At the beginning of this unit I tell students that at the end of the unit they will be solving problems similar to the following:
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?
There are a total of 180 blue, green and white beads in the ratio of 5:2:3, respectively. How many of each are there?
Ratios and proportions are perhaps one of the most significant part of seventh grade math. In this unit students learn the basics of algebraic and other techniques to solve various word problems. These algebraic basics for ratios and proportion are then put to use in the next unit on percents.
Under the Common Core standards, many concepts about ratios and proportions are covered in the interest of teaching students to make connections between these various topics. I cover the topics included in the Common Core standards, but give some less emphasis than others when it comes to proportional reasoning. I will provide more detail about this in a later section, but my rationale is that a solid foundation in ratios and proportions allows for building on that base in algebra, in which concepts such as unit rates, constant of proportionality, direct variation, slope, and proportions expressed in equation form are addressed.
My experience in teaching seventh graders is that students find the various topics confusing with the result that the basic ideas about proportional reasoning occasionally get lost in the shuffle. Therefore, my approach is to provide students with very basic procedures for the concepts that students will cover later when they have complete mastery of the fundamentals upon which to build.
Warm-Ups. Today’s warm-ups continue with a problem about averages, and some problems that will segue into the day’s lesson.
1. The average of Ryan and David’s weights is 78 lbs. If Ryan weighs 60 lbs how much does David weigh? Answer: Total weight of Ryan and David = 78 x 2 = 156. If Ryan weighs 60 lbs then David’s weight = 156 – 60 = 96.
2. Find the sum. -5 -9 -4 +3 – (-13) Answer: -18 +3+14 = -2
3. Write as an inequality and solve. The sum of three times a number and 17 is less than -98. Find the number. Answer: 3n + 17 < -98; 3n < -87; n < -29
4. Solve. (-2/5)x> 1/3 Answer: x < 1/3 ∙ (-5/2); x < -5/6
5. A box of fruit contains 5 apples and 7 oranges. Apples are what fraction of the total amount of fruit? Answer: Total fruit = 12. Fraction of apples: 5/12
Problem 1 extends the average problem from the previous warm up. Students now know that for an average m for n items, the total is mn. Given one of the weights of the boys, that weight when subtracted from the total equals the weight of the other boy.
Problem 5 sets the stage for a discussion that will occur in this lesson. Students have learned that fractions can express what a part is of a whole, which they can then apply to this problem.
What is a Ratio. Students have had some exposure to ratios in sixth grade, so it is not a foreign concept. To put it in context, however, I talk about how subtractions is one method for comparing quantities. For example, if I have twelve oranges and my friend has 15, subtraction tells me my friend has three more oranges than I have. He has 3 more oranges than I do.
But sometimes, subtraction is not a good way to compare. If a recipe for paint calls for 2 parts yellow to 3 parts blue, then if I use subtraction, that means I always use 1 part more of blue than yellow. I will ask the class: If I use 10 gallons of yellow, should I use 11 gallons blue?” The answer usually comes back “No”. Sometimes division is the way to compare; that method of comparison is called a ratio.
The definition is written on the board: A ratio is a comparison of two quantities by division.
Ratios can be expressed in various ways. For the paint recipe of 2 parts yellow to 3 parts blue, it can be written as 2:3, 2/3 or 2 to 3.
The Surf City Ratio. In introducing ratios, I often ask if the students have heard of “pi”, which most have, though most don’t know what it is. “We’ll be talking about what pi is in a later lesson, but it is in fact a ratio—a very famous one. It is the ratio of the circumference of a circle to the diameter of the circle. Ratios are used all the time in math. Now there’s one you probably haven’t heard of, but which is also quite famous, which I call the Surf City Ratio.”
No seventh grader I’ve met has heard of the song “Surf City” by Jan and Dean which was a hit in 1963. I take full advantage of this. “Surf City was a song that came out long before you were born, and the first line of the song describes a ratio.” Though I don’t divulge in fancy audio-visual presentations, this is one time that I pull out all the stops, and I will play the first line of the song: “Two girls for every boy!”
