• Use the number-line drawing activities (questions 1 and 2) on the exit ticket (M-6-1-1_Exit Ticket and KEY.docx) to determine if students have mastered the number line concepts.
• Observation of student interactions during activities, class discussion, and throughout the lesson may be used to assess student understanding.

The goal of this lesson is for students to understand that negative numbers are a logical extension of the positive number line, and they represent measurements below a designated baseline. The activities are designed to show students how to extend the number line and how negative numbers are used in the real world. The lesson uses group activities and discussions to allow students to explore and learn the material. You act as facilitator throughout the lesson. This way, students can practice applying the material while it is being presented.

Begin by asking, “What is the coldest outdoor temperature you can ever remember?” Let a few students answer; one or more students will likely give a negative number answer or state the answer as a number below zero. Make sure students understand that a number below zero can be referred to as a negative number. “As this question shows, we cannot assume that positive numbers will always work. Sometimes we need to use negative numbers instead. In today’s lesson we will discuss what negative numbers represent and learn more about how to work with them.”

Example 1

For this activity, have students work in pairs. Ask students to draw a positive number line from zero to ten. If necessary, assist students in drawing the number line, particularly making sure they include arrows at either end. Have pairs work together to locate the following numbers on their number lines:

• 1
• 5
• 9
• 0
• −1

Students should be able to locate all the numbers on their number line except for −1. Ask, “Why could you not find −1 on this number line?” (Answers might include “It wasn’t on the number line.”)

“Despite the fact that you could not plot −1 on your number line, it is a valid number, so there should be some way to represent it. If we want to represent −1 on this number line, how do you think we might do that?” Allow time for students to respond or to brainstorm ideas together. If needed, provide a hint that the arrows on the ends of a number line show that it extends forever in both directions. This may help students explore for themselves what the line left of zero might look like.

After students have had time to express their ideas, show them that −1 can be represented by extending the number line to the left of the zero to include negative numbers. On the board, draw a number line extending from −10 to +10 but only label −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 as 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 show that these numbers are negative, a negative sign is placed in front of the numbers.” While explaining the similar pattern of positive and negative numbers on the number line, model how to fill in the missing negative numbers. Point out the fact that numbers indicate distance from zero, so numbers count up moving right and count back moving left.

Prior to this part of the activity, record rational numbers on sticky notes. (Examples can include: −4, −2.5, 0, 1, , 4.) Use a variety of rational numbers. Put a number line similar to the one below on the board or use the number line template (M-6-1-1_Number Line Template.docx) and project it on the board.

Remind students that a rational number is a fraction, mixed number, or decimal that can be either positive or negative. Divide the class into five groups and then give each group a sticky note with a rational number written on it. Have students think-pair-share about where the 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 order the sticky notes from least to greatest values. Have the class discuss order and make necessary changes. Then have students place the numbers correctly on the number line. Discuss the placement of the sticky notes and confirm accuracy, clarifying any misconceptions. Repeat using different examples of rational numbers. Ask questions similar to 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 place ? Does it go to the left or to the right of the number 1?”
• “Does anyone know a number we can put between  and 2? If we made that number negative would it be between 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

Using the student pairs from Example 1, have students draw a number line from −3 to 3, counting by ones. Make sure everyone has the right idea and knows where the negative numbers belong on the number line. Once you are satisfied that students have mastered the concept, proceed to Activity 2.

Activity 2: Locate It

For this activity, divide the class into two groups or have students work with a partner. Explain that the class will be playing a couple of games involving negative and positive numbers. For the first game, draw a number line going across the board, from −10 to 10, but only label −10, 0, and 10 on the line (make graduated marks for the other missing whole-number integers). You will need to print and cut up the Number Card Sheet (M-6-1-1_Number Card Sheet.docx) and mix up the cards in a bowl for this activity.

For this game, begin with Team 1. Have a team member select a number card from the bowl and read aloud the number. Remind students to use appropriate mathematical terms, such as negative eight rather than minus eight. The team members should discuss among themselves where that number should go on the line and then send a representative to draw the number at the appropriate spot on the number line. If the team supplies the correct answer and plotted point, the play moves to Team 2. If Team 1’s answer is incorrect, let the team try again. Repeat the process until the team plots the number in the correct spot; then go 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 plotting a number correctly and −1 for plotting a number incorrectly. Each team should only have one chance to plot a number correctly. Return the cards with numbers that were plotted incorrectly to the bowl to be drawn again later. This will give the class a practical demonstration of an application for negative numbers. If you are using this activity for proficient students, include some fractions and decimals in the cards to make the game more challenging.

Example 2

This example requires two (or more) nondigital (analog) Celsius thermometers capable of negative readings. Pass the thermometers around the class and explain that these thermometers include negative numbers, but instead of the number line going from left to right, it goes up and down. Show the class where zero is on a thermometer and indicate the negative and positive numbers on both sides of zero.

“As you can see from this example, negative numbers exist below the baseline of zero. In the case of Celsius temperatures, the baseline is the freezing point of water, which is 0 degrees C. Temperatures that are colder than the baseline are said to be negative. The temperatures continue to decrease below the baseline as necessary.”

