Once again the composing and decomposing of numbers in addition and subtraction are directly related to each other and students need to see the similarities. If I add
I can immediately see that now I have more than 10 in the ones space. If I demonstrate with base ten blocks students should be able to decompose the 12 tens into 1 ten and 2 ones. Having done this they can now add the ones column and figure the answer to be 96. Now when subtraction with renaming is introduced I need to fall back on previously learned ideas about addition and subtraction. If I now use the same example but structure it as
I begin to decompose the 9 tens into 8 tens and 10 ones; I can add those ones to the 2 already in the ones column and have 12. When I now subtract I have 8 in the ones column and 6 in the tens column, or an answer of 64. Looking at the two problems students can see that these two examples are like part/whole problems. I put two numbers or parts together and now I separated one part from the other. Usually when we begin subtraction with renaming students are also taught how to check their subtraction with addition. The two go hand in hand and the stronger the relationship students have with proving and verifying these principles the better understanding they will have of subtraction with renaming.
Addition with regrouping plays an important part in students seeing what is happening in subtraction with renaming. If we add 14 +8 = 22 students have an easier time seeing that there is more than 9 in the ones column so that ten must be put over into the tens column making it now 2 tens or 20. If I now show the students 22 - 8 they can see that where I added 4+8 =12 in the addition problem I am now reversing that by taking a ten back and making the 2 ones 12. In the addition problem my first calculation was 4+8 =12, while in the subtraction problem the first calculation becomes 12 -- 4 =8. It becomes obvious that working with an eye toward having students see these relationships can lead to a deeper understanding of the subtraction algorithm and saves it from being a mere set of procedures.
A student might also expand 14 to be 10+4 and so the problem becomes 10+4+8. In this case some children if they understand the zero might find it easier to combine 10+8=18, and then add on the 4 which equals 22. In the subtraction problem 22 -- 8, they could expand 22 to (10 +10 +2) -- 8. They then could subtract 10-8 =2 and add that to the other 10+2 and reach the same conclusion of 14.4.
Throughout the literature the research stresses the need for students to constantly talk and write down their thoughts about what they are doing and why. Students need to work together and talk out what they are doing. Students should have math journals or notebooks where they write about what they are learning in math, or describe how they did a certain problem, or how they know that 5324 is larger than 3259.It is part of the teacher’s role to provide the atmosphere where children can explore and share their ideas- right or wrong. The teacher needs to question the child but most of all listen to their explanations and discoveries.
Following are a group of problems that support the different levels I have identified in my package for learning subtraction with renaming.