Subtracting with Base-Ten Pieces

Est. Class Sessions: 2

Developing the Lesson

In this lesson, students solve several subtraction problems with base-ten pieces. They begin with two-digit problems and then move to three- and four-digit problems. The purpose of the activity is to help students develop a concrete understanding of the trades involved in subtracting as a foundation for their work with a standard paper-and-pencil algorithm in the next lesson.

Be aware that there is a fundamental difference between modeling addition with base-ten pieces and modeling subtraction. When modeling addition, you begin by putting down pieces to represent each number you are adding (the addends). But when modeling subtraction, you only put down pieces that represent the first number in the problem (the minuend). You then show subtraction by taking away pieces that represent the second number (the subtrahend). See Figure 1. As students work with the problems, be sure they do not begin the problem by laying out pieces for both numbers.

Solve Problems with No Trades. As a warm up, ask students to solve a few problems that do not involve trading. Ask volunteers to show their solutions with a display set of base-ten pieces as the other students solve the problems with their own pieces. If you have enough pieces, it is best for each student to work with his or her own pieces.

Begin with a problem such as 84 − 63 as shown in Figure 1.

Ask students to represent the number 84 with base-ten pieces. Then ask:

What is the value of the 8 in 84? (80)
Which pieces represent the 8? (8 skinnies)
If we remove pieces that represent 63 pieces, what problem will we solve? (84 − 63)
After removing pieces that show 63, what number is left? (21)

Ask students to work on other problems that do not involve trades, for example:

42 − 11 68 − 43

See the Sample Dialog 1 box for dialog taken from classroom video in which a student solves 68 − 43.

Here is a sample dialog taken from classroom video in which a student solves 68 − 43.

Teacher: Who's ready? Marco, go ahead. It was 68 minus 43.

Marco: [Represents the number 68 with 6 skinnies and 8 bits on the display.] 68.

Teacher: How did you make 68?

Marco: 6 skinnies.

Teacher: Uh-huh.

Marco: And that's 60 and 8 bits. That's 68.

Teacher: Very good.

Marco: Subtract—minus 43. [Removes 4 skinnies, from the 6 he originally had, as well as three bits from the eight bits he originally had.]

Teacher: 43. How many skinnies are you taking?

Marco: 4. And it equals 25.

Teacher: Can you count them for us, point to them and count them?

Marco: 10, 20, 21, 22, 23, 24, 25.

Teacher: Very good.

For the first few subtraction problems students solve, delay mentioning the second number until it is time to subtract. This will ensure that students do not put down pieces to represent both numbers at the beginning of the problem. For example, do not begin by saying, “We are going to solve 84 − 63.” Instead, say, “Please represent the number 84 with base-ten pieces.” Then say, “Now remove pieces that represent 63.”

When solving subtraction problems, students should manage three piles of pieces—the “bank,” where they can go to trade pieces or get new pieces; the “work pile” that first shows the pieces they start with and then, after the problem is solved, shows the answer; and the “take-away pile” that contains the pieces they have taken away. They can push the take-away pile to the side of their workspace. One way for students to check their work is for students to add their “take-away pile” to the pieces in their answer to see if they get the minuend.

Solve Problems with One Trade. After solving a few problems that do not involve trading, students can begin exploring problems that involve trading.

For example, ask:

At the beginning of the day, there were 57 Chocos at the TIMS Candy Company. Customers bought 39. How many were left?
Which pieces should we use to represent the number of Chocos at the beginning of the day? (5 skinnies and 7 bits)
How many pieces total do we need to take away? (39 pieces)
What problem are we solving? (57 − 39)

This problem is shown in Figure 2. As students work on the problem, circulate and see how they are dealing with the need to trade. If there are several students who do not know what to do, work together as a class to solve the problem. Otherwise, let students talk in their groups. Then ask students to show their work to the class using a display of base-ten pieces.

As students show their work, check their understanding of trading. Make sure they understand that trading does not change the number. It just changes the way it is partitioned. See the Sample Dialog 2 box.

Ask:

What happens when you are subtracting and you don't have enough bits? (You take one of the skinnies away and trade it for 10 bits.)
Does trading a skinny for bits change the value of your number? (No, 10 bits is the same as 1 skinny, so the value stays the same.)

