Exploring Integers Exploring Integers With ManipulativesWith Manipulatives

“Witches Brew” Model“Witches Brew” Model

Exploring Integers Exploring Integers With ManipulativesWith Manipulatives

“Witches Brew” Model“Witches Brew” ModelMTA October 2008MTA October 2008LOIS BOUDREAULOIS BOUDREAU

AVRSB Math MentorAVRSB 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

• Two Sided Counters

• Cube – a- Links

• Number Lines

1 -1

1 -1

1 -1

Hot coalIce cube

INTRODUCING “WITCHES BREW”

Cauldron

1 -1

One hot coal represents

positive one(+1)

One ice cube represents

negative one(-1)

Model the following Temperatures in your

Cauldron…

• 3° C• -2° C• 0° C

INTRODUCING “Zero Principle”

Cauldron

1 -1

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).

Begin by adding the first integer into the cauldron - hot coals for positive and ice cubes for negatives.

Adding Integers

Ex. 5 + (-2)

1 111 1

Adding Integers

5 + (-2)

When adding, add the second integer into the Cauldron

1 111 1

-1-1

Adding Integers

5 + (-2)

Now combine the “zeros”

The Temperature in the Cauldron is the final answer = 3 or +3

1 111 1

-1-1

Adding Integers

With a partner, do the following questions using the Cauldron and draw a

picture of each step.

• (-4) + (-5)• (6) + (-10)

Subtracting IntegersEx. (-2) – (-4)

Begin by adding the first integer into thecauldron - hot coals for positive and ice

cubes for negatives

-1 -1

Subtracting IntegersEx. (-2) – (-4)

You have to “take out” 4 ice cubes so you must add Some zeros to your cauldron to have enough to remove.

-1 -1-1

1

-1

1

Subtracting IntegersEx. (-2) – (-4)

-1-1

1

-1

1

-1

The temperature of the Cauldron once youRemove 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 IntegersEx. (+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

• The temperature of the Cauldron once you add the ice cubes is the answer = -12

Multiplying Integers

1 1 1 1

1 1

1 1 1

1 1

-1

Multiplying IntegersEx. (-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 IntegersEx. (-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

11

1 1 1 1

1 1 1

1

11

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

11

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 IntegersEx. (-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 IntegersEx. (+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

11

1 1 1

1

1

1

1

Dividing IntegersEx. -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

11

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