1. Sorcha Moran sorcha.moran@gmail.com
Add1ng Num8er5 t0 L1fe
Multiply the possibilities
Sorcha Moran

2. Education System failed
Learners lack self-belief in ability to learn maths
Perception of intelligence

3. Understanding is the key to successful learning

4. ‘Rules’ are shortcuts – NEVER begin teaching using ‘rules’
No Shortcuts!

5. Nurture your learners Ensure they understand the basic concepts
Visuals & manipulatives for understanding & acceptance
Real-life material for recognition & affirmation
Link to previous knowledge for making connections
Present in many ways for confidence
Link maths symbols to language for proficiency

6. Lost in Translation Maths is not just a jumble of numbers
Order of numbers combined with symbols gives them meaning

7. Replace Fears with Confidence

8. What makes learning fractions so difficult?

9. Local v. Global
The brain recognises the values of 3 and 4 (local values); but the fraction has a global value of 0.75

10. So many rules!! Fractions addition - common denominator
Fractions addition - different denominators
Finding equivalent fractions
Simplifying fractions
Comparing fractions
Fraction multiplication
Fraction division
Convert a mixed number to a fraction
Convert an improper number to a mixed fraction
Converting fractions to decimals

11. Multiply the numerators and the denominators.
Convert the fractions so they have a common denominator. Then compare the numerators
Multiply the numerators and the denominators.
Divide using long division or a calculator
First find a common denominator by taking the least common multiple of the denominators. Then convert all the addends to have this common denominator. Then add using the rule number 1
Multiply the whole number part by the denominator and add the numerator to get the numerator. Use the common denominator as in the fractional part of the mixed number.
Find the (greatest) common divisor of the numerator and denominator, and divide both by it.
Divide the numerator by the denominator to get the whole number part. The remainder will be the numerator of the fractional part. Denominator is the same.
Multiply both the numerator and denominator by a same number.
Add the numerators, use the common denominator
Find the reciprocal of the divisor, and multiply by it

12. Students can become blind followers of the rules, getting answers without having any idea if they are reasonable or not

13. So, what can we do to help?

14. Use Visual or Manipulatives
Fractions become something concrete to the student, and not just a number on top of another without a meaning

15. Think through pictures/exploration Discover the rules

16. By using visual models extensively at the beginning stages, the rules will make more sense

17. Some ideas...

18. Introduction - What are Fractions?

19. What are Fractions - Activity (Playdough)
Other options – lego, lengths string, strips of paper, etc

20. Talking Fractions - Visual
1. Shape your playdough into any regular shape you like e.g. a
rectangle, a ‘cake’, or a ‘sausage’.
2. Cut your playdough into as many equal size portions you like.
3. Take some portions away.
4. How do you describe to me what you have taken away, or what you have left?

21. Talking Fractions Begin to use Fractions Terminology
Don’t correct their answer in simplified fractions
Write replies on the board so they see the notation we use in fractions
Bottom number describes how many pieces would be in the whole thing
Top number describes how many of those pieces we have
Same number on the top and bottom if we have 1 whole

22. Give plenty of opportunities to play around with the concept
Express the shaded part of a shape as a fraction
Draw and shade a shape to match a fraction
Discuss where they see fractions every day - what do they mean?
Change a sentence into a fraction
*Language - ‘out of’
Give plenty of opportunities to play around with the concept
Use correct language
No need to simplify at this stage

23. Equivalent Fractions - Visual

24. Equivalent Fractions - Terminology
When we have two fractions that are actually the same size portion, we call them equivalent fractions (equal).
It is usually best to show an answer using the simplest fraction ( 1/2 in this case ). That is called Simplifying, or Reducing the Fraction

25. Equivalent Fractions - Visual

26. Equivalent Fractions - Visual

27. Equivalent Fractions - Visual
...Learner creates some equivalent fraction shapes

28. Equivalent Fractions - Activity (Exploration)

29. Simplifying Fractions - Visual

30. Simplifying Fractions - Visual

31. Simplifying Fractions
Equivalent Fractions
Simplifying Fractions
Equivalent fractions, but in the other direction

32. Simplifying Fractions - Visuals
Using the visuals and giving the learners a chance to explore with the playdough, makes it clear why we have to make the same change above and below the line. Now they understand the concept instead of trying to remember a rule.

33. Fractions of Whole Numbers - Visual

34. Fractions of Whole Numbers - Activity
1. Make 20 small balls of playdough
2. Divide them into 4 equal bundles
3. Count how many balls you have in “3 out of 4” bundles (3/4)

35. of
always translates to
Maths is a language!

