The Elevator Model

A Context to help Develop the Concept of Integers

Once upon time there was an owner of a large apartment building with several floors of apartments and several floors in the underground parking garage. The ground floor of the building was labeled 0 on the elevator; the elevator went up to floors with numbers 1, 2, 3, etc. or down to parking levels with numbers –1, -2, -3, etc.

Draw a vertical number line that would represent the first six floors above and below ground. Don’t forget the ground floor!

One day the elevator stopped working and he could not afford to make the repairs. He noticed, however, that each helium balloon he had would take the elevator up one floor, and each brick he had would take the elevator down one floor, no matter how many people were on it! He decided that he could leave these balloons and bricks in the hallways outside the elevator doors or inside the elevator itself. He decided to label each balloon “positive 1” using +1 and each brick

“negative 1” using –1.

What do you think would be in the elevator if it was on the ground floor?

We could use two-colour counters to model the elevator activities by assigning red to the balloons (+1) and white to the bricks (-1). Model some combinations of balloons and bricks that could be inside the elevator if:

If it is on the ground floor (0) ________________________________________

What do you notice about the number of bricks and balloons on the elevator when it is on floor 0?________________________________________This relationship is called the zero principle. Write your own definition for the zero principle and check it with your partner.

Zero Principle

• The sum of opposite integers is zero.

3 + -3 = 0

-8 + 8 = 0

Model some more combinations of balloons and bricks that could be inside the elevator if:

It is on floor –3 ________________________________________

It is on floor + 6 ________________________________________

What floor is the elevator on for each of these combinations of balloons and bricks?

A)

B)

C)

Addition of Integers:

All the activity on this elevator can be represented by writing number sentences. For example, if someone is on floor 5 and adds a brick to take them to floor 4, we could represent this as 5 + (-1) = 4. If someone is on floor –2 and adds a balloon to take them to floor –1, we could represent this as –2 + (+1) = -1.

Model , illustrate, and write the number sentence for each elevator activity:(when illustrating, let shaded counters be positive and unshaded be negative.)

Starts on floor 6 and adds 2 balloons: Starts on floor 3 and adds 3 balloons: Starts on floor -5 and adds 2 bricksStarts on floor -3 and adds 4 bricks.Starts on floor 3 and adds 5 bricksStarts on floor -2 and adds 8 balloons

Subtraction of Integers

When David got off the elevator on the fifth floor, he took 2 balloons with him.

Model, illustrate, and write a number

sentence for this situation.

Model, illustrate and write the number sentence for each elevator activity.

Starts at floor 3 and removes 4 balloons: ______________________

Starts at floor 2 and removes 3 bricks: ________________________

Starts at floor -5 and removes 4 bricks: ______________________

Starts at floor -3 and removes 2 balloons: ______________________

What activity has occurred on the elevator according to the illustration below

Model each of the following number expressions using two-colour counters and find the answer.A) 3 + (+2)

B) –2 + (+3)

C) 4 + (-3)

D) –1 + (-2)

E) –4 – (+2)

F) –3 – (-2)

Explain two ways to make the elevator go from floor –2 to floor 8, from floor 7 to floor 3, and from floor 1 to floor –3. For each way, write the corresponding number sentence.

Multiplication of Integers

The elevator was on the ground floor when

3 people got on each carrying 2 bricks.

Model, illustrate and write the number sentence

for this elevator activity.

Model, illustrate and write the number sentence for each ground floor activity .

A) Two people got on the elevator each carrying 3 bricks

B) Four people got on the elevator each carrying 2 bricks

C) Three people got off the elevator each carrying 3 balloons

D) 5 people got off the elevator each carrying 1 brick

Burt explained that “3 + 4(-2)” could represent what happens when the elevator is on floor 3 and four people get on, each carrying two bricks. With 8 more bricks on the elevator, it will go down 8 floors to floor –5. What would Burt’s explanations be for each of the following:A) –1 + 3(+3)B) 12 + 2(-3)C) 10 – 2(+3)D) 9 – 2(-2)

Division of Integers

Suppose some people are handling the balloons

and bricks in the hallway.

Model, illustrate, and write a number sentence for

the handling of these materials.

-8 10 -6

4 2 -3

At floor 5, 3 people share the 12 bricks that are in the hall. One of these people gets on theelevator. What floor will the elevator go to?

B) Does this number expression tell the same story ?

5 + (-12 ) 3Explain.

Tell the elevator story for each of the following expressions and record the answers.

4 + (+6) 8 + (-16) 2 – (-8) 2 2 4

Jeri explained how she modeled the integer expression she was given: First, I put out 7 white counters (negative) and 9 red counters (positive). Then I removed 3 white counters, and made pairs of white and red counters. I examined what I had left to get my answer of positive 5.

What number sentence was Jeri modeling?