There are many different ways to understand multiplicative expansion (the word "growth" can become confusing when students begin to multiply fractions and integers). One way is to see it as a series of addition problems, such as:
2and so on. As we move down each row, the multiplier increases by 1 so that:
1x2 is 2You can also say it as "3 two's make 6" or, "3 groups of 2 equals 6".
2x2 is 2+2
3x2 is 2+2+2
4x2 is 2+2+2+2
5x2 is 2+2+2+2+2
Another way to represent this is to use grouping. Students can make piles of manipulatives such that each pile has the same number of items in it. This time, let's use multiples of 7. In this case, each pile would have 7 pieces. Let's say they wanted to know how many 4 groups of 7 are, or, 4 x 7. Students can use various strategies to determine the total.
- count up the total number of pieces by either counting each individual item in all of the piles
- start counting-on using their understanding that the first pile has 7, then continuing counting the remaining piles from 8 onward
- count each group "by 7's", also known in some circles as "skip counting"
- count the number of groups and use their remembered answer for 4x7
Each of these stages show a different level of mastery of the concept.
However, piles of manipulatives, or circled pictures of groups of objects on a worksheet have a limited usefulness when it comes to visualizing the patterns that are common to multiplication.
For this reason, we can try and move to an area model as shown below. It is called the area model, because the solution to the multiplication problem also represents the value of the area of the rectangle. Area is another way in which multiplication can be visualized, and it also shows a practical application of the concept.
In our example the number of units in each row is 7, while the number in each column is 6. We have 6 rows of 7, or 6 x 7 units in the rectangle.
The grid lines in these pictures don't have to be there for the model to work. Simply knowing the base and the height of the rectangle gives enough information so that the multiplication problem can be solved, and the area found.
Adding the grid lines helps us see the groupings involved, and makes the visual representation of the problem clearer, particularly for students who have not yet reached mastery. Approaching the same problem using groups of rows and repeating it using groups of columns helps reinforce the key principle of commutativity, in which the order of the numbers multiplied does not change the final product.
Once the multiplication concept has been explored, students will eventually need to learn to access those facts quickly. There are a number of options that can help with this including:
Each of these has its place at various times, however, students who can spend less effort to retrieve these facts do better as more concepts are introduced in higher grades.
- learning to skip-count (counting by a number, such as 3-6-9-12-15-18-21-24-27-30-33-36-39-42-45 etc.
- classroom games
- finding number patterns to follow (even numbers for 2's, ends in 0 for 10's, digits add to 9 for 9's etc.)
- recognizing patterns in daily life (eggs come in 2x6=12; a case of canned vegetables has 4x5=20 cans, etc.)
- intensive answering, such as with regular timed tests, Mad Minutes, etc.
- rote repetition
- written tables such as the one to the right, which are given blank for students to complete and mark for patterns
- musical chants /songs
- classroom charts
The over-use of rote methods, and the dawn of Bloom's Taxonomy, have made the task of having students memorize their times-tables unpopular in the classroom. This is slowly changing.
The problem is not so much that students spend time memorizing these facts, which is admittedly a lower-level task, but that if they do so without an understanding of how multiplication actually works, the knowledge of "facts" will have limited value as more complex mathematics are introduced. Students who understand the language, the commutative property, the groupings, and that multiplication is an advanced form of addition, can show how multiplication patterns continue, and likewise, how using the inverse operation of division causes the pattern to reverse, will be well equipped to apply it to fractions, decimals, integers, algebra, etc. They will also be better able to handle related concepts such as area and volume.