Lesson 8
Different Representations of Tens and Ones
Warmup: Estimation Exploration: How Many? (10 minutes)
Narrative
The purpose of an Estimation Exploration is to practice the skill of estimating a reasonable answer based on experience and known information. When students make estimates based on the number of tens they see and mentally organize the ones into groups of 10, they look for and make use of structure (MP7).
Launch
 Groups of 2
 Display the image.
 “This diagram shows a collection of connecting cubes.”
 “What is an estimate that’s too high? Too low? About right?”
 1 minute: quiet think time
Activity
 “Discuss your thinking with your partner.”
 1 minute: partner discussion
 Record responses.
 “Let’s look at another image of the same collection.”
 Display the image.
 “Based on the second image, do you want to change your estimates?”
Student Facing

How many do you see?
Record an estimate that is:
too low about right too high \(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\) \(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\) \(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\) 
How many do you see?
Record an estimate that is:
too low about right too high \(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\) \(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\) \(\phantom{\hspace{2.5cm} \\ \hspace{2.5cm}}\)
Student Response
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Activity Synthesis
 “Did anyone change their original ‘about right’ estimate? Why did you change it?” (I changed it because I see there are at least 40 cubes because there are 4 towers.)
 “Let’s look at our revised estimates. Why were these estimates more accurate the second time?” (Some of the cubes are organized.)
Activity 1: Compare Representations of a Collection (10 minutes)
Narrative
The purpose of this activity is for students to interpret different baseten representations of a twodigit number. Students compare drawings to words (_____ tens _____ ones) and an expression showing the value of the tens and ones digits as a sum. One of the representations shows the ones to the left of the tens so that students think about the value of the units. One of the representations shows a baseten diagram with the towers of ten labeled “10” rather than representing all ten cubes in each tower. It is important that students who use this type of drawing can explain that the unit of ten contains 10 ones.
Advances: Representing, Conversing
Required Materials
Materials to Gather
Launch
 Groups of 2
 Give students access to connecting cubes in towers of 10 and singles.
Activity
 Read the task statement.
 5 minutes: independent work time
 3 minutes: partner discussion
 Monitor for students who:
 draw each representation as tens and ones
 use numbers and words to connect each representation
Student Facing
Each student counted and showed a collection.
 Clare drew
 Han drew
 Kiran wrote 3 ones and 7 tens.
 Priya wrote \(70 + 3\).
How do you know?
Show your thinking using drawings, numbers, or words.
Student Response
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Activity Synthesis
 Invite previously identified students to share.
 “What connections do you see between the representations?” (Each representation shows the same amount. Each representation shows tens and ones.)
 “How is Han’s representation the same as Clare’s? How is it different?" (They both show 73. Han’s shows tens and ones, but the tens don’t show all the ones. Instead he labels it as 10 or 1.)
 “Han’s representation is a new representation that you may want to try using to show your thinking.”
Activity 2: Card Sort: Baseten Representations (15 minutes)
Narrative
The purpose of this activity is for students to match cards that show different baseten representations. The representations are baseten diagrams, ___ tens ___ ones, and addition expressions that show the value of the tens and ones digits. Some cards represent the ones to the left of the tens to encourage students to focus on the units and the meaning of any digits (MP2, MP7). As students work, encourage them to refine their descriptions of representations and how they match using more precise language and mathematical terms (MP6).
Supports accessibility for: Organization, Attention
Required Materials
Materials to Gather
Materials to Copy
 Representations of Tens and Ones
Required Preparation
 Create a set of cards from the blackline master for each group of 2–4.
Launch
 Groups of 24
 Give each group a set of cards and access to connecting cubes in towers of 10 and singles.
 Display the student workbook page.
 “Today we are going to sort cards into groups that show the same twodigit number. For example, look at these three cards. Which two representations show the same twodigit number? Why doesn’t the other one belong?” (The first two cards both show 4 tens and 1 one or 41. The last card isn't the same because it only shows 1 ten. It has the same digits, but they mean something different.)
 30 seconds: quiet think time
 1 minute: partner discussion
 Share responses.
Activity
 “This set of cards includes baseten diagrams, words that show ___ tens and ___ones, expressions, and twodigit numbers. Find all the cards that show the same number. Each group should have three cards that match, but they might be different kinds of representations."
 “Work with your partner to explain how the cards match or why some cards don’t belong with a group.”
 8 minutes: partner work time
 As students work, consider asking:
 “Why did you match these cards together? How do they match?”
 “Would Card __ match with this group of cards? Why or why not?”
 “How could you explain how your cards match using the words tens and ones?”
 “How are the cards in this group the same? How are they different?”
Student Facing
Your teacher will give you a set of cards that show different representations of a twodigit number. Find the cards that match. Be ready to explain your reasoning.
Student Response
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Activity Synthesis
 Invite students to share the matches they made and how they know those cards go together.
 Attend to the language that students use to describe their matches and the representations, giving them opportunities to describe the representations more precisely with attention to how they show amounts of tens and ones.
 Highlight the use of terms like tens, ones, and digits.
Activity 3: Introduce Grab and Count, Ones Cubes (15 minutes)
Narrative
The purpose of this activity is for students to learn a new center called Grab and Count. Students grab a handful of ones cubes and put them together with their partner’s. They estimate how many cubes there are and then count the cubes. Students record their estimate and the actual number of cubes on the recording sheet. Since students have not written twodigit numbers yet, students may attempt to record the twodigit number or use other, more familiar, representations to show the amount of cubes.
Required Materials
Launch
 Groups of 2
 Give each group two recording sheets and access to ones cubes.
 “We are going to learn a new center called Grab and Count.”
 “Both partners grab a handful of cubes. Put your cubes together. Then estimate how many cubes you have and record your estimate on your recording sheet. Then count the cubes in any way that makes sense to you. Record the number of cubes you counted.”
Activity
 10 minutes: partner work time
Activity Synthesis
 “How did you keep track of the cubes as you counted?” (We made groups of ten. We moved the cubes as we counted them.)
Lesson Synthesis
Lesson Synthesis
Display cards J, E, and S from Activity 2.
“Today we found different representations of the same number. Do all of these cards show the same number? Why or why not?” (No, J and S match because they show 3 tens and 2 ones. E shows 3 ones and twenty or 3 ones and 2 tens.)
Display card V or write 95.
“What would you look for on other cards to know if it matched this number?” (I’d look for a diagram that showed 9 towers of ten and 5 singles. I’d look for something that said 9 tens and 5 ones.)
Cooldown: Unit 4, Section B Checkpoint (0 minutes)
CoolDown
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