# Lesson 17

Sums of Tenths and Hundredths

## Warm-up: Which One Doesn’t Belong: Tenths and Hundredths (10 minutes)

### Narrative

This warm-up prompts students to carefully analyze and compare the features of four fractions. They may consider size (of the fraction, the numerator, or the denominator), equivalence, relationship to benchmark numbers, and more. The reasoning here will be helpful later in the lesson, as students classify sums of fractions by their size and relationship to 1.

### Launch

• Groups of 2
• Display the four expressions.
• “Pick one that doesn’t belong. Be ready to share why it doesn’t belong.”
• 1 minute: quiet think time

### Activity

• 2–3 minutes: partner discussion
• Share and record responses.

### Student Facing

Which one doesn't belong?

A.

$$\displaystyle \frac{48}{100}$$

B.

$$\displaystyle \frac{8}{10}$$

C.

$$\displaystyle \frac{120}{100}$$

D.

$$\displaystyle \frac{70}{100}$$

### Activity Synthesis

• Consider asking: “Let’s find at least one reason why each one doesn’t belong.”
• “Are any of these equal to 1?” (No)
• “Which of these fractions are greater than 1? How do you know?” ($$\frac{120}{100}$$, because it is greater than $$\frac{100}{100}$$.)

## Activity 1: Card Sort: Less Than, Equal to, or Greater Than 1? (20 minutes)

### Narrative

The purpose of this activity is for students to practice adding tenths and hundredths, by sorting a set of expressions based on whether their values are less than, equal to, or greater than 1. A sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP7). They decide whether it is necessary to write equivalent fractions, and if so, whether to use tenths or hundredths.

Through repeated reasoning, students build their ability to compare the size of hundredths to tenths and to 1 (MP8). They also have an opportunity to look for and make use of structure (MP7). For instance, students may conclude that certain expressions are greater than 1 by noticing that one of the addends is greater than 1.

Here is a list of the expressions on the blackline master, for reference:

A.

$$\frac{10}{100} + \frac{8}{10}$$

B.

$$\frac{80}{100} + \frac{2}{10}$$

C.

$$\frac{20}{10} + \frac{30}{100}$$

D.

$$\frac{7}{10} + \frac{8}{100}$$

E.

$$\frac{22}{100} + \frac{8}{100}$$

F.

$$\frac{12}{10} + \frac{8}{100}$$

G.

$$\frac{12}{100} + \frac{12}{10}$$

H.

$$\frac{73}{100} + \frac{3}{10}$$

I.

$$\frac{150}{100} + \frac{1}{10}$$

J.

$$\frac{9}{10} + \frac{11}{100}$$

K.

$$\frac{10}{100} + \frac{9}{10}$$

L.

$$\frac{6}{10} + \frac{39}{100}$$

MLR8 Discussion Supports. Students should take turns deciding where to place their card and explaining their reasoning to their group. Display the following sentence frame for all to see: “I noticed ___ , so I placed the card . . .” Encourage students to challenge each other when they disagree.

### Required Materials

Materials to Gather

Materials to Copy

• Card Sort: Less Than, Equal to, or Greater Than 1

### Required Preparation

• Create a set of cards from the blackline master for each group of 2-4 students.

### Launch

• Groups of 2–4
• Give each group one set of cards from the blackline master and a couple of sticky notes.

### Activity

• “Work with your group to sort the cards into three groups, based on whether the expressions are less than 1, equal to 1, or greater than 1.”
• “Be prepared to explain or show how you know where each sum belongs.”
• 8–10 minutes: group work time
• Monitor for:
• the ways students decide whether to write equivalent fractions in tenths or hundredths
• students who estimate the value of the expressions by looking at the relative size of the addends, without finding the sum
• “When you finish, visit another group’s sorted collection. Examine it and leave a note about any questions you have.”
• 3–4 minutes: Visit another group.

### Student Facing

1. Sort the cards from your teacher based on whether the value of the expression is less than 1, equal to 1, or greater than 1.

When done, make a quick list of which expressions you have in each group.

2. Visit the sorted collection of another group.

• Did they sort the cards the same way?
• Select 1–2 cards that you have a question about or whose placement you disagree with.
• Leave a note for the group members to discuss.

• Discuss any notes that are left for your group, or revise your sorting decision based on what you learned from another group.
• Record the expressions here.
less than 1 equal to 1 greater than 1

$$\phantom{\frac{\huge0}{\huge0}}$$

### Student Response

Some students may have a better intuition for how tenths or hundredths relate to $$\frac{1}{2}$$ than how they relate to one another, or how they relate to 1. Consider asking: “Which fraction in the pair is close to $$\frac{1}{2}$$?” and “Can this help you determine how this expression relates to 1?”

### Activity Synthesis

• Discuss questions such as:
• “How did you decide where each expression should go? Did you always write an equivalent fraction?”
• “When was it necessary to write an equivalent fraction? When was it not?”
• “Were there expressions you were able to sort without rewriting any fractions or adding anything? What was it about those expressions that made that possible?”

## Activity 2: What’s Missing? (15 minutes)

### Narrative

In previous activities, students learned to combine tenths and hundredths. In this activity, students complete addition equations to make them true. To do so, they rely on a range of understandings and skills: how to write equivalent fractions, how to add fractions, and how to decompose a fraction into a sum. Though many of the equations involve an unknown addend, students are not expected to find them by subtraction.

