Transcript The_Elevator_Model[1]

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 bricks
Starts on floor -3 and adds 4 bricks.
Starts on floor 3 and adds 5 bricks
Starts 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
4
10
2
-6
-3
At floor 5, 3 people share the 12 bricks that are
in the hall. One of these people gets on the
elevator. What floor will the elevator go to?
B) Does this number expression tell the same
story ?
5 + (-12 )
3
Explain.
Tell the elevator story for each of the following
expressions and record the answers.
4 + (+6)
2
8 + (-16)
2
2 – (-8)
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?