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.


Mathematical concept/skill options: 
A. Write and interpret numerical expressions.
B. Generate and analyze patterns.
C. Analyze patterns and relationships.
D. Use properties of operations to generate equivalent expressions.
E. Use the four operations with whole numbers to solve problems.
F. Work with radicals and integer exponents.
G. Solve systems of equations.
H. Represent and solve problems involving multiplication and division.
I. Perform arithmetic operations on polynomials.
J. Understand the relationship between zeros and factors of polynomials.
K. Understand properties of multiplication and the relationship between multiplication and division.
L. Solve real-life and mathematical problems using numerical and algebraic expressions and equations.
M. Solve problems involving the four operations, and identify and explain patterns in arithmetic.
N. Use polynomial identities to solve problems.
O. Apply and extend previous understandings of arithmetic to algebraic expressions.
P. Rewrite rational expressions.
Q. Interpret the structure of expressions.
R. Add and subtract within 20.
S. Analyze and solve linear equations and pairs of simultaneous linear equations.
T. Represent and analyze quantitative relationships between dependent and independent variables.
U. Solve equations and inequalities in one variable.
V. Create equations that describe numbers or relationships
W. Reason about and solve one-variable equations and inequalities.
X. Represent and solve equations and inequalities graphically.
Y. Understand solving equations as a process of reasoning and explain the reasoning.
Z. Understand and apply properties of operations and the relationship between addition and subtraction.
AA. Gain familiarity with factors and multiples.
AB. Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.
AC. Work with addition and subtraction equations.
AD. Multiply and divide within 100.
AE. Work with equal groups of objects to gain foundations for multiplication.
AF. Represent and solve problems involving addition and subtraction.
AG. Write expressions in equivalent forms to solve problems.
AH. Understand the connections between proportional relationships, lines, and linear equations.


Thought: This task teaches students to understand addition as putting together counters, or adding to already placed counters.
Answer: AB

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
$$-\frac{1}{4}$$


Mathematical concept/skill options: 
A. Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.
B. Reason quantitatively and use units to solve problems.
C. Apply and extend previous understandings of numbers to the system of rational numbers.
D. Represent complex numbers and their operations on the complex plane.
E. Perform arithmetic operations with complex numbers.
F. Compute fluently with multi-digit numbers and find common factors and multiples.
G. Represent and model with vector quantities.
H. Perform operations on vectors.
I. Use properties of rational and irrational numbers.
J. Use complex numbers in polynomial identities and equations.
K. Extend the properties of exponents to rational exponents.
L. Know that there are numbers that are not rational, and approximate them by rational numbers.
M. Perform operations on matrices and use matrices in applications.
N. Apply and extend previous understandings of multiplication and division to divide fractions by fractions.


Thought: This problem involves understanding fractions, multiplication, and division.
Answer: A

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


Mathematical concept/skill options: 
A. Prove and apply trigonometric identities.
B. Model periodic phenomena with trigonometric functions.
C. Interpret functions that arise in applications in terms of the context.
D. Build a function that models a relationship between two quantities.
E. Define, evaluate, and compare functions.
F. Use functions to model relationships between quantities.
G. Understand the concept of a function and use function notation.
H. Analyze functions using different representations.
I. Extend the domain of trigonometric functions using the unit circle.
J. Interpret expressions for functions in terms of the situation they model.
K. Build new functions from existing functions.
L. Construct and compare linear and exponential models and solve problems.


Thought: This problem applies trigonometric identities, including the angle addition formula for cosine.
Answer: A

Now, assign the following problem or activity to one or more relevant mathematical concepts/skills it teaches.
shot_info: im_task_001477

K

fl_problem_001097

7

fl_problem_001894

HS_0
options: Interpret the structure of expressions.

Apply and extend previous understandings of arithmetic to algebraic expressions.

Add and subtract within 20.

Understand and apply properties of operations and the relationship between addition and subtraction.

Solve systems of equations.

Gain familiarity with factors and multiples.

Reason about and solve one-variable equations and inequalities.

Use polynomial identities to solve problems.

Solve real-life and mathematical problems using numerical and algebraic expressions and equations.

Understand solving equations as a process of reasoning and explain the reasoning.

Solve equations and inequalities in one variable.

Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.

Represent and analyze quantitative relationships between dependent and independent variables.

Represent and solve equations and inequalities graphically.

Work with radicals and integer exponents.

Represent and solve problems involving multiplication and division.

Generate and analyze patterns.

Analyze and solve linear equations and pairs of simultaneous linear equations.

Multiply and divide within 100.

Represent and solve problems involving addition and subtraction.

Work with addition and subtraction equations.

Write expressions in equivalent forms to solve problems.

Use the four operations with whole numbers to solve problems.

Understand properties of multiplication and the relationship between multiplication and division.

