Eureka Lesson for 7 th Grade Unit ONE Zero Pairs & Using Number Lines to Add Integers These 2 lessons can be taught in 2 class periods – Or 3 with struggling learners. Challenges: We (middle school teachers) are more comfortable teaching rules, but allowing students to see adding integers work will lead to the rule and they’ll remember it because they’ll know WHY. These 2 lessons are well laid out with step-by-step instructions to make it easier. Please familiarize yourselves with the lesson before using it. Page 2 Overview & 3 Minute Collaborative Pairs Review Pages 3-7 Lesson 1 Teachers’ Detailed Instructions Pages 8-11 Exit Ticket w/ solutions for Lesson 1 Pages 12-15 Student pages for Lesson 1 Pages 16-20 Lesson 2 Teachers’ Detailed Instructions Pages 21-25 Exit Ticket w/ solutions for Lesson 2 Pages 26-31 Student pages for Lesson 2
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Eureka Lesson for 7th Grade Unit ONE
Zero Pairs & Using Number Lines to Add Integers
These 2 lessons can be taught in 2 class periods – Or 3 with struggling learners.
Challenges: We (middle school teachers) are more comfortable teaching rules, but allowing students to
see adding integers work will lead to the rule and they’ll remember it because they’ll know
WHY. These 2 lessons are well laid out with step-by-step instructions to make it
easier. Please familiarize yourselves with the lesson before using it.
Lesson 1: Opposite Quantities Combine to Make Zero
Student Outcomes
Students add positive integers by counting up and negative integers by counting down (using curved arrows on the number line).
Students play the Integer Game to combine integers, justifying that an integer plus its opposite add to zero.
Students know the opposite of a number is called the additive inverse because the sum of the two numbers is zero.
Lesson Notes There is some required preparation from teachers before the lesson begins. It is suggested that number lines are provided for students. However, it is best if students can reuse these number lines by having them laminated and using white board markers. Also, the Integer Game is used during this lesson, so the teacher should prepare the necessary cards for the game.
Classwork
Exercise 1 (3 minutes): Positive and Negative Numbers Review
In pairs, students will discuss “What I Know” about positive and negative integers to access prior knowledge. Have them record and organize their ideas in the graphic organizer in the student materials. At the end of discussion, the teacher will choose a few pairs to share out with the class.
Exercise 1: Positive and Negative Numbers Review
With your partner, use the graphic organizer below to record what you know about positive and negative numbers. Add or remove statements during the whole class discussion.
Negative Numbers Positive Numbers
They are to the left of 𝟎𝟎 on a number line and get smaller going to the left.
They can mean a loss, drop, decrease, or below sea level.
They look like −𝟕𝟕,−𝟖𝟖.
They are opposites of positive numbers.
They are to the right of 𝟎𝟎 on a number line and get larger going to the right.
They can mean a gain, increase, or above sea level.
They don’t have a sign.
They are opposites of negative numbers.
Both
are
on
a nu
mbe
r lin
e.
Scaffolding:
Laminate (or place in sheet protectors) 1-page of number lines (vary blank and numbered) for individual use with white board markers.
Create a number line on the floor using painters tape to model the “counting on” principle.
Provide a wall model of the number line at the front of the room for visual reinforcement.
Lesson 1: Opposite Quantities Combine to Make Zero
Example 1 (5 minutes): Introduction to the Integer Game
Read the Integer Game outline before the lesson. The teacher selects a group of 3 or 4 students to demonstrate to the whole class how to play the Integer Game.1 The game will be played later in the lesson. The teacher should stress that the object of the game is to get a score of zero.
Example 2 (5 minutes): Counting Up and Counting Down on the Number Line
Model a few examples of counting with small curved arrows to locate numbers on the number line, where counting up corresponds to positive numbers and counting down corresponds to negative numbers.
Example 2: Counting Up and Counting Down on the Number Line
Use the number line below to practice counting up and counting down.
Counting up corresponds to positive numbers.
Counting down corresponds to negative numbers.
a. Where do you begin when locating a number on the number line?
Start at 𝟎𝟎.
b. What do you call the distance between a number and 𝟎𝟎 on a number line?
The absolute value.
c. What is the relationship between 𝟕𝟕 and −𝟕𝟕?
Answers will vary. 𝟕𝟕 and −𝟕𝟕 both have the same absolute values. They are both the same distance from zero, 𝟎𝟎, but in opposite directions; therefore, 𝟕𝟕 and −𝟕𝟕 are opposites.
