Math for Parents

Math for Parents
Tim Whiteford PhD

Multiplying and Dividing

Multiplication is more than 16 x 6.
Division is more than 96 ¸ 6.

Multiplication concepts or ideas

1. Repeated addition

             Combining groups that have the same number.
             An extension of addition.
             You know the number in each group and the number of groups.

Using repeated addition to develop the idea of Multiples and
multiplication facts.

Multiplication fact square. An alternative to multiplication tables.

2. Area or Array Concept

     An array is the grouping of objects to form a rectangle or square.
               In area the unit changes from linear to square units.

               Square and Triangular Numbers

3. Multiplicative comparison.

              Comparing two sets or groups using multiplication.
              Mary has 7 times as many books as Peter. If Peter has 4 books
              how many books does Mary have?

4. Combinations (Cartesian Product or the Ben and Jerry’s concept)

              Uses only two sets.
              Defines the possible number of outcomes or possibilities.
              Each “one” of the product is unique.

Division concepts or ideas

1. Fair sharing

When you know the total and the number of groups but you do not
know the number in each group.

2. Repeated subtraction

When you know the total and the number in each group but you do not
know the number of groups.

3. Division Comparison

The opposite of the multiplication comparison idea.

The “remainder” is what is left over after everything has been shared out. The nature of the remainder depends on which concept is being applied.

Every time you multiply you can divide too; and vice versa.

Computation procedures

Multiplication

e.g. 68 x 5

Alternative methods help students understand the standard procedure

68 x 5 = (60 x 5) + (8 x 5)
68 x 5 = 34 x 10
68 x 5 = (68 x 10) ¸ 2 = 680 ¸ 2.

        68
      x 5
        40
     300
      340

or the standard procedure

     ·        5 x 8 = 40
·        regroup the 40 ones as 4 tens.
·        place a small 4 just above the 6.
·        there are no ones so place a 0 in the ones place below the line
·        multiply the 6 tens by 5 and add in the 4 tens to get 34 tens.
·        put the 34 below the line in the 100s and tens place because it is
         really 34 tens.

For 12 x 12 think of a square that is 10 + 2 by 10 + 2. Divided the square like this:

You now have a 10 x 10 square, a 2 x 2 square and two 2 x 10 rectangles. Add the areas together and you get 144.

                12
              x12
                 4       2 x 2
                20       2 x 10
                20     10 x 2
              100     10 x 10
              144

Division

e.g. 235 ¸ 5

Try 100 fives - 500 – too many.
Try 50 fives - 250 - still too many

Try 20 fives – 100 – OK - how many left? – 135                   100 = 20 x 5
Lets do 5 x 20 again - OK -                                               100 = 20 x 5
How many left? – 35
Let’s try 7 fives                                                                       35     7 x 5
Aha, that works
Add the number of fives we’ve have used                                        47
So, 235 divided by 5 is 47
Apply to standard form:           47
                                           5/235
                                               200   40 x 5
                                                 35     7 x 5

More math at home

* Things to combine in equal groups.
          * Buying several things costing the same price|
          * Multiplication fact games
          * Things that come in six-packs or fives etc.
          * Counting in multiples of two, three, four etc.
          * How many options
          * Sharing out candy, cookies etc.
          * Playing with calculators
          * Computer games

There are some neat interactive computer activities at the National Library for Virtual Manipulatives .

For more information contact Tim Whiteford PhD