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: 
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.

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

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 and division and of fractions to multiply and divide rational numbers.

Answer: 
Yes. 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).

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


Concept/skill: 
Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

Answer: 
No. This problem does not directly teach how side ratios in right triangles lead to definitions of trigonometric ratios. Instead, it involves solving a trigonometric function.

Now, does the problem or activity shown below enable students to learn the full intent of the following concept or skill?

shot_info: im_task_001477_pos_0

K

fl_problem_001097_pos_0

7

fl_problem_001894_neg_0

HS_0
standard_description: Understand that a function from one set (called the domain) to another set (called the range) assigns to each element of the domain exactly one element of the range. If f is a function and x is an element of its domain, then f(x) denotes the output of f corresponding to the input x. The graph of f is the graph of the equation y = f(x).
problem_activity: Activity
The purpose of this activity is for students to work with a function where either variable could be the independent variable. Knowing the total value for an unknown number of dimes and quarters, students are first asked to consider if the number of dimes could be a function of the number of quarters and then asked if the reverse is also true. Since this isn't always the case when students are working with functions, the discussion should touch on reasons for choosing one variable vs. the other, which can depend on the types of questions one wants to answer.
Identify students who efficiently rewrite the original equation in the third problem and the last problem to share during the discussion.
Launch
Arrange students in groups of 2. Give students 3–5 minutes of quiet work time followed by partner discussion for students to compare their answers and resolve any differences. Follow with a whole-class discussion.
Representation: Internalize Comprehension.
Provide appropriate reading accommodations and supports to ensure students access to written directions, word problems and other text-based content.
Supports accessibility for: Language; Conceptual processing
Reading: MLR6 Three Reads.
Use this routine to support reading comprehension and to set students up to interpret the representations of the situation provided in the task statement (an equation) and discussion (function diagrams). In the first read, students read the information with the goal of comprehending the situation (e.g., Jada has dimes and quarters). In the second read, ask students to identify important quantities. Listen for, and amplify, naming of the quantities that vary in relation to each other in this situation: number of dimes, total value of dimes, number of quarters, total value of quarters. After the third read, ask students to discuss possible strategies to answer the questions that follow, paying attention to the different coin values, and how each question is phrased. This will help students comprehend the problem and make sense of important quantities and variables when working with a function in which either variable could be the independent variable.
Design Principle(s): Support sense-making
Student Facing
Jada had some dimes and quarters that had a total value of
$
12.50. The relationship between the number of dimes,
\(d\)
, and the number of quarters,
\(q\)
, can be expressed by the equation
\(0.1d + 0.25q = 12.5\)
.
If Jada has 4 quarters, how many dimes does she have?
If Jada has 10 quarters, how many dimes does she have?
Is the number of dimes a function of the number of quarters? If yes, write a rule (that starts with
\(d = \)
...) that you can use to determine the output,
\(d\)
, from a given input,
\(q\)
. If no, explain why not.
If Jada has 25 dimes, how many quarters does she have?
If Jada has 30 dimes, how many quarters does she have?
Is the number of quarters a function of the number of dimes? If yes, write a rule (that starts with
\(q=\)
...) that you can use to determine the output,
\(q\)
, from a given input,
\(d\)
. If no, explain why not.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
Some students may be unsure how to write rules for the number of dimes as a function of the number of quarters and vice versa. Prompt them to use the provided equation and what they know about keeping equations equal to create the new equations.
Activity Synthesis
Select previously identified students to share their rules for dimes as a function of the number of quarters and quarters as a function of the number of dimes, including the steps they used to rewrite the original equation.
Tell students that if we write an equation like
\(d = 125 - 2.5q\)
, this shows that
\(d\)
is a function of
\(q\)
because it is clear what the output (value for
\(d\)
) should be for a given input (value for
\(q\)
).
Display the diagrams for all to see:
[IMG]
When we have an equation like
\(0.1d + 0.25q = 12.5\)
, we can choose either
\(d\)
or
\(q\)
to be the independent variable. That means we are viewing one as depending on the other. If we know the number of quarters and want to answer a question about the number of dimes, it is helpful to write
\(d\)
as a function of
\(q\)
. If we know the number of dimes and want to answer a question about the number of quarters, it is helpful to write
\(q\)
as a function of
\(d\)
.
Ensure students understand that we can’t always do this type of rearranging with equations and have it make sense because sometimes only one variable is a function of the other, and sometimes neither is a function of the other. For example, students saw earlier that while squaring values is a function, the reverse—that is, identifying what value was squared—is not. We will continue to explore when these different things happen in future lessons.


aligns: no