Example 3 (5 minutes): Using the Integer Game and the Number Line
The teacher leads the whole class using a number line to model the concept of counting on (addition) to calculate the value of a hand when playing the Integer Game. The hand’s value is the sum of the card values.
1 Refer to the Integer Game outline for player rules.
A positive 𝟕𝟕 is 𝟕𝟕 units to the right of 𝟎𝟎. |𝟕𝟕| = 𝟕𝟕 A negative 𝟕𝟕 is 𝟕𝟕 units to the left of 𝟎𝟎. |−𝟕𝟕| = 𝟕𝟕
Lesson 1: Opposite Quantities Combine to Make Zero
Start at 0 and end up at positive 5. This is the first card drawn, so the value of the hand is 5.
Second Card: −5
Start at 5, the value of the hand after the first card; move 5 units to the left to end at 0.
Third Card: −4
Start at 0, the value of the hand after the second card; move 4 units to the left.
8
Fourth Card: 8
Start at −4, the value of the hand after the third card; move 8 units to the right.
What is the final position on the number line?
The final position on the number line is 4.
What card or combination of cards would you need to get back to 0?
In order to get a score of 0, I would need to count down 4 units. This means, I would need to draw a −4 card or a combination of cards whose sum is −4, such as −1 and −3.
We can use smaller, curved arrows to show the number of “hops” or “jumps” that correspond to each integer. Or, we can use larger, curved arrows to show the length of the “hop” or “jump” that corresponds to the distance between the tail and the tip of the arrow along the number line. Either way, the final position is 4 units to the right of zero. Playing the Integer Game will prepare students for integer addition using arrows (vectors) in Lesson 2.
Card 1: Count up 5
Card 2: Count down 5 Card 3: Count down 4
Card 4: Count up 8
Move 4 units to the left to get back to 0.
Lesson 1: Opposite Quantities Combine to Make Zero
Example 3: Using the Integer Game and the Number Line
What is the sum of the card values shown? Use the counting on method on the provided number line to justify your answer.
𝟒𝟒
a. What is the final position on the number line? 𝟒𝟒
b. What card or combination of cards would you need to get back to 𝟎𝟎? −𝟒𝟒 or −𝟏𝟏 and −𝟑𝟑
Exercise 2 (5 minutes): The Additive Inverse
Before students begin, the teacher highlights part of the previous example where starting at zero and counting up five units and then back down five units brings us back to zero. This is because 5 and −5 are opposites. Students work independently to answer the questions. At the end of the exercise questions, formalize the definition of additive inverse.
Exercise 2: The Additive Inverse
Use the number line to answer each of the following questions.
a. How far is 𝟕𝟕 from 𝟎𝟎 and in which direction? 𝟕𝟕 units to the right
b. What is the opposite of 𝟕𝟕? −𝟕𝟕
c. How far is −𝟕𝟕 from 𝟎𝟎 and in which direction? 𝟕𝟕 units to the left
d. Thinking back to our previous work, how would you use the counting on method to represent the following: While playing the Integer Game, the first card selected is 𝟕𝟕, and the second card selected is −𝟕𝟕.
I would start at 𝟎𝟎 and count up 𝟕𝟕 by moving to the right. Then, I would start counting back down from 𝟕𝟕 to 𝟎𝟎.
e. What does this tell us about the sum of 𝟕𝟕 and its opposite, −𝟕𝟕?
The sum of 𝟕𝟕 and −𝟕𝟕 equals 𝟎𝟎.𝟕𝟕 + (−𝟕𝟕) = 𝟎𝟎.
𝟓𝟓 −𝟓𝟓 −𝟒𝟒 𝟖𝟖
Lesson 1: Opposite Quantities Combine to Make Zero
For all numbers 𝒂𝒂 there is a number −𝒂𝒂, such that 𝒂𝒂 + (−𝒂𝒂) = 𝟎𝟎.
The additive inverse of a real number is the opposite of that number on the real number line. For example, the opposite of −𝟑𝟑 is 𝟑𝟑. A number and its additive inverse have a sum of 0. The sum of any number and its opposite is equal to zero.
f. Look at the curved arrows you drew for 𝟕𝟕 and −𝟕𝟕. What relationship exists between these two arrows that would support your claim about the sum of 𝟕𝟕 and −𝟕𝟕?
The arrows are both the same distance from 𝟎𝟎. They are just pointing in opposite directions.
g. Do you think this will hold true for the sum of any number and its opposite? Why?
I think this will be true for the sum of any number and its opposite because when you start at 𝟎𝟎 on the number line and move in one direction, moving in the opposite direction the same number of times will always take you back to zero.
