1. Fractions, Decimal Fractions and Percentages - a closer look.
Jill Smythe
Numeracy Facilitator
(Based on work by Peter Hughes)

2. Fractions:

3. Pirate Problem
(While you are waiting - there is a prize for whoever gets the answer first!)
Three pirates have some treasure to share. They decide to sleep and share it equally in the morning.
One pirate got up at at 1.00am and took 1/3 of the treasure.
The second pirate woke at 3.00am and took 1/3 of the treasure.
The last pirate got up at 7.00am and took the rest of the treasure.
Do they each get an equal share of the treasure?
If not, how much do they each get?
Rope Activity afterwards.

4. The Rope Activity:
Pegs
2/5

5. Objectives:
Identify the progressive strategy stages of fractions, decimal fractions, percentages and ratios.
Further develop teacher’s confidence and content knowledge of fractions, decimal fractions, percentages and ratios.
Explore key ideas, equipment and activities used to teach knowledge and strategy of the above.

6. Developing Proportional Thinking:
Order the scenarios.
How well did you do? Check by using The Number Framework: Book 1 (Pg 15).
Highlight all the fractional knowledge across the stages (Pg18-22).

7. Fraction Knowledge Test:
Draw 2 pictures: (a) one half (b) one eighth
Mark 5 halves on a number line
12 is three fifths of what number?
What is 3 ÷ 5?
Draw a picture of 7 thirds
Write one half as a ratio.
The ratio of kidney beans to green is 3:4. What fraction of the beans are green?
Order these fractions:
2/4, 3/4, 2/5, 7/16, 2/3, 6/49
Now include these % and decimals into your order
30%, 75%, 0.38, 0.5

8. Group discussion Fraction questions Why do we have fractions?
Shape on board - divided in 3. Are they thirds?
Shape divided in 3. Are they thirds?
Is a fraction greater than zero or less than zero.
If a students folds a strip of paper in three, but in finding this difficult, cuts a piece off to fit. Is that a problem?

9. Views of Fractions: What does this fraction mean?
What language should we use!!
Suggest ways of saying this fraction.

10. Views of Fractions: 3 ÷ 7 3 out of 7 3 : 7 3 over 7 3 sevenths
Does this fraction mean?
3 ÷ 7
3 out of 7
3 : 7
3 over 7
3 sevenths
What language should we use!!
1 out of 2 might work but does this help with
*7 out of 3 or 7/3
1 out of 2 of 24
Wrapped blocks - starting with whole - tenths, sixths
Problem with language

11. The Problem with Language:
Use words first before using the symbols:
e.g. one half not 1 out or 2
How do you explain the top and bottom numbers? 1 2
The number of parts chosen
The number of parts the whole has been divided into
Start with words rather than symbols which are very abstract

12. Models of Fractions:
Show 3 quarters using any of the materials on your table.
Discussion within groups

13. Continuous Model:
Models where the object can be divided in any way that is chosen.
e.g. ¾ of this line and this square are blue. 1

14. Discrete Model: Discrete: Made up of individual objects.
e.g. ¾ of this set is blue

15. Whole to Part:
Most fraction problems are about giving students the whole and asking them to find parts.
Show me one quarter of
this circle?
Before next slide:-
Triangle shape – this is a quarter of a whole shape – show the whole

16. Part to Whole: We also need to give them part to whole problems, like:
5 is a quarter of this
number.
What is the number?
Make reference to book 7, and fair shares activity (Pg.11) showing discrete and continuous, part to whole and whole to part.
Show triangle – this is a quarter – what is the whole?

17. Teaching Fractions:
What do you see as some of the confusions associated with the teaching and understanding of fractions?
Discussion at tables - feedback

18. Misconceptions with Fractions:
Charlotte believes that one eighth is bigger than one half.
1/2  1/3  1/4  1/8
Why do you think Charlotte has this misunderstanding?
How would you address this misconception?
What equipment would you use?

19. Misconceptions with Fractions:
Jenna says the following:
¼ + ¼ + ¼ = 3/12
Why do you think Jenna has this misunderstanding?
How would you address this misconception?
What equipment would you use?

