Regrouping to Multiply and Divide
One of the keys to success in regrouping to multiply and divide is mastery of the basic multiplication and division facts. Use a variety of activities to reinforce the facts daily. Draw upon your own resources as well as those offered in Using a Multiplication Table and Multiplication Tables and Fact Families.
Mental math will play a major role in introducing your students to multiplying two- and three-digit numbers by one-digit numbers. They will use basic facts and patterns of zero to multiply multiples of 10, 100, and 1,000.
Notice that in the examples following the basic facts, the products have the same number of zeros as the basic fact plus the number of zeros in the factors. In each example, the second factor is a multiple of ten and has one more zero than the factor above, so there is an additional zero in each of the subsequent products.
Manipulatives will help your students grasp the concept of multidigit multiplication. The base-ten blocks below show 3 groups of 32, or
There are 9 tens blocks and 6 ones blocks, so
Base-ten blocks will help your students see when it is necessary to regroup. The following model shows
There are 4 tens blocks and 12 ones blocks, so 10 ones need to be regrouped as 1 ten.
Now there are 5 tens blocks and 2 ones blocks, so
When your students are experienced with modeling regrouping in multiplication, introduce them to the multiplication algorithm. Go through each step carefully, giving students ample time to see the relationship between the model and the process.
Allow students to use base-ten blocks as they progress to regrouping twice in multiplication, and when they multiply money, encourage them to use play money. Students can break away from using a model to multiply when they feel ready to; there's no need to rush them.
Manipulatives also play an important role in division. The first step that students take beyond basic division facts is division with remainders. Counters work well for modeling this concept. This model shows 19 counters divided into 3 groups, or 19 ÷ 3.
There are 6 counters in each group. One counter is left over. The amount left over in division is the remainder.
Base-ten blocks work well for modeling two- and three-digit quotients. The model below shows
Divide the tens first. There is 1 ten in each group.
Divide the ones next. There are 2 ones in each group and 1 one left over.
Models become even more important when students learn regrouping in division. In this example, tens are regrouped as ones.
Divide the tens. There are 2 tens in each group and 1 ten left over.
Regroup the 1 ten as 10 ones to divide. There are 16 ones.
Divide the ones. There are 8 ones in each group.
As you did with multiplication, provide students with many opportunities to model division before introducing the algorithm. Don't be discouraged if students progress slowly with this concept. Remember, this is the first time most students have ever encountered it. Take the time necessary to encourage students as they move through the process.