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: Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.

Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate.

Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

Understand ordering and absolute value of rational numbers.
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