Students will learn about negative numbers and how they can be considered an extension of the positive number line. Students will: 
- learn about the extended number line, which includes all integers. 
- use the number line to find both negative and positive numbers. 
- identify real-world scenarios that can be represented by negative numbers. 

- How can mathematics help to quantify, compare, depict, and model numbers? 
- How can mathematics help us communicate more effectively? 
- How are relationships represented mathematically?
- How are expressions, equations, and inequalities used to quantify, solve, model, and/or analyze mathematical problems?
- What makes a tool and/or strategy suitable for a certain task?

- Integer: A real number that does not include a fractional part.
- Negative Number: A number with a value less than zero.
- Positive Number: A number with a value greater than zero.

- bowl 
- copies of Number Card Sheet (M-6-1-1_Number Card Sheet) 
- two or more nondigital (analog) Celsius thermometers 
- copies of Match Game cards (M-6-1-1_Match Game) 
- copies of the Exit Ticket (M-6-1-1_Exit Ticket and KEY) 
- copy of Number Line Template (M-6-1-1_Number Line Template)—optional 
sticky notes for student use 
- piece of string—optional to show absolute value 
- decks of playing cards—optional for Extension activity 
- Absolute Value worksheet (M-6-1-1_Absolute Value and KEY)

- Use the number-line drawing activities (questions 1 and 2) on the exit ticket (M-6-1-1_Exit Ticket and KEY) to see if students understand the number line concepts. 
- Observing student interactions during activities, class discussions, and throughout the lesson can help assess student understanding. 

Scaffolding, Active Engagement, Modeling, and Explicit Instruction 
W: Introduce the concept of negative numbers and apply them to a number line. 
H: Use negative numbers to represent temperature. Because this is likely a familiar example for students, it should help to engage students in the learning process. 
E: Have students work in pairs to design a number line with only positive numbers and locate provided numbers on it. Make one of the numbers a negative value. 
R: Confirm with students that they couldn't locate any negative numbers on their number line. Extend the number line to −10 to locate the negative number. This activity could be used as a review of lesson concepts for students. 
E: Label the number line with students to ensure the numbers are listed in the proper order. 
T: The Extension section offers suggestions for adapting the lesson to the requirements of the students. The Routine section includes suggestions that can be used throughout the year to remind students of lesson concepts. The Small Group section includes activities for students who could benefit from more teaching or practice. The Expansion section provides a way to accommodate students who are ready to advance beyond the level of the standard. 
O: This lesson aims to teach students that negative numbers are a logical extension of the positive number line, representing measurements below a designated baseline. The lesson is structured so that it begins with a simple example and builds on that knowledge to gain a better understanding of what it means for a number to be negative. 

The purpose of this lesson is to help students realize that negative numbers are a logical extension of the positive number line, representing measures below a designated baseline. The activities are intended to teach students how to extend the number line and how to apply negative numbers in the real world. The lesson uses group activities and discussions to encourage students to explore and learn the concept. You act as a facilitator throughout the lesson. This way allows students to practice applying the concept as it is being taught.

Begin by asking, "What is the coldest outdoor temperature you can ever remember?" Allow a few students to respond; one or more will most likely answer with a negative number or a number less than zero. Make sure students understand that a number less than zero is considered negative number. "As this question shows, we cannot assume that positive numbers are always valid. Sometimes we have to use negative numbers instead. In today's lesson, we'll talk about what negative numbers are and how to deal with them.”

Example 1

For this activity, students should work in pairs. Ask students to draw a positive number line from zero to ten. If required, assist students in drawing the number line, making careful to add arrows at both ends. Pairs should work together to locate the following numbers on their number lines:

1
5
9
0
−1
Students should be able to find all numbers on their number line, except for −1. Ask, "Why could you not find −1 on this number line?" (Answers could include, "It wasn't on the number line.")

"Although −1 cannot be shown on your number line, it is still a valid number and should be represented somehow. If we want to represent -1 on this number line, how would you think we might do that?" Allow students time to respond or brainstorm ideas together. If necessary, explain that the arrows at the ends of a number line indicate that it extends indefinitely in both directions. This may allow students to investigate for themselves what the line to the left of zero may look like.

After students have expressed their ideas, demonstrate that −1 can be represented by extending the number line to the left of the zero to include negative numbers. Draw a number line on the board from -10 to +10, labeling only -10, -1, 0, 1, and 10. Ask students to explain how they can fill in the missing numbers on the right side of the number line. Model on the board when a student explains how to fill in the missing numbers between 1 and 10. Then explain to students, "The negative numbers, those numbers to the left of 0, follow the same pattern. To indicate that these numbers are negative, a negative sign is inserted in front of them." Explain the similar pattern of positive and negative numbers on the number line, and then demonstrate how to fill in the missing negative numbers. Point out the fact that numbers represent distance from zero, so numbers count up moving right and count back moving left. 

