Use the Fewest Pieces Rule. Students should be ready to solve problems that require trading of base-ten pieces relatively quickly.
Present the following problem:
- Eric made 36 Chocos and Maya made 29. Find how many Chocos they made altogether.
As students work, ask questions that promote connections between the base-ten pieces and the numbers in their number sentences. Encourage students to make trades in order to give their answer using the Fewest Pieces Rule. Remind students to check their answers. See Figure 4.
- I see that you combined your skinnies and bits for both numbers. How many total skinnies do you have? How many total Chocos does that represent? (5 skinnies or 50 Chocos)
- How many total bits do you have? (15 bits)
- How many total Chocos do your base-ten pieces represent? How do you know? (65 Chocos; I started at 50 and counted on the bits—51, 52, 53 … 65.)
- Have you used the Fewest Pieces Rule? If not, what will you trade so that you can show your answer using the Fewest Pieces Rule? (I can trade ten bits for one skinny. Then I will have 6 skinnies and 5 bits.)
- How many Chocos is that? (65 Chocos)
- How can you solve the problem a second way to check your answer? (Possible strategy: I start at 36 on my desk number line. I can count on 3 tens for 30 more and land at 66. Since 30 is one more than 29, I go back one to 65. The answers match.)
- Should we add that to our collection of strategies?
- Does anyone have a different way to use mental math to check the answer? (Possible strategy: I think of 30 instead of 29. I add 36 + 30 in my head to get 66 and then take the one away that I added to 29 in the beginning.)
Ask a student to show how he or she solved the problem with base-ten pieces. Then ask the student to show the addends in base-ten shorthand by writing numbers in the appropriate columns on a display of the Base-Ten Recording Sheets 2 Master. See Figure 5.
- If you do not make any trades, have you shown the Fewest Pieces Rule? (No, we had to make trades when we used the base-ten pieces.)
- If you did not have the columns, would 515 be a reasonable answer? (No, 515 would have 5 flats, it would be too big. Our number only has skinnies.)
Help the students show their trades using base-ten shorthand. Then ask them to record their total using the Fewest Pieces Rule in the columns as shown in Figure 5.
- Show us how your base-ten shorthand matches the number you wrote for the total. (Students can connect the 6 skinnies to the 6 in the tens column and the 5 bits to the 5 in the ones column.)
- Do 5 skinnies and 15 bits represent the same number of Chocos as 6 skinnies and 5 bits? How do you know? (Possible response: Yes, because 5 skinnies is the same as 50 and 50 + 15 is 65.)
Provide more practice solving problems that involve one trade using a similar procedure. Students first solve the problem with base-ten pieces and then record their work on a base-ten recording sheet. Sample problems are shown in Figure 6. Remind students to check their work with a second method.
Solve Problems with Multiple Trades. Present problems that involve multiple trades.
- Eric made 58 Chocos and Maya made 65. Use mental math to estimate about how many Chocos they made altogether. (Possible strategy: Both 58 and 65 are close to 60. 60 doubled is 120.)
Remind students to keep this estimate in mind when they are checking the reasonableness of their answer. Next have students use base-ten pieces to work toward finding an exact answer.
This problem will involve trading twice to find the fewest pieces. When using base-ten pieces, it does not matter whether you start trading on the left or on the right. Sample trades with base-ten pieces are shown in Figure 7. As students work, ask them to explain how they found the total number of Chocos using the base-ten pieces. Then ask a student to show his or her work on a display of the Base-Ten Recording Sheets 2 Master using base-ten shorthand and numbers. Figure 8 shows sample student work.
- Think back to your estimation. Does the answer 123 seem reasonable? (Yes, because we estimated a sum of 120.)
Challenge students to solve some problems with two trades on their own. Sample problems are shown in Figure 9. Each time students should use base-ten pieces, record their work on a base-ten recording sheet, and check with a second method as before.
- Use mental math to solve 49 + 55 another way. (Possible strategy: I add 49 + 1 + 54. 50 + 54 = 104.)
- Solve 67 + 53 a second way. (Possible strategy: I add the tens and get 110. I add the ones and get another ten. 110 + 10 = 120.)
- Should we add these strategies to our collection?
Add to Collection of Addition Strategies. Use additional sheets of chart paper to collect additional addition strategies. Students may want to expand the collection of ways to use base-ten pieces. Students may also want to represent other mental math strategies in this collection. Differentiate these strategies with a cloud or thought bubble.
For each set of problems, students can vary the way they solve the problems and show their work. Use these problems as opportunities to assess students' understanding of addition. An indicator of understanding is the ability to make connections between representations of the same problems—with base-ten pieces, in pictures, or in symbols.
Students who are still struggling with two-digit addition or who have developed “buggy algorithms” may need more experiences just using the base-ten pieces in order to develop an understanding of the trading process. Ask these students to explain their trades orally as they use the pieces.
For one or two problems, ask students to show their trades using base-ten shorthand on the left of the base-ten recording sheet while using numbers on the right, so they can practice using pictures to record their work and make connections between the base-ten pieces and the symbols.
For some problems, students can use the base-ten pieces and record their work with numbers only on the chart. Alternatively, students can just use base-ten shorthand to make trades and then record their work in the chart.
