This post provides details about an artifact that has a manipulative and visual representation of tax rate and discount rate. These contexts are used as an introduction to percent change. The manipulatives are presented on a Google Jamboard.
The price or original price is presented as dollar bill. The bill is cut into proportional pieces to show the increase or decrease amount, visually, as a part of the original amount. The pieces can be moved around the Jamboard and replaced by other denominations.
The slides are presented in the slide show below. They are arranged in the following order. the slides show the different positions of the manipulatives, e.g., how the $20 bill is cut into discount and sales price.
Slide 1: 5% Tax Rate for $20 price – compute the total to pay
Slide 2: 20% off discount for $20 original price – compute the sales price
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.
Percents is a very challenging concept for students often because it is presented in symbolic terms, e.g. 40% of a 25 is what? Students typically understand what 100% and 50% mean. The concept of percent can build on this intuitive understanding. Here is my approach using a Concrete-Representation-Abstract approach (from a lesson taught today).
First, I ask the students to match common terms used for various percents on a percent number line. The students typically understand the first four terms.
I build on their knowledge of 50% first by having them cut (see below) a pizza into 50% then 25%. Then they cut $200 into 50% then 25% and then count the money.
Many struggle with 1/4 and 3/4 (25% and 75%). The little less than half is the building block because it is the first step away from 0%, 50% and 100%. Here’s how I use this step (below). The students compute 50% or half of the $200 then estimate 40% which is “a little less than 50%.” If a student struggles with this step, we go back to the cutout money and they cut a little off of the 50% and guess how much money is left.