
This unit is about multiplication and division. The main ideas here are to reinforce (if your child was in Portland Public Schools, he or she was exposed to multiplication in third grade, in detail) the idea that multiplication is repeated addition and division is repeated subtraction. Knowing more than one strategy to solve problems is a major emphasis.
Homework will include the idea of clustering multiplication problems, which means taking a big multiplication problem and breaking it into smaller, easier problems. Other ways of solving problems are shown, such as using tables, making pictures, using manipulatives, skip counting and traditional "Old School" are taught. Also, understanding the notion of remainders in division is important, as seen in problems about taking kids in vans to events. Can we use half a van, or do we need to use a whole van with empty seats? That type of thinking is part of the unit.
Vocabulary introduced here will include arrays, which are either rectangles or squares filled with square areas inside; multiples, which are the numbers we say when we count by a number, or the answers to multiplication problems (also known as products); and factors, the numbers used in a multiplication problem. A simple way to remember this is the following equation, F x F = M. This simply means factor (F) times factor (F) equals Multiple (M), as in 2 x 3 = 6, with 2 and 3 being factors of 6, or seen a different way, 6 is a multiple of 2 and 3. The array for 6 can be made with a shape which is 3 squares wide by 2 squares deep. These are the dimensions of the array and can be shortened with symbols A for area of an array = Length x Width of the sides. Length = 3 squares and Width = 2 squares, so A = 3 x 2 = 6
Understanding groups of numbers is also important. Numbers with only 1 array (or 2 factors) are called prime, including 2, 3, 5, 7, 11 and so on. Numbers with 2 or more arrays are called composite, including 4, 6, 8, 9, 10 and so on. Another way of looking at numbers is through factor pairs, such as the number 2 which has a factor pair of (1, 2) which is the same as (2, 1), so is only counted as one factor pair. This allows us to see composite numbers as having 2 or more factor pairs. The number 10 for example has factor pairs (2, 5) and (10, 1).
The notion of factor pairs also allows us to see the number 1 as different from all other numbers, since it has only 1 factor, itself, and thus called "unique". 1 is also a square number, like 4, 9, and 16 which are arrays made with equal sides. Picture a square with 2 squares across and 2 squares going down. This array is square in shape and makes the number 4. Similarly, 9, 16, 25 and so can be made by forming arrays with equal dimensions, or side measurements.
Computation focuses on two main ideas, fluency, which is just old fashioned learning the times table; and clustering, which is breaking a big problem into smaller, easier problems. Let's look at the problem 23 x 6. First, can we break 23 into easier numbers to work with? Yes, let's make the problem into 10 x 6 and 10 x 6 and 3 x 6, finally adding the pieces. 60 + 60 + 18 = 138. We will also show them the "Old School" method most adults learned, which is harder to show in this format, but goes something like this:
A. 3 x 6 = 18 then B. we regroup the 18 into 8 ones and 1 ten, carrying the 1 ten above the tens place in 23. C. We multiply 6 x 20 = 120 and add the carried ten, shown as 1 to get 130
1 1 A. 23 B. 23 C. 23 X 6 X 6 X 6 _____ _____ _____ 8 8 138 Yes, there are a few little tricks we show them, such as multiplying by 10 can be thought of as sticking an extra "0" behind a number. For example 17 x 10 = 170 which is just 17 with a "0" behind it. This makes multiplying by 100 and 1000 and so on much easier as keeping track of zeroes is not hard for most students.