Lesson 5

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.

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.

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 \(0.007=m\).

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

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.

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

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 \(b\).’” 

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. 

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.

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.)

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