A pseudo- concrete representation of a sales price problem is shown below. This is what I use as an entry point for teaching these problems.
The entire shape represents the total price of $80. This is 100%, which in student language is “the whole thing.”
The discount rate is 25%. Cut with scissors to lop off the 25% which also lops off $20, which is the actual discount. Explain to the student that this 25% is part of the “whole thing.”
What remains is 75% or $60. This is the “new price” which is called the sales price.
In special education there is a tool called a task analysis. It is a formal approach of identifying the steps taken to demonstrate mastery of a skill. For example, putting on shoes with Velcro straps involves the following steps: get shoes, sit on chair, match shoes with feet (right to right), insert foot into respective shoe etc.
I have applied this approach to general curriculum math topics from counting money to solving using the Quadratic Formula. Below are the iterations of my task analysis for the objective count TENs, a FIVE and ONES (dollar bills) to pay a given price. The first shows a rough draft of notes I took as I actually counted out the money, going through each little step. The second shows the steps written out on a task analysis table I created. The third shows the final, typed version.
The table is used for assessment, collection of data and progress monitoring. The steps that are problematic can be targeted individually, e.g. skip counting by 10s.
This is a snippet of a handout that is a follow up to a previous percent activity that I posted. It walks students through the mathematical steps including subtracting the price. I found that the cutting part of the previous activity was effective but the steps were not as clear. This handout is an attempt to clarify the steps.
What is surprising is how confused and challenged students are by simply computing the new price. Part of it is working memory and part of it, I believe, is learned behavior. Students are trained in math to simply follow steps without understanding the meaning of the steps. I ask them what the 6 represents in the problem above and many are unclear.
Here are the full handouts worked out by this student. The tax part of the first handout has the discount rate and not the tax rate listed – error. I corrected this after the lesson.
BTW, this student has autism and struggles to complete work in class. He took a long time to complete it. One problem is that he doesn’t have the personal experience and prior knowledge about discounts and taxes. To supplement this he would be given money and an item to buy and act out the actual purchase of the item to see how he pays less money.
I continue to be surprised at how much of a challenge computing percent discount is for students. It’s prior knowledge. If you ask them to explain what a discount is in their own words you’ll get a response like “it makes something cost less.” The students may even have mastery of computing a percent of a number. I believe that in part this is a working memory issue – extra step to process is a little too much.
The photo shows a scaffolded handout (links below) I created to help with the conceptual development of the steps for computing the new, discounted price along with the actual mathematical steps. In another post I showed the use of ten-dollar bills to conceptualize percents. This handout builds on that activity and the scaffolding makes it easier for the students to access the concept. The students have an item that costs $250 and is on sale for 60% off  (allows for nice “round” numbers). The students cut out the discount and have the new price in their hand. When a student has trouble with the mathematical steps at the bottom of the page I review the act of cutting to help with concrete understanding.
On a follow-up pop quiz the vast majority of students were able to compute the new discounted price.
money handout (print in black and white per legal restrictions)
