- Construct a mathematical argument
- Compare and multiply fractions
- Critique others’ mathematical arguments
- Farming Fractions Student Worksheet printable
- Whiteboard and markers
- Computer and projector or chart paper
Make a class set of the Farming Fractions Student Worksheet printable.
Step 1: Engage students in the lesson by presenting a faulty argument on the board, such as: “I read that each student should have 15 minutes of homework a night. Since there are 20 students in our class, I should be assigning 5 hours of homework each night.” (You can customize this example to meet your students’ interests.)
Step 2: Ask students to turn and talk with a partner about whether the statement is valid and why. Lead a discussion about which students’ arguments are most convincing.
Step 3: Review with students the guidelines for creating an argument. For example:
- Saying “because I think so” is not enough to construct an argument.
- A strong argument includes an explanation of the correct process.
- Using examples and drawings or models will help support your argument.
- When critiquing an argument, asking questions in your mind about how others reached their conclusions is a good way to start.
- You can look for errors in others’ logic or calculations and contrast it with the correct process to critique their arguments.
- It’s important to be polite when critiquing an argument.
Step 4: Explain that the class will now apply this strategy to a mathematical argument. Share the following example using a projector or chart paper: “Tina and Henry are having a snack. Tina pours 1/8 of a carton of milk into a red cup and 1/6 of a carton of milk into a blue cup. Henry wants the cup with the most milk, so he chooses the blue cup. Did he choose the right cup?” (Answer: Yes; 1/6 > 1/8).
Use the following questions to guide your class’s discussion:
- What is the process for comparing fractions?
- How could a drawing of the situation help us?
- Did Henry make any errors in his calculations?
Step 5: Present another argument to the class: “Hassan needs 1/3 cup of olive oil for a recipe. He has 1/4 cup of olive oil in his kitchen. Does he have enough olive oil to make the recipe?” (Answer: No; 1/4 < 1/3) Ask students to work with a partner to determine whose reasoning is correct.
Step 6: Invite pairs to share their reasoning with the class.
Step 7: Have students complete the Farming Fractions Student Worksheet printable to practice constructing and critiquing arguments.
1a. No, the friend is wrong because 1/2 is actually bigger than 1/4. The denominator tells how many total equal pieces the whole is broken into. A bigger denominator means the whole is broken into more pieces, so each piece is smaller than if the whole were broken into fewer pieces.
1b. Grass is the largest part of the cows’ diet because it’s more than 1/2, so any fraction that’s left would have to be less than 1/2 (because if you were to add 1/2 to more than 1/2, you get a total greater than 1, but the cow only has one diet). Therefore, no other fraction of the diet can be bigger than the grass portion.
1c. 1/2 > 1/4
2b. 1/5 is less than 1/4 because 5 is the bigger denominator (see 1a, above), so 1/5 has to be a smaller amount than the 1/4 maximum.
2c. 1/5 < 1/4
3a. Nevaeh’s estimate of 5 homes is closer.
3b. 1/15 of the energy for one home multiplied by 80 cows equals 5.33.