Alternative Example 2 Idea: Visit www.weather.com and enter a region into the window to check the weather for Barrow, Alaska; Ulan Bator, Mongolia; or some other area that tends to be cold. Get the upcoming forecasts for these regions and show them to the class. It is likely that there will be at least one negative temperature in one of the forecasts. Use this forecast to begin a discussion about what a negative temperature represents.

Activity 3: Below a Point

For this activity, the class will explore the use of negative numbers below a baseline and what the baselines represent for some common situations. Demonstrate with the following example (which relates 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 class into groups of three or four. Give each group one of the words listed below and ask students to state what the value of zero means in relation to this word, and what negative and positive numbers represent in relation to this word. Concepts to have the groups discuss are listed below. Examples of answers are provided, but student answers 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 no money in the account; a positive number means 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 elevation of the sea or the baseline; a positive number represents the distance above the elevation of the sea; and a negative number represents the distance below the elevation of the sea.

After each group has had time to discuss all three concepts, ask a few groups to share their answers. If other groups have different ideas not mentioned by the first group, invite them to share their ideas as well. If some time remains, invite each group to give additional examples of situations in which negative numbers are used. (Possible other answers may include points lost in a game, debits/credits from an account, etc.).

Reinforce an understanding of real-world application of positive and negative numbers by having students review the Match Game cards (M-6-1-1_Match Game.docx). Working in pairs, have students cut apart the pre-made cards. To begin play, students turn the cards face down. Each player takes a turn by turning over two cards. If the two cards match (the real-world situation and the number that represents that situation) that player keeps the cards and continues play. If the two cards do not match, that player puts the cards back into play face down in their previous locations. Then it is the next player’s turn. The player with the most cards at the end of the game is the winner. Students can make additional cards following a similar format using the Blank Card Template for Match Game (see page 2 of M-6-1-1_Match Game.docx).

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

[Alternative Lesson Idea: It is also possible to do this lesson in the following order: Example 2, Activity 3, Example 1, Activity 1, Activity 2, Exit Ticket worksheet. This alternative order may work better with a class that would benefit from seeing the practical uses of mathematics earlier in the lessons. Also, this alternative works 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 mention when they’ve seen a negative number used in the school newspaper, the local newspaper, or in magazines. Consider having students cut out examples and post them in a particular location. Negative numbers are all around and students will understand that more easily the more they see them used in daily life.
• Small Groups: Pull students who could benefit from additional instruction into a small group or groups. Have one student in each group draw a number line labeling zero using a whole sheet of paper. Write down a few rational numbers on sticky notes and have the group work together to decide where it goes on the line. The group may choose additional numbers or you can give additional numbers. Use this as an opportunity to further explain the use of number lines and how negative numbers fit into the scheme of things.
• Expansion: Option 1—Negative Fractions and Decimals: For students who demonstrate proficiency, present the opportunity to plot  on the number line. This will help them think about how  relates to the sequence and will allow them to see the number line as one continuous group of numbers. If desired, expand this idea by presenting the class with more fractions and decimals to plot on the line.

Option 2—Ordering Rational Numbers: Using a deck of playing cards, use the number cards (2–10) for the red suit (hearts) and black suit (clubs). The black cards represent positive numbers. The red cards represent negative numbers. Students can work in pairs or in triads. Each player is dealt three to five cards, depending on the proficiency. Then students order the cards from least to greatest values. Students check one another’s work. If students are correct, they get a point for each card dealt to them. Students keep track of how many points they earn for each round. Cards are returned to the pile and reshuffled. Play continues until a player earns 50 points.

• Sample set of playing cards dealt to a student: 4♥, 4♣, 1♥, 9♣, 10♥. If ordering the cards from smallest to the largest the correct order would be: 10♥, 4♥, 1♥, 4♣, 9♣. The values represented by these cards are (−10), (−4), (−1), 4, 9.

Option 3—Absolute Value: Explain to students that the absolute value of a number is the distance the number is from zero on a number line. If we think of measuring this distance with a piece of string, students can visually see that the piece of string is the distance regardless of whether it spans above or below zero. Model a few examples to show this concept.

• The absolute value of (4) can be shown by holding a piece of string from 0 to 4 on a number line. This distance is four units. Show how to write the absolute value of (4) symbolically: |4| = 4.
• The absolute value of (−4) can be shown by holding a piece of string from –4 to 0 on a number line. This distance is four units. Point out to students that this distance is equivalent to the absolute value of (4). The distance is represented by the string and cannot be negative. Show how to write the absolute value of (−4) symbolically: |−4| = 4.

Demonstrate a few other examples, emphasizing to students that absolute value is how far a number is from zero. Then explain the following absolute value rule: When adding numbers with different signs, find 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| = ___. This reads the absolute value of 9 minus the absolute value of (−4) equals ___; 9 – 4 = 5. The number with the greater absolute value is 9, and it is positive. So the answer is positive.
• Example 2: (−15) + 2 = ___. |−15| − |2| = ___. This reads the absolute value of (−15) minus the absolute value of 2 equals ___; 15 − 2 = 13. The difference between the two numbers is 13. The number with the greater absolute value is 15, and it is negative. So the answer is negative, −13.

Students then can complete the Absolute Value worksheet (M-6-1-1_Absolute Value and KEY.docx).