Provide several more problems that involve trading once, such as the problems in Figure 3.

Teacher: So it's 57 minus 39. Okay, who thinks they can come up and explain to us how you're going to solve this? What are you going to do to get the right answer? John, come on up. Watch what he's doing. If he needs help you need to help him.

John: I made 57.

Teacher: How did you make 57?

John: By 5 skinnies and 7 bits.

Teacher: Can you count the 5 skinnies for us?

John: 1, 2, 3, 4, 5.

Teacher: Can you count by tens?

John: 10, 20, 30, 40, 50.

Teacher: Very good.

John: And then I, um, I can't take away 9 from 7 because I have, I don't have enough so I take away one skinny and trade it for ten.

Teacher: Ten what?

John: Bits.

Teacher: Ten bits, right. Then how many bits do you have? Can you count them out loud for us, John?

John: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17.

Teacher: This is the question I kept asking people at Table 1. After everyone at Table 1 traded their skinny for ten bits, I asked them, do you still have 57? What do you think? Do you still have 57?

Students: [Some say “No,” some say “Yes.”]

Teacher: Raise your hand if you think yes, you have 57. John, you think yes?

John: Yes.

Teacher: Prove to us that you still have the number 57.

John: Because, because you have 4 skinnies and 17 bits—that's 57. So it doesn't change.

Teacher: 57. Very good. You still have 57. See how the number doesn't change?

Third Student: So that's the answer—57?

Teacher: No, he's just showing us 57. Now he has to subtract 39. So what do you do?

John: 1, 2, 3, 4, 5, 6, 7, 8, 9. [counts up nine bits and pushes them off to the side] and then I take away 3 [referring to the 3 skinnies, which he puts to the side]. That leaves 1 skinny and 8 bits [counts the bits]. That's 18.

Teacher: And it equals 18. Very good, John. Good job. How many of you got 18?

Solve Problems with Multiple Trades. Provide problems that involve trading more than once, such as the problems in Figure 4. Display the first problem 563 − 265 and ask students to use base-ten pieces to solve it independently. Again, make sure they are placing pieces to represent the number 563 only, and then removing the pieces that represent 265. Students may solve this problem in a variety of ways. Figure 5 shows one way to solve it, where students begin on the right by trying to subtract 5 ones from 3 ones, or take 5 bits from 3 bits. However, some students may begin the problem from the left instead. Others might find it easier to do all the trading first and then subtract. Discuss various approaches.

Ask:

Show us how you solved 563 − 265.
Who can show us another way to solve it?
Which method is easier? Which is more efficient? Why?

Display 205 − 76. This problem involves a zero in the tens place.

As you work through the problem together, ask:

Can I subtract 6 bits from 5 bits? (no)
Can I trade a skinny for 10 bits? (No, there aren't any skinnies.)
Is there a way I can get a skinny? (You can trade a flat for ten skinnies.)
Now what pieces do I have? (1 flat, 10 skinnies, and 5 bits)
How many Chocos do those pieces represent? (205 Chocos)
Can I trade for bits now? (yes)
Now what pieces do I have? (1 flat, 9 skinnies, and 15 bits)
How many Chocos do those pieces represent? (205 Chocos)
Now I can take away 6 bits. How many are left? (9 bits)

Encourage students to solve the problem a second way as a check. For example, they can use a number line and count up:

4 + 20 + 100 + 5 = 129

Figure 6 shows more subtraction problems that can be solved with the whole class together, in groups, or individually. The amount of guidance and practice needed will depend on students' experience. Do not expect mastery immediately. See Content Note.

Requiring children to do too many of these problems may result in boredom and frustration. It is best to assign a few problems a day over a longer period of time.

Note that the last three problems in Figure 6 involve regrouping several times. Students often have trouble with problems such as the last one. Point out that the problems could be done by alternative methods that may be more efficient. Display 3000 − 524 and ask students to demonstrate some different solution strategies. For example, 3000 − 524 can be solved by first subtracting 2999 − 524 and then adding 1 to the answer. Using this method, no regrouping is required. 3000 − 524 can also be done mentally by first subtracting 500 (2500), then subtracting 20 (2480), and finally subtracting 4 to get 2476.

After students have shared a variety of solution strategies, ask: