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: 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.)

Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers.

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.
problem_activity: Activity
The purpose of this activity is to use the new skills of multiplying and dividing rational numbers to represent and solve problems in a new context (MP4). In this activity students are using using multiplication and division of negatives and working with a proportional relationship with a negative constant of proportionality. Students should use what they know about proportional relationships to help them (for example; points lie on a straight line, line passes through zero, there is a point
\((1, k)\)
that lies on the line).
Digital
Launch
Remind students that we can model positions below the surface with negative values, so drilling 30 feet down is represented with -30 feet.
Ask students what they remember about proportional relationships. Examples:
Often represented with an equation in the form
\(y=kx\)
.
The constant of proportionality, often called
\(k\)
, is the change in
\(y\)
for a change by 1 in
\(x\)
.
A graph representing a proportional relationship is a line through (0,0) and
\((1,k)\)
Arrange students in groups of 3 during the discussion.
Representation: Internalize Comprehension.
Provide appropriate reading accommodations and supports to ensure students have access to written directions, word problems and other text-based content.
Supports accessibility for: Language; Conceptual processing
Student Facing
A water well drilling rig has dug to a height of -60 feet after one full day of continuous use.
Assuming the rig drilled at a constant rate, what was the height of the drill after 15 hours?
If the rig has been running constantly and is currently at a height of -147.5 feet, for how long has the rig been running?
Use the coordinate grid to show the drill’s progress.
At this rate, how many hours will it take until the drill reaches -250 feet?
[IMG]
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
Remind students that we can model positions below the surface with negative values, so drilling 30 feet down is represented with -30 feet.
Ask students what they remember about proportional relationships. Examples:
Often represented with an equation in the form
\(y=kx\)
.
The constant of proportionality, often called
\(k\)
, is the change in
\(y\)
for a change by 1 in
\(x\)
.
A graph representing a proportional relationship is a line through (0,0) and
\((1,k)\)
Arrange students in groups of 3 during the discussion.
Representation: Internalize Comprehension.
Provide appropriate reading accommodations and supports to ensure students have access to written directions, word problems and other text-based content.
Supports accessibility for: Language; Conceptual processing
Student Facing
A water well drilling rig has dug to a height of -60 feet after one full day of continuous use.
Assuming the rig drilled at a constant rate, what was the height of the drill after 15 hours?
If the rig has been running constantly and is currently at a height of -147.5 feet, for how long has the rig been running?
Use the coordinate grid to show the drill’s progress.
[IMG]
At this rate, how many hours will it take until the drill reaches -250 feet?
[IMG]
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 struggle to get the length of time out of the phrase "one full day of continuous use," wondering whether the drill was running at night. Let them know the drill has been going for 24 hours.
Activity Synthesis
Have students to share their solutions with each other in groups of three and work to come to agreement.
To wrap up, emphasize that because we can now complete any calculation with any rational number, we can extend constants of proportionality
\(k\)
to include negative values. Because the operations are now well-defined, we can answer questions using such a constant.
Speaking: MLR8 Discussion Supports.
Use this routine to support whole-class discussion. Invite select groups to share their solution and reasoning with the class. After each group presents, ask students to restate and/or revoice what they heard using mathematical language. Consider providing students time to restate what they heard to a partner, before selecting one or two students to share with the class. Ask the original group if this accurately reflects their thinking. This will provide more students with an opportunity to describe what they have learned about proportional relationships with a negative constant of proportionality.
Design Principle(s): Support sense-making


correct_option_index: 1