A Direct Counting Approach For Fractions in PSLE Math

“The esscence of mathematics lies in its freedom”

- Georg Cantor

A Direct Counting Approach For Fractions in PSLE Math

Ho Soo Thong

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Copyright © June 2015 AceMath, Singapore. 

Abstract
This article shows how fractions evolved from the part-whole concept and how a bar modelling approach with Euclidean Algorithm can be used to obtain fractions in a counting approach. Some examples similar to recent PSLE questions will be used to illustrate this approach

Fractions with Part-Whole Concept

Figure 1 shows a scenario where there are 30 marbles in total and 12 of the marbles
are red in colour.

1.1

In terms of marbles, we have

2.1a

If we separate the marbles into five boxes (parts) of equal size of 6 marbles each, a
bar model for the concrete view is shown in Figure 2.

2.1

In terms of boxes,

2.2a

 

A fraction (from Latin fractus, “broken” ) represents a part of a whole or, alternatively,
any number of equal parts of a whole.
Figure 3 shows a bar model for the above mathematical situation.

3.1

In terms of an equal part (common part) in the whole, a fraction is given by

3.2a

 

Bar Modelling of Fractions with Ratios

Fractions can represent ratios and vice versa. Thus the fraction 2/5 (of the whole) is also represented by the ratio 2 : 5 (of the part to the whole).

If a part is 3/7 of a whole, then the ratio of the part to the whole is part : whole = 3 : 7

The part consists of 3 equal parts and the whole consists of 7 equal parts and as shown in Figure 4.

4

 

In general, if a part is p/q of a whole where q > p > 0, then we construct a bar model consisting of q equal parts as the whole and indicate p equal parts as the fraction of the whole as shown in Figure 5.

5

 

Problems with directly related parts

We will illustrate the bar modelling approach with some examples which are similar to some PSLE questions (See [1].[2]).

Example 1

A coffee house provides brown sugar and white sugar for coffee drinkers. After drinkers consumed an equal amount of brown and white sugar, 3/8 of the brown sugar and 2/3 of the white sugar were left.
What fraction of the sugar was brown sugar at first?

Solution

Figure 7 shows a bar model with 8 equal parts for the whole of brown sugar . 3 parts for the leftoverand 5 equal parts for the consumed brown sugar.

Similarly, the other bar model shows are 3 equal parts. 2 parts for the leftover and 1 part for the consumed white sugar.
Noting that

1 equal part of white sugar = 5 equal parts of brown sugar

We choose an equal part of brown sugar as one unit, then

7

At first, there are 3×5 + 8 = 23 units of all sugar (whole) and there are 8 units of brown sugar.
Therefore, the fraction of brown sugar at first = 8/15 .

 

Example 2

Ah Hua sells yoghurt in cups, each cup is topped with an equal amount of almond or cashew nuts.
In a single day, twice as many almond topped cups as cashew topped cups are sold. 1/3 of the almonds and 5/8 of the cashew nuts are left at the end of the day.
What is the fraction of the nuts was used?

Solution

In Figure 8, the whole of the almonds is 3 equal parts and the customers consumed 2 equal parts. For the cashew nuts, the whole is 8 equal parts and the customers consumed 3 equal parts.

9

Then we choose the smaller part as the common unit.

Number of units in the whole is 9 + 8 = 17.
Number of units left is 3 + 5 = 8.

Therefore, fraction of nuts left =8/17.

Note that each of the above two examples have two sets of equal parts and one of them is a multiple of the other. The smaller one is chosen as a common unit for counting.

 

Algebraic Symbol for Greatest Common Unit

Algebraic symbols are avoided in the bar model approach in PSLE related books, however algebraic symbols are used for algebraic expressions (see [3]).In a spiral curriculum, more complex problems will gradually appear at upper levels and pupils will need to use effective problem solving skills.

Use of an algebraic symbol is the first step into the word of algebra and it can be helpful for a clearer presentation form. We will begin with an algebraic symbol U for the greatest common unit in the Bar Model Approach.
As in another example, then the bar models in Figure 10 (Example 2 ) becomes

10

 

Total amount of nuts 9U + 8U = 17U.
Amount of nuts left is 3U + 5U = 8U.
Therefore, fraction of nuts left = 8/17 .

In general, Figure 12 shows the visualised Greatest Common Unit (part) Procedure

mX = nY

where m and n are relatively prime, that is m and n have no common factor other than 1.

12

where U represents the greatest common part of X and Y.

 

Example 3

Two Primary 6 classes A and B have an equal number of boys.
The fraction of boys in class A is 2/5 and the fraction of boys in class B is 3/7 .
What is the fraction of boys in both classes?

Solution

First, we construct a comparison bar model for the boys in both classes as shown in Figure 13.
Apply the Common Unit Procedure to the parts for the boys, there is a greatest common unit U as shown in Figure 14.

14

Total number of boys = 6U + 6U= 12U.
Total number of all students = 15U + 14U = 29U.
Fraction of boys in the two classes = 12/29 .

 

Example 4

A rich man made a will. In it, he decides that 2/5 of his fortune will be donated to an charity organisation. The remaining portion will be shared equally by his 4 children.
What fraction of the fortune will each child get?

Solution

First we construct a bar model with the fortune divided into 5 equal parts. Then 3 equal parts represent 3/5 of the whole fortune which will be shared by his 4 children as shown in Figure 15.

15

Next, we apply the Common Unit Procedure for a common unit U as shown in Figure 16.

16

Therefore, each child gets 3/20 of the fortune.

Note : The model explains the arithmetic process for finding the value of 1/5 of 3/4. This also illustrate the multiplication of two fractions 15 and 34 , that is 1/5×3/4=1×3/5×4=3/20

The final example will illustrate again the use of the greatest common unit procedure for various scenarios in [2], [3] and [4].

 

Example 5

A special cereal mix contains wheat, cashews and almonds.
The ratio of cashews to wheat is 1 : 3 and the ratio of almonds to wheat is 2 : 7.
What is the fraction of the wheat in the cereal?

Solution

In Figure 17, both bar models for the ratios have wheat as a common part. Considering the part for the wheat in two scenarios, we apply the Common Unit Procedure as shown.

17

Fraction of Wheat in the mixture is 21/34 .

Note : In a traditional approach, a smart student can give an answer with a pictorial view of the bar model in Figure 18.

18

Counting Problems at Primary Olympiad Level.
The greatest common unit procedure has been used to solve many challenging counting problems at Primary Olympiad level (see [4], [5],[6] and [7]).

 

References
[1] PSLE Examination Questions 2005 – 2009, Educational Publishing House Pte Ltd
[2] PSLE Examination Questions 2011 – 2013, Educational Publishing House Pte Ltd
[3] Kho Tek Hong, Yeo Sue Mei, James Lim The Singapore Model Method for Learning Mathematics, EPB Pan Pacific, Singapore 2009.
[4] Ho Soo Thong, Ho Shuyuan, Bar Model Method for PSLE and Beyond, AceMath, Singapore 2011
[5] Ho Soo Thong, Advanced Bar Model Method for Counting Problems, Asia Pacific Mathematics News letter Volume 2 Number 1, January 2012
[6] Ho Soo Thong, Bar Model method for Job Problems, AceMath, Singapore 2013
[7] Ho Soo Thong, Bar Model Approach to Linear Diophantine Equations, AceMath, Singapore 2013