Educators teaching math typically start with the “mathy” stuff first. For example, for finding the sales price teachers may start with showing students the steps to calculate (photo below).

I start with the concept, either with a pictorial representation or actual objects to represent the underlying concept. In the photo above, I show an object (related to the student’s interest – this student is into weight training) on sale. The $50 circled in yellow represent the original price. I explain the concept of being on sale and discount and show that 20% is $10 to take away (marked out). This leaves $40 (in green) which is the sales price. This allows for conceptual understanding before showing him the “mathy” way of doing the problem.

I am following the blog Adventures at the Pond (author has some great stuff!) and the following photo was posted on this blog.

This is an excellent alternate representation of a concept. I do not know how the author would present this in class but here’s what I would do. I would give the students the number line with the initial information and ask them to estimate the total minutes – no calculation.

I would then explain the factor in the percent part (multiply by 5 to get to 100%) and lead the students to carry this factor over into the minutes:This would lead into proportions nicely. I would present the number lines and have the students work out problems using this approach first (Universal Design for Learning) then redo the problems using proportions.

I have a student with autism who loves comic books, especially DC and Marvel, and is often disinterested in the math I have to offer. In an activity to shop for clothes using a $200 gift card, 40% discount and 6% sales tax I gave this student a print out of comic books for his shopping – see photo below for part of what he was given. He immediately worked through the problem.

This is a photo of matching cards for the topic of percent discount and percent tax.

Students are given a card with an item to buy, a percent discount and our state tax rate of 6% (not listed). We’ve identified 4 steps for to find the total amount to pay the cashier: compute discount, compute new price, compute tax, compute the total cost. There are two cards for each step, one that shows the step completed correctly and one with an incorrect step. Students are to identify the correct steps and arrange them in order.

This activity is useful for a variety of reasons.

By having the steps listed there are no calculations. The students can focus on the steps and the concept.

To figure out the correct order students engage in higher level thinking. They analyze and justify their choices.

Because the problems are worked out and students simply focus on choosing cards they can do much more on their own. This is engaging.

While they are engaged the teacher can circulate and ask leading questions.

It can be very surprising how poorly understood discount and tax can be for students yet most have bought something and paid tax, even items on sale. As such this is an important topic.

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)

I gave a student a pop quiz on percent change – find discount and tax – after she completed independent practice on this objective. The student followed the steps very well, as you can see in the photo (I added the ink later) up to the very end. At that point she added the tax to 6 for a total cost of $9.36 – good deal for an $80 coat!

This is hardly uncommon if not typical in education both at the secondary and post-secondary level. The focus is largely on steps and not concept building. I asked her follow up questions (as opposed to telling her what to do to fix it) to flesh out the concept. “What did you do first and why?” “What does discount mean?” “What did you do here (pointing to the 6/100 part)?” “What do you pay the cashier?” She was confused because she wasn’t thinking real life situation but got caught in the weeds of the math steps.

What often happens is that students give up when pressed to think. I often have students tell me “can’t you just tell me the answer?” and “you’re the teacher, you’re supposed to teach me.” I respond “Yes, I can show you and I already taught you; now it’s your turn.”