# Lesson 8

Different Representations of Tens and Ones

## Warm-up: 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

• 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

1. How many do you see?

Record an estimate that is:

too low about right too high
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2. How many do you see?

Record an estimate that is:

too low about right too high
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### Student Response

For access, consult one of our IM Certified Partners.

### 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 base-ten representations of a two-digit 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 base-ten 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.

MLR7 Compare and Connect. Synthesis: Lead a discussion comparing, contrasting, and connecting the different representations. Ask, “How are the representations the same? How are they different?” and “How do tens and ones show up in each representation?”

### Launch

• Groups of 2
• Give students access to connecting cubes in towers of 10 and singles.

### Activity

• 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 $$70 + 3$$.

Did the students count the same number of objects?
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: Base-ten Representations (15 minutes)

### Narrative

The purpose of this activity is for students to match cards that show different base-ten representations. The representations are base-ten 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).

Engagement: Develop Effort and Persistence. Chunk this task into more manageable parts. Give students a subset of the cards to start with and introduce the remaining cards once students have completed their initial set of matches.
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 2-4
• 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 two-digit number. For example, look at these three cards. Which two representations show the same two-digit 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 base-ten diagrams, words that show ___ tens and ___ones, expressions, and two-digit 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 two-digit number. Find the cards that match. Be ready to explain your reasoning.

### Student Response

For access, consult one of our IM Certified Partners.

### 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 two-digit numbers yet, students may attempt to record the two-digit number or use other, more familiar, representations to show the amount of cubes.

### Required Materials

Materials to Gather

Materials to Copy

• Grab and Count Stage 2 Recording Sheet

### 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.)

## Cool-down: Unit 4, Section B Checkpoint (0 minutes)

### Cool-Down

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