20. Misconceptions with Fractions:
A group of students are investigating the books they have in their homes.
Steve notices that of the books in his house are fiction books, while Andrew finds that of the books his family owns are fiction.
Steve states that his family has more fiction books than Andrew’s.
Consider….
Is Steve necessarily correct?
Why/Why not?
What action, if any, do you take?

21. The size of the fraction depends on the size of the whole.
Key Idea:
The size of the fraction depends on the size of the whole.
Steve is not necessarily correct because the amount of books that each fraction represents is dependent on the number of books each family owns.
For example: of 30 is less than of 100.

22. Misconceptions with Fractions:
Anna says is not possible as a fraction.
Consider…..
Is possible as a fraction?
Why does Anna say this?
What action, if any, do you take?

23. Key Idea: A fraction can represent more than one whole.
Can be illustrated through the use of materials and diagrams.
Question students to develop understanding:
Show me 2 thirds, 3, thirds, 4 thirds…
How many thirds in one whole? two wholes?
How many wholes can we make with 7 thirds?

24. Misconceptions with Fractions:
You observe the following equation in Bill’s work:
Consider…..
Is Bill correct?
What is the possible reasoning behind his answer?
What, if any, is the key understanding he needs to develop in order to solve this problem?

25. Key Idea: No he is not correct. The correct equation is
To divide the number A by the number B is to find out how many lots of B are in A. When dividing by some unit fractions the answer gets bigger!
No he is not correct. The correct equation is
Possible reasoning behind his answer:
1/2 of 2 1/2 is 1 1/4.
He is dividing by 2.
He is multiplying by 1/2.
He reasons that “division makes smaller” therefore the answer must be smaller
than 2 1/2.

26. Fractional Knowledge:
Having secure fractional knowledge is the key to operating successfully with fractions…
Stage 4: Identifies and writes common fractions.
Stage 5: Orders unit fractions.
Stage 6: Identifies any fraction including improper fractions.
Stage 7: Knows equivalent fractions for halves, thirds, quarters, fifths and tenths.
Stage 8: Orders fractions, decimals and percentages.
(Taken from IKAN assessment)

27. Exploring Book 7 & FIO:
Folding Fractions Figure it Out Number Lev.2 Pg.18
Stage 5-6: Birthday Cakes Page 26
Focus on the following :
What key knowledge is required before beginning this stage.
Highlight the important key ideas at this stage.
The learning intention of the activity.
Work through the teaching model (materials, imaging, number properties).
Possible follow up practice activities.
Link to the planning units and Figure It Out.
Split in 2 groups:_
Jill – Folding fractions
Fiona - Birthday Cakes
Each of the two groups share back

28. Fractions as Operators
Sione has 35 marbles.
At the end of play he only has 3 sevenths of his marbles left.
At the end of lunch he has only two fifths of the marbles he started lunchtime with.
How many marbles does he have left?
Can you draw a grid to show your thinking?

29. Fractions as Operators
So we are finding out a fraction of a fraction.
2/5 of 3/7 29

30. Fractions as Operators
So we are finding out a fraction of a fraction.
2/5 of 3/7
We start with the 3/7 of 35
Which is 15.

31. Fractions as Operators
So we are finding out a fraction of a fraction.
2/5 of 3/7
We start with the 3/7 of 35
Which is 15.
Now we need to work out
2/5 of 15
Which is 6. 31

32. Multiplying with Fractions
When you multiply by some fractions the answer gets smaller
This is ⅓ of one whole strip.
If it is cut into quarters, four equivalent pieces, what will each new piece be called?
1/3
1/12
Before next slide - read Knowledge aspect of Bk1 and highlight all relating to fractions
Read strategy

33. Summary of key ideas: Fractional language - emphasise the “ths” code.
Fraction symbols - use symbols with caution. Combine language, symbols and a visual representation to consolidate understanding.
Continuous and discrete models - use both.
Go from Part-to-Whole as well as Whole-to-Part.
Fractions are always relative to the whole.
Begin to teach fraction when children are at stage 2!!
Use the teaching model to develop conceptual understanding.

34. Introducing Decimal Fractions:
Present the class with unifix cubes that have been
wrapped in tens. Each represents one bar of
chocolate with 10 pieces. How many pieces/tenths
would each person get?
2 ÷ 5 = ?