36. Mixed and Improper Fractions - Visual
Improper: How many pieces?
Mixed: How many pies?

37. Mixed and Improper Fractions - Activity
Mark measurements on a ruler (in/cm)
Measure a line
Measure everyday items: pencil, eraser, finger, etc

38. Adding Fractions, Same Denominator

39. Adding Fractions, Different Denominators - Visual
Back to the playdough!
Present ½ of a ‘pie’, and ⅛ of a ‘pie’
Ask the learners how much pie you have
The only way we can explain how much we have accurately is to make all the pieces the same size, and then count them.

40. Adding Fractions, Different Denominators - Explore
If a learner asks why ½ + ⅛ is not just 2/10 (adding top, adding bottom). Ask them to explore it themselves with the playdough:
Cut a shape the same as yours into 10 equal pieces
Are 2 of those pieces a similar size to your ½ and ⅛ together?

41. Adding Fractions, Different Denominators - Visual

42. Multiplying Fractions
Top by Top, and Bottom by Bottom
Multiply the top numbers
Multiply the bottom numbers
Simplify if possible

43. Multiplying Fractions - Visual
Use the playdough to represent ⅖
Remind the learners that ‘of’ in English translates to multiply in maths, and visa versa
So, ½ x ⅖ is the same as “half of two fifths”
Remove half of the pieces
We are left with ⅕ - same as above - Magic!
Point out to the learners that they can make life a little easier by dividing before they multiply - i.e. cancel a top with a bottom before they calculate.

44. Dividing Fractions ½ ÷ ⅙ = “one half divided by one sixth”
Begin by doing division with integers (something familiar).
Take, 20 ÷ 4. We can ask “What is 20 divided by 4?”, or we can ask the same question in another way - “How many 4s in 20?”
This is the way we need to think about division with fractions:
“How many ____
will fit in _____?”

45. Dividing Fractions - Visual
Use the playdough to visualise:
Now it’s not so odd that dividing two fractions can result in a whole number!

46. Dividing Fractions - Visual
Try the same with dividing a small fraction by a larger one e.g. ⅙ ÷ ½
Use the playdough to visualise.
Only part of the half will fit in to the sixth...how much of it?
Here we are trying to fit a bigger piece into a smaller piece, so only a portion of the piece will fit in. Hence, we can expect our answer to be a fraction.

47. Dividing Fractions So, how do we do the same on paper?
Turn the second fraction upside down and multiply
Now, that’s just strange!

48. Dividing Fractions Again, bring it back to the familiar! 8 ÷ 2
If we want to share 8 sweets between 2 children,
they get ½ of the 8 sweets each
So, we can rewrite ‘8 ÷ 2’ as ‘½ x 8’ or ‘8 x ½’
See what just happened!!
Turn the second fraction upside down and multiply

49. Dividing Fractions - Visual
½ ÷ ⅛
1. Make a regular shape with your playdough e.g. a circle, a rectangle, etc
2. Divide your shape into 8 equal pieces, i.e. ⅛ ths
3. Check how many ⅛ ths are in half of your shape
Check the answer mathematically

50. Making Sense of Algebra

52. Replace Fear with Confidence

53. Fill the Gaps

54. Learn the Language
= x ’s & y’s / ()

55. No Shortcuts

56. Layout on Page

57. Success! Remove the fear Fill the gaps Learn the language
Don’t skip steps
Layout on the page
When do we ever use algebra???

58. Fill the Gaps Order of operations Distributive property
Divisibility/Factors
Working with negative numbers

59. Learning the Language What does ‘=’ mean? 2+3=5 ☐ + 2 = 6
☐ + ☐ = 6, why we use letters
Operators in algebra
Why we don’t use x for multiply

60. Learning through discovery

61. Trial and Error

62. Trial and Error

63. Removal of Blocks

64. Removal of Numbers

65. On Paper – Emptying the Box!
4x + 2 = 3x + 9
5x + 2 = 2x + 14
2x + 4 = 10

66. On Paper – Layout is Important!
Always work down the page
Only one change on a line
Use colour
4x + 2 = 3x + 9
4x = 3x + 7
x =
-3x x

67. Remove Blocks as Part of Set-Up
5x - 3x + 2 = x + 5

68. Multiplying out Brackets
2(x + 3) = x + 8
Looking at 2(x + 3) we read it as “two
times ‘x + 3’”, or, in other words, ‘x + 3’ twice. So we
place it on the scales like this:
“two times ‘x + 3’” is the same as “two times x” plus “two times 3”, or 2(x + 3) is the same as 2(x) + 2(3)

69. Multiplying out Brackets
2(x + 3) = x + 8
2(x) + 2(3) = x + 8
2x = x + 8
-x x
x =
x =

70. Mystery Solved!

71. Individuality

80. Sorcha Moran sorcha.moran@gmail.com
Add1ng Num8er5 t0 L1fe
Multiply the possibilities
Thank You!
Sorcha Moran