1. Teaching Math Conceptually

2. Students who cannot easily memorize rules tend to not do as well in math.
Students who can memorize rules tend to do better in math, but do not always understand HOW or Why the math works.
What are some rules that we teach students?

3. Intergers – add the opposite, two negative make a positive (but not always)
Dividing Fractions – Keep Change Flip
Proportions – Cross Multiplying
Converting Mixed Numbers and Improper Fractions

4. How and Why
Eventually, they will probably use the “rule” because it is faster, but they will make fewer mistakes if they truly understand how the math is working.

5. Turn to your neighbor and discuss how you usually teach adding, subtracting, multiplying and dividing integers.

6. When I start teaching integers, I always tell my students that after we start learning this there will be some students that miss 3+5 and 9-7 because they are so concerned about the rules instead of THINKING about how the numbers make SENSE. They think they will never miss such easy questions, but it happens every year!!

7. Integers with a Number Line or Thermometer
+ number walk forward (to the right)
- numbers walk backward (to the left)
“Plus” go in same direction
“minus” turn around

8. One Way to Add Integers Is With a Number Line
When the number is positive, count
to the right.
When the number is negative, count
to the left. + 1 2 3 4 5 6 -1 -2 -3 -4 -5 -6

9. To add a positive integer we move forwards up the number line.
Examples
We can use a number line to help us add positive and negative integers.
–2 + 5 =
= 3 -2 3
How can we use the number line to work out –2 + 5?
Start at –2 (click to highlight the –2 on the number line) and count forwards 5.
On a number line we move to the right for forwards (in a positive direction) and to the left for backwards (in a negative direction).
Explain that the answer can be positive or negative depending on the starting point. It is important to stress this fact because when pupils learn the rules for multiplying and dividing negative numbers they often confuse these rules for the rules for addition and subtraction.
Remember:
To add a positive integer we move forwards up the number line.

10. Examples -6 -5 -4 -3 -2 -1 1 2 3 4 5 6

11. –3 + (–4) = = –7 Examples Remember:-7 -3
Remember:
To add a negative integer we move backwards down the number line.
How can we use the number line to work out –3 + -4?
Explain that when we add a negative number we have to move backwards down the number line. The answer can be positive or negative depending on the starting point.
Stress that adding a negative number is equivalent to subtracting the positive value of that number.
is the same as

12. Examples -6 -5 -4 -3 -2 -1 1 2 3 4 5 6

13. This works with Subtraction also. Students know that – means opposite.
Examples
-5-3 = = (-3)=

14. Integers Using Algebra tiles
This represents a negative one.
This represents a positive one.
A positive one and a negative one cancel each other and make a zero pair.

15. 2 + 4=6
Positives plus some MORE positives equal lots of POSITIVES.

16. =-6
Negatives plus some MORE Negatives equal lots of Negatives.

17. = 2
Cross out the zero pairs and whatever left is the answer. Students quickly figure out that if you are adding a positive and a negative it is actually a SUBTRACTION problem

18. Multiplying and Dividing
2 x 3……….This problem means “two groups of three” =6
2 x - 3……..This problem means “two groups of negative 3” = -6

19. Multiplying and Dividing
- 2 x 3……..This problem means “the opposite of two groups of three” = .) - 2 x - 3…….This problem means “the opposite of two groups of negative three”

20. Subtracting
Subtracting with the tiles is a little more difficult, but it is a good way for kids to visually see why subtracting involves adding sometimes.
We will be saying “take away” for the subtraction sign and will be using zero pairs.

21. 5 – 2 = 3 Positive 5 take away positive 2
That leaves us with positive 3

22. -5 – (- 2) = -3 Negative 5 take away negative 2
That leaves us with negative 3

23. 5 – (-2) = 7???
I don’t have any negatives to take away. I can add as many zeros as I want without changing the number.
Now I have two negatives to take away and I am left with 7 positives.

24. -5 – 2 = -7
I don’t have any positives to take away. I can add as many zeros as I want without changing the number.
Now I have two positives to take away and I am left with 7 negatives.

25. Multiplying Fractions
In Primary school students associate multiplying with numbers getting larger and dividing with numbers getting smaller. Multiplying proper fractions is probably the first time the answer to a multiplying question is LESS than what they started with.

26. Why bother with pictures?
• Because we won’t help make mathematics work for a much larger proportion of the student body until and unless we recognize that abstractions like 3⁄4 or rules for dividing fractions work fine for some students but must be grounded in pictures and models for others.
(Leinwand, pp )

27. 1/2 x 1/3
How would you
model this?

28. 1/2 x 1/3
You had a birthday party and had ½ of a cake left over.

29. 1/2 x 1/3
Your brother ate 1/3 of the left over cake.

30. 1/2 x 1/3 How much of the whole cake did your brother eat?
1/6 of the whole cake is shaded twice.

31. 3/5 x 3/4

32. 3/5 x 3/4
9 out of 20 are double colored
9/20

33. Try it #1

34. Try it #2

35. How would you model this?

45. Try it #1
Use the blocks (or draw) to show

46. Try it #2

47. Try it #3
Use the blocks (or draw) to show

48. Try it #4

49. How would you model this?

54. This is a student’s example

58. Try it #1

59. Try it #2

60. Converting Mixed Numbers and Improper Fractions
What does this mean?

61. The denominator is 4 which means our object (candy bars) is cut into fourths. We have 2 whole candy bars and ¼ of another. How many total fourths do we have?

62. The denominator is 5 which means our object (candy bars) is cut into fifths. We have 1 whole candy bar and 2/5 of another. How many total fifths do we have?

63. How many Wholes and how many thirds will this
make? 2 whole candy bars and 2/3 of
another.

64. The site to create models
This Glencoe site allows the user to explain their thinking using manipulatives.