Engagement: Develop Effort and Persistence. Differentiate the degree of difficulty or complexity. Some students may benefit from starting with whole numbers to recall how they might decompose a sum to find a missing addend. For example, invite students to articulate strategies to find the missing number in $$10 + \underline{\hspace{0.5in}} = 30$$. Make sure to elicit responses that do not require subtraction.
Supports accessibility for: Conceptual Processing, Organization, Social-Emotional Functioning

### Launch

• Groups of 2
• Display these equations:
• $$\frac{1}{2} + \frac{4}{2} = 2$$
• $$\frac{9}{10} + \underline{\hspace{0.5in}} = 1$$
• “Are these equations true? Take a minute to think about it.”
• 1 minute: quiet think time
• Discuss responses.
• “Why is the first equation not true?” (The sum of the fractions on the left is $$\frac{5}{2}$$, which does not equal 2.)
• “Why are $$\frac{1}{10}$$ and $$\frac{10}{100}$$ both true for the last equation?” (They are equivalent, so when added to $$\frac{9}{10}$$ both result in 1.)
• “Let’s find some other fractions that would make equations true.”

### Activity

• “Work independently to complete at least three equations from the first problem and three from the second before discussing with your partner.”
• 6–7 minutes: independent work time
• 3–4 minutes: partner work time
• Monitor for the equations that seem to be challenging to many students or to be prone to errors. Discuss them during synthesis.

### Student Facing

1. Each equation is missing a fraction in hundredths. Find the fraction that makes each equation true.

1. $$\frac{10}{100} + \underline{\hspace{.5in}} = \frac{30}{100}\\$$
2. $$\underline{\hspace{.5in}} + \frac{2}{10} = \frac{80}{100} \\$$
3. $$\frac{7}{10} + \underline{\hspace{.5in}} = \frac{94}{100} \\$$
4. $$\frac{9}{100} + \underline{\hspace{.5in}} = \frac{8}{10} \\$$
5. $$\frac{16}{100} + \frac{4}{10} = \underline{\hspace{.5in}} \\$$
6. $$\underline{\hspace{.5in}}+\frac{14}{10} = \frac{172}{100} \\$$
2. Each equation is missing a fraction in tenths or hundredths. Find the fraction that makes each equation true.

1. $$\frac{20}{100} + \underline{\hspace{.5in}} = \frac{28}{100}\\$$
2. $$\frac{110}{100} + \underline{\hspace{.5in}} = \frac{15}{10}\\$$
3. $$\frac{61}{100} + \frac{3}{10} = \underline{\hspace{.5in}}\\$$
4. $$\frac{9}{10} + \underline{\hspace{.5in}} = \frac{170}{100}\\$$
5. $$\underline{\hspace{.5in}}+\frac{72}{100} = \frac{102}{100}\\$$
6. $$\frac{15}{100}+\underline{\hspace{.5in}} =1\frac{55}{100}$$

### Activity Synthesis

• “Which equations were difficult to complete? What about the given fractions made it hard to find the missing numbers?”
• “Which did you find more challenging: finding missing tenths or missing hundredths? Why might that be?”

## Activity 3: Fraction Action: Tenths, Hundredths [OPTIONAL] (20 minutes)

### Narrative

This optional activity allows students to practice adding tenths and hundredths (and to reinforce their ability to compare fractions) through a game. Students use fraction cards to play a game in groups of 2, 3, or 4. To win the game is to draw pairs of cards with the greater (or greatest) sum, as many times as possible.

Consider arranging students in groups of 2 for the first game or two (so that students would need to compare only 2 sums at a time), and arranging groups of 3 or 4 for subsequent games. Before students begin playing, ask them to keep track of and record pairs of fractions that they find challenging to add.

### Required Materials

Materials to Copy

• Fraction Action: Tenths, Hundredths

### Required Preparation

• Create a set of cards from the blackline master for each group of 2.

### Launch

• Groups of 2 for the first game or two, then groups of 3–4 for subsequent games, if time permits
• Give each group one set of fraction cards from the blackline master.
• Tell students that they will play one or more games of Fraction Action.
• Demonstrate how to play the game. Invite a student to be your opponent in the demonstration game.
• Read the rules as a class and clarify any questions students might have.

### Activity

• “Play one game with your partner.”
• “As you play, you may come across one or more pairs of fractions whose sums are hard to find. Record those fractions. Be prepared to explain how you eventually figured out which sum is greater.”
• “If you finish before time is up, play another game with the same partner, or play a game with the players from another group.”
• 15 minutes: group play time

### Student Facing

Play Fraction Action with 2 players:

• Shuffle the cards from your teacher. Place the cards in a stack, face down.
• Each player turns 2 cards over and adds the fractions on the cards.
• Compare the sums. The player with the greater sum wins that round and keeps all four cards.
• If the sums are equivalent, each player turns one more card over and adds the value to their sum. The player with the greater new sum keeps all cards.
• The player with the most cards wins the game.

Play Fraction Action with 3 or 4 players:

• The player with the greatest sum of fractions wins the round.
• If 2 or more players have the greatest sum, those players turn two more cards over and find their sum. The player with the greatest sum keeps all the cards.

Record any pair of fractions whose sum is challenging to find here.

________ and ________

________ and ________

________ and ________

________ and ________

### Activity Synthesis

• Invite groups to share some of the challenging expressions they recorded and how they eventually determined the sums.
• As one group shares, ask others if they have other ideas about how the fractions could be added.

## Lesson Synthesis

### Lesson Synthesis

“Today we practiced adding fractions and finding missing fractions that would make equations true.“

“What strategies did you find helpful when adding tenths and hundredths and writing true equations with both tenths and hundredths?”

“Was there an error that you made or something that was missed multiple times? What was it? Why might it be an easy error to make or an easy thing to miss?”