Lesson 5

A New Way to Interpret $a$ over $b$

5.1: Recalling Ways of Solving (5 minutes)

Warm-up

The purpose of this warm-up is to apply what students have learned to some equations. Note that ${0.07}\div {10}$ and $10.1-7.2$ should be easy to evaluate given that work with fluently computing with decimals precedes this unit.

Launch

Ask students to summarize what they learned in the previous lessons before setting them to work on this warm-up. Allow 1-2 minutes quiet think time, followed by a whole-class discussion.

Student Facing

Solve each equation. Be prepared to explain your reasoning.

\(0.07 = 10m\)

\(10.1 = t + 7.2\)

Student Response

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

Activity Synthesis

At the conclusion of the previous lesson, students should have seen that we can approach solving any equation of the form $px=q$ (where $p$ and $q$ are rational numbers and $x$ is unknown) by dividing each side by $p$. Also, we can approach solving any equation of the form $x+p=q$ by subtracting $p$ from each side. Discussion should focus on given $0.07=10m$, we can write $0.07 \div 10 = 10m \div 10$ and then $0.007=m$.

5.2: Interpreting $\frac{a}{b}$ (15 minutes)

Activity

Students solve more equations of the form \(px=q\) while interpreting the division as a fraction.

Launch

Arrange students in groups of 2. Give 5–10 minutes of quiet work time and time to share their responses with a partner, followed by a whole-class discussion.

Representation: Internalize Comprehension. Activate or supply background knowledge about division involving decimals and fractions. Allow students to use calculators to ensure inclusive participation in the activity.
Supports accessibility for: Memory; Conceptual processing
Conversing, Representing, Writing: MLR2 Collect and Display. As share their responses with a partner, circulate and listen to their conversations. Collect and display any vocabulary or representations students use (e.g., reciprocal, dividing, multiplying) to describe how to solve each equation. Continue to update collected student language once students move on to the activity. Remind students to borrow language from the display as needed. This will help student to use academic mathematical language during paired and group discussions to connect fractions with division.
Design Principle(s): Maximize meta-awareness; Support sense-making

Student Facing

Solve each equation.

  1. \(35=7x\)

  2. \(35=11x\)

  3. \(7x=7.7\)

  4. \(0.3x=2.1\)

  5. \(\frac25=\frac12 x\)

Student Response

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

Student Facing

Are you ready for more?

Solve the equation. Try to find some shortcuts.

\(\displaystyle \frac{1}{6} \boldcdot  \frac{3}{20} \boldcdot  \frac{5}{42} \boldcdot  \frac{7}{72} \boldcdot x = \frac{1}{384}\)

Student Response

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

Anticipated Misconceptions

Monitor for students who want to turn \(\frac{35}{11}\) into a decimal, and reassure them that \(\frac{35}{11}\) is a number.

Activity Synthesis

Define what \(\frac{a}{b}\) means when a and b are not whole numbers. Tell students, “In third grade, when you saw something like \(\frac25\), you learned that that meant ‘split up 1 into 5 equal pieces and take 2 of them.’ But that definition only makes sense for whole numbers; it doesn't make sense for something like \(\frac{2.1}{0.3}\) or \(\frac{\frac25}{\frac12}\). From now on, when you see something like \(\frac25\), you'll know that that means the number \(\frac25\) that has a spot on the number line, but it also means ‘2 divided by 5.’ The expression \(\frac{2.1}{0.3}\) means ‘the quotient of 2.1 and 0.3,’ the expression \(\frac{\frac25}{\frac12}\) means ‘the quotient of two fifths and one half,’ and generally, the expression \(\frac{a}{b}\) means ‘the quotient of \(a\) and \(b\)’ or  ‘\(a\) divided by \(b\).’” 

5.3: Storytime Again (15 minutes)

Activity

This is a continuation of the activities Storytime and More Storytime from previous lessons. Over time in this unit, we are reminding students of work they should have done in previous grades with expressions that represent particular, concrete relationships. In grade 6, students are working toward producing such expressions themselves to represent a context. 

