All-Partials Method. Write the problem 33 + 49 on a display of the Base-Ten Recording Sheets 2 Master as shown in Figure 10.
- How can we find numbers that are easier to use to estimate this sum? Let's call these convenient numbers “friendly” numbers. (Possible strategy: I use friendly numbers like 30 and 50 to estimate a sum of 80.)
Place the base-ten pieces for the problem on display and ask students to total the number of skinnies and bits on a recording sheet.
- Can we just drop the columns and write 712? Why or why not? (Possible response: 712 would have 7 flats and there are just 7 skinnies.)
- What do the numbers stand for? (7 skinnies and 12 bits)
- How can you say that using tens and ones? (That's 7 tens and 12 ones.)
- What is the problem with writing 712 when we do not have columns to separate the skinnies and bits? (You can't have 12 in the ones places. That is the same as 1 ten and 2 ones.)
To present the first paper-and pencil method, model solving the problem using the method shown in Figure 11. This method totals the value of each type of base-ten piece (or totals the value of each digit) instead of trading.
- Is 82 a more reasonable answer than 712 for the sum of 33 and 49? Why or why not? (Possible response: 49 is one less than 50, so adding 33 + 50 is very close, so 82 is more reasonable.)
- Why did I write 70 under the problem? What did I add? (You added 30 + 40.)
- How is the 70 related to the base-ten pieces? (70 is like the 7 skinnies.)
- Why did I write the 12 under the 70? How is 12 related to the base-ten pieces? (Altogether there are 12 bits and you need to add them to the 70 from the skinnies to get the total number.)
Convenient or “Friendly” Numbers. Rounding numbers to the nearest multiple of 10, 100, 1000, etc., gives one type of convenient number for estimating sums and differences. For example, the sum 98 + 25 can be easily estimated as 100 + 25 if we are rounding to the nearest 10.
However, other types of convenient numbers might be chosen for estimating as well. Students may refer to these numbers as “friendly” numbers. For example, 32 + 35 can be estimated as 33 + 33, a double which is easy to mentally compute. Or, 24 + 78 can be estimated as the sum of convenient numbers such as 25 + 75. Usually, there are several different choices of convenient numbers that make sense for a problem. It is important to help students learn to choose convenient numbers that will help them perform mental math and make logical estimates.
Ask students to solve the problems in Figure 12 using the method you modeled above. As students work, ask questions that help them make connections between the base-ten pieces and the numbers in problems.
Return to the problem in Figure 10. Remind students that they agreed that they could not write 712 without the columns. Ask students to make trades to show the total using the Fewest Pieces Rule.
Write the problem vertically again and ask:
- How can I use paper and pencil and make trades in my head, so that I am using the Fewest Pieces Rule? (Trade ten bits for 1 skinny in your head.)
Compact Method. For ease of reference, we refer to the traditional or standard algorithm for addition taught in the United States as the compact method.
Compact Method. Review the compact method as another paper-and-pencil method where you record your trades. In Figure 13, 33 was added to 49. If the bits are added first, we get 12 bits. However, this does not follow the Fewest Pieces Rule. So arrange 12 bits as 1 skinny and 2 bits. Record 2 bits in the bits column. To keep track of a new skinny, put a small 1 in the skinnies column. Add up the skinnies to get 8 skinnies. The answer is 82.
Try other problems such as those in Figure 14. As students work, ask them to explain their paper-and-pencil methods and to connect them to the base-ten pieces.
For example, when solving 25 + 37, ask:
- Why did you write a 6 in the tens place? (That shows 2 skinnies + 3 skinnies + the 1 other skinny I made when I combined 5 bits with 7 bits.)
- How did you show the 12 bits? (I put a little 1 over the tens place to remind me that I made another ten and a 2 in the ones place to show the two bits left over.)
Make sure students can explain that when they trade bits (ones) for a skinny (ten) and then add it to the number of skinnies in the tens place, that they are adding tens.
Understanding and Using Paper-and-Pencil Methods for Addition. Although many of your students will likely be proficient with the compact paper-and-pencil algorithm for addition, take some time to be sure that they can use and understand both the compact method and the all-partials method. During our research, student interviews revealed that many students could complete the compact algorithm, but they did not understand it well enough to connect it to the base-ten pieces. When asked to explain their addition, they often did not refer to the digits in the tens place as tens and they often talked of “carrying a 1” instead of a ten. It is not clear that students fully understood what the symbols in the algorithm represented because they could not make connections between the base-ten pieces and the written symbols.
The All-Partials Method. The all-partials method is transparent in that it is very easily understood and explained by students. It helps students understand what they are actually adding when they are adding digits in the tens place. To complete the problem in Figure 11, they understand that they are adding 30 + 40 when they combine the digits in the tens place. Using this method will increase students' understanding of the addition process. Asking them to explain it and connect it to the base-ten pieces will assess that understanding. The method will be used with larger numbers in Lesson 5 and can help some students who have trouble keeping track of their trades. It can also serve as a second method for checking answers.
