[math technique] About Those Percentages…


During my first year of teaching Math, I could not make my students break down “word problems” with percent and set up an equation as I learned how to do it when I was in school. Even though I tried to re-explain them in several ways they would not “get it”.

percent-symbols-pile

 

Then I realized that they could not remember or made confusions when they had to translate words such as: “of”, “out of”, or “is”  into mathematical symbols as part of the equation.
One day, I walked into the room of a veteran teacher and all of the sudden I started starring for a few minutes at one of her classroom walls. When I saw the percent symbol “%” I could not believe that the solution to my problem was written right there on that classroom wall!

girl-eureka

She explained to me that this was a technique she learned in school when she grew up and not too many teachers are using it nowadays. She didn’t know its source but it was good enough to her SOLVE ANY KIND OF PERCENT PROBLEM. I tried to find where it was coming from or who came up with this idea but could not find any information. I started using this technique right away and was surprised at the results! It consists of a triangle that is split in three parts and depending on what you needed to find you would set up an equation as a proportion that normally would get solved using cross multiplication. After a few exercises, you would notice  how the variable moved from one section of the triangle to another depending on the problem. Students immediately noticed that there were only three possible problems with percents. This diagram is different from the one that uses the “part/whole/percent” technique. My students stated that sometimes they would become confused about what number was the “part” and what number was the “whole” and switched them by mistake. This technique uses the same concept of setting up a proportion in the same way according to the location of the variable. My students got excited because they found a “cool” method that finally worked for them. They admitted that they had never seen it before and were glad that their “percent nightmare” was over.

percent-1

They created posters with their own problems that they either solved themselves or gave them to another peer to solve. Later on, I have created a “foldable” that my students pasted it inside their notebook or kept it in a separate envelope attached to their notebook so they could take it out any time they needed it.

Making the foldable: 

This is actually a base fold used in origami that is called “waterbomb base” (“The Origami Handbook” by Rick Beech).

Step 1: Take a sheet of paper, preferably colored paper, 8 1/2 x 11 inches in size. Hold one corner with two fingers and bring it to the opposite long side where you are going to align it with and fold the paper forming a triangle.

Step 2: Open the paper flat. Take the upper corner where the creased was formed and repeat step 1 in the opposite direction. When you open it you will see three congruent triangles and a pentagonal shape that contains another triangle of the same size. Take a ruler and draw a line to separate the triangle from the rectangle that was part of the pentagon. Cut along the line using scissors. You will have another congruent triangle. All four triangles will form a square. Fold the square on both creases inward and outward to have flexibility on both sides of the paper.

Step 3: Fold the square forming two perpendicular lines of symmetry that cross half of the square. Again, fold inward and backward to have flexibility on both sides of the paper.

Step 4: Have the paper “standing” on two equal sides like a “tent”. Push inward the right and left sides along the crease forming a pyramid.

Working With The Diagram:

Squash the pyramid with two small triangles up and write the name of the technique: “The Triangle Diagram” on one small triangle. Draw the actual diagram with all its parts on the adjacent small triangle and separate the two triangles with a straight line.

First type of percent problem

Flip the right side of the large triangle to the left like you are turning the pages of a book. On the left side, you will see the words of the problem and the workout using a proportion and solving the equation for “x”. On the right side, you will find the diagram for the problem. You will read the problem starting with the “x” at the top counterclockwise: “WHAT is 30% of 120?” Because “x” is at the top of the diagram, it will be placed at the top of the fraction. Below “x” you will place 120 and 30% will be written as a fraction “out of a hundred”. x/120 = 30/100. Solve using cross multiplication. Your equation will be: x = 30 x 120 ÷ 100 = 36.

Second type of percent problem:

You will move “x” counterclockwise where the “%” symbol is and fill the other two spaces with two numbers. The problem will read: “28 is WHAT percent of 112?” You will set up the proportion IN THE SAME ORDER: 28/112 = x/100. Solve using cross multiplication. Your equation will be: x = 28 x 100 ÷ 112 = 25 %.

Third type of percent problem:

We will move “x”counterclockwise where the word “of” is placed. We will fill the other two spaces with two numbers. The problem will read: “42 is 28% of WHAT number?” You will set up the proportion IN THE SAME ORDER: 42/x = 28/100. Solve using cross multiplication. Your equation will be: x = 42 x 100 ÷ 28 = 150.

Why do I like this technique?

  1. Students will identify the words “of” and “is” and the “%” symbol in a word problem and find the numbers that are placed next to these words and % symbol and write them in the corresponding places on the diagram.
  2. The set up of the problem is the same no matter the position of “x”. Students will notice very quickly that they will find TWO numbers in any word problem that they will place inside the diagram and the space without a number will represent “x”.
  3. This diagram will help students that have a difficulty breaking down a word problem with percent to practice until they get comfortable understanding the meaning of percent in different situations and finding their own ways to solve these types of problems.
  4. This technique is similar to the “part/whole/percent” technique. However, students may get confused about what number is the “part” and what number is the “whole” assuming that “part” is always “smaller” than the whole. The set up of the problem being the same, students will be able to find values of percent greater than 100% in problems such as: “What percent of 112 is 286?”
  5. Anyone, regardless of their level of understanding word problems with percent, will be able to solve these problems because the technique is the same for all three types of problems assuming that they know that the meaning of percent is “out of a hundred” or “/100”, how to set up a proportion, and find the value of “x” using cross multiplication.
  6. Because the foldable has the shape of a pyramid, it could be placed standing up on students’ desk as they work on problems having a model available for each type of problem with percent.

*** The three photos with girls and lots of percent symbols are from http://www.pixabay.com

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