Lesson 4

Working with Fractions

These materials, when encountered before Algebra 1, Unit 5, Lesson 4 support success in that lesson.

4.1: Math Talk: Subtracting from 1 (5 minutes)

Launch

Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.

Student Facing

Evaluate mentally:

\(1 - \frac12\)

\(1 - \frac{1}{10}\)

\(1-\frac{3}{10}\)

\(1-\frac{5}{17}\)

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Ask students to share their strategies for each problem. Record and display their responses for all to see. To focus the discussion for the rest of the lesson, consider asking: To involve more students in the conversation, consider asking:

  • “Who can restate \(\underline{\hspace{.5in}}\)’s reasoning in a different way?”
  • “Did anyone have the same strategy but would explain it differently?”
  • “Did anyone solve the problem in a different way?”
  • “Does anyone want to add on to \(\underline{\hspace{.5in}}\)’s strategy?”
  • “Do you agree or disagree? Why?”

4.2: Partway There (20 minutes)

Activity

This activity is an opportunity to talk about subtracting fractions from 1, multiplying by fractions, and the distributive property, using the number line as a representation for sense-making. In the last question, students are asked to express regularity in repeated reasoning (MP8).

Monitor for students who write different, equivalent expressions to help them reason. In the synthesis, you will want to draw connections between these different expressions.

Launch

Ensure students understand that the provided number line represents the 60-mile drive in the first question. It would be best if students work through this problem without a calculator, to keep the focus on reasoning.

Student Facing

Suppose a driver is traveling from one city to another. A diagram is provided to help with the first question. Create additional diagrams as needed. Be prepared to explain your reasoning.

Number line from 0 to 60. 7 evenly spaced tick marks.

  1. The distance between the cities is 60 miles and the driver has driven \(\frac13\) of the way.
    1. How many miles has she driven?
    2. How many miles remain?
  2. She has driven \(\frac25\) of the way.
    1. How many miles has she driven?
    2. How many miles remain?
  3. The distance between the cities is 300 miles and she has driven \(\frac16\) of the way.
    1. How many miles has she driven?
    2. How many miles remain?
  4. A trip is \(x\) miles long, and the driver has gone \(\frac14\) of the way. Write an expression to represent how many miles remain in her trip.

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

By the end of this synthesis, you want to be displaying a diagram that looks something like this, either with \(x\) for the last question, or using only numbers for one of the other questions.

Diagram. Line labeled x. Three points plotted. Left side, 1 fourth x. Right side, x minus 1 fourth x.
 

Capitalize on any expressions students wrote while they were working, either numerical expressions or those for the last question containing an \(x\). Invite students to share their reasoning, and ensure students understand why the three expressions representing the remaining part of the drive are equivalent.

4.3: Distribute and Subtract and Multiply! (15 minutes)

Activity

This activity is an opportunity to practice creating and identifying equivalent expressions, using the properties that come into play in the associated Algebra 1 lesson. The expressions with exponents allow for review of performing operations in the conventional order (exponent before multiply), as well as the convention for raising a number to the 0 power.

Launch

Arrange students in groups of 2 or allow them to work individually.

Using a calculator for this activity would move the work away from reasoning about equivalent expressions and operations and properties to learning in what order to press buttons. Encourage students to reason about equivalent expressions and perform calculations without a calculator.

Student Facing

  1. Explain why each pair of expressions is equal.
    1. \((1 - \frac15) \boldcdot 20\) and \(\frac45 \boldcdot 20\)
    2. \(24 - \frac13 \boldcdot 24\) and \(24(1 - \frac13)\)
    3. \(64 - \frac14 \boldcdot 64\) and \(\frac34 \boldcdot 64\)
  2. Match each expression in List A with an equal expression in List B.

List A

\(\frac14 \boldcdot 80\)

\(\frac34 \boldcdot 80\)

\(80 \left(1 - \frac{5}{8}\right)\)

\(80 - \frac18 \boldcdot 80\)

\(\frac{3}{10} \boldcdot 80\)

\(\frac{7}{10} \boldcdot 80\)

\(80\left(\frac14\right)^2\)

\(80\left(\frac12\right)^3\)

\(80 \left(\frac34\right)^0\)

List B

\(80 - \frac58 \boldcdot 80\)

20

\(80 \boldcdot \left(\frac{1}{16}\right)\)

\(\left(1 - \frac14\right) \boldcdot 80\)

56

70

80

\(\left(1-\frac{7}{10}\right) \boldcdot 80\)

10

Student Response

Student responses to this activity are available at one of our IM Certified Partners

Activity Synthesis

Display the correct matches and encourage students to ask questions about any that they had a hard time figuring out. Spend a little time analyzing the expressions that use exponents, reminding students that the conventional order is to evaluate the exponent first, and then multiply. Students may need help understanding how to evaluate a fraction raised to a power.