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.


Your response:
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.


Your response:
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.


Your response:
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: Perform operations on vectors.

Extend the properties of exponents to rational exponents.

Represent complex numbers and their operations on the complex plane.

Apply and extend previous understandings of numbers to the system of rational numbers.

Use properties of rational and irrational numbers.

Represent and model with vector quantities.

Compute fluently with multi-digit numbers and find common factors and multiples.

Know that there are numbers that are not rational, and approximate them by rational numbers.

Use complex numbers in polynomial identities and equations.

Perform arithmetic operations with complex numbers.

Apply and extend previous understandings of multiplication and division to divide fractions by fractions.

Reason quantitatively and use units to solve problems.

Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers.

Perform operations on matrices and use matrices in applications.
problem_activity: Activity
The purpose of this task is for students to understand that for a given number, numbers to the left are always less than the number, and numbers to the right are always greater than the number. The precise use of the term “absolute value” is not expected at this time.
Teacher Notes for IM 6–8 Accelerated
During the activity synthesis, introduce the terms
opposite
and
rational number
. Two numbers that are the same distance from zero on the number line are called opposites. Rational numbers are fractions and their opposites.
Digital
Launch
Allow 10 minutes quiet work time followed by whole-class discussion.
Students using the digital materials can graph the points and check them with the applet. Marks at each half, quarter, and eighth of a unit can be shown to help plot the points or to self-check for accuracy.
Representation: Internalize Comprehension.
Activate or supply background knowledge about equivalent fractions. Some students may benefit from access to pre-prepared number lines to compare fractions for the final two problems.
Supports accessibility for: Conceptual processing; Organization
Student Facing
Plot the numbers -2, 4, -7, and 10 on the number line. Label each point with its numeric value.
[IMG]
Decide whether each inequality statement is true or false. Be prepared to explain your reasoning.
\(\text-2 < 4\)
\(\text-2 < \text-7\)
\(4 > \text-7\)
\(\text-7 > 10\)
Drag each point to its proper place on the number line. Use your observations to help answer the questions that follow.
Andre says that
\(\frac14\)
is less than
\(\text{-}\frac {3}{4}\)
because, of the two numbers,
\(\frac14\)
is closer to 0. Do you agree? Explain your reasoning.
Answer each question. Be prepared to explain how you know.
Which number is greater:
\(\frac14\)
or
\(\frac54\)
?
Which number is farther from 0:
\(\frac14\)
or
\(\frac54\)
?
Which number is greater:
\(\text{-}\frac {3}{4}\)
or
\(\frac58\)
?
Which number is farther from 0:
\(\text{-}\frac {3}{4}\)
or
\(\frac58\)
?
Is the number that is farther from 0 always the greater number? Explain your reasoning.
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Print
Launch
Allow 10 minutes quiet work time followed by whole-class discussion.
Students using the digital materials can graph the points and check them with the applet. Marks at each half, quarter, and eighth of a unit can be shown to help plot the points or to self-check for accuracy.
Representation: Internalize Comprehension.
Activate or supply background knowledge about equivalent fractions. Some students may benefit from access to pre-prepared number lines to compare fractions for the final two problems.
Supports accessibility for: Conceptual processing; Organization
Student Facing
Plot the numbers -2, 4, -7, and 10 on the number line. Label each point with its numeric value.
[IMG]
Decide whether each inequality statement is true or false. Be prepared to explain your reasoning.
\(\text-2 < 4\)
\(\text-2 < \text-7\)
\(4 > \text-7\)
\(\text-7 > 10\)
Andre says that
\(\frac14\)
is less than
\(\text{-}\frac {3}{4}\)
because, of the two numbers,
\(\frac14\)
is closer to 0. Do you agree? Explain your reasoning.
Answer each question. Be prepared to explain how you know.
Which number is greater:
\(\frac14\)
or
\(\frac54\)
?
Which is farther from 0:
\(\frac14\)
or
\(\frac54\)
?
Which number is greater:
\(\text{-}\frac {3}{4}\)
or
\(\frac58\)
?
Which is farther from 0:
\(\text{-}\frac {3}{4}\)
or
\(\frac58\)
?
Is the number that is farther from 0 always the greater number? Explain your reasoning.
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 have difficulty comparing numbers on the negative side of the number line. Have students plot the numbers on a number line from least to greatest. It may be helpful to provide an example that students can use as a visual aid while they are working independently.
Some students may have difficulty comparing fractions in question 4. Remind them that comparing fractions is easier using a common denominator. Suggest that they subdivide the interval from 0 to 1 into fourths or eighths.
Activity Synthesis
The key takeaway from the discussion is that now that we have numbers on both sides of 0, distance from 0 isn’t enough to compare two numbers. Instead, we call numbers farther to the left on the number line “less” and numbers farther to the right “greater.” Display the following number line for all to see:
[IMG]
One inequality at a time, ask students to indicate whether they think each of the following is true or false:
\(\frac54 > \text{-}\frac {3}{2}\)
(true)
\(\frac54\)
is farther from 0 than
\(\text{-}\frac {3}{2}\)
(false)
\(\text{-}\frac {3}{2} < \text{-}\frac {3}{4}\)
(true)
\(\text{-}\frac {3}{2}\)
is farther from 0 than
\(\text{-}\frac {3}{4}\)
(true)
Invite students to share their reasoning. Here are some sentence frames that might be helpful:
“__ is greater (less) than __ because __.”
“__ is farther from 0 than __ because __.”
Speaking: MLR2 Collect and Display.
During the class discussion, record and display words and phrases that students use to explain why they decided certain inequality statements are true or false. Highlight phrases that include a reference to “to the right of,” “to the left of,” and a distance from zero. If students use gestures to support their reasoning, do your best to connect words to the gestures.
Design Principle(s): Optimize output (for explanation); Support sense-making


correct_option_index: 3