Create equations that describe numbers or relationships

Solve problems involving the four operations, and identify and explain patterns in arithmetic.

Work with equal groups of objects to gain foundations for multiplication.

Use properties of operations to generate equivalent expressions.

Understand the connections between proportional relationships, lines, and linear equations.

Understand the relationship between zeros and factors of polynomials.

Write and interpret numerical expressions.

Analyze patterns and relationships.

Perform arithmetic operations on polynomials.

Rewrite rational expressions.
problem_activity: Activity
Up to this point in the unit, students have written and interpreted equations representing exponential change which were meaningful for non-negative input values. In this lesson, they interpret the meaning of a negative value in a context. The independent variable is time,
\(t\)
, so negative values of
\(t\)
refer to times
before
\(t\)
is 0.
For the last question, to find the time when the coral had the given volume, students may:
Calculate the values of
\(y\)
when
\(t\)
is -3, -4, -5, etc. (i.e., extend the table of values, if they previously created one) until they reach 37.5 or a value close to it.
Repeatedly divide by 2 (or multiply by
\(\frac12\)
) the value of
\(y\)
when
\(t\)
is -2 until it approaches or hits 37.5.
Use the equation
\(1,\!200 \boldcdot 2^t = 37.5\)
to find the value of
\(2^t\)
and then reason about
\(t\)
from there.
Look for students using these or other strategies so they can share later.
Making graphing technology available gives students an opportunity to choose appropriate tools strategically (MP5).
Launch
Arrange students in groups of 2. Encourage them to think quietly about the questions before discussing with their partner. Creating a table or spreadsheet may help students organize the work in the second question. If needed, encourage students to do so.
Reading, Conversing, Writing: MLR5 Co-Craft Questions.
To help students make sense of the language of exponential situations, start by displaying the task statement “A marine biologist estimates that a structure of coral has a volume of 1,200 cubic centimeters and that its volume doubles each year.” 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 activity’s questions. Listen for and amplify any questions referring to the value of
\(t\)
, especially those wondering about time in the past. This will build student understanding and interpretation of the language used to represent negative exponents.
Design Principle(s): Maximize meta-awareness; Cultivate conversation
Representation: Internalize Comprehension.
Begin the activity with concrete or familiar contexts. Encourage students to create a timeline next to their tables, selecting a starting point year for zero and then labeling the years correlated with the positive exponents prior to labeling years associated with negative exponents.
Supports accessibility for: Conceptual processing; Memory
Student Facing
A marine biologist estimates that a structure of coral has a volume of 1,200 cubic centimeters and that its volume doubles each year.
[IMG]
Write an equation of the form
\(y=a \boldcdot b^t\)
representing the relationship, where
\(t\)
is time in years since the coral was measured and
\(y\)
is volume of coral in cubic centimeters. (You need to figure out what numbers
\(a\)
and
\(b\)
are in this situation.)
Find the volume of the coral when
\(t\)
is 5, 1, 0, -1, and -2.
What does it mean, in this situation, when
\(t\)
is -2?
In a certain year, the volume of the coral is 37.5 cubic centimeters. Which year is this? Explain your reasoning.
Student Response
Teachers with a valid work email address can
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for free access to Student Response.
Anticipated Misconceptions
Students may struggle to think of how to start finding the values for
\(a\)
and
\(b\)
. Suggest that they make a table of corresponding values of
\(t\)
and
\(y\)
, and think about starting with
\(t\)
and calculating
\(y\)
.
Activity Synthesis
Review how students found an equation,
\(y = 1,200 \boldcdot 2^t\)
, to represent the volume of coral. Make sure that they recall that the 1,200 represents the 1,200 cubic centimeters of coral when it was first measured and the 2 indicates the doubling of its volume each year.
Select students to share their interpretations of negative values of
\(t\)
and how they found the value of
\(t\)
when given a volume. Ask questions such as:
“In this situation, what does it mean to say that when
\(t\)
is -3,
\(y\)
is
\(1,200 \boldcdot 2^{\text-3}\)
or 150?” (3 years before the coral was measured to be 1,200 cubic centimeters, its volume was 150 cubic centimeters).
“How did you estimate or find when the volume of the coral was 37.5 cubic centimeters?”
It may be helpful to display a graph of
\(y = 1,200 \boldcdot 2^t\)
, such as the one embedded in the digital version of these materials. Display the graph (either with or without the points from the table plotted) and ask students to identify the volume of coral when
\(t\)
is -3 and the time when the volume of coral is 37.5 cubic centimeters.
If using the digital applet, consider initially turning off the function and the point, and hiding the expression list. Ask students to gesture at points on the graph corresponding to questions in the activity. Then, turn on the function and the point for demonstration. Remember that if you click on a plotted point, its coordinates are revealed.


correct_option_index: 24