Prior to this part of the activity, write rational numbers on sticky notes. (Examples can include: −4, −2.5, 0, 1, \(1 {1 \over 2} \) , 4.) Use a range of rational numbers. Place a number line like the one below on the board, or use the number line template (M-6-1-1_Number Line Template) to project it on the board.

Remind students that a rational number might be a fraction, mixed number, or decimal that can be positive or negative. Divide the class into five groups and assign each group a sticky note with a rational number written on it. Encourage students to think-pair-share about where their group's number should be placed on the number line. Then, have a representative from each group come to the front of the class and arrange the sticky notes from lowest to highest value. Have the class discuss the sequence and make any necessary modifications. Then, have students correctly position the numbers on the number line. Discuss the placement of the sticky notes and confirm accuracy, clarifying any misconceptions. Repeat with various examples of rational numbers. Ask questions like those listed below.

"How do we know whether to place a number on a number line to the left or right of the 0?" 
"Why aren't -4 and 4 in the same place on the number line?" "How are they similar?" (They are the same distance from 0.) 
"How do we know where to put \(1 {1 \over 2} \)?" Does it go to the left or right of the number 1?" 
"Does anyone know what number we may use between \(1 {1 \over 2} \) and 2? If we made that number negative, would it be between -\(1 {1 \over 2} \) and -2? Explain." 
"What do we know about positive numbers that get farther to the right of 0?" 
"What do we know about negative numbers that get farther to the left of 0?" 

Activity 1: Left and Right

Use the student pairs from Example 1 to draw a number line from -3 to 3, counting by ones. Ensure that everyone gets the right idea and understands where the negative numbers are on the number line. When you are confident that students understand the concept, move on to Activity 2.

Activity 2: Locate It

For this activity, divide the class into two groups or have students collaborate with a partner. Explain that the class will play a few games with negative and positive numbers. For the first game, draw a number line going across the board, from -10 to 10, but label only −10, 0, and 10 on the line. (make graduated marks the remaining missing whole-number integers). For this activity, print and cut the Number Card Sheet (M-6-1-1_Number Card Sheet), then mix the cards in a bowl.

For this game, start with Team 1. Have a team member select a number card from the bowl and read it aloud. Remind students to use appropriate mathematical terms, such as negative eight instead of minus eight. The team members should discuss where that number should go on the line, and then send a representative to draw the number in the correct spot on the number line. If the team provides the correct response and plotted point, the play advances to Team 2. If Team 1's answer is incorrect, let them try again. Repeat the process until the team has plotted the number in the correct spot; then move on to Team 2. Repeat until all the numbers have been drawn. 

Alternative Activity 2 Idea: Keep score by giving a team one point for correctly plotting a number and -1 for incorrectly plotting. Each team should only get one chance to plot a number accurately. Return the cards with incorrectly plotted numbers to the bowl and draw again later. This will allow students to see a practical application for negative numbers. If you're using this activity with advanced students, place some fractions and decimals on the cards to make the game more difficult.

Example 2:

This example requires two (or more) nondigital (analog) Celsius thermometers with negative values. Pass the thermometers around the class and explain that the numbers on the thermometers are negative, but the number line moves up and down rather than left to right. Show the class where zero is on a thermometer, and label the negative and positive values on either side of zero. 

"This example demonstrates that negative numbers occur below the baseline of zero. In the case of Celsius temperatures, the baseline is the freezing point of water, which is 0 degree C. Colder temperatures than the baseline are referred to as negative. The temperatures continue to fall below the baseline as necessary."

Alternative Example 2 Idea: Go to www.weather.com and enter a region to check the weather in Barrow, Alaska, Ulan Bator, Mongolia, or another cold-weather area. Get the upcoming forecasts for these places and present them to the class. At least one of the forecasts is expected to include a negative temperature. Use this forecast to begin a discussion about what a negative temperature means. 

Activity 3: Below a Point 

For this activity, students will investigate the use of negative numbers below a baseline, as well as what the baselines represent in certain common situations. Demonstrate with the following example (which is related back to Example 2).

"When it comes to temperature, 0 degrees Celsius represents the freezing point of water, so negative temperatures on the Celsius scale mean it is colder than the freezing point of water, and positive temperatures mean it is warmer than the freezing point of water." 

Divide the students into groups of three or four. Give each group one of the following words and ask students to explain what the value of zero means in regard to this word, as well as what negative and positive numbers represent. The following concepts should be discussed by the groups. Examples of answers are provided, but student responses will vary.

A Football Play: Zero represents no gain on a play; a positive number represents gaining yards on a play; and a negative number represents losing yards on a play. 