Launch

Remind students of work they did previously to match a situation with an equation. For example, they matched the equation \(x+5=20\) with the situation “After Elena ran 5 miles on Friday, she had run a total of 20 miles for the week. How many miles did she run before Friday?” In this activity, they come up with their own situations that can be represented by equations.

Keep students in the same groups. Clarify that for each equation, each partner will come up with a story, and one of those stories is chosen. Give students 5–10 minutes to work with their partner, followed by a whole-class discussion.

Student Facing

Take turns with your partner telling a story that might be represented by each equation. Then, for each equation, choose one story, state what quantity \(x\) describes, and solve the equation. If you get stuck, consider drawing a diagram.

\(0.7 + x = 12\)

\(\frac{1}{4}x = \frac32\)

 

Student Response

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

Anticipated Misconceptions

For students with limited fraction and decimal understanding, coming up with a reasonable story where the numbers are not whole can be daunting. You might suggest that students imagine stories with similar structures that involve whole numbers, and then tweak the stories toward using the numbers given in the problems. Remind them that using fractions and decimals has to make sense in the situations, and encourage them to think about what kinds of situations those might be (measurement situations will usually work while those that involve counting discrete objects won't.)

Activity Synthesis

Invite students to share their stories. Ask each student to interpret the solution in terms of their situation. 

Writing, Speaking: MLR1 Stronger and Clearer Each Time. Use this routine with successive pair shares to give students a structured opportunity to revise and refine their writing. For this activity, students should use the story for the equation they chose to solve. Ask each student to meet with 2–3 other partners in a row for feedback. Provide students with prompts for feedback that will help teams strengthen their ideas and clarify their language (e.g., “How are the parts of the equation represented in your story?”, “Can you say more about how your solution fits in your story?”). Provide students with time to complete a final draft based on the feedback they receive about language and clarity.
Design Principle(s): Optimize output (for comparison)

Lesson Synthesis

Lesson Synthesis

Ask students to work with their partner. Each partner writes a number that is in fraction or decimal form. Have them choose one number to be the coefficient in an equation of the form \(px=q\) and the second number the quantity on the other side of the equal sign. They then work together to write and evaluate the solution of the equation. Complete multiple rounds as time allows. 

5.4: Cool-down - Choosing Solutions (5 minutes)

Cool-Down

Cool-downs for this lesson are available at one of our IM Certified Partners

Student Lesson Summary

Student Facing

In the past, you learned that a fraction such as \(\frac45\) can be thought of in a few ways. 

  • \(\frac45\) is a number you can locate on the number line by dividing the section between 0 and 1 into 5 equal parts and then counting 4 of those parts to the right of 0.
  • \(\frac45\) is the share that each person would have if 4 wholes were shared equally among 5 people. This means that \(\frac45\) is the result of dividing 4 by 5.

We can extend this meaning of a fraction as a quotient to fractions whose numerators and denominators are not whole numbers. For example, we can represent 4.5 pounds of rice divided into portions that each weigh 1.5 pounds as: \(\frac{4.5}{1.5} = 4.5\div{1.5} = 3\). In other words, \(\frac{4.5}{1.5}=3\) because the quotient of 4.5 and 1.5 is 3.

Fractions that involve non-whole numbers can also be used when we solve equations. 

Suppose a road under construction is \(\frac38\) finished and the length of the completed part is \(\frac43\) miles. How long will the road be when completed?

We can write the equation \(\frac38x=\frac43\) to represent the situation and solve the equation.

The completed road will be \(3\frac59\) or about 3.6 miles long.

\(\displaystyle \begin {align} \frac38x&=\frac43\\[5pt] x&=\frac{\frac43}{\frac38}\\[5pt] x&=\frac43\boldcdot \frac83\\[5pt] x&=\frac{32}{9}=3\frac59\\ \end {align}\)