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 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.

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 ordering and absolute value of rational numbers.

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.
problem_activity: Activity
The purpose of this task is to understand that there are natural mathematical questions about certain contexts for which there are no answers if we restrict ourselves to positive numbers. The idea is to motivate the need for negative numbers and to see that there is a natural representation of them on the number line. This task is not about operations with signed numbers, but rather why we extend our number system beyond positive numbers. Students reason abstractly and quantitatively when they represent the change in temperature on a number line (MP2).
Digital
Launch
Display this image for all to see.
[IMG]
Tell students, “The thermometer showed a temperature of 7 degrees Celsius one morning. Later, the temperature increased 4 degrees. We can represent this change in temperature using a number line, as shown in the picture.”
Arrange students in groups of 2. Give students 2 minutes of quiet work time for question 1. Give students 2 minutes to compare their responses to their partners and to work on question 2.
Students using the digital version of the curriculum can explore the changes in temperature with the dynamic applet.
Reading: MLR6 Three Reads.
Use this routine to support students’ comprehension of the temperature changes for the first situation. In the first read, students read the situation with the goal of comprehending the text (e.g., the thermometer was at a certain temperature earlier in the day and then changed later in the day). Delay asking “What was the temperature late in the afternoon?”. In the second read, ask students to analyze the text to understand the quantities (e.g., at noon, the temperature was 7 degrees Celsius; it increases 6 degrees by late afternoon). In the third read, direct students’ attention to the question and ask students to brainstorm possible mathematical solution strategies to answer the question.
Design Principle(s): Support sense-making
Student Facing
1.  Here are three situations involving changes in temperature. Represent each change on the applet, and draw it on a number line. Then, answer the question.
At noon, the temperature was 5 degrees Celsius. By late afternoon, it has risen 6 degrees Celsius. What was the temperature late in the afternoon?
The temperature was 8 degrees Celsius at midnight. By dawn, it has dropped 12 degrees Celsius. What was the temperature at dawn?
Water freezes at 0 degrees Celsius, but the freezing temperature can be lowered by adding salt to the water. A student discovered that adding half a cup of salt to a gallon of water lowers its freezing temperature by 7 degrees Celsius. What is the freezing temperature of the gallon of salt water?
2.  Discuss with a partner:
How did you name the resulting temperature in each situation? Did both of you refer to each resulting temperature by the same name or different names?
What does it mean when the resulting temperature is above 0 on the number line? What does it mean when a temperature is below 0?
Do numbers below 0 make sense outside of the context of temperature? If you think so, give some examples to show how they make sense. If you don’t think so, give some examples to show otherwise.
Student Response
Teachers with a valid work email address can
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for free access to Student Response.
Print
Launch
Display this image for all to see.
[IMG]
Tell students, “The thermometer showed a temperature of 7 degrees Celsius one morning. Later, the temperature increased 4 degrees. We can represent this change in temperature using a number line, as shown in the picture.”
Arrange students in groups of 2. Give students 2 minutes of quiet work time for question 1. Give students 2 minutes to compare their responses to their partners and to work on question 2.
Students using the digital version of the curriculum can explore the changes in temperature with the dynamic applet.
Reading: MLR6 Three Reads.
Use this routine to support students’ comprehension of the temperature changes for the first situation. In the first read, students read the situation with the goal of comprehending the text (e.g., the thermometer was at a certain temperature earlier in the day and then changed later in the day). Delay asking “What was the temperature late in the afternoon?”. In the second read, ask students to analyze the text to understand the quantities (e.g., at noon, the temperature was 7 degrees Celsius; it increases 6 degrees by late afternoon). In the third read, direct students’ attention to the question and ask students to brainstorm possible mathematical solution strategies to answer the question.
Design Principle(s): Support sense-making
Student Facing
Here are three situations involving changes in temperature and three number lines. Represent each change on a number line. Then, answer the question.
At noon, the temperature was 5 degrees Celsius. By late afternoon, it has risen 6 degrees Celsius. What was the temperature late in the afternoon?
The temperature was 8 degrees Celsius at midnight. By dawn, it has dropped 12 degrees Celsius. What was the temperature at dawn?
Water freezes at 0 degrees Celsius, but the freezing temperature can be lowered by adding salt to the water. A student discovered that adding half a cup of salt to a gallon of water lowers its freezing temperature by 7 degrees Celsius. What is the freezing temperature of the gallon of salt water?
[IMG]
Discuss with a partner:
How did each of you name the resulting temperature in each situation?
What does it mean when the temperature is above 0? Below 0?
Do numbers less than 0 make sense in other contexts? Give some specific examples to show how they do or do not make sense.
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 representing change on the number line. Students sometimes count tick marks rather than counting the space between tick marks when working on a number line. For example, in the original problem image, the arrow on the number line represents a change of 4 degrees. Some students may begin at tick mark 7 and count the tick marks to yield a temperature change of 5 degrees. When reviewing that task with the whole class, be sure to make this important point and demonstrate counting on a number line by highlighting the space between the tick marks while counting out loud.
Activity Synthesis
Some students will use the phrase “degrees below zero.” Use this activity to introduce the term
negative
as a way to represent a quantity less than zero. In contrast, ask students how they would describe a quantity that is greater than zero on the number line. Some students will have a pre-existing understanding of positive and negative numbers. Discuss the use of + and - as symbols to denote positive and negative numbers. Notation will be important throughout the rest of this unit. Students should understand that +7 and 7 both represent positive 7. Negative 7 is represented as -7.
Representation: Develop Language and Symbols.
Create a display of important terms and vocabulary. Invite students to suggest language or diagrams to include that will support their understanding of: negative.
Supports accessibility for: Conceptual processing; Language; Memory


correct_option_index: 0