive definition for
\(S\)
and how they created their graph for Steps 1 to 7. Invite previously identified groups to share how they created their definitions, making sure to display for all to see any additional representations they used to help their thinking. After these students have shared, ask "Did anyone use a different strategy for writing their definition?" and invite any new students to share their thinking.
Conclude the discussion by reviewing graphing strategies. Select 2–4 students to share how they created their sketch of
\(S\)
. In particular, focus on how the scale on each axis was chosen.


correct_option_index: 1