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: Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

Distinguish between situations that can be modeled with linear functions and with exponential functions.

Observe using graphs and tables that a quantity increasing exponentially eventually exceeds a quantity increasing linearly, quadratically, or (more generally) as a polynomial function.

For exponential models, express as a logarithm the solution to ab^(ct) = d where a, c, and d are numbers and the base b is 2, 10, or e; evaluate the logarithm using technology.
problem_activity: Activity
In a previous lesson, students used equations defining functions to reason about graphs. Here they work in the opposite direction, analyzing graphs and creating corresponding equations. In the first question, the coordinates given are not of two consecutive values of
\(x\)
, so students need to reason about the decay factor indirectly. They may:
Try different factors that, when multiplied by
\(f(0)\)
twice, give
\(f(2)\)
.
Use the graph to estimate
\(f(1)\)
, and then use that estimate to find a reasonable decay factor.
Find the factor that takes
\(f(0)\)
to
\(f(2)\)
(or
\(f(2)\)
to
\(f(4)\)
), recognize that number as
\(b^2\)
, and then find
\(b\)
.
The second question has no context and is thus more abstract. This prepares students to interpret graphs in the next activity, where minimal information is given.
Launch
Arrange students in groups of 2. Ask students to take a few minutes of quiet think time on each question before discussing their thinking with a partner.
Reading, Conversing, Writing: MLR5 Co-Craft Questions.
To help students make sense of an exponential situation in which the graph is given before the equation, start by displaying only the first task statement and its graph. Give students 1–2 minutes to write their own mathematical questions about the situation. Invite students to share their questions with the class, then reveal the questions. Listen for and amplify any questions seeking to write an equation, interpret the meaning of various points, or predict future trends. This will help students create questions and use language that demonstrates what they know is true about a situation given a graph first.
Design Principle(s): Maximize meta-awareness; Cultivate conversation
Student Facing
Here is a graph representing an exponential function
\(f\)
. The function
\(f\)
gives the value of a computer, in dollars, as a function of time,
\(x\)
, measured in years since the time of purchase.
[IMG]
Based on the graph, what can you say about the following?
The purchase price of the computer
The value of
\(f\)
when
\(x\)
is 1
The meaning of
\(f(1)\)
How the value of the computer is changing each year
An equation that defines
\(f\)
Whether the value of
\(f\)
will reach 0 after 10 years
Here are graphs of two exponential functions. For each, write an equation that defines the function and find the value of the function when
\(x\)
is 5.
[IMG]
[IMG]
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?
Consider a function
\(f\)
defined by
\(f(x)=a\boldcdot b^x\)
.
If the graph of
\(f\)
goes through the points
\((2,10)\)
and
\((8,30)\)
, would you expect
\(f(5)\)
to be less than, equal to, or greater than 20?
If the graph of
\(f\)
goes through the points
\((2,30)\)
and
\((8,10)\)
, would you expect
\(f(5)\)
to be less than, equal to, or greater than 20?
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Extension Student Response.
Anticipated Misconceptions
If students have trouble getting started, ask them to consider the graph and say what they know is true about the situation that the graph represents. Most likely, they will answer some parts of the first question by doing this, and will be in a better position to figure out any questions they didn't answer.
Activity Synthesis
Select students to share their responses to the first question. Focus the discussion on how students found the decay factor for the computer. Make sure students understand that the key is to notice that every 2 years, the computer's value is multiplied by
\(\frac{1}{4}\)
. This means that the annual decay factor is
\(\frac{1}{2}\)
, since
\(\left(\frac{1}{2}\right)^2 = \frac{1}{4}\)
.
For the last graph, discuss questions such as:
“What information is needed to describe an exponential function?” (the factor of growth or decay and the initial value)
“Where on the graph do we see the initial value?” (the
\(y\)
-intercept of the graph)
“What points on the graph can we use to help us find the growth factor?” (the points
\((\text-1, 40)\)
and
\((0,20)\)
, or
\((0,20)\)
and
\((2,5)\)
)
“Why might
\((\text-1,40)\)
and
\((0,20)\)
be a more strategic choice than
\((0,20)\)
and
\((2,5)\)
for finding the factor of growth?” (The
\(x\)
coordinates of first two points differ by 1, so the quotient of the
\(y\)
coordinates of those points can be used to find the factor of decay. It is a more direct way to find the factor of decay, which is the factor by which the output of
\(f\)
changes when the input
\(x\)
changes by 1. In the second pair, the
\(x\)
coordinate changes by 2.)
Representation: Internalize Comprehension.
Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, display the equation
\(100 = 400\boldcdot (b)^2\)
and use color or annotations to illustrate how each value or variable is represented graphically.
Supports accessibility for: Visual-spatial processing; Conceptual processing


correct_option_index: 0