Primary Mathematics/Multiplying numbers
The Teaching of Multiplication
Manipulatives and Models
In the early and middle primary grades, the operation of multiplication is first denoted by the symbol "x", and is normally first taught as repeated addition, where multiplying the numbers "3" and "5" (3 x 5) is the same as adding the number three a total of five times: 3 + 3 + 3 + 3 + 3 = 15, so 3 x 5 = 15. Modern curriculums initially teach this concept with manipulatives, or toys and objects that serve to model, and enable students to visualize what multiplication is. For instance, a teacher might model the problem 4 x 6 by putting 4 jelly beans each in 6 bags. Subsequent models become more abstract, but the model that all students eventually learn to understand and use is called an area rectangle:
The above area rectangle models the problem 9 x 4, which equals 36. Note that there are 4 rows and 9 columns and that the height of the rectangle is 4 units and the width (or base) is 9 units. Students find that when they count the 9 unit rows four times, they will count a total of 36 units. Soon, they come to the realization that if they were to count the 4 unit columns nine times, they arrive at the same answer. This becomes one of the ways that they learn the commutative property: a x b = b x a (See:Wikipedia:Commutativity) Eventually, students should memorize all of combinations of multiplying single digit numbers, but it should be noted that just because a student has memorized their multiplication facts that they necessarily understand at an abstract level what multiplication represents. Note that the above rectangle has a total of 36 square units, which serves to introduce students to the geometric axiom that area of a rectangle is equal to the base (9 in this case) times the height(4).
By working with area rectangles, students make connections to geometry that serve to strengthen their understanding of multiplication. Eventually students no longer need to see the interior square units and soon come to the understanding that one or both of the lengths of these area rectangles can be expressed as the sum of two lengths. The following area diagram models the problem 16 x 3 :
Because 16 = 10 + 6, students can visualize that the length of the above rectangle is still 16. They then can see that the larger rectangle can be expressed as the sum of the two smaller area rectangles: (3 x 10) + (3 x 6).
In upper elementary grades students are introduced to a more standard mathematical nomenclature where the mulitpication symbol "x" is no longer necessary, permitting them to notate an equation for this model as: 3(16) = 3(10) + 3(6). This is the primary model used to teach the distributive property: a(b + c) = a(b) + a(c) (See: Wikipedia:Distributivity)
The area model can be used to show the distributive property at an even more complex level. Consider the problem 155x360:
The Use of Different Algorithms to Solve Multiplication Problems
Note that in the above model, the height and width (base) of the larger area rectangle has been broken up into smaller area rectangles based on the place holder of each digit in each number, making it a representation of the standard algorithm for multiplication:
This standard algorithm can be further modified to make it more efficient.
In the example above, multiplication is the same, but uses carried digits and integrates them into the same row.
The Lattice Method
Some students find an algorithm known as the lattice method easier to use because it involves the use of lines that guide the eventual values of the final place holders. It should be noted that students using this algorithm generally take more time to complete a problem. Additionally, this algorithm tends to become confusing for students when it is used with problems involving decimals. For these reasons, the use of the lattice method can become a liability in later grades. It is recommended that as soon as this model is mastered by the student, it becomes advisable to teach and encourage them to use the standard model.
In the left image, the two numbers are arranged at the edges of a grid. In each cell of the grid, the two numbers in the row/colum are multiplied together to get a two digit number; the tens place is placed in the upper-left triangle in the grid, and the ones in the bottom-right corner.
In the right image, each diagonal is summed to produce a number on the bottom; digits are carried to the diagonal on the left.
Multiplication with decimals
When decimals appear in a multiplication problem, they can be hidden during the multiplication. For example, when looking at the problem 21.4 x 5.63 you would ignore the decimals and perform the operation for 214 x 563. When the computation is complete and you have the answer of 120482, count the digits (or place values) to the right of the decimal in each of the original two factors. The first factor, 21.4, has one digit to the right of the decimal and the second factor 5.63 has two digits to the right of the decimal. Next, take the total of these two counts, which in this case is 3, and include the same number of digits to the right of the decimal point within the product. This makes the answer 120.482
Another way to determine where the decimal point belongs in your answer is to make an estimate. For example if you once again have the problem 21.4 x 5.63 you could make a quick estimate of 20 x 5 which is 100, so you know your answer will have to be near 100. If you complete the multiplication computation, once again ignoring the decimals (214 x 563), your answer will be 120482. The only reasonable place to insert the decimal point, based on our estimate of 100, is between the 0 and 4, making the answer 120.482