We recently came across two entirely different question types and solved them using the exact same bar model.
Geometry Type of Question
Here’s the first question:
The figure below is made up of two overlapping shapes, a triangle and a square. The ratio of the area of the triangle to the area of the square is 5:2. The ratio of the unshaded part of the triangle to the unshaded part of the square is 5:1. Find the area of the shaded part if the length of each side of the square is 4 inches.
How would you solve this using bar model? Well, here’s our solution:
First, from the bottom pair, we see that since the ratio between the full triangle and full square is 5:2, the difference is 3 / 2 of the area of the square, i.e. (3/2)*16 = 24 sq. inch.
Now, referring to our definition in the top pair of bar models, 1 unit is the area of the unshaded portion of the square. Hence the difference in area is represented by 4 units.
Combining these two snippets of information, we have 4 units = 24 sq. inch., 1 unit = 6 sq. inch..
Since the area of the square = unshaded part of square + shaded part, we have:
Area of shaded part = 16 – 6 = 10 sq. inch.
Standard Type of Question
Now, try this simpler question:
Tom has 5 times as many pencils as his sister, Sally. After their parents gave each of them an equal number of pencils, Sally has 16 pencils and the ratio of Tom’s pencils to Sally’s becomes 5:2. How many pencils did their parents give each kid?
Here is our bar models:
Look familiar? They are the same bar models!
(We don’t have to solve this again – just replace “sq. inch.” with “pencils” from the solution above).
Getting Used to Bar Models
Of course, these two problems are related. We made up the second question just to illustrate the point.
Both these questions are fundamentally about ratios. Getting kids used to
I do not understand where the 3/2 come into play. How did you come up with 3/2? Please help me as I am a math teacher searching for alternative methods to teaching.
Hi Myeesha,
The area of the square is 16 sq inch (4×4). The ratio of the area of triangle to the area of square is 5:2. So for this ratio, the area of square is 2 units and the difference (area of triangle minus area of square) is 3 units (total 5 units). Then 1 unit is 16/2 (since 16 = 2 units) and the difference (3 units) is 3 * (16/2), i.e. (3/2)*16=24.
Hope this helps!
Do you mean that Sally has 16 pencils after she received pencils from her parents?
The problem states that Tom has 16 pencils after he received pencils form his parents.
but the bar model shows that a constant value + 1 unit = 16 for Sally after.
Yes! Edited the typo. Thanks for pointing out!