### Teaching Fractions Developmentally

Teaching of Fractions Developmentally.

Bruce Williams. CreatingRealMathematicians.com

It is important to realise fractions are not a new topic for students to learn, but another way of representing an understanding they already possess

For example, for the array     0 0 0

0 0 0

We can describe this numerically as 3 + 3 = 6, 2 + 2 + 2 = 6, 2 x 3 = 6, 3 x 2 = 6, as well as 6 :/ 3 = 2 and 6 :/ 2 = 3.

Practice with students adding to their understanding of arrays and Fact Families the recording of Fraction number sentences and turn around facts.

Finding Fractions of 24.

What does the Top Number mean? What does the Bottom Number mean?

Learning Intentions:

To understand what the Denominator and Numerator in a fraction mean.

To be able to find a unit fraction of a group of items.

To visualize the relative size of different fractions.

To identify the patterns in fractions of the same denominator while the numerator increases.

To explain the relationship between a fraction and its related multiplication (and division) fact.

To explain why when the denominator increases the fraction becomes smaller.

To develop a fraction chart based on fractions of 24.

To see the relationship between equivalent fractions.

Appropriate for Years 3/4 and above.

(For Year 3s I might stick to the unit fractions (fractions with 1 as the numerator).

Summary:

Students find 1/2, 1/3, 1/4, 1/6, 1/8, 1/12 & 1/24 of 24 sheep for the farmer.

Introduction:

“I was talking to a farmer on the weekend who had difficulty working out how many sheep he had to shear. His wife told him to shear 1/2 of his sheep. He told me he had 6 sheep. How many sheep should he shear?”

Students will know he should shear 3 sheep, but insist they explain why it is 3. Keep questioning until students explain:

a) He needs to split them into two groups.

b) The denominator (or bottom number) tells us the number of groups to make.

c) Both groups should be the same size.

d) He takes one of these groups.

e) The Numerator (or top number) tells us the number of groups to take.

Write the number sentence “1/2 of 6 is 3”.

Run through what each of the digits in the number sentence represent. (I.e., 6 is the total number of sheep, 3 is the number of sheep to shear, 2 is the number of equal groups to make, 1 is the number of groups to take (shear)).

Have students explain again the denominator tells us the number of equal groups to make. The 1 the number of groups to take.

“1/3 of 6 is 2”

Discuss “What about 2/3 of 6?”                  (Make three equal groups, take two groups.)

“2/3 of 6 is 4”.

Activity:

Provide students with 24 counters to represent their own flock of sheep. They need to work out for the farmer 1/2, 1/3, 1/4, 1/6, 1/8, 1/12 & 1/24 of the 24 sheep for the farmer. They make and record the number sentence for each as modelled.

Once students complete these fractions, have them work out as many of the following as they can.

1/2 of 24 is

1/3 of 24 is      2/3 of 24 is

1/4 of 24 is      2/4 of 24 is      3/4 of 24 is      4/4 of 24 is

1/6 of 24 is      2/6 of 24 is      3/6 of 24 is      4/6 of 24 is      5/6 of 24 is      6/6 of 24 is

1/8 of 24 is      2/8 of 24 is      3/8 of 24 is      4/8 of 24 is      5/8 of 24 is      6/8 of 24 is      7/8 of 24 is      8/8 of 24 is

1/12 of 24 is     2/12 of 24 is    3/12 of 24 is   4/12 of 24 is    5/12 of 24 is    6/12 of 24 is   7/12 of 24 is     8/12 of 24 is      9/12 of 24 is    10/12 of 24 is   11/12 of 24 is      12/12 of 24 is

1/24 of 24 is     2/24 of 24 is    3/24 of 24 is   4/24 of 24 is    5/24 of 24 is    6/24 of 24 is   7/24 of 24 is      8/24 of 24 is      9/24 of 24 is    10/24 of 24 is   11/24 of 24 is      12/24 of 24 is      13/24 of 24 is     14/24 of 24 is    15/24 of 24 is   16/24 of 24 is    17/24 of 24 is    18/24 of 24 is   19/24 of 24 is      20/24 of 24 is      21/24 of 24 is    22/24 of 24 is   23/24 of 24 is    24/24 of 24 is

Discuss the results with the students as you write their solutions on the board. Discuss “Why do they appear to be in skip counting patterns?”

E.g., 1/6 of 24 is 4    2/6 of 24 is 8     3/6 of 24 is 12      4/6 of 24 is 16     5/6 of 24 is 20     6/6 of 24 is 24. (Because you are adding another group of four each time).

On completion of this task, have students fill in the following table. They are making a Fraction Wall with 24 being the whole, (whereas most Fraction Walls have one as the whole).

