Activity 1
Have 10 students raise their hands. Choose randomly using, for instance, 2 rows of 5 randomly seated students, etc. Ask students, “What fraction of the students with their hands raised are girls?”
Remind students that the denominator of the fraction should represent the whole—in this case, the number of students with their hands raised (10). The numerator should represent the part we’re interested in—in this case, the number of girls.
Write down the fraction of the students who are girls.
“What fraction of the students with their hands raised are boys?” Write this down as well, and the students can lower their hands.
“Fractions are one way to compare amounts; they compare a part to a whole. We can use another comparison, ratios, to compare amounts as well. Ratios often compare a part to another part. For instance, we might talk about the ratio of girls to boys in the 10 students who had their hands raised.”
Write the ratio of girls to boys on the board using a colon, such as 7:3. Don’t tell students what the ratio represents.
“What do you think this represents, thinking back to how many boys and girls had their hands raised?” Students should recognize that one number represents the number of boys and the other represents the number of girls. Identify which number represents which quantity and label each number appropriately.
“This is a ratio of the number of girls to the number of boys. The order in which we read a ratio is important. If the ratio compares the number of girls to the number of boys, it means the first number represents girls and the second represents boys. How would I write the ratio of the number of boys to the number of girls?” (3:7)
“These two ratios compare parts of the whole. We can compare other quantities as well. How would I write the ratio of the number of boys to the total number of students who raised their hands?” (3:10)
Come up with other ratios students can create—for instance, the number of boys and girls in the entire class, different hair colors, etc.
Activity 2
“In addition to writing ratios like 3:8, using a colon to separate the two numbers, we can also write them as fractions. For example, we could write the ratio 3:8 as . It means exactly the same thing as 3:8, and it’s important to specify that it’s a ratio. Usually, ratios compare a part to another part. For instance, what part of the class is boys and what part is girls? A fraction typically compares a part to a whole. For instance, how many boys are in the class compared to how many students in total are in the class? So, while they look the same, it’s important to understand the difference between them.”
“However, ratios can be reduced, just like fractions. Suppose there are 4 boys and 8 girls. What is the ratio of boys to girls?” (4:8) “We can reduce that, just like we would if it were the fraction 4/8, to 1:2. Now, does that mean there is 1 boy and there are 2 girls?” (No) “So what does it mean?” Guide students toward the understanding that it means there is 1 boy for every 2 girls. “In this sense, a ratio is kind of like a rate; for every boy we have, we know we have 2 girls.”
“So, using this ratio of 1 boy for every 2 girls, suppose there are 6 boys in a group that has this ratio. How many girls must there be in the group?” (12)
“Suppose there are 20 boys; how many girls must there be?” (40)
“Now, we’ll use it the other way. Suppose there are 18 girls in the group. How many boys must there be?” (9)
Have students work in pairs on the Using Ratios in Problem Solving sheet (M-6-7-1_Using Ratios in Problem Solving and KEY.docx).
Activity 3
Divide the class into groups depending on how many sets of playing cards (or other similar objects) have been prepared. Give each group a set of 20 playing cards.
“Each group has a pool of objects to work with. You’re going to determine how many different cards in total there could be in a set made from the cards you have using the given ratio. For example, if the given ratio of red to black cards is 7:12, how many different groups of cards can be made with that ratio? You can certainly make a group of 7 red cards and 12 black cards. Can you reduce the ratio?” (No)
“Can you use more objects? For example, can you multiply each number in the ratio by 2 to make it 14:24? Can you make a set of 14 red objects and 24 black objects?” (No) “Why not?” Students should note that they don’t have 14 red objects (or 24 black objects).
“As another example, how about a ratio of 3 red cards to every 5 black cards. Can you make a group of 3 red cards and 5 black cards?” (Yes) “Can you reduce the ratio 3:5?” (No) “Can you double each number in the ratio and make a group with 6 red cards and 10 black cards?” (Yes) “Can you triple the ratio and make a group with 9 red cards and 15 black cards?” (No) “So, for the ratio of 3:5, there are two possible sets you can make: one with 8 cards in total and one with 16 cards in total.”
Give each group a copy of the Sets with Equal Ratios sheet (M-6-7-1_Sets With Equal Ratios and KEY.docx).
The lesson begins by connecting new knowledge to existing knowledge. The lesson contains a class discussion portion, a small group activity, and a larger-group activity. It also incorporates verbal portions, mathematical portions, and a concrete, hands-on activity (although the cards are not essential.)
Extension:
Use the following strategies to tailor the lesson to meet the needs of your students throughout the year.
- Routine: Ratios show up frequently in real life, so real-life examples can be used to keep students “fresh” with ratios. The topic can also be combined with other subjects; for instance, examining the ratio of males and females in the U.S. Senate, the ratio of state names that start with a vowel, the ratio of countries in Europe to countries in South America, and so on.
- Small Group: Use this activity for students who may benefit from additional practice. Students can create different groups by combining all the cards together in Activity 3 and then selecting a random assortment of 20 cards (rather than the predetermined distribution of 8 red cards and 12 black cards). Students can explore ratios in which there are an equal number of cards of each type, as well as ratios in which there are a minimal number of cards of one type (such as 1 or 0 red cards).
- Expansion: Use this suggestion for students who are ready for a greater challenge. Students can explore multipart ratios (such as 1:2:5). They can also begin to incorporate algebra into problems with ratios; for instance, if a ratio is 1:2:5 and there are 124 objects in total, how many objects are there of the second type. (Set up 1x + 2x + 5x = 124, solve for x, and then multiply by 2.)
Students can also explore ratios in which one of the quantities (typically the first) is always reduced to 1 (to form a unit ratio). For instance, the ratio of a person’s “wingspan” to his/her height might be expressed as 1:1.2.
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