A Bank Account: Zero represents there is no money in the account; a positive number suggests there is money in the account; and a negative number means the account holder owes the bank money (because s/he likely spent more money than s/he had). 

Sea Level: Zero represents the sea's elevation or baseline; a positive number represents the distance above the sea's elevation; and a negative number shows the distance below the sea's elevation.

After each group has had a chance to discuss all three concepts, invite a few groups to share their responses. If other groups have additional ideas that were not mentioned by the first group, invite them to share them as well. If time permits, ask each group to provide more instances of situations in which negative numbers are used. (Other possible answers include game points lost, account debits/credits, and so on). 

By reviewing the Match Game cards (M-6-1-1_Match Game), students can reinforce their understanding of how positive and negative numbers are used in real life. Students should work in pairs to cut apart the premade cards. To begin play, students turn the cards face down. Each player takes their turn by turning over two cards. If the two cards match (the real-world situation and the number representing that situation), the player keeps the cards and continues playing. If the two cards do not match, the player returns the cards to play face down in their original locations. The following player then takes their turn. The player with the most cards at the end of the game wins. Students can use the Blank Card Template for Match Game to create extra cards with a similar format (see page 2 of M-6-1-1_Match Game). 

After all of the activities and examples have been completed, give students a copy of the exit ticket (M-6-1-1_Exit Ticket and KEY), and have them solve the problems. If students run out of time, give this as homework for the day.

[Alternative Lesson Idea: This lesson can also be done in the order listed below: Example 2, Activity 3, Example 1, Activity 1, Activity 2, Exit Ticket worksheet. This alternative order may be more effective with a class that would benefit from seeing practical applications of mathematics earlier in the lessons. Furthermore, this option is better if time is limited, as Activity 2 can be shortened by using fewer numbers (−5 to 5 instead).]

Extension: 

Routine: During the school year, ask students to recall examples where they saw a negative number used in the school newspaper, the local newspaper, or magazines. Consider having students cut out and place examples in a specific location. Negative numbers are all around, and students will understand this more easily when they see them used in everyday situations.

Expansion: Option 1 – Negative Fractions and Decimals: Give students who demonstrate proficiency the option to plot –\(1 \over 2\) on the number line. This will help them understand how –\(1 \over 2\) relates to the sequence and allow them to see the number line as one continuous group of numbers. If desired, expand this idea by providing the class with additional fractions and decimals to put on the line. 

Option 2 – Ordering Rational Numbers: Using a deck of playing cards, use the number cards (2-10) to represent the red suit (hearts) and black suit (clubs). The black cards indicate positive numbers. Red cards indicate negative numbers. Students may work in pairs or in triads. Each participant is handed three to five cards, depending on their level of proficiency. Students then sort the cards by value, from least to greatest value. Students check each other's work. If students are correct, they receive one point for each card given to them. Students keep note of how many points they earned in each round. The cards are returned to the pile and shuffled. Play continues until a player reaches 50 points. 

A set of playing cards dealt to a student: 4♥, 4♣, 1♥, 9♣, 10♥. If ordering the cards from smallest to largest, the right order is 10♥, 4♥, 1♥, 4♣, 9♣. These cards represent the values (−10), (−4), (−1), 4, 9.

Option 3 – Absolute Value: Explain to students that a number's absolute value is its distance from zero on a number line. When we consider measuring this distance using a piece of string, students can see that the string represents the distance regardless of whether it spans above or below zero. Model a few examples to demonstrate this concept. 

Holding a string from 0 to 4 on a number line reveals the absolute value of (4). The distance is four units. Show how to express the absolute value of (4) symbolically: |4| = 4.
Holding a string from -4 to 0 on a number line will show the absolute value of (−4). The distance is four units. Explain to students that this distance is equivalent to the absolute value of (4). The distance is represented by a string, which cannot be negative. Show how to write the absolute value of -4 symbolically: |-4| = 4. 
Provide a few more examples, emphasizing to students that absolute value is the distance a number is from zero. Then, describe the following absolute value rule: When adding numbers with different signs, calculate the difference of the absolute values of the numbers. The sum has the sign of the number with the greater absolute value. Model a few examples to show this concept.

Example 1: 9 + (−4) = ___; |9| − |−4| = ___. The absolute value of 9 minus the absolute value of (−4) equals ___. 9 − 4 = 5. The number with the greater absolute value is 9, which is positive. So, the answer is positive. 
Example 2: (−15) + 2 = ___; |−15| − |2| = ___. The absolute value of (−15) minus the absolute value of 2 is ___; 15 − 2 = 13. The difference between these two numbers is 13. The number with the greater absolute value is 15, which is negative. So the answer is negative, −13. 
Students can next fill out the Absolute Value worksheet (M-6-1-1_Absolute Value and KEY).