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

On completion, is should look like below:

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 1/2 1/3 2/3 3/3 1/4 2/4 3/4 4/4 1/6 2/6 3/6 4/6 5/6 6/6 1/8 2/8 3/8 4/8 5/8 6/9 7/8 8/8 1/ 12 2/ 12 3/ 12 4/ 12 5/ 12 6/ 12 7/ 12 8/ 12 9/ 12 10/12 11/12 12/12 1/ 24 2/ 12 3/24 4/ 24 5/ 24 6/ 24 7/ 24 8/ 24 9/ 24 10/24 11/24 12/24 13/24 14/24 15/24 16/24 17/24 18/24 19/24 20/24 21/24 22/24 23/24 24/24

Discuss why some fractions appear more than once, such as “How come 1/2 of 24 is 12, but so is 2/4 of 24, 3/6 of 24, 4/8 of 24, 6/12 of 24 and 12/24 of 24? How can all these different fractions equal the same thing?” (Equivalent fractions. Actually are not different fractions but different ways of saying the same thing).

Reflection:

3 Things I learnt.

2 Things I am better at.

I Thing I am unsure about.

Enjoy :)

Teaching Operations with Fractions Visually. (Appropriate for Year 5 and above).

There are two things I remember from my time in Primary School regarding learning maths. The first was solving fractions, which I never understood, but worked out that quite often the answer was the denominator in the question (E.g., 1/2 of 4 = 2). This obviously did not always work, but gave me enough correct to satisfy the teacher that maybe I had at least learned something, when in actual fact I had no clue as to what was going on.

Fractions are still a mystery to many students, resulting in many adults thinking similarly, and hence the thinking that all mathematics is a mystery and only accessible to the mathematically gifted. I want to allow access and success for all students in all of mathematics, including the “mystery” of fractions.

Firstly, if we go back to first principles and look at the simple array, such as a 2x3 array block of chocolate.

We can describe this array in a variety of mathematical ways.

Most simply is represents 3+3=6 or 2+2+2=6 as Repeated Addition.

It also represents respectively 2x3=6 and 3x2=6 Multiplicatively.

Furthermore it also shows 6÷2=3 and 6÷3=2 as Division.

These representations link to ½ of 6 is 3 and 1/3 of 6 is 2.

It can also therefore also show 2/2 of 6 is 6 as well as 2/3 of 6 is 4 and 3/3 of 6 is 6.

In other words, they are all different ways of saying the same thing. Importantly they are not different topics in mathematics but are interchangeable and can all described in the one simple array. Progression from Foundation to Year 6 should be a flowing understanding of how to represent arrays in the ways described above.

I presented the following problem to my class.

“Last night I bought home some chocolate for my two children. My daughter ate 1/2 of the whole block and my son ate 1/3 of the whole block. How much of the whole block was eaten?”

After allowing a couple of minutes for children to contemplate the problem and discuss among themselves, when they and their partner have agreed how much was eaten we share solutions. Over 90% of all Year 3/4 students correctly describe my daughter ate 3 pieces (1/2 of the block which is 3 pieces out of the 6) and my son ate 2 pieces (1/3 of the block which is 2 pieces out of the 6), therefore between them ate 5 out of the 6 pieces (3 pieces plus 2 pieces), or 5/6 of the block.

I modeled the formal representation of the original question as 1/2 + 1/3 =5/6 for them.

The representations can be modelled as

Problem:             1/2         +             1/3         =             ?

Solution:              3/6         +             2/6         =             5/6

Labelling the diagram in 1/2’s and 1/3 also helps visualisation.

1/3             2/3               3/3

1/2

2/2

Incredibly they have shown they can already understand and solve this without any formal introduction to fractions.

When students were given a further problem they applied this understanding to more challenging situations.

Again,

Problem:            1/3         +             1/5         =             ?

Solution:              5/15       +             3/15       =             8/15

I.e., 5 pieces out of the 15 and 3 pieces out of the 15 makes 8 out of the 15 pieces.

I allowed the students to now make up a problem of their own using unit fractions (with one as the numerator). Students drew, wrote then shared their solutions with the rest of the class to solve. This will allow strengthening of their understanding of adding fractions.

Note that this method of developing fractions is all visual. This is deliberate with students developing a real understanding of how fractions work without teaching of any of the “rules” of fractions. Hence there is no need to teach Greatest Common Factor (GCF) or Lowest Common Multiple (LCM) that always causes much confusion and frustration in the classroom.

Students can then work out the shortcuts for adding two fraction with unlike denominators.

Students can quickly make assumptions as to how to find a rule without drawing the diagram. They are in effect deriving their own rule for finding the solution, but have invented it themselves, and if they forget, can always go back to first principles and draw the diagram. And they can explain why their rule works!