35. Discuss:
Why should we introduce decimal fractions by division of whole numbers rather than telling students that that the places after the decimal point represent tenths, hundredths, thousandths and so on?

36. Reason:
The whole must be broken into ten parts – canon of place value (ten for one)
Teaching division of whole numbers leads to fractional answers
E.g 3 ÷ 7 = 3 sevenths of one whole
This is an essential precursor to decimal fractions

37. Solve these using Materials:
7 ÷ 2 = ? Wholes and ? Tenths
8 ÷ 5 = ? Wholes and ? Tenths
6 ÷ 4 = ? Wholes and ? Tenths
5 ÷ 2 = ? Wholes and ? Tenths
2 ÷ 4 = ? Wholes and ? Tenths
Candy Bars Bk.5 Add/Sub

38. Why have we used words? Words emphasise what fractions really are.
Stops students thinking of fractions as division, out of, over or ratios.
Develops a deeper understanding of what fractions represent.
The number 3.5 should be read as 3 and 5 tenths rather than 3 point 5.

39. Solve this:
4.5 ÷ 3 =
4 wholes ÷ 3 = 1 whole for each share with 1 whole left over. This 1 whole is split into ten tenths – so there are now fifteen tenths altogether. 15 tenths ÷ 3 = 5 tenths for each share. Each share is 1 whole and 5 tenths (1.5)

40. I Whole - take one large piece of paper
Make 1 tenth
Make 1 hundredth

41. Becoming familiar with the equipment:
Decimal Fraction Mats (book 4, page 8 and book 7 pg 45).
Prove that 1.7 is bigger than 1.68 using one of the above materials.

42. Misconceptions with Decimal Fractions:
Jill believes that 0.45 is less than 0
Why do you think Jill has this misunderstanding?
How would you address this misconception?
What equipment would you use?

43. Misconceptions with Decimal Fractions:
John says the following:
“one half is the same as 0.2”
Why do you think John has this misunderstanding?
How would you address this misconception?
What equipment would you use?

44. Key Idea:
Decimals are special cases of equivalent fractions in that they always involve tenths, hundredths, or thousandths, etc.
John is not correct. ½ is 0.5 ( 5 tenths) and 0.2 is two tenths
Practise in needed in order to convert fractions to equivalent decimals

45. Misconceptions with Decimal Fractions:
Mary has to order the following fractions from smallest to largest:-
She orders them as follows:-
Consider….
Is Mary correct?
Why/Why not?
What are the misconceptions?
What action, if any, do you take?

46. Misconceptions with Decimal Fractions:
Jacob added the following decimals together:-
= 4.12
Why do you think Jacob has this misunderstanding?
How would you address this misconception?
What equipment would you use?

47. The Connection between Fractions and Percentages:
What does % mean?
In mathematics, a percentage is a way of expressing a number as a fraction of 100 (per cent meaning "per hundred"). It is often denoted using the percent sign, % For example, 45% (read as "forty-five percent") is equal to 45 hundredths or 0.45.
What do we need to do to fractions so that it can be read as a percentage?
What key mathematical knowledge do children need
to be able to do this?

48. Book 7 P 47 - 49 Extending Hot Shots P 56 - 60 Hot Shots 48
Drawing attention to 2 units of teaching - Numeracy book 7 48

49. % Problems 20% of 150 is 20% of is 30 % of 150 is 30 49 Big point -
we often ask questions like the No 1 question here - 20% of 150 is……
We don’t ask enough of other types of questions
20% of … is 30
Or what% of 150 is 30
What else do you know goes with this?
What else would it look like ?
Have to do all sorts of problems - in many different ways 49

50. Before they attempt the question:
Question (in context)
The local dairy farmer is selling 20% of his herd of 150 cows. How many is he selling?
Rewrite in maths language
20% of 150 is  
150
0% % %
Before they attempt the question:
Can’t underestimate the mathematising.. Is the critical step.
Many teachers go from problem to materials and miss out the important step - mathematising - eg 20% of 150 is ….. It is the maths language
Materials is the support to get students to think about algebraic understanding.
P 57 top quote!!!!
And giving students opportunity to form questions of the same type.
Many teachers say % - that symbol % just means divide by 100 ……. That is why write in a diff form - stops procedural / rule
From a problem,
NOW do problem 50

