Teaching Tips – Percent Calculations
No matter which method your students use to solve percent problems, they should always start with a picture.
Here are three examples to illustrate this:
1.Harrison saved $12 when he bought a pink tutu that was discounted by 15%. What was the original price?
2.A swan gave birth to 5 baby swans. Two of them were male. What percent were male?
3.A 20 pound turkey lost 15% of its weight during cooking. How many kilograms were lost?
There are two different pictures you can draw – a percent number line, or a vertical rectangle. I am partial to the vertical rectangle, for a reason I will outline below.
Percent Number Line
Always start with the following empty number line. Amounts are written above the number line, and percents are written below the number line.
In a typical percent problem, you are given two quantities and have to find the third. Read the question carefully, and decide where to put the two quantities. Then decide which quantity you are looking for, and put the variable there. In question 1, we are looking for the original amount (also called the whole amount), so the variable is written opposite 100%.
Three methods of solving percent problems
1.Set up a proportion - notice how a proportion just leaps out at you!
Now solve the usual way.
2.You can solve it by reasoning:
“From the picture, I see that $12 = 15% of the whole amount.
To find 1%, I have to divide by 15.
12 15 = 0.8
To find 100%, I have to multiply by 100.
0.8 x 100 = 80.
The original cost of the tutu was $80.”
3.You can solve it by writing the resulting equation and solving it. The percent equation for all percent problems is
part = percent x whole.{where the percent is written as a decimal}
I will use the flowchart method to solve the equation:
part = percent x whole
12 = 0.15 xw
The vertical rectangle
A teacher I know brings in a measuring cup to illustrate this method. He fills it three-quarters full with water and asks “How much water do I have?” The answer is either ¾ cup or 6 ounces, depending on which scale used. The point of this exercise is the same amount of water can be represented by two equivalent quantities.
Here is the picture for Question 1:
As you can see, this picture is equivalent to the percent number line. Conceptually, the difference is that here you are thinking of filling up a cup. For some students, filling up a cup may be easier to visualize.
Question 2
Here is the percent number line
Here is the vertical rectangle:
The easiest way to find p is to think “100 is 5 x 20, so p is 2 x 20, or 40. So 40% of the baby swans were male.” The vertical rectangle gives us space to write “x 20”, which helps the students see what they have to do. This is why I like the vertical rectangle.
Question 3
Here is the percent number line
Here is the vertical rectangle:
This can be solved by thinking “20 times 5 is 100, so p times 5 is 15; so p is 3. The turkey lost 3 kilograms.” Writing “x 5” on the vertical rectangle helps us decide how to proceed.
Another way to think about the problem is to add 10% to the picture and think “If 100% is 20, then 10% is 2, so 15% is 3.”
Summary
Irrespective of the method used to solve the problem, drawing a picture helps students visualize the problem.
In general, students should use number sense (i.e. proportional reasoning) where possible. This is possible when the numbers are “nice”.
Start every percent problem by drawing a picture.
If the numbers aren’t nice, then
- setting up and solving a proportion, or
- setting up and solving an equation
are the alternative methods.
A picture is worth a thousand words.
Some students may not yet be able to think proportionally. Such students typically think in terms of adding, rather than multiplying. I am not too sure how to overcome this, other than having them see that their method of solution leads to obviously incorrect answers. The resulting “cognitive dissonance” may help them revise their reasoning.
Did I mention the value of drawing a picture first?