If

1/2+ 1/3 = 5/6

and

1/3 + 1/5 = 8/15

Students can quickly assume multiplying the denominators produces the denominator in the solution (2 x 3 = 6 and 3 x 5 = 15 respectively). This always works because students have made a 2 x 3 array in the top example, or a 3 x 5 array in the second example.

They also find adding the denominators give the numerator in the solution (2 + 3 = 5 and 3 + 5 = 8 respectively), because this is the number for pieces of each.

Students can test this with any unit fractions of their own, and check their solution by making the diagram. E.g., 1/5 + 1/6 should equal 11/30. (5 x 6 = 30, and 5 + 6 = 11).

Check

1/5            2/5            3/5          4/5           5/5

1/6

2/6

3/6

4/6

5/6

Problem:             1/5         +             1/6         =             ?

Solution:              6/30       +             5/30       =             11/30

I.e., 6 pieces out of the 30 and 5 pieces out of the 30 makes 11 out of the 30 pieces.

Subsequent lessons can look similarly at other fractions with numerators greater than one, which leads to solutions with improper fractions. But drawing these solutions out clearly models improper fractions can be drawn as mixed numbers.

1/3 + 2/5 = ?

1/5             2/5            3/5            4/5           5/5

1/3

2/3

3/3

Again,

Problem:            1/3         +             2/5         =             ?

Solution:              5/15       +             6/15       =             11/15

This even works with a mixed number as the solution.

1/2 + 3/4 = ?

1/4         2/4         3/4         4/4

1/2

2/2

Problem:             1/2         +             3/4         =             ?

Solution:              4/8         +             6/8         =             10/8        or       1 and 2/8          or       1 and 1/4.

So, if

1/3 + 2/5 = 11/15

and

1/2 + 3/4 = 10/8,  what are the shortcuts to find the solution without all the diagrams?

Using their originally developed rule of multiplying the denominators gives the denominator in the solution (again because it forms an array) still applies.

Multiplying the numerator of each fraction with the denominator of the other fraction (cross multiplication) gives us the two numerators to add to find the numerator of the solution.

1/2 + 3/4 = 10/8 Denominator: 2 x 4 = 8                   Numerator: (1 x 4) + (3 x 2) = 10

2/5 + 3/7 = 29/35              Denominator: 5 x 7 = 35                Numerator: (2 x 7) + (3 x 5) = 29

Using these shortcuts, what would 2/3 + 5/9 = ?

Denominator: 3 x 9 = 27. Numerator: (2 x 9) + (3 x 5) = 33. Solution 33/27 or 1 and 5/27.

I do not expect students to remember these shortcuts. If they can that’s a bonus, but I still expect them to be able to explain why 1/2 + 3/4 = 1 and 1/4.

This strategy of firstly drawing the two fractions in the one array can similarly be used for the subtraction, multiplication and division of fractions. No rules. No formulae. No forgetting. All understanding. If we build our foundations on understanding, students progress quickly and confidently and with a respect for mathematics that remembering rules cannot.

The second thing I remember from my time in Primary School regarding learning maths is a blue and a white make an orange. But at least this maths made sense! (If you don’t know what this means, ask one of your more experienced co-workers about Cuisenaire rods).

Fractions with Pattern Blocks.  (Appropriate for Year 5 and above)

Provide students with multiples of the Yellow Hexagon, Red trapezium, Blue Rhombus and Green Triangle. Pose the following question.

“If the Yellow Hexagon has a value of One, what are the values of the other shapes?”

Students quickly work out through playing with the relationships the following.

Yellow Hexagon                =             1

Red Trapezium                  =             1/2

Blue Rhombus                   =             1/3

Green Triangle                  =             1/6

Discuss how they know these are correct. Once this is estasblished, ask student to record all the ways to make One with a number sentence. E.g., 1/2  + 1/2 = 1

Students will come up with solutions such as

1/2 + 1/3 + 1/6 = 1

1/3 + 1/3 + 1/6 + 1/6 = 1    etc.

Share and discuss these with the class. Ask students if there is another way to write the number sentence.

1/3 + 1/3 + 1/6 + 1/6 = 1

Students should be able to explain using multiplication.

2 x 1/3 + 2 x 1/6 = 1

You may need to model the use of brackets to keep separate and so you know what goes with what.

( 2 x 1/3 ) + ( 2 x 1/6 ) = 1

In my experience I have found this will make perfect sense to the students.

Finally have the students make a picture with the Pattern Blocks, using only those four shapes we have been working with. Ask student what is the value of their picture and write a corresponding number sentence. Ask students to pile the pattern blocks on top of each other in hexagons (ones) make finding the total very easy.

Example 1.

Which can be rearranged into

( 3 x 1/2 )+ ( 2 x 1/3 ) + ( 4 x 1/6 ) = 2  and 5/6

Example 2.

1 + ( 3 x 1/2 ) + ( 2 x 1/3 ) + ( 3 x 1/6 ) = 3 and 4/6  or 3 and 2/3.