Example 4 (5 minutes): Modeling with Real-World Examples
The purpose of this example is to introduce real-world applications of opposite quantities to make zero. The teacher holds up an Integer Game card, for example −10, to the class and models how to write a story problem.
How would the value of this card represent a temperature?
−10 could mean 10 degrees below zero.
How would the temperature need to change in order to get back to 0 degrees?
Temperature needs to rise 10 degrees.
With a partner, write a story problem using money that represents the expression 200 + (−200).
Answers will vary. Timothy earned $200 last week. He spent it on a new video game console. How much money does he have left over?
Students share their responses to the last question with the class.
Exercise 3 (10 minutes): Playing the Integer Game Exercise 3: Playing the Integer Game
Play the Integer Game with your group. Use a number line to practice counting on.
Students will play the Integer Game in groups. Students will practice counting using their number lines. Let students explore how they will model addition on the number line. Monitor student understanding by ensuring that the direction of the arrows appropriately represents positive or negative integers.
MP.6 &
MP.7
Lesson 1: Opposite Quantities Combine to Make Zero
Students will discuss the following questions in their groups to summarize the lesson.
How do you model addition using a number line? When adding a positive number on a number line, you count up by moving to the right. When adding a
negative number on a number line, you count down by moving to the left.
Using a number line, how could you find the sum of (−5) + 6?
Start at zero, then count down or move to the left five. From this point, count up or move to the right six.
Peter says he found the sum by thinking of it as (−5) + 5 + 1. Is this an appropriate strategy? Why do you think Peter did this?
Peter did use an appropriate strategy to determine the sum of (−5) + 6. Peter did this because 5 and −5 are additive inverses, so they have a sum of zero. This made it easier to determine the sum to be one.
Why is the opposite of a number also called the additive inverse? What is the sum of a number and its opposite?
The opposite of a number is called the additive inverse because the two numbers’ sum is zero.
Exit Ticket (5 minutes)
Lesson Summary
Add positive integers by counting up, and add negative integers by counting down. An integer plus its opposite sum to zero. The opposite of a number is called the additive inverse because the two numbers’ sum is zero.
Lesson 1: Opposite Quantities Combine to Make Zero
Lesson 1: Opposite Quantities Combine to Make Zero
Exit Ticket 1. Your hand starts with the 7 card. Find three different pairs that would complete your hand and result in a value of
zero.
2. Write an equation to model the sum of the situation below.
A hydrogen atom has a zero charge because it has one negatively charged electron and one positively charged proton.
3. Write an equation for each diagram below. How are these equations alike? How are they different? What is it about the diagrams that lead to these similarities and differences?
Diagram A:
Diagram B:
7
7
7
Lesson 1: Opposite Quantities Combine to Make Zero
1. Your hand starts with the 𝟕𝟕 cards. Find three different pairs that would complete your hand and result in a value of zero.
Answers will vary. (−𝟑𝟑 and −𝟒𝟒), (−𝟓𝟓 and −𝟐𝟐), (−𝟏𝟏𝟎𝟎 and 𝟑𝟑)
2. Write an equation to model the sum of the situation below.
A hydrogen atom has a zero charge because it has one negatively charged electron and one positively charged proton.
(−𝟏𝟏) + 𝟏𝟏 = 𝟎𝟎 or 𝟏𝟏 + (−𝟏𝟏) = 𝟎𝟎
3. Write an equation for each diagram below. How are these equations alike? How are they different? What is it about the diagrams that lead to these similarities and differences?
A: 𝟒𝟒 + (−𝟒𝟒) = 𝟎𝟎
B: −𝟒𝟒+ 𝟒𝟒 = 𝟎𝟎
Answers will vary. Both equations are adding 𝟒𝟒 and −𝟒𝟒. The order of the numbers is different. The direction of A shows counting up 𝟒𝟒, then counting down 𝟒𝟒. The direction of 𝑩𝑩 shows counting down 𝟒𝟒, then counting up 𝟒𝟒.
Students may also mention that both diagrams demonstrate a sum of zero, adding opposites, or adding additive inverses.
7
7
7
Lesson 1: Opposite Quantities Combine to Make Zero
Problem Set Sample Solutions The Problem Set will provide practice with real-world situations involving the additive inverse such as temperature and money. Students will also explore more scenarios from the Integer Game to provide a solid foundation for Lesson 2.
For Problems 1 and 2, refer to the Integer Game.
1. You have two cards with a sum of (−𝟏𝟏𝟐𝟐) in your hand.
a. What two cards could you have?
