MTH 489 – Mathematics for Elementary and Middle School Teachers
Summer 2005
Grid Rectangle Problem Discussion
Class #2
How do you know when you have all of the grid rectangles?
Wendy’s method: Go through the divisors of the product starting with 1, and get that divisor and its factor pair. Continue to half the number until you get the reverse factor pair or a square.
e.g., when you get to 25, you can stop at 5
Dawn’s method: Start with the number and find its factors and then takes the reverse factor pair. (Building factor pairs).
Don’t have to go all the way to half the number: Start with 1, continue checking each number until you reached a pair you’ve already used in reverse.
Claim: If a number is a perfect square, then you can stop checking at the square root of the number.
perfect square – the product you get from multiplying a number (whole number?, integer?) by itself. It is called a square number because it makes a square (geometrically).
E.g., 6x6 = 36, 36 is the perfect square, 6 is the square root
Is 0 a square number? yes
Is 0 is a number? -- yes
Other mathematical terms:
multiples of a number are produced by multiplying the original number by an integer
Is 0 is a multiple of 3? Yes! 0 is in fact a multiple of every number.
some multiples of 3: 3, 6, 9, … (1x3, 2x3, …)
factor – A factor is an integer multiplied with another integer to get another number (e.g., 2x2=4, 2 is a factor of 4, 4 is the product)
(Note: If we don’t say integer, numbers such as 2 ½ could be factors…)
Are whole numbers and integers synonyms?
whole number – non-negative integers (0, 1, 2, 3, …)
integers: (…, -3, -2, -1, 0, 1, 2, 3, …)
could make an constructive definition of the integers: e.g., Start with 0; the integers are all of the numbers you get by adding and subtracting 1.
synonym for factor – divisor (all of the numbers that evenly divide another number)
factoring – to find all of a number’s factors
e.g., The factors of 21 are 1, 3, 7, and 21. The divisors of 21 are 1, 3, 7, and 21.
Other terms we are interested in defining
even number
reciprocal
square root
prime numbers
multiplicity
additive
integers
Some patterns observed in the grid rectangles problem:
1. Each number that is a perfect square has an odd number of grid rectangles.
2. Numbers with square roots are the only numbers that have an odd number of grid rectangles.
3. All prime numbers create exactly two grid rectangles.
4. The numbers with odd numbers of rectangles have an even number of numbers between them, starting with 2 and increasing by 2 with each one. e.g., 1, 4, 9, 16 have odd numbers of rectangles, and there are 2 numbers between 1 & 4, 4 numbers between 4 & 9, etc.
5. Numbers that have 3 grid rectangles are squares of numbers with 2 grid rectangles. (e.g., 9 has 3 grid rectangles, and 9 is the square of 3 which has 2 grid rectangles)
6. If the number of tiles is a multiple of 6, then create a group of it and the five numbers preceding it.
Number of tiles / Number of grid rectangles1 / 1
2 / 2
3 / 2
4 / 3
5 / 2
6 / 4
7 / 2
8 / 4
9 / 3
10 / 4
11 / 2
12 / 6
The group of 6 tiles will have the maximum number of grid rectangles for that group; it also will either match or exceed the maximum number of grid rectangles up to that point.
7. All numbers that have 6 grid rectangles are even. Numbers with 4 grid rectangles can be even or odd.
8. The first number (1) and the last number (36), have the least and most grid rectangles respectively. But this is a feature of our particular choice of numbers. For example, if we would have gone to 37, this wouldn’t be true.