Fun, easy and makes total sense to students. No need for complicated fraction definitions ( such as for Equivalent Fractions) or rules to remember. Just playing with the blocks. I have modelled this lesson with numerous classes and never had any difficulty with the students not being able to explain what is happening and why. Enjoy!

Fraction Action. A game to teach visualisation of Fractions.

Rationale:

7/8 + 12/13 approximately equals a) 1,  b) 2,  c) 19,  d) 20.

The most common response for 4th graders? C) 19.

The most common response for 8th graders? C) 19.

 Results of Estimating a Fraction Sum Estimate the answer to 12/13 + 7/8. You will not have time to solve the problem using paper and pencil. Responses Percent responding Age 13 1 7 2 24 19 28 21 27 I don't know 14

(Post, T. (1981, May). Fractions: Results and Implications from National Assessment. The Arithmetic Teacher, 28(9), 26-31.)

Why are our students so poor at working with fractions? Why after another four years of schooling being taught fractions do most students still think the answer is c) 19? As a consequence of reading numerous research papers detailing similar findings detailing the way we teach fractions is failing our students, the following game was developed. Students answering the above question before and after the game displayed a huge turnaround in making sense of fractions, with a majority of students after playing the game for no more than 10 minutes were confident in explaining why the answer must be b) 2. The response was sustained even 2 two weeks after playing the game.

Playing Fraction Action

Draw 6 circles, one of each showing 1, 1/2, 1/3, 1/4, 1/5 and 1/6.

Roll two six sided dice and make a fraction equivalent to or less than one. Discuss this with students why, for example, 3/4 is less than one, while 4/3 is greater than one. Ensure the students are explaining, not you the teacher.

Colour in that exact amount in one of your circles.

Pass the dice to your partner who does similar with their dice roll and own circles.

The first player to have all their circles completely filled is the winner.

Equivalent fractions are allowed if they can explain why that should be allowed.

The exact amount must be coloured in each roll and must be in the one circle.

If you cannot colour in, forgo your turn and pass the dice to your partner.

Effective questions to ask at the conclusion of the game include “Which is the easiest circle to complete?” and “Which is the hardest circle to fill?” The answers to these questions may not be as obvious as first thought. Students often believe the “whole” (1) is the hardest to fill, but there are six ways to complete this circle, (1/1, 2/2, 3/3, 4/4, 5/5, 6/6). When we look at equivalent fractions there are a number of ways to fill each, but the fifths have no equivalents involved within the game using a six sided die and hence generally are the last to be filled.

Enjoy and watch as your students quickly are able to form mental visualisations of fractions, which is a vital stage in developing a working understanding of fractions.

Fractions, Decimals & Percents. (Appropriate for Year 5 and above)

Finding Fraction / Decimal / Percentage Equivalents of 100

E.g.1, Sharing 1 metre of bubblegum or liquorice with 2,3,4,5...friends. Provide students with a metre ruler or measuring tape to work out. Students can use MAB 10's and 1's to assist by lining up ten MAB 10's to begin with, then physically share between the required number of people. Express your solutions in fraction form, (for 2 friends, ½ each), cm, (50cm each), metres (0.5 metres each). Draw findings into a table. After draw student attention to the Percentage and Decimal terminology.

 Number of People Fraction Centimetres Metres 2 1/2 50 0.5 3 1/3 33 1/3 0.33 4 1/4 25 0.25 5 1/5 20 0.2 6 1/6 16 2/3 0.16 7 1/7 14 2/7 0.14 8 1/8 12 1/4 0.12 9 1/9 11 1/9 0.11 10 1/10 10 0.1 FRACTION PERCENT DECIMAL

Discuss in share relationship between cm and percentage, and metres and decimals.

E.g.2, Sharing an MAB chocolate block with 2,3,4,5,... friends. The flat (100) is the whole block, how many pieces will each friend receive? Express your solutions in fraction of whole block (2 friends, ½ the block), Number of pieces each received (50), and as a decimal (0.5). Students model and draw each solution also. Requires trading of the flat for longs and longs for ones.

Both above examples have fractions with 1 as the numerator (1/2, 1/3, 1/4, 1/5,...). Students then need to move to using numbers greater than 1 as the numerator.

Sharing the price and quantity of different foods between a picnic of zoo animals. (As the food will not be shared fairly). E.g., An Elephant, a Lion and a Penguin went on a picnic and shares the cost relative to the amount each ate. The Lion ate twice as much as the Penguin, and the Elephant ate twice as much as the Lion. (If you think that’s too hard, try the penguin ate 1/10, the lion ate 3/10 and the elephant 6/10 of each item). Select items from the supermarket catalogue and calculate the fraction each ate, how much (in volume or mass) each animal ate and how much each should pay for their share.