Answers will vary. (−𝟔𝟔 and −𝟔𝟔)
b. You add two more cards to your hand, but the total sum of the cards remains the same, (−𝟏𝟏𝟐𝟐). Give some different examples of two cards you could choose.
Answers will vary, but numbers must be opposites. (−𝟐𝟐 and 𝟐𝟐) and (𝟒𝟒 and −𝟒𝟒)
2. Choose one card value and its additive inverse. Choose from the list below to write a real-world story problem that would model their sum.
a. Elevation: above and below sea level
Answers will vary. (A scuba diver is 𝟐𝟐𝟎𝟎 feet below sea level. He had to rise 𝟐𝟐𝟎𝟎 feet in order to get back on the boat.)
b. Money: credits and debits, deposits and withdrawals
Answers will vary. (The bank charges a fee of $𝟓𝟓 for replacing a lost debit card. If you make a deposit of $𝟓𝟓, what would be the sum of the fee and the deposit?)
c. Temperature: above and below 𝟎𝟎 degrees
Answers will vary. (The temperature of one room is 𝟓𝟓 degrees above 𝟎𝟎. The temperature of another room is 𝟓𝟓 degrees below zero. What is the sum of both temperatures?)
d. Football: loss and gain of yards
Answers will vary. (A football player gained 𝟐𝟐𝟓𝟓 yards on the first play. On the second play, he lost 𝟐𝟐𝟓𝟓 yards. What is his net yardage after both plays?)
3. On the number line below, the numbers 𝒉𝒉 and 𝒌𝒌 are the same distance from 𝟎𝟎. Write an equation to express the value of 𝒉𝒉 + 𝒌𝒌. Explain.
𝒉𝒉 + 𝒌𝒌 = 𝟎𝟎 because their absolute values are equal, but their directions are opposite. 𝒌𝒌 is the additive inverse of 𝒉𝒉, and 𝒉𝒉 is the additive inverse of 𝒌𝒌 because they have a sum of zero.
𝒉𝒉 𝒌𝒌 𝟎𝟎
Lesson 1: Opposite Quantities Combine to Make Zero
4. During a football game, Kevin gained five yards on the first play. Then he lost seven yards on the second play. How many yards does Kevin need on the next play to get the team back to where they were when they started? Show your work.
He has to gain 𝟐𝟐 yards. 𝟓𝟓 + (−𝟕𝟕) + 𝟐𝟐 = 𝟎𝟎, 𝟓𝟓+ (−𝟕𝟕) = −𝟐𝟐, and −𝟐𝟐+ 𝟐𝟐 = 𝟎𝟎.
5. Write an addition number sentence that corresponds to the arrows below.
𝟏𝟏𝟎𝟎+ (−𝟓𝟓) + (−𝟓𝟓) = 𝟎𝟎.
Lesson 1: Opposite Quantities Combine to Make Zero
Lesson 1: Opposite Quantities Combine to Make Zero
Classwork
Exercise 1: Positive and Negative Numbers Review
With your partner, use the graphic organizer below to record what you know about positive and negative numbers. Add or remove statements during the whole class discussion.
Negative Numbers Positive Numbers
Lesson 1: Opposite Quantities Combine to Make Zero
For all numbers 𝑎𝑎 there is a number −𝑎𝑎, such that 𝑎𝑎 + (−𝑎𝑎) = 0.
The additive inverse of a real number is the opposite of that number on the real number line. For example, the opposite of −3 is 3. A number and its additive inverse have a sum of 0. The sum of any number and its opposite is equal to zero.
d. Thinking back to our previous work, how would you use the counting on method to represent the following: While playing the Integer Game, the first card selected is 7, and the second card selected is −7.
e. What does this tell us about the sum of 7 and its opposite, −7?
f. Look at the curved arrows you drew for 7 and −7. What relationship exists between these two arrows that would support your claim about the sum of 7 and −7?
g. Do you think this will hold true for the sum of any number and its opposite?
Exercise 3: Playing the Integer Game
Play the Integer Game with your group. Use a number line to practice counting on.
Lesson 1: Opposite Quantities Combine to Make Zero
Problem Set For Problems 1 and 2, refer to the Integer Game.
1. You have two cards with a sum of (−12) in your hand.
a. What two cards could you have? b. You add two more cards to your hand, but the total sum of the cards remains the same, (−12). Give some
different examples of two cards you could choose.
