Lesson 17
Sums of Tenths and Hundredths
Warmup: Which One Doesn’t Belong: Tenths and Hundredths (10 minutes)
Narrative
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
 “Discuss your thinking with your partner.”
 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.
\(\displaystyle \frac{70}{100}\)
Student Response
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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.
\(\frac{6}{10} + \frac{39}{100}\)
Advances: Speaking, Writing, Conversing, Representing
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 24 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.
 “Return to your collection. Address any questions left for you or revise your thinking. Then, record what’s in each group.”
Student Facing

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.

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.

Return to your collection.
 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
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Advancing Student Thinking
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.
Supports accessibility for: Conceptual Processing, Organization, SocialEmotional 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

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

Each equation is missing a fraction in tenths or hundredths. Find the fraction that makes each equation true.
 \(\frac{20}{100} + \underline{\hspace{.5in}} = \frac{28}{100}\\\)
 \(\frac{110}{100} + \underline{\hspace{.5in}} = \frac{15}{10}\\\)
 \(\frac{61}{100} + \frac{3}{10} = \underline{\hspace{.5in}}\\\)
 \(\frac{9}{10} + \underline{\hspace{.5in}} = \frac{170}{100}\\\)
 \(\underline{\hspace{.5in}}+\frac{72}{100} = \frac{102}{100}\\\)
 \(\frac{15}{100}+\underline{\hspace{.5in}} =1\frac{55}{100}\)
Student Response
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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 ________
Student Response
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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?”
Cooldown: Missing Fractions (5 minutes)
CoolDown
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