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: Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.

Understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output. (Function notation is not required in Grade 8.)

Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
problem_activity: Activity
The purpose of this activity is for students to make connections between different representations of functions and start transitioning from input-output diagrams to other representations of functions. Students match input-output diagrams to descriptions and come up with equations for each of those matches. Students then calculate an output given a specific input and determine the independent and dependent variables.
Launch
Arrange students in groups of 2. Give students 3–5 minutes of quiet work time and time to share their responses with their partner and come to agreement on their answers. Follow with whole-class discussion.
Student Facing
Record your answers to these questions in the table provided.
Match each of these descriptions with a diagram:
the circumference,
\(C\)
, of a circle with
radius
,
\(r\)
the distance in miles,
\(d\)
, that you would travel in
\(t\)
hours if you drive at 60 miles per hour
the output when you triple the input and subtract 4
the volume of a cube,
\(v\)
given its edge length,
\(s\)
Write an equation for each description that expresses the output as a function of the input.
Find the output when the input is 5 for each equation.
Name the
independent
and
dependent variables
of each equation.
[IMG]
|              description               | a | b | c | d |
|----------------------------------------|---|---|---|---|
|                diagram                 |   |   |   |   |
|                equation                |   |   |   |   |
| input = 5                   output = ? |   |   |   |   |
| independent                   variable |   |   |   |   |
|  dependent                   variable  |   |   |   |   |
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Student Facing
Are you ready for more?
Choose a 3-digit number as an input.
Apply the following rule to it, one step at a time:
Multiply your number by 7.
Add one to the result.
Multiply the result by 11.
Subtract 5 from the result.
Multiply the result by 13
Subtract 78 from the result to get the output.
Can you describe a simpler way to describe this rule? Why does this work?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Activity Synthesis
The goal of this discussion is for students to describe the connections they see between the different entries for the 4 descriptions. Display the table for all to see and select different groups to share the answers for a column in the table. As groups share their answers, ask:
“How did you know that this diagram matched with this description?” (We remembered the formula for the circumference of a circle, so we knew description A went with diagram D.)
“Where in the equation do you see the rule that is in the diagram?” (The equation is the dependent variable set equal to the rule describing what happens to the independent variable in the diagram.)
“Explain why you chose those quantities for your independent and dependent variables." (We know the independent variable is the input and the dependent variable is the output, so we matched them up with the input and output shown in the diagram.)
Speaking: MLR8 Discussion Supports.
As students describe the connections they noticed in the table across from the different entries for the four descriptors, revoice student ideas to demonstrate mathematical language use. In addition, press for details in students’ explanations by requesting that students challenge an idea, elaborate on an idea, or give an example. This will help students to produce and make sense of the language needed to communicate their own ideas about functions and independent and dependent variables.
Design Principle(s): Support sense-making; Optimize output (for explanation)


correct_option_index: 0

1

2