2. Choose one card value and its additive inverse. Choose from the list below to write a real-world story problem that would model their sum. a. Elevation: above and below sea level
b. Money: credits and debits, deposits and withdrawals
c. Temperature: above and below 0 degrees
d. Football: loss and gain of yards
3. On the number line below, the numbers h and k are the same distance from 0. Write an equation to express the value of ℎ + 𝑘𝑘. Explain.
4. During a football game, Kevin gained five yards on the first play. Then he lost seven yards on the second play. How
many yards does Kevin need on the next play to get the team back to where they were when they started? Show your work.
5. Write an addition number sentence that corresponds to the arrows below.
Lesson Summary
Add positive integers by counting up, and add negative integers by counting down. An integer plus its opposite sum to zero. The opposite of a number is called the additive inverse because the two numbers’ sum is zero.
𝒉𝒉 𝒌𝒌 𝟎𝟎
Lesson 1: Opposite Quantities Combine to Make Zero
Lesson 2: Using the Number Line to Model the Addition of Integers
31
7•2 Lesson 2
Lesson 2: Using the Number Line to Model the Addition of
Integers
Student Outcomes
Students model integer addition on the number line by using horizontal arrows; e.g., an arrow for −2 is a
horizontal arrow of length 2 pointing in the negative direction.
Students recognize that the length of an arrow on the number line is the absolute value of the integer.
Students add arrows (realizing that adding arrows is the same as combining numbers in the Integer Game).
Given several arrows, students indicate the number that the arrows represent (the sum).
Classwork
Exercise 1 (5 minutes): Real-World Introduction to Integer Addition
Students answer the following question independently, as the teacher circulates around
the room providing guidance and feedback as needed. Students focus on how to
represent the answer using both an equation and a number line diagram. They will be
able to make the connection between both representations.
Exercise 1: Real-World Introduction to Integer Addition
Answer the questions below.
a. Suppose you received $𝟏𝟎 from your grandmother for your birthday. You spent $𝟒 on snacks. Using addition, how
would you write an equation to represent this situation?
𝟏𝟎 + (−𝟒) = 𝟔.
b. How would you model your equation on a number line to show your answer?
Real-world situations can be modeled with equations and represented on a real number line. In this exercise, positive
ten represents the “$10 given as a birthday gift” because it is a gain. Negative four represents the “$4 spent on snacks”
because it is a loss. Gaining $10 and then taking away $4 will leave you with $6.
Example 1 (5 minutes): Modeling Addition on the Number Line
The teacher models addition on a number line using straight arrows (vectors) to find the sum of −2 + 3. Elicit student
responses to assist in creating the steps. Students record the steps and diagram.
Scaffolding:
Create an anchor poster for the additive inverse to help access prior knowledge of number line features including arrow placement and direction and ordering of positive and negative numbers.
Lesson 2: Using the Number Line to Model the Addition of
Integers
Classwork
Exercise 1: Real-World Introduction to Integer Addition
Answer the questions below.
a. Suppose you received $10 from your grandmother for your birthday. You spent $4 on snacks. Using addition, how would you write a number sentence to represent this situation?
b. How would you model your equation on a number line to show your answer?
Example 1: Modeling Addition on the Number Line
Complete the steps to finding the sum of −2 + 3 by filling in the blanks. Model the number sentence using straight arrows called vectors on the number line below.
a. Place the tail of the arrow on ________.
b. Draw the arrow 2 units to the left of 0, and stop at ________. The direction of the arrow is to the ________ since
you are counting down from 0.
c. Start the next arrow at the end of the first arrow, or at ________.
d. Draw the second arrow ________ units to the right since you are counting up from −2.
e. Stop at ________.
Lesson 2: Using the Number Line to Model the Addition of Integers
Problem Set For Questions 1–3, represent each of the following problems using both a number line diagram and an equation.
1. David and Victoria are playing the Integer Card Game. David drew three cards, −6, 12, and −4. What is the sum of the cards in his hand? Model your answer on the number line below.
2. In the Integer Card Game, you drew the cards, 2, 8, and −11. Your partner gave you a 7 from his hand.
a. What is your total? Model your answer on the number line below.
b. What card(s) would you need to get your score back to zero? Explain. Use and explain the term “additive inverse” in your answer.
3. If a football player gains 40 yards on a play, but on the next play, he loses 10 yards, what would his total yards be for the game if he ran for another 60 yards? What did you count by to label the units on your number line?
Lesson Summary
On a number line, arrows are used to represent integers; they show length and direction. The length of an arrow on the number line is the absolute value of the integer. Adding several arrows is the same as combining integers in the Integer Game. The sum of several arrows is the final position of the last arrow.
Lesson 2: Using the Number Line to Model the Addition of Integers