51. How do we use the lines to get the answer?
0% % % 
150 divided by 5 = 30
20 x 5 = 100
Find 10% : divided by 10
0% % % 
So 10% = 15
So 20% =30 51

52. 15 x 2
15 x 10
0% % % 
10 x 2
10 x 10 52

53. What is the maths? (Mathematize it)
There are 30 students in Room % are girls.
How many girls are there in the class?
What is the maths? (Mathematize it)
Mathematize
Double no line is a material
This is a scaffold to assist chrens learning
And we all know that t some time, the scaffolding has to come down!!!!
If don’t mathematize it means nothing 53

54. Before they attempt the question:
40% of 30 is  30 
__________________________________________
0% % %
Before they attempt the question:
Can’t underestimate the mathematising.. Is the critical step.
Many teachers go from problem to materials and miss out the important step - mathematising - eg 20% of 150 is ….. It is the maths language
What different ways can we write it.
Materials is the support to get students to think about algebraic understanding.
P 57 top quote!!!!
And giving students opportunity to form questions of the same type.
Many teachers say % - that symbol % just means divide by 100 ……. That is why write in a diff form - stops procedural / rule
From a problem,
NOW do problem 54

55. How do we use the lines to get the answer?
30 divided by 5 = 6
0% % % % 
20 x 5 = 100
20% = 6
So 40% = 12
Find 20% : divided by 5
0% % % % 
20% = so 40% =12 55

56. 3 x 10
3 x 4 
% %
10 x 4
10 x 10
This is the proportional thinking 56

57. So instead of paying 100% she only paid?
Sarah went shopping for a new bike which cost $350. When she got to town there was a sale and she got 20% off the price. What did she pay?
Did she pay more or less?
How much less?
So instead of paying 100% she only paid?
Show all this on the number lines
$350 0%
80%
100%
Level 5
Increasing and decreasing
5.3 understand operations on %
5.5 Know common % conversions - GST
Curr 5 - key one for the above
Level 4
Na 4.3 find % of amounts 57

58. Additional problems using the Double Number line!!!
Emily’s team won the basket ball game Emily shot 60% of the goals. How many goals did Emily get?
John scored 104 runs in a one day cricket match, that was 40% of the teams total. How many runs did his team score altogether?
In a bike race, 30% of cyclists drop out. 42 riders finish the race. How many cyclists started the race?

59. Ratios:
In the rectangle below, what is the ratio of green to blue cubes?
What is the fraction of blue and green cubes?
Can you make another structure with the same ratio? What would it look like?
What confusions may children have?

60. More on Ratios….
Divide a rectangle up so that the ratio of its blue to green parts is 7:3.
Think of other ways that you can do it.
What is the fraction of each colour?
If I had 60 cubes how many of them will be of each colour?

61. A Ratio Problem to Solve:
There are 27 pieces of fruit. The ratio of fruit that I get to the fruit that you get is 2:7. How many pieces do I get?
How many pieces would there have to be for me to get 8 pieces of fruit?
What key mathematical knowledge is required?

62. Which recipe will make the darkest green?
A: One part of blue with three parts yellow (1:3)
B: Four parts of blue with eight parts yellow (4:8)
C: Three parts of blue with five parts of yellow (3:5)

63. A diagnostic question
A recipe needs 6 eggs, 4 kg of flour and 8 teaspoons of flavouring.
What would the recipe be if 6 kg of flour is used?
What would a successful answer look like?

64. Thought for the day: Smart people believe only half of what they hear.
Smarter people know which half to believe.

65. Materials Images Knowledge Teaching progression Start by:
Using materials, diagrams to illustrate and solve the problem
Progress to:
Developing mental images to help solve the problem
Extend to:
Working abstractly with the number property
Teaching progression
Materials
Images
Knowledge
Even though we are working at stages 7 & 8 - Years 7 and 8, we will still be using materials.
However at stage 7/8, the use of materials will be reducing.
Number lines are materials! 65