Pct 13 Traditional Math: Bar Modeling Approach for Solving Percent Problems; Part I
Seventh Grade
In previous lessons, students have learned to solve problems involving percentages: percent of change, and discounts and mark-ups, finding the amount of the reduction or the mark-up, as well as the final price, and original amount. All such problems rely on the part/whole relationships represented using proportions or directly translated into algebra from English. I have emphasized that students need to practice these problems in mixed (interleaved) settings.
I have found that the bar model approach is a useful way to cement in the part/whole relationships and proportions we have been working with. I happened to use this approach for percentages when teaching remotely during the Covid-related school shutdowns in 2020. I introduce it here as an option to be considered.
In view of introducing new information slowly so as not to burden students (in keeping with Cognitive Load Theory) I believe the bar model approach should be introduced after presentation of the various methods discussed in previous lessons. (This was the case when I taught it remotely.)
It is a pictorial method employed in Singapore’s Primary Math Textbook 6A (U.S. Edition). The bar model is in essence a diagram of the unitary approach. I structure the graphic a bit differently than is done in the textbook, but it is essentially the same approach which allows students to establish the proportion equations. Because of its pictorial nature, it also reinforces the concepts of what is the part and what is the whole, particularly when working with percentages greater than 100%.
Warm-Ups. In this set of Warm-Ups, there are problems that are the same but are stated in different form. This ties in directly with the particular lesson.
1. Solve by equation or proportion. 25 is 40% of what number? Answer: 25 = 0.4x; x = 62.5; OR 40/100 = 25/x; 0.4/1 = 25/x; 0.4x = 25; x = 62.5
2. A kayak is discounted 20%. The amount of the discount is $90. How much is the original price of the kayak? Answer: 20/100 = 90/x; 0.2x = 90; x = $450
3. For problem 2, what is the sales price of the kayak? Answer: $450 - $90= $360
4. A sweater is discounted 40%. The reduction from the original price is $25. What was the original price? Answer: 40/100 = 25/x or 0.4x = 25; x = $62.50
5. A bicycle store marks up the cost of bicycles by 200% and sells it for $4,590. What was the original cost of the bike that the store paid? Answer: (100+200)/100 = 4590/x; 300/100 = 4590/x; 3 = 4590/x; 3x = 4590; x = $1,530.
Problems 1 and 4 represent the same equation which I point out when discussing the Warm-Up answers. Ultimately the goal is for students to translate into algebra which can be done by proportion or by direct translation of English to algebraic expressions/equations. Problem 3 can be solved without calculating the original price of the kayak which this particular lesson will show. In Problem 5, the mark-up multiplier is 3, which students by now should have encountered in homework problems.
Using Bar Models for Discount (Decrease) Problems. I start with one of the warm-up problems; specifically Problem 1:
“A sweater is discounted 40%. The reduction from the original price is $25. What was the original price?”
We have already gone over the answer to this during the Warm-Up discussion. Now I present how to solve it using bar models.
“We start with a rectangle and we make a mark on the top part of the bar to indicate where the discount rate, 40%, is located. We’ll write it as 40 like we’ve done when we’ve solved these problems before. And the number that the 40% relates to in the problem goes on the bottom. What number is that?” Hearing “25” I write both on the diagram.
Having done that we now indicate where the 100% line should be. “We’ll let the entire bar represent 100%, so it goes at the very end. And this is going to represent the original price. Do we know what that is? So how do we represent that?”
We end up with the following:
“This diagram is a map that gives us directions for writing the proportion we’ve been using for problems like this. The numbers at the top become the ratio that you’ve been writing in your proportion equations. We want part over whole, which in this case is discount rate/original rate. Which is the discount rate? Which is the original rate? Rather than write 40%/100% we write 40/100, which we also write how?”
I’m hoping to hear 0.4/1.
“The ratio of values is written by using the bottom numbers. What is that ratio? What does the equation look like?”
It is now the proportion equation that students have been working with:
Students are familiar with this form and already saw it in the Warm-Ups.
I present a discount problem in which we are given the original cost and want to find the sales price. I remind students that for such problems we still subtract the discount rate from 100%.
A watch normally sells for $180. It is discounted 10%. What is the sales price of the watch?
“If it’s discounted 10%, that means you pay what percent?” In other words: Subtract 10% from 100% to get 90%. “Where does the $180 go? And x?”
The proportion equation is:
Using Bar Models for Mark-up (Increase) Problems. I introduce mark-up problems that they’ve seen before, and show how to solve them using bar modeling.
“A cell phone is marked up 75% by the dealer. The sales price is $1,050. What is the original cost the dealer paid?”
(It is important to get across that in doing these diagrams that the same rules apply as in the proportion equations. For increases such as mark-ups we add the percent mark-up to 100%. This is the advantage of doing the bar models after having practiced with the proportion equations—they are simply applying the same rules in drawing the diagrams.)
We start out with the rectangle but since we are dealing with an increase, we know that the original cost is 100%. “We are adding 75% to the original cost which is 100%; what does that come to?”
Since it comes to 175%, that is the value at the end of the rectangle at the top. “What percent represents the original cost?”
I should hear someone say 100%. After putting these values at the top of the rectangle as shown below, I ask “What is the value that corresponds to 175%? How about 100%?” After discussion with the class, the diagram on the board (and in their notebooks) looks like this:
“Now we want part/whole again, which if you remember from doing mark-up problems is mark-up rate/over original rate. How do we write that?”
Hoping to hear 1.75/1 (but likely hearing 175/100) I ask them to write the proportion equation in their notebooks and walk around checking, answering questions and otherwise providing guidance to ensure they get this:
Some will have it written like this:
This is not wrong; the proportion equation will come out exactly the same way. In fact, the ratios can also be written by top number/bottom number:
This is not surprising given that the proportion a/b = c/d is equivalent to a/c = b/d. I choose to use the method above because it is consistent with the proportion approach which has been introduced previously. (Some programs such as JUMP Math teach this equivalency. I have chosen not to pursue it given that the information in the units on proportions and percents contain much information to digest and master. Also, students will encounter this equivalency relationship in algebra in more detail.)
No matter what proportion is used, the resulting equation comes out the same; that is:
With mark-ups, students may find it confusing that percents greater than 100 are considered “part”. I remind them that we are comparing the mark-up rate to the whole which is 100 percent of the original value. I also remind them that we can have percents that are greater than one hundred.
I present a discount problem. Like with increases, the same rules apply for discounts as in the proportion equations. That is, we still subtract the discount rate from 100%.
A watch normally sells for $180. It is discounted 10%. What is the sales price of the watch?
“If it’s discounted 10%, that means you pay what percent?” In other words: Subtract 10% from 100% to get 90%. “Where does the $180 go? And x?”
The proportion equation is:
THIS UNIT IS CONTINUED IN PART II.