I stop the song right there, and write the first line on the board which students without hesitation and without being so directed write down in their notebooks. Students tend to have an innate sense of what is important. (All that said, I offer this accounting of the Surf City Ratio only as an example of how to introduce ratios. I’m sure there are equally effective and compelling ways.)
I then write the ratio on the board in the three ways: 2:1, Two to one, 2/1
Part to Part Ratios. The Surf City Ratio gives me an entre to talk about the two types of ratios, “part to part” and “part to whole”.
The Surf City Ratio is an example of a ‘Part to Part’ ratio. When we compare the parts that make up the whole, the ratio is part to part. Girls to boys in a classroom, or in a city represents a part to part ratio. For a box of candy, the ratio of chocolate to non-chocolate candies in the box is part-to-part.
“So if there were 2 boys in Surf City, there would be 4 girls. The ratio is 4/2. But when we write ratios, we write it in simplest form, just like we write fractions in lowest terms. What is the 4/2 ratio in simplest terms?”
Hearing 2/1, or 2:1, I continue.
“Now, let’s say we have a classroom where the ratio of girls to boys is in the Surf City ratio. If there are 10 girls, how many boys are there?”
I usually get a flurry of correct answers: 5.
“What about if there were 10 boys; how many girls would there be?”
Again, I usually get the correct answer: 20
Part to Whole Ratios. This is sometimes left out of textbooks in the introductory lessons on ratio, but I think it is important to distinguish between the two types. Both Singapore’s Primary Math series and JUMP Math include it in their initial lessons.
When we compare a part of something to the entire quantity, that’s part to whole. “If we have a classroom with a surf city ratio: 2 girls: 1 boy, what is the entire amount? 2 + 1 = 3. What is the ratio of girls to total students in such a classroom?” The answer is 2:3. Of importance is this ratio holds no matter how many students are in the classroom. The ratio of 2:3 is in simplest terms, but if there are 9 students in the classroom then there are 6 girls, since 6:9 = 2:3. In both cases, the ratio of part to whole represents the fraction of students in the class who are girls.
Examples to do in class: (These are based on questions from JUMP Math).
There are 8 boys and 5 girls in a class. What is the part:part ratio (boys: girls)?\ Answer: 8:5
What is the ratio of girls to the entire class? Answer: 5:13
A team has 2 wins for every loss. What is the ratio of wins to total games? Answer: 2:3
What is the ratio of losses to total games? Answer: 1:3
There are 7 girls in a class of 20 students. What is the ratio of girls: boys?
Answer: 7:13
What is the ratio of boys to total students? Answer: 13:20
There are 2 boys in a class of 11 students. What is the ratio of girls to total students?
Answer: 9:11
What is the fraction of girls in a class where the ratio of girls:boys is 6:7? Answer: 6/13
Different Units. Ratios involving measurements should have the same units. For example 1 foot: 1 yard is 1 ft:3 ft or 1:3. I’ll ask what the ratio is for 4 quarts to 3 gallons. Answer: Since there are 4 quarts to a gallon the ratio is 4:12 which simplifies to 1:3. (Because units of pints, quarts and gallons are not emphasized as much in early grades as in previous eras, many students do not know these. I will sometimes write the conversions on the board.)
More examples: 2 pounds to 6 ounces. Answer: 32:6, simplified to 16:3
9 inches to 1 yard: Answer: 9:36 = 1:4
10 yards to 10 feet: Answer: 30:10 = 3:1
Homework: I usually supplement the textbook in use with questions from JUMP Math and/or Singapore’s Primary Math series. For JUMP, the relevant material is in Assessment and Practice, 7.1, p. 12-13.
Problem 10 on p. 13 is an effective challenge problem which in my experience students enjoy solving:
In Mr. X’s class, 2/5 of the students are girls. In Ms Y’s class, 5/8 of the students are girls.
a) What is the ratio of girls to boys in each class? Mr. X’s class: Ms Y’s class:
b) Whose class has more girls than boys? How can you tell from the fraction? How can you tell from the ratio?