standard_description: Multiply a vector (regarded as a matrix with one column) by a matrix of suitable dimensions to produce another vector. Work with matrices as transformations of vectors.
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
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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


aligns: no