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.


Your response:
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.


Your response:
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.


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

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 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 see how to tell, from the equation, whether the sum will be positive, negative, or 0, without having to draw a number line diagram every time.
In this activity, students return to the context of height and depth to continue making sense of adding signed numbers. They use the structure of number line diagrams (MP7) to examine a variety of situations, which include starting in the positives, starting in the negatives, moving away from 0, moving towards 0, crossing over 0, and ending exactly on 0. From this variety of situations, students see that the lengths of the arrows in the number line diagrams give important information about the situation. This opens up the discussion of comparing the magnitude of the addends. When the addends have opposite signs, the longer arrow (the number with the larger magnitude) determines the sign of the sum.
\(400 + (\text-150) = 250\)
and
\(\text-200 + 150 = \text-50\)
.
This activity purposefully gives many problems to figure out, but only asks students to draw a diagram for 3 rows. Watch for students who think they need to draw a number line for every problem. Encourage them to look for and generalize from regularity in the problems (MP8) so that they don't need to always draw a diagram.
For students using the digital version of the materials, be sure they do not use the applet to represent and answer every question and bypass good thinking. Encourage students to figure out an answer for each question first, and then using the applet to check their work.
In preparation for dealing with subtraction of signed numbers, some of the problems involve determining how much the temperature changed, given the initial and final temperatures.
Digital
Launch
Explain that a
mountaineer
is someone who climbs mountains, and a
spelunker
is someone who explores caves and caverns. Arrange students in groups of 2. Give students 3 minute of quiet work time followed by partner discussion after the first question. Then have students work on the second question and follow with whole-class discussion.
Representation: Internalize Comprehension.
Represent the same information through different modalities by using physical objects to represent abstract concepts. Create an interactive display that allows students to experience physically moving an object (mountaineer) up and down a number line. Highlight connections between the act of physically moving the object up and down with arrows on a number line diagram.
Supports accessibility for: Conceptual processing; Visual-spatial processing
Student Facing
Explore the applet and then answer the questions.
A mountaineer is climbing on a cliff. She is 200 feet above the ground. If she climbs up, this will be a positive change. If she climbs down, this will be a negative change.
[IMG]
Complete the table.
|   | starting elevation                                   (feet) | change                                   (feet) | final elevation                                   (feet) |
|---|-------------------------------------------------------------|-------------------------------------------------|----------------------------------------------------------|
| A |                                                             |                                                 |                                                          |
| B |                                                             |                                                 |                                                          |
| C |                                                             |                                                 |                                                          |
| D |                                                             |                                                 |                                                          |
Write an addition equation and draw a number line diagram for B. Include the starting elevation, change, and final elevation in your diagram.
A spelunker is down in a cave next to the cliff. If she climbs down deeper into the cave, this will be a negative change. If she climbs up, whether inside the cave or out of the cave and up the cliff, this will be a positive change.
Complete the table.
|   | starting elevation                                   (feet) | change                                   (feet) | final elevation                                   (feet) |
|---|-------------------------------------------------------------|-------------------------------------------------|----------------------------------------------------------|
| A |                                                             |                                                 |                                                          |
| B |                                                             |                                                 |                                                          |
| C |                                                             |                                                 |                                                          |
| D |                                                             |                                                 |                                                          |
| E |                                                             |                                                 |                                                          |
Write an addition equation and draw a number line diagram for C and D. Include the starting elevation, change, and final elevation in your diagram.
​What does the expression
\(\text-45 + 60\)
tell us about the spelunker? What does the value of the expression tell us?
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
Explain that a
mountaineer
is someone who climbs mountains, and a
spelunker
is someone who explores caves and caverns. Arrange students in groups of 2. Give students 3 minute of quiet work time followed by partner discussion after the first question. Then have students work on the second question and follow with whole-class discussion.
Representation: Internalize Comprehension.
Represent the same information through different modalities by using physical objects to represent abstract concepts. Create an interactive display that allows students to experience physically moving an object (mountaineer) up and down a number line. Highlight connections between the act of physically moving the object up and down with arrows on a number line diagram.
Supports accessibility for: Conceptual processing; Visual-spatial processing
Student Facing
A mountaineer is climbing on a cliff. She is 400 feet above the ground. If she climbs up, this will be a positive change. If she climbs down, this will be a negative change.
Complete the table.
|   | starting                                         elevation                                         (feet) | change                                         (feet) | final                                         elevation                                         (feet) |
|---|-----------------------------------------------------------------------------------------------------------|-------------------------------------------------------|--------------------------------------------------------------------------------------------------------|
| A |                                                                                                           |                                                       |                                                                                                        |
| B |                                                                                                           |                                                       |                                                                                                        |
| C |                                                                                                           |                                                       |                                                                                                        |
| D |                                                                                                           |                                                       |                                                                                                        |
[IMG]
Write an addition equation and draw a number line diagram for B. Include the starting elevation, change, and final elevation in your diagram.
[IMG]
​
A spelunker is down in a cave next to the cliff. If she climbs down deeper into the cave, this will be a negative change. If she climbs up, whether inside the cave or out of the cave and up the cliff, this will be a positive change.
Complete the table.
|   | starting elevation                                   (feet) | change                                   (feet) | final elevation                                   (feet) |
|---|-------------------------------------------------------------|-------------------------------------------------|----------------------------------------------------------|
| A |                                                             |                                                 |                                                          |
| B |                                                             |                                                 |                                                          |
| C |                                                             |                                                 |                                                          |
| D |                                                             |                                                 |                                                          |
| E |                                                             |                                                 |                                                          |
Write an addition equation and draw a number line diagram for C and D. Include the starting elevation, change, and final elevation in your diagram.
[IMG]
​
[IMG]
​
What does the expression
\(\text-75 + 100\)
tell us about the spelunker? What does the value of the expression tell us?
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 with working backwards to fill in the change when given the final elevation. Ask them "What do you do to 400 to get to 50?" for example.
Activity Synthesis
The most important thing for students to get out of this activity is how to tell from the equation whether the sum will be positive, negative, or 0, without having to draw a number line diagram every time.
Write all of the equations where the two addends have the same sign. The ask, “What happens when the two addends have the same sign?”
Write all of the equations where the two addends have the opposite sign. The ask, “What happens when the two addends have opposite signs and . . .
the number with the larger magnitude is positive?”
the number with the larger magnitude is negative?”
Ask, “How can you tell when the sum will be zero?”
Speaking, Listening, Writing: MLR1 Stronger and Clearer Each Time.
To provide students with an opportunity to generalize about the sum of two addends, ask students to draft an initial response to the question “How can you tell from the equation whether the sum will be positive, negative, or 0?”. Ask students to meet with 2–3 partners for feedback. Provide students with prompts for feedback that will help each other strengthen their ideas and clarify their language (e.g., “What if the signs of both numbers are the same/different?” or “How do you know if the sum will be positive/negative?”, etc.). Students can borrow ideas and language from each partner to refine their explanation. This will help students to use mathematical language to generalize about the sums of rational numbers.
Design Principle(s): Optimize output (for generalization)


correct_option_index: 2