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: Find inverse functions.

Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Understand the inverse relationship between exponents and logarithms and use this relationship to solve problems involving logarithms and exponents.
problem_activity: Activity
Continuing the thinking from the warm-up, students now compare three functions whose graphs are vertical translations from one another and generalize their observations of the graphs and data tables into equations where one function is defined in terms of another.
Monitor for students making connections between representations as they write an equation for
\(g(x)\)
in terms of
\(h(x)\)
. For example, students may:
compare the graphs of
\(g\)
and
\(h\)
, perhaps sketching them on the same axes and then measuring the vertical distance between them
calculate the difference between the outputs from the table
combine the equation they wrote for
\(g(x)\)
in terms of
\(f(x)\)
and
\(h(x)=f(x)-2.5\)
to rewrite
\(g(x)\)
in terms of
\(h(x)\)
Launch
Representation: Internalize Comprehension.
Chunk this task into more manageable parts. Check in with students after the first 2–3 minutes of work time to look for students who notice the constant difference between the outputs of
\(f(x)\)
and
\(g(x)\)
. Check to make sure that students are filling out the
\(g(x)\)
column of the table correctly.
Supports accessibility for: Conceptual processing; Organization
Student Facing
The graph of function
\(g\)
is a vertical translation of the graph of polynomial
\(f\)
.
[IMG]
[IMG]
Complete the
\(g(x)\)
column of the table.
If
\(f(0) = \text-0.86\)
, what is
\(g(0)\)
? Explain how you know.
Write an equation for
\(g(x)\)
in terms of
\(f(x)\)
for any input
\(x\)
.
The function
\(h\)
can be written in terms of
\(f\)
as
\(h(x)=f(x)-2.5\)
. Complete the
\(h(x)\)
column of the table.
| \(x\) | \(f(x)\) | \(g(x)\) | \(h(x)=f(x)-2.5\) |
|-------|----------|----------|-------------------|
|  -4   |    0     |          |                   |
|  -3   |   -5.8   |          |                   |
| -0.7  |    0     |          |                   |
|  1.2  |   -3.3   |          |                   |
|   2   |    0     |          |                   |
Sketch the graph of function
\(h\)
.
[IMG]
Write an equation for
\(g(x)\)
in terms of
\(h(x)\)
for any input
\(x\)
.
Student Response
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Activity Synthesis
A key takeaway from this discussion is that students understand that a vertical translation from one graph to another corresponds to constant differences between the outputs of the functions for the same input. We can then generalize these point by point comparisons using function notation, where adding or subtracting from the output of the function resulting in a vertical translation of its graph.
Select previously identified students to share the connections they made between the representations in the order listed in the Activity Narrative. If time allows and no students used the equations for
\(g(x)\)
and
\(h(x)\)
in terms of
\(f(x)\)
to write a new equation for
\(g(x)\)
in terms of
\(h(x)\)
, invite them to do so now.


correct_option_index: 1