Primary mathematics/Multiplying numbers
Manipulatives and ModelsEdit
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 curriculum initially teach this concept with manipulatives and models, which 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:w: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 does not necessarily conclude that they 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 multiplication 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: w: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 ProblemsEdit
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 is then further modified to make it more efficient.
The Lattice MethodEdit
Some students initially 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.
Multiplication and DecimalsEdit