Workshop: Subtraction

Est. Class Sessions: 2

Developing the Lesson

Part 1. Develop Subtraction Strategies Menu

Students frequently do not understand that when they make trades (or regroup) in order to subtract they are just partitioning the top number in a different way. It is important for them to realize that the new representation should have the same value. For example, in Whitney's problem in Figure 1, 4 hundreds, 11 tens, and 14 ones is not equal to 514.

Look at Solutions. Ask students to solve 514 − 267.Then write the incorrect solution shown in Figure 1 on the board or overhead. Tell students that a third-grader named Whitney solved the problem this way.

Ask:

What do you think about Whitney's solution?
Is it correct? How do you know?
How does it compare to your strategy and solution?
What would you tell Whitney to help her? (Possible response: The regrouping was not done correctly. 400 + 110 + 14 = 524 and does not equal 514. Always check to see that when you trade, you don't change what the top number is worth. Also be sure to keep track of your trades.)

Ask a few students to share what they think and what they would say to Whitney about this incorrect solution. Allow students to decide if they agree or disagree and to explain why.

Discuss the incorrect trades in the problem shown in Figure 1. Remind students that trading does not change the value of the number; it just partitions the number in a different way so it is easier to subtract. To correctly subtract 267 from 514, it is necessary that 514 be represented correctly. Encourage students to look back at the trades, and if necessary, write a number sentence for the new partitions to check that it adds up to the original number. Students may need to represent the problem using base-ten pieces in order to understand how to trade correctly. Ask students to help you partition 514 correctly.

Ask:

Write a number sentence for 514 in expanded form. (500 + 10 + 4 = 514)
If Whitney were using base-ten pieces, what pieces would she start with? (5 flats, 1 skinny, and 4 bits)
How many flats, skinnies, and bits would she have after making her incorrect trades? Show the trading with the pieces. (4 flats, 11 skinnies, and 14 bits)
What number do those pieces represent? Is it 514, as it should be? (No, it's 524.)
Write a number sentence for the way that Whitney showed her trades. (400 + 110 + 14 = 524)
Rewrite the problem and help Whitney make the correct trades. How many flats, skinnies, and bits would you have? Show me with the pieces. Write a number sentence that shows the value of those pieces. Is it 514, as it should be?

A correct solution is shown in Figure 2.

Assign Questions 1–6 on the Workshop: Subtraction pages in the Student Guide. Students should work on these problems with a partner to encourage discussion of ideas. As students work, use discussion prompts similar to those used with Whitney's example to encourage students to look carefully at the trades to see if they are correct. When appropriate, ask students to support their thinking with base-ten pieces. You may choose to discuss the problems and solutions after each set of problems (1–3 and 4–6).

Use Questions 1–6 in the Student Guide to assess students' ability to apply place value concepts to make connections among representations of numbers [E1].

Record Different Strategies. Ask students to solve the problem in Question 7 in the Student Guide. Remind students that in this unit they have explored several different subtraction strategies. Identify several students in the room who have used different strategies. Ask these students to record their strategies on chart paper for all the class to see. Then ask students to discuss and answer Questions 8 and 9 with a partner.

Look at the strategies we have recorded on the chart paper.

Are any of the strategies in Question 9 the same?
Which strategies are different?
Does anyone have a different strategy to share?
Talk to your partner. Which strategy do you think works best for this problem (301–189)? Which strategy is most efficient? (Possible response: I like Jerome's strategy. It is easy, and I do not have any of that regrouping to keep track of.)
Do you think this would be a good strategy for any subtraction problem? (Possible response: No, I think this works best for this problem, but some problems are easier if I use the compact method or expanded form.)
Which strategy do you think would work best to solve 293 – 301? (Possible response: Any strategy will work, but counting up 8 from 293 to 301 or counting back 8 from 301 to 293 are probably the most efficient.)

Look at the Subtraction Strategies Menu. Display the Subtraction Strategies Menu from the Student Guide Reference section. Refer students to the Subtraction Strategies Menu in the Student Activity Book. Ask students to compare the strategies that are recorded on the chart paper to the strategies recorded on the Subtraction Strategies Menu.

Ask:

Are there any strategies on the menu that are not recorded on our chart paper?
Which are they?
Do you think this an efficient strategy to solve 293 − 301?
Are there any strategies on our chart paper that are not recorded on the menu?
Which are they?
Are these strategies similar to one of the strategies on the menu?

Ask students to record strategies not represented on the menu on the back of the Subtraction Strategies Menu page in the Student Activity Book. To better acquaint students with the strategies on the menu, ask students to use strategies from the Subtraction Strategies Menu to solve 553 – 467.

Ask several students to record their strategies on the board.

Ask: