Lesson 10

Place Value and the Number Line

Warm-up: Notice and Wonder: Base-ten and the Number Line (10 minutes)

Narrative

The purpose of this warm-up is for students to connect a base-ten diagram to addition on the number line. This will support their work later in the lesson when they connect place value methods to representations of addition and subtraction on the number line.

Launch

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

Activity

• 1 minute: partner discussion
• Share and record responses.

Student Facing

What do you notice? What do you wonder?

Activity Synthesis

• “What addition equation could be represented here?” ($$28 + 10 + 5 = 43$$ or $$28 + 15=43$$)
• “We are going to keep thinking about what is the same and what is different between base-ten diagrams and number lines.”

Activity 1: Compare Representations (20 minutes)

Narrative

The purpose of this activity is for students to connect a subtraction method based on place value to representations on the number line. Students compare representations of a subtraction method using base-ten diagrams, equations, and number lines (MP2). They notice that, just like with base-ten blocks, they can think about subtracting or counting by tens first or by ones first on the number line.

Representation: Develop Language and Symbols. Support understanding of the problem, by inviting students to act it out. For example, create a number line on the ground or across a large white board in the front of the classroom. Allow students to physically move on the number line.
Supports accessibility for: Conceptual Processing

Required Materials

Materials to Gather

Launch

• Groups of 2
• Display image of Clare’s base-ten diagram.

Activity

• “Clare subtracted and represented her thinking with a base-ten diagram.”
• “What does this diagram tell us?” (She started with 46. She took away 35. She has 1 ten and 1 one left.)
• 30 seconds: quiet think time
• 1 minute: partner discussion
• Share responses
• “You are going to write an equation to represent Clare’s work. Then you will represent Clare's method on a number line.”
• “Work with your partner to decide how to best represent what you think Clare did. You may discuss where to start, how many jumps you should draw, how long each jump should be, and where to land.”
• 5 minutes: partner work time
• “Now try one on your own.”
• 8 minutes: independent work time
• Monitor for students who:
• start at 58 and jump 20 and then 4
• start at 58 and jump 4 and then 20
• start at 58 and jump 10, 10, and then 4
• start at 58 and show a jump for each ten and each one (6 total jumps)

Student Facing

Clare subtracted and represented her work with a base-ten diagram.
1. Write an equation to represent Clare's work.

2. Represent Clare’s method on the number line.

3. Find the value of $$58 - 24$$.

Show your thinking using a base-ten diagram.

4. Represent how you found the value of $$58-24$$ on the number line.

Student Response

If students represent their thinking using a base-ten diagram, but the number line doesn't match their thinking, consider asking::
• “Can you tell me more about your number line? How did you decide where to start?”
• “How does the number line connect to your base-ten diagram?”

Activity Synthesis

• Invite previously identified students to share.
• Display or record their methods.
• “How did you decide where to start, how many jumps to make, and the length of each jump?”
• “How are these methods the same? How are they different?” (They started at 54. Some show subtracting tens first, some show ones first. Some show subtracting the value of all the tens or all the ones. Some show subtracting each ten or each one.)
• “How does the number line help you see how these methods are the same?” (It helps you see that it doesn't matter if you subtract tens first or ones first. They both show subtracting 24.)

Activity 2: On the Number Line (15 minutes)

Narrative

The purpose of this activity is for students to represent addition and subtraction within 100 on a number line. Students make connections to strategies based on counting on or back by place. The numbers in each subtraction equation are designed to elicit methods that do not require students to explicitly decompose a ten. For example, when finding the value of $$50 - 32$$, students may first add on to make a ten ($$32 + 8 = 40$$), then add on more tens to reach the total ($$40 + 10 = 50$$). Others may see they can count back 2, then subtract the tens. In the synthesis, students share their thinking and discuss how the number line helps see how they can use what they know about the structure of counting sequence and what they know about tens and ones to add and subtract (MP7).

MLR8 Discussion Supports. Synthesis: For each comparison that is shared, invite students to turn to a partner and restate what they heard using precise mathematical language. Ask, “Who can restate what ____ shared using the place value language?”

Required Materials

Materials to Gather

Launch

• Groups of 2
• Display images of Diego’s number line.
• “Diego found the value of $$33 + 45$$. He used a number line to represent his thinking.”
• “Where do you see 33 and 45 on his number line?” (On the number line there is a point on 33 and 4 jumps of 10 and 1 jump of 5.)
• 30 seconds: quiet think time
• Share responses

Activity

• “You will be finding the value of expressions and representing your thinking on the number line.”
• “Draw base-ten diagrams if it helps.”
• 810 minutes: independent work time

Student Facing

Diego is finding the value of $$33 + 45$$. He says he can count on by tens, then by ones. He used a number line to show what he means.

1. Write an equation to show the sum for Diego’s work.

2. Find the value of $$23 + 24$$.

Represent your thinking on the number line.

3. Find the value of $$50 - 32$$.

Represent your thinking on the number line.

4. Find the value of $$40 - 26$$.

Represent your thinking on the number line.

Student Response

If students use base-ten diagrams or blocks to show decomposing a ten to subtract, validate their reasoning and consider asking:

• “After you decomposed a ten, did you subtract tens or ones first?”
• “Locate 50 (or 40) on the number line. If you used the number line to show counting back, would you count back by tens first or ones first? Why?”
• “What should you do next?”
• “How is this method like what you did with the block (or base-ten diagram)? How is it different?”

Activity Synthesis

• Invite 23 previously selected students to share.
• “How did you decide where to start?”
• “How did you decide how much to add/subtract first?”
• “How does your number line show the value of the difference?”
• “What other questions do you have about _____'s number line?”
• “How does the number line help you make sense of _____'s method?”

Lesson Synthesis

Lesson Synthesis

“Today we learned that some of the methods we use to add or subtract can be represented on the number line. We saw you can add or subtract the tens first and then the ones or the ones first and then the tens. We saw methods for subtraction that counted back by tens and ones from the larger number and those that showed counting on by tens or ones from the smaller number.”

“Did you prefer showing your thinking with base-ten diagrams, the number line, or another way? Was it the same for addition and subtraction? Explain.”