Repack the same objects
Four small bags each hold six counters. We want to make them easier to count without changing how many counters we have.
- Empty the first two bags into one larger bag. Six counters and six more make twelve.
- Do the same with the other two bags. That gives you another bag of twelve.
- You now have two bags of twelve: twenty-four counters altogether.
So 4 × 6 and 2 × 12 have the same answer: 24.
There are half as many bags, with twice as much in each bag. No counters have been added or taken away.
Try it yourself
Draw six groups of five dots. Pair the groups, circling each pair together. You now have three groups of ten without adding or removing a dot.
For 6 × 5, pair the six groups into three groups of ten. Three tens are thirty, so 6 × 5 is also thirty. The second arrangement may be easier to count even though both show the same objects.
Why both numbers must change
The two numbers you multiply are called factors. In 4 × 6, the factors are 4 and 6.
Here, one number counts the bags and the other counts the counters in each bag. To keep the same total, both must change together: half as many bags, twice as much in each.
If you doubled the counters in each bag but kept all four bags, you would double the total instead.
For now, use an even number of bags so that you can pair them without one being left over.
Try it with groups of five
Try 16 × 5: sixteen bags with five counters in each.
- Pair the bags. Sixteen small bags become eight larger bags.
- Each larger bag holds ten counters: five from each small bag.
- Eight groups of ten make eighty.
So 16 × 5 = 80. Counting tens made the job easier.
Use this method when the new groups feel easier to count. You do not have to use it for every multiplication.