1. Exploring Integers With Manipulatives “Witches Brew” Model MTA October 2008 LOIS BOUDREAU AVRSB Math Mentor

3. 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

4. Contextual Where can we find integers in our everyday life?

5. Manipulatives for Integers Alge-Tiles Two Sided Counters Cube – a- Links Number Lines 1 1 1 Hot coal Ice cube

6. INTRODUCING “WITCHES BREW” Cauldron 1 One hot coal represents positive one(+1) One ice cube represents negative one(-1)

7. Model the following Temperatures in your Cauldron… 3° C -2° C 0° C

8. INTRODUCING “Zero Principle” Cauldron 1 One hot coal (positive one) neutralizes one ice cube (negative one) which keeps the temperature of the cauldron at zero degrees.

9. 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).

10. Begin by adding the first integer into the cauldron - hot coals for positive and ice cubes for negatives. Adding Integers Ex. 5 + (-2) 11111

11. Adding Integers 5 + (-2) When adding, add the second integer into the Cauldron 11111

12. Adding Integers 5 + (-2) Now combine the “zeros” The Temperature in the Cauldron is the final answer = 3 or +3 11111

13. Adding Integers With a partner, do the following questions using the Cauldron and draw a picture of each step. (-4) + (-5) (6) + (-10)

14. Subtracting Integers Ex. (-2) – (-4) Begin by adding the first integer into the cauldron - hot coals for positive and ice cubes for negatives

15. Subtracting Integers Ex. (-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

16. Subtracting Integers Ex. (-2) – (-4) 1 1 The temperature of the Cauldron once you Remove the ice cubes is the answer = 2 or +2

17. 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)

18. Multiplying Integers Ex. (+3) x (-4) Remember, the temperature in the cauldron always starts at 0 0. The +3 indicates you are adding 3 groups of -4 into the cauldron. Therefore you would add 12 ice cubes (-12) into the cauldron. 1111 11 111 11

19. The temperature of the Cauldron once you add the ice cubes is the answer = -12 Multiplying Integers 1111 11 111 11

20. 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. 111 11 1 1 1 11 1 1

21. Multiplying Integers The temperature of the Cauldron once you remove the hot coals or positives is the answer = -6 111 111

22. 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. 111 1 11 1111 111 1 11 1111

23. Multiplying Integers The temperature of the Cauldron once you remove the ice cubes or negatives is the answer = +10 111 1 11 1111

24. 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)

25. 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. ?

26. Dividing Integers Remembering the temperature in the cauldron starts at 0 0, 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 1111 1111

27. 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. ?

28. Dividing Integers Again, because the temperature in the cauldron starts at 0 0 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 11 111 1 1 1 1

29. 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. ?

30. Dividing Integers You would first have to add 4 zeros to the cauldron to show a temperature of 0 0. 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 11 1 1 1 1

31. 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