A Direct Bar Model Approach to Number Problems

“The esscence of mathematics lies in its freedom”

- Georg Cantor

A Direct Bar Model Approach to Number Problems

Ho Soo Thong

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Copyright © June 2016, AceMath, Singapore. 

Abstract
This article illustrates a direct use of simple arithmetic operations in the bar model approach to two simple Number problems

Bar Model Method uses pictorial view of mathematical situations based on the Part-Whole and Comparison concepts. In an early article “Bar Model Method for Averages of Numbers in Statistics”, we visualise the meaning and the implication of averages (arithmetic means) in part-whole bar models and deal directly with them as mathematical quantities for problem solving at primary school level.

In the following, we will apply a mathematical implication of Distributive Law and the use of simple arithmetic operations to illustrate a direct approach to two number problems at Primary Olympiad level.

Example 1
There are three integers X, Y and Z.
The average of Y and Z is 16 lower than the average of X, Y and Z.
The average of X and Y is 14 higher than the average of X, Y and Z.
If Y is 160, what is the value of Z and of X?

Solution
In Figure 1,the first bar model depicts the sum in terms of the average of X, Y and Z.
Next, we construct part-whole bar models for the two given scenarios and compute the differences
Finally, we construct a bar model for the sum of the three numbers with Y = 160 and compute the value of Z and of X as shown in Figure 1..

1

 

The next problem involves indeterminate equations.

 

Example 2
There are four distinct integers.
The sum of the smallest integer and the average of the remaining three numbers is 15.
The sum of the largest integer and the average of the remaining three numbers is 29.
What is the greatest possible value of the largest integer?

Solution
1st bar model displays b, the largest number, and m, the average of the remaining three smaller integers.
2nd bar model displays a, the smallest integer, and m + d , the average of the remaining three larger integers.
By equating the two expressions for a common difference, we find the value of d.

2.1

With d = 7, we have

2.2

 

Since b is largest when m is smallest, that is when m = 5 and a = 3,
b = 29 − 5 = 24.

Note : The four integers are 3, 5, 7 and 24.

Reference
[1] Ho Soo Thong, Bar Model Approach for Averages of Numbers in Statistics, barmodelhost.com, Nov 2015.