Exploring Integers
With Manipulatives
“Witches Brew” Model
MTA October 2008
LOIS BOUDREAU
AVRSB Math Mentor
Five Representations of
Learning
• Concrete- use of manipulatives
• Pictorial – charts, pictures, diagrams,
graphs,…
• Symbolic- use of numbers and algorithms
• Verbal – explanation of a concept or task
• Contextual- a real world connection – word
problems
Contextual
Where can we find
integers in our everyday
life?
Manipulatives for
Integers
• Alge-Tiles
1
-1
• Two Sided Counters
• Cube – a- Links
1
Hot coal
• Number Lines
1
-1
Ice cube
-1
INTRODUCING
“WITCHES BREW”
One hot coal
represents
positive one(+1)
1
-1
Cauldron
One ice cube
represents
negative one(-1)
Model the following
Temperatures in your
Cauldron…
• 3° C
• -2° C
• 0° C
INTRODUCING
“Zero Principle”
1
-1
Cauldron
One hot coal
(positive one)
neutralizes one
ice cube
(negative one)
which keeps the
temperature of
the cauldron at
zero degrees.
Adding/Subtracting/
Multiplying and Dividing
Integers
It’s important to remember that the
temperature in the Cauldron always
begins with zero degrees.
Remember zero means equal number of ice
cubes and hot coals (zero principle).
Adding Integers
Ex. 5 + (-2)
Begin by adding the first integer into
the cauldron - hot coals for positive
and ice cubes for negatives.
1
1
1
1
1
Adding Integers
5 + (-2)
1
1
-1
-1
1
1
1
When adding, add the second
integer into the Cauldron
Adding Integers
5 + (-2)
1
1
-1
-1
1
1
1
Now combine the “zeros”
The Temperature in the Cauldron is the
final answer = 3 or +3
Adding Integers
With a partner, do the
following questions using
the Cauldron and draw a
picture of each step.
• (-4) + (-5)
• (6) + (-10)
Subtracting Integers
Ex. (-2) – (-4)
Begin by adding the first integer into the
cauldron - hot coals for positive and ice
cubes for negatives
-1
-1
Subtracting Integers
Ex. (-2) – (-4)
-1
-1
1
1
-1
-1
You have to “take out” 4 ice cubes so you must add
Some zeros to your cauldron to have enough to remove.
Subtracting Integers
Ex. (-2) – (-4)
-1
-1
1
1
-1
-1
The temperature of the Cauldron once you
Remove the ice cubes is the answer = 2 or +2
Subtracting Integers
With a partner, do the
following questions using
the Cauldron and draw a
picture of each step.
• (5) – (8)
• (-7) – (4)
• (+3) – (-6)
Multiplying Integers
Ex. (+3) x (-4)
• Remember, the temperature in the
cauldron always starts at 00.
• The +3 indicates you are adding 3 groups
of -4 into the cauldron. Therefore you
would add 12 ice cubes (-12) into the
cauldron.
1
1
1
1
1
1
1
1
1
1
1
-1
Multiplying Integers
• The temperature of the Cauldron
once you add the ice cubes is the
answer = -12
1
1
1
1
1
1
1
1
1
1
1
-1
Multiplying Integers
Ex. (-2) x (+3)
• The -2 indicates you are subtracting 2
groups of +3 from the cauldron.
Therefore you would subtract 6 hot coals
(+6) from the cauldron. You must add 6
zeros first to be able to do this.
1
1
1
1
1
1
1
1
1
1
1
1
Multiplying Integers
• The temperature of the Cauldron
once you remove the hot coals or
positives is the answer = -6
1
1
1
1
1
1
Multiplying Integers
Ex. (-5) x (-2)
• The -5 indicates you are subtracting 5
groups of -2 from the cauldron.
Therefore you would subtract 10 ice
cubes(-10) from the cauldron. You must
add 10 zeros first to be able to do
this.
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
Multiplying Integers
• The temperature of the Cauldron
once you remove the ice cubes or
negatives is the answer = +10
1
1
1
1
1
1
1
1
1
1
Multiplying Integers
With a partner, do the
following questions using the
Cauldron and draw a picture
of each step.
• (+4) x (-2)
• (-3) x (-3)
• (-2) x (+2)
Dividing Integers
Ex. (-8) divided by (+2)
• “Think Multiplication” by asking
yourself “ If I add 2 groups of ____
into the cauldron, I will get -8.
?
Dividing Integers
• Remembering the temperature in the
cauldron starts at 00, you would have
to add ice cubes (negatives) to get
the temperature to drop to -8. You
would add 2 groups of -4 (8 ice
cubes) to get -8. Therefore -8
divided by +2 = -4
1
1
1
1
1
1
1
1
Dividing Integers
Ex. (+6 )divided by (-3)
• Think Multiplication by asking
yourself “ If I subtract 3 groups of
____ from the cauldron, I will get a
temperature of +6.
?
Dividing Integers
• Again, because the temperature in the
cauldron starts at 00 and you are being
asked to subtract 3 groups of something
to end up with a temperature of +6, you
would have to add 6 zeros and subtract 3
groups of -2 (6 ice cubes) to get the
temperature to rise to +6. Therefore +6
divided by -3 = -2.
1
1
1
1
1
1
1
1
1
1
1
1
Dividing Integers
Ex. -4 divided by -2
• Think Multiplication by asking
yourself … “ If I subtract2 groups of
____ from the cauldron, I will get a
temperature of -4.
?
Dividing Integers
• You would first have to add 4 zeros to the
cauldron to show a temperature of 00. You
would have to subtract 2 groups of +2 (4
hot coals, to end up with a temperature of
-4. Therefore -4 divided by -2 = +2
1
1
1
1
1
1
1
1
Dividing Integers
• With a partner, do the following
questions using the Cauldron and
draw a picture of each step.
• -8 divided by -4
• +6 divided by -3
• -10 divided by +5