# Lesson 16

Tenths and Hundredths, Together

## Warm-up: Notice and Wonder: Shaded Rectangles and Squares (10 minutes)

### Narrative

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.

### Launch

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

### Activity

• 1 minute: partner discussion
• Record responses.

### Student Facing

Each large square represents 1.

What do you notice? What do you wonder?

### Student Response

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### Activity Synthesis

• “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.)

## Activity 1: Tenths and Hundredths (15 minutes)

### Narrative

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.

Representation: Access for Perception. Invite students to examine a meter stick, and identify correspondences between this and the number line: One centimeter is one hundredth of a meter and ten centimeters is one tenth of a meter (called a decimeter). Clearly mark decimeters on the meter stick, and invite students to come back to reference this concrete representation as they work on the task.
Supports accessibility for: Conceptual Processing, Visual-Spatial Processing

### Launch

• 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?”

### Activity

• “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

### Student Facing

1. Complete the table with equivalent fractions in tenths or hundredths. In the last row, write a new pair of equivalent fractions.
tenths     hundredths
a. $$\frac{1}{10}$$
b. $$\frac{4}{10}$$
c. $$\frac{6}{10}$$
d. $$\frac{50}{100}$$
e. $$\frac{90}{100}$$
f. $$\frac{12}{10}$$
g. $$\frac{200}{100}$$
h. $$2\frac{3}{10}$$
i. $$\frac{125}{100}$$
j. $$\phantom{\frac{\0}{\0}}$$
2. Name some fractions that are:

1. between $$\frac{50}{100}$$ and $$\frac{60}{100}$$
2. between $$\frac{3}{10}$$ and $$\frac{4}{10}$$

Be prepared to explain your reasoning.

### Student Response

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### Activity Synthesis

• 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.

## Activity 2: Walk, Stop, and Sip (20 minutes)

### Narrative

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).

MLR1 Stronger and Clearer Each Time. Synthesis: Before the whole-class discussion, give students time to meet with 2–3 partners to share and get feedback on their response to ”the total distance Noah has walked”. Invite listeners to ask questions, to press for details and to suggest mathematical language. Give students 2–3 minutes to revise their written explanation based on the feedback they receive.

### Launch

• Groups of 2
• 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.)

### Activity

• “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

### Student Facing

Noah walks $$\frac{2}{10}$$ kilometer (km), stops for a drink of water, walks $$\frac{5}{100}$$ kilometer, and stops for another sip.

1. Which number line diagram represents the distance Noah has walked? Explain how you know.

2. The diagram that you didn’t choose represents Jada’s walk. Write an equation to represent:

1. the total distance Jada has walked

2. the total distance Noah has walked

3. Find the value of each of the following sums. Show your reasoning. Use number lines if you find them helpful.

1. $$\frac{5}{10} + \frac{1}{10}$$

2. $$\frac{50}{100} + \frac{10}{100}$$

3. $$\frac{5}{10} + \frac{30}{100}$$

4. $$\frac{15}{100} + \frac{4}{10}$$

### Student Response

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### Activity Synthesis

• 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 $$\frac{20}{100} + \frac{5}{100}$$.

## Lesson Synthesis

### Lesson Synthesis

“Today we learned to find the sum of tenths and hundredths. We used what we know about equivalent fractions and what we know about adding fractions with the same denominator.”

“How do we find the sums of tenths and hundredths when the denominators are different?” (Either think about tenths in terms of hundredths or hundredths in terms of tenths. Then, add them together.)

Discuss the last two sums: $$\frac{5}{10}+\frac{30}{100}$$ and $$\frac{15}{100}+\frac{4}{10}$$.

“In each case, how do we know whether to rewrite the tenths as hundredths, or to write hundredths as tenths?” (Sample response:

• For $$\frac{5}{10}+\frac{30}{100}$$, either way works. $$\frac{5}{10}$$ is equivalent to $$\frac{50}{100}$$ and $$\frac{30}{100}$$ is equivalent to $$\frac{3}{10}$$.
• For $$\frac{15}{100}+\frac{4}{10}$$, we’d write in hundredths, because $$\frac{4}{10}$$is equivalent to $$\frac{40}{100}$$ but $$\frac{15}{100}$$ has no equivalence in tenths.)

## Cool-down: Some Sums (5 minutes)

### Cool-Down

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