Having covered the basic structure of ratios, this lesson provides a problem solving heuristic for ratios, which the Common Core standards refers to as “tape diagrams” and which Singapore’s math books call “bar models”.
Tape diagrams are a pictorial method that is used in JUMP Math, which is my source for this technique. Also as addressed in JUMP, tape diagrams serve as an entry point to algebraic solutions, which are addressed in the subsequent lesson. Students generally enjoy the pictorial method, providing them a sense of accomplishment in solving problems that many previously considered to lie well outside of their abilities.
Warm Ups. The warm-ups continue with an extension of working with averages, as well as a multi-step problem and work with ratios.
1. Tim’s average score on the last four math tests is 92. If he gets a 97 on his next test, what is his average score? Answer: 92 x 4 = 368 total score for the 4 tests. 368+95 = 465 total score for 5 tests. Average score for 5 tests = 465/5 = 93.
2. Harry bought 155 oranges for $35. He found that 15 of them were rotten. He sold all the remaining oranges at 7 for $2. How much money did he make? Answer: 155-15 = 140 good oranges. 140 ÷ 7 = 20 groups of 7 oranges. $20 x 2 = $40. $40 - $35 = $5 which is amount of money made.
3. Mark makes 5 gallons of purple paint by mixing red and blue paint. He uses 2 gallons of red paint. What is the ratio of red to blue paint in his mixture? Answer: 2:3
4. Write the sentence as an inequality and solve. “A number a divided by 2 is no more than 6.” Answer: a/2 ≤ 6 or (1/2)a ≤ 6; a ≤ 6∙2; a ≤ 12
5. At the Cypress school there are three hundred students. Two hundred of them are girls. What is the ratio of girls:boys in simplest terms? Answer: 2:1
For the first problem, students use what they know about finding the total amount prior to finding the given average. In this problem they then have to add the fifth test score to obtain the total for five tests, and then divide by five to get the new average.
Problem 2 is a multi-step problem which is taken from Singapore’s Primary Math series. My experience in giving this problem is that seventh graders enjoy solving it, and look at it as a puzzle.
Using Tape Diagrams (Bar Models). Taking off from the previous lesson, I use the Surf City ratio (2 girls: 1 boy). Students have seen that the number of girls is 2 times the number of boys and as expressed is a part to part ratio. A part to whole ratio would be the number of either boys or girls to the total number. So the ratio of girls to total would be 2:3.
I give the students a problem. If the total number of students in a school is 600 and we have a 2:1 (Surf City) ratio of girls to boys, how many girls and boys attend the school?
To do this we draw boxes representing the 2:1 ratio as indicated:
Each box can represent any amount of students as long as all the boxes represent the same number. If a box = 50 then there are 100 girls and 50 boys. For this problem we know the total number of students (the whole) equals 300. The whole is represented by the total number of boxes—in this case, there are three.
“Since the number in each box must be equal, how do we find the number in each box if the total for all three boxes is 300?” I will ask. And I will generally get an answer immediately: 100.
I will ask how they calculated it and am satisfied if the answer resembles “Divide the total number by the number of boxes to get the number in each box.”
Now we can easily find the number of girls and boys by adding the numbers: 2 boxes of 100 each equals 200 girls, and one box of 100 is 100 boys.
How many total boxes? Do they have the same number of marbles in each box?
Examples: I provide examples for them to work in their notebooks or on mini-whiteboards.
I have red and green marbles in the ratio of 5:2, and a total of 28. How many of each do I have?
There is a total of 7 boxes, and a total of 28 marbles. 28 ÷ 7 = 4. 5 red boxes with 4 per box means 5 x 4 or 20 red marbles and 2 green boxes with 4 in each means 8 green marbles.
Variations: The total is 35 marbles. How many of each? Answer: 5 in each box—25 and 10 red and green, respectively.
I ask students if there is a more efficient way of finding the total number of boxes than by counting them. Usually someone will say that the sum of the two numbers in the ratio gives us the total. And on a good day, someone will volunteer that answer without my asking.
This next example varies in that the total is not given. Students are told there are 50 red marbles. How many green are there?
“Any ideas of how to solve this one?” If I need to give a hint I will say “What do we know about the number of red boxes?” This is usually enough to get them going. Since there are 50 red marbles and 5 boxes representing red, there are 50÷5 = 10 marbles per box. Thus there are 20 green marbles.
Similar problem: Ratio of girls : boys is 3:5. There are 30 boys. How many girls are there? Answer: 30 boys, 5 boxes = 6 per box. Girls = 3 x 5 = 15.
More Than Two Ratios. We can compare more than two things using ratios. Let’s say we have blue, green and white beads in the ratio of 5:2:3. I have the students draw the diagrams for the beads:
The total is 180 beads; how many of each are there? “We solve it the same way as when there were just two quantities; who can do this?” Generally, students can make the leap, but if not, a prompt like “How many boxes are there?” gets them going. Since there are six boxes, then 180 ÷ 10 = 18 beads per box.
Blue: 90, Green: 36, White: 54
I remind students that I mentioned this problem at the start of this unit and said they would be able to solve problems like it. Usually they are amazed at this. But in all honesty, sometimes they are not. You have to roll with whatever comes, sometimes.
Examples:
There are 3 apples, and 2 oranges for each plum at a fruit stand. There are 420 fruits altogether. How many of each fruit are there? Answer: Total of 6 boxes; 420/6 = 70, so there are 3 x 70 = 210 apples, 70 x 2 = 140 oranges, and 70 plums.
To make purple paint, the colors red, blue and white are mixed as follows: Two times as much blue as white. Three times as much red as white. If we want 30 gallons, how much of each of the colors do we mix?
Since the problem is stated differently than the others I may give them a prompt such as “How do we draw the boxes for blue and white?”
Next, are red boxes with respect to white:
Now the problem is easy. Answer: 30 ÷ 6 = 5 per box. White: 5, Blue: 10, Red: 15
More of One Thing than Another. The last structure I cover are problems which indicate how much more of one thing we have than another. I will pose a question: “Suppose we have the red and green marbles in a ratio of 5:2 again, but this time I tell you that there are 12 more red marbles than green marbles.”
Dead silence.
“What do we do first?”
If someone doesn’t suggest to draw the boxes then I will:
“Since the top row of boxes represent red marbles, how many more red boxes do we have than green boxes?”
Hearing “Two”, I ask “Those two boxes represent how many marbles?”
Usually this is all they need to hear for them to figure out that 12 ÷ 2 or 6 goes in all the boxes, giving us an answer of 30 red and 18 green.
“Now suppose I tell you that there are 40 more red marbles than green marbles. How many of each do I have? Raise hands when you have an answer.” I usually wait until I have at least three hands up before I call on someone. If the same people are raising hands, I will sometimes take a chance on someone who I feel pretty sure will have the correct answer but doesn’t want to risk it. This question isn’t too hard, since they have already seen the pattern. 40/2 = 20 per box giving us 100 red and 60 green.
Examples:
The ratio of roses to tulips is 2:3. There are ten more tulips than roses. How many of each are there? Answer: Since the tulips are one box more than roses, each box = 10, and there are 20 roses and 30 tulips.
The last example is a problem I gave at the start of the unit on ratios:
At Tidewater Tech, the ratio of full-time teachers to part-time teachers is 5:3. There are 18 more full-time teachers than part-time. How many of each are there? Answer: There are two more full-time teacher boxes than part-time, so each box = 18/2 = 9. There are 45 full time teachers and 27 part time teachers.
Homework. I use problems that are in JUMP Math’s Assessment and Practice books, a page of which I have reproduced below. (Used with permission from JUMP Math.):