Teaching some math concepts can be difficult. If you have read this blog much, you know that for Math skills, I like to use the concrete-pictorial-abstract pattern for every skill. Some basic math, though, needs a creative method to be able to make an idea "stick." Regrouping, in both addition and subtraction, is one of those skills.
Regrouping is a better word to describe what we used to call "borrowing" or "carrying over." It is used in both adding and subtracting where you have to exchange values.
One thing I learned early on when I was a teacher is that children have to value something before they want to learn about it. I have also found that children (as well as adults) value money. I use money often as a manipulative to teach difficult skills. As much as possible, I use real coins. This was very difficult to do when I had 27 students, as it was hard to gather enough coins needed and not worry about theft, but as a homeschooler or as a tutor, I just grab some coins out of our "change cup" which we keep on the refrigerator. (My husband will never miss anything except the quarters! And besides, they usually go right back into the cup when the lesson is over) I discovered that before the child sees the skill itself as important, s/he will value the coins and/or bills that you are using to teach the concept. Either way, it gets their attention and, therefore, makes the skill easier to teach. Also, when I had to reteach this skill to 4th or 5th graders, they did not feel embarrassed that they did not understand something which they should have picked up in 1st or 2nd grades. In fact, the students who did not need to relearn this concept were often jealous that they were not included in this lesson!
Before using these coins in your Regrouping lesson, make sure you have taught your child that a penny is worth one cent and a dime is worth ten cents. Also help them understand that you can exchange ten pennies for one dime. They don't have to fully understand that, as this lesson will have the added benefit of helping them understand that also, but it helps if you are not teaching two different new skills at once. They also need to understand how to add two digit numbers without regrouping.
THE LESSON - Addition with Regrouping
Write the problem on a piece of paper:
Then pull out dimes and pennies. I usually make sure I have 20 pennies and 10 dimes (per child) as a general rule.
Have the child show you 34 cents using dimes and pennies. Have some divider to put the dimes on the left side and the pennies on the right side. (I use the line down the center of my expandable table, but if you don't have that you can use a ruler, a licorice stick, a long toy- anything that is straight).
About an inch below those coins, using the same dividing line as the 34, have your child show you 28 cents.
Explain that they just showed you the problem. To add them, push the pennies together and the dimes together.
Now ask if they have enough pennies to trade you for one dime (which you take from the "extras" pile). They should carefully count 10 pennies and push them to the penny "extra" pile. You give them a dime, but put it on the dime side.
Tell your child that you have changed the number of pennies and dimes you have, so you need to change the problem to show what you now have. You added a dime, so you have to write a 1 to the dime side of the problem to show what you just did.
Now count how many pennies are left. Have your child write the 2 under the 8. (for younger children, or even some older children, they have had hard time lining things up. Draw a line on the problem to separate the ones from the tens, extending it into the answer space. This will resemble the separating line on the table, which helps). Tell them you now have 2 "ones" because pennies are worth one cent. (When I say an important lesson, I usually make them repeat it verbally a few times to make sure they heard me and know it's important).
Now remind the child that they added a dime and point again to the 1 he added on the paper. Then have him count the dimes. Have him write the "6" in the tens column. Tell him he has six "tens" because each dime is worth ten cents.
Have him tell you how many "tens" he has. Then tell you how many "ones" he has. Then have him read the answer. Many children, especially at first, will say something like "six and two." Help them see that if you take away the line, it is "sixty-two." You write the word "sixty-two" under the answer.
Now go back and solve the problem you just did together. I use "touch math," so at this point, I or my child writes the dots on the digits of the problem. (you can skip that step if you do not want to use touch math. If you don't know about it, look it up. It is a fabulous method of teaching counting, and it ends up being much more accurate than fingers. They have some free materials so that you can see where to put the dots, which for me is all that I used from the program).
Then have them add the ones. I usually say this several times, trading out the words "pennies" and "ones" so they can connect them in their brain. Do NOT use the actual pennies at this stage unless your child is completely lost, in which case you probably need to spend more time counting single digits instead of going on to double digits. Remind them you traded ten pennies for one dime, so they should trace over the 2 in the answer place and the one that you added above the dimes. Get them to explain to you why that "1" was added. (Even if you just said the answer, the more your child can explain, the more you know he understood the concept). We put the dot in the center of that "1" for touch math.
Next, add the tens. Talk about what they are doing, using the words "dimes" and "tens" often and interchangeably.
Have them read the answer (which you already wrote in words). Then have them copy the words below where you wrote them.
Do 3 - 4 more problems, then stop for the day. You want to give your child time to "stew over" the method of solving the problem. This should be done 3 - 5 times per week for at least 2 weeks (about 5 problems per day) before you use problem with hundreds in them. After a few days, you can start having problems with a hundred in the answer, but you will need to have a dollar to trade for ten dimes. Use the pennies and dimes every day, but as your child understands the concept better, you will find that they are using them less and less. It's okay for them to like playing with the manipulatives (the dimes and pennies), as long as they are using them correctly to solve the problems. He will stop playing with them as he gets excited about solving the problem correctly. It shouldn't be rushed because this will actually hurt the learning process.
Don't use the virtual money (meaning the programs on the internet) until AT LEAST 2 weeks of successfully using the real or play pennies and dimes. Anything online requires a higher level of understanding and your child will lose a lot of the value of this type of learning if you ignore the physical money and go straight to the virtual world.