Lesson 16

The purpose of this warm-up is to elicit observations about fractions in tenths and hundredths and about equivalence, which will be useful when students find sums of tenths and hundredths later in the lesson. While students may notice and wonder many things about these diagrams, focus the discussion on the relationship between tenths and hundredths and how we might express equivalent amounts.

• Groups of 2
• Display the diagrams.
• “What do you notice? What do you wonder?”
• 1 minute: quiet think time

• “Discuss your thinking with your partner.”
• 1 minute: partner discussion
• Record responses.

• “What fraction does each part in the first diagram represent?” (One tenth or \(\frac{1}{10}\))  “What about in the second diagram?” (One hundredth or \(\frac{1}{100}\))
• “Can you see tenths in both diagrams? Where?” (Yes. Each rectangle in A is a tenth. Each group of small squares in B is a tenth.)
• “Can you see hundredths in both diagrams? Where?” (No, only in B. Each square is a hundredth.)
• “Do you think the shaded parts of the two diagrams represent the same fraction or different fractions? Which fraction(s)?” (The same fraction, \(\frac{15}{100}\). Different fractions: The second one represents \(\frac{15}{100}\), but I'm not sure about the first one.)

In this activity, students refresh what they know about equivalent fractions in tenths and hundredths. Students are given fractions in tenths and are to write equivalent fractions in hundredths, and vice versa. In one case, they encounter a fraction in hundredths that cannot be written as tenths and consider why this might be. The work here reminds students of the relative sizes of tenths and hundredths and prepares students to add such fractions in upcoming activities.

• Groups of 2
• Consider asking students some of these questions:
  • “What do you know about 1 tenth? What about 1 hundredth?”
  • “Which is greater: 1 tenth or 1 hundredth?”
  • “How many hundredths are in 1 tenth?”

• “Work independently on the activity for 5 minutes. Then, share your responses with your partner.”
• 5 minutes: independent work time
• 2–3 minutes: partner discussion

• Select students to share their responses and reasoning for the first set of problems. Display or record their responses.
• Discuss the fraction \( \frac{125}{100}\) and what students wrote for its equivalent in tenths.
• Invite students to share the fractions they thought of for the last set of problems. Focus the discussion on how they know what fractions  are between \(\frac{3}{10}\) and \(\frac{4}{10}\). 
• If not mentioned in students' explanations, ask: “Could that number be expressed in tenths? Why or why not?” (No, because there is not a whole number between 3 and 4.)
• Highlight explanations that show how expressing the \(\frac{3}{10}\) and \(\frac{4}{10}\) in hundredths would allow us to name the fractions between these given two fractions.

In this activity, students use jumps on number lines to visualize addition of tenths and hundredths and find the values of such sums. Using diagrams helps to reinforce the relative sizes of tenths and hundredths. It provides a visual reminder that all tenths can be expressed in terms of hundredths, and that some hundredths can be written in tenths, which can in turn help with addition of these fractions.

This is the first activity in which students are to write expressions and equations to represent sums of fractions with different denominators. Initially, students will likely find it helpful to write equivalent fractions in the same denominator. Later, as students become more fluent in expressing tenths in hundredths and vice versa, they may perform the rewriting mentally rather than on paper. When students create and compare their own representations for the context, they reason abstractly and quantitatively (MP2).

• Groups of 2
• Ask students what they know about kilometers.
• If not mentioned by students, explain that, just like mile (which may be more familiar), it is a unit of length, used to measure long distances.
• If students wonder how it is related to meter, explain that 1 kilometer is 1,000 meters. (Kilometers will be explored more closely in a future unit.)

• “Take a few quiet minutes to work on the first two problems. Then, share your responses with your partner.”
• 5 minutes: independent work time
• 2 minutes: partner discussion
• Monitor for the ways students think about the total distance Noah has walked (third problem) given a fraction in tenths and one in hundredths.
• “Now try finding the values of the sums in the last problem.”
• 5 minutes: independent or partner work time

• Invite students to share how they know which diagram represents Noah’s walk and their equations for the distances Noah and Jada walked.
• Given the number line diagram for support, students are likely to write \(\frac{2}{10} + \frac{5}{100} = \frac{25}{100}\). Discuss why this is true.
• “How do you know that the sum of \(\frac{2}{10}\) and \(\frac{5}{100}\) is \(\frac{25}{100}\)?”
• Highlight that \(\frac{2}{10}\) is equivalent to \(\frac{20}{100}\) and another \(\frac{5}{100}\) makes \(\frac{25}{100}\).
• Consider displaying a number line that is partitioned into tenths and hundredths and shows \(\frac{25}{100}\) as \(\frac{20}{100} + \frac{5}{100}\).