Lesson 5
A New Way to Interpret $a$ over $b$
5.1: Recalling Ways of Solving (5 minutes)
Warmup
The purpose of this warmup is to apply what students have learned to some equations. Note that ${0.07}\div {10}$ and $10.17.2$ 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 warmup. Allow 12 minutes quiet think time, followed by a wholeclass 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 wholeclass discussion.
Supports accessibility for: Memory; Conceptual processing
Design Principle(s): Maximize metaawareness; Support sensemaking
Student Facing
Solve each equation.

\(35=7x\)

\(35=11x\)

\(7x=7.7\)

\(0.3x=2.1\)
 \(\frac25=\frac12 x\)
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 wholeclass 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.
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: Cooldown  Choosing Solutions (5 minutes)
CoolDown
Cooldowns 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 nonwhole 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.
\(\displaystyle \begin {align} \frac38x&=\frac43\\[5pt] x&=\frac{\frac43}{\frac38}\\[5pt] x&=\frac43\boldcdot \frac83\\[5pt] x&=\frac{32}{9}=3\frac59\\ \end {align}\)