Don’t I Already Know This Fact?
Although we are told that squares have more area than rectangles (where the average of the length and width is equal to the side length of the square), why does this actually occur? More importantly, one would assume that the area should stay the same as you are increasing and decreasing the length of the square to form the new dimensions of the rectangle. However, this is not the case:
Thus, let’s understand what’s happening behind the scenes. In addition, let’s see how we can use this additional knowledge to solve other problems very quickly.
What’s Really Happening?
When we look at the 7 by 3 rectangle, what we actually realize is that the dimensions are formed by either adding or subtracting 2 from the side length (of the square) 5. Now, if we remember back to our grade 10 unit regarding quadratic equations, you will remember a special binomial called the ‘Difference of squares’. Sure, but how does this help us? To best the connection of the difference of squares to the comparison of the square and the rectangle, let’s convert the dimensions of our rectangle:
Hence, there is a direct correlation to our area comparison and the grade 10 math logic. The reason being, the square will always have more area as the difference of squares is being applied. Meaning, we can actually make a claim on the product of numbers: As the numbers come closer together, their product will increase. For instance, 6 times 4 is 24, whereas 7 times 3 is 21. Note that both of these pairs of numbers average out to 5, yet their products are different. Not only that, but if we look at the values around the square numbers in a multiplication table, their values will decrease. More importantly, it will first be: -1, -4, -9, -16, -25 and so on. Thus, it is exponentially decreasing.
In conclusion, it is evident that the square will always have more area than the rectangle. Furthermore, the mathematical proof behind this is that: In a rectangle, we are removing a square number from the total area. However, with a square, every portion of the area stays:
Wow! This really makes a lot more sense as we can now mathematically prove why the rectangle has less area. Not to mention, we saw the exact same trend in the multiplication table, which means we have found a valid pattern. Now, how exactly can we apply this logic to other problems?
What Is The Application Of This Knowledge?
If I asked you a question like 25 x 15 or 51 x 49, do you think you could you could find the product mentally? If not, then let me help you solve it: First, 25 x 15 can be represented as (20 + 5) (20 – 5), which implies that the product should be 20^2 – 5^2. This means the product of 25 and 15 is 375. Next, for 51 x 49, we can follow the same steps. 51 x 49 can be rewritten as (50+1)(50-1), which means our final product will be 50^2 – 1^2. As result, 51 x 49 is 2499. Now, I have another question for you: What is 99^2? If you’re not sure, let’s break down how you can solve this problem from a logical perspective, while also using the new knowledge we learned today. To begin, if the problem was 98 x 100, what would the solution be?
Thus, by just doing a little analysis and rearrangement of the equation, we were able to find the value of 99^2. The reason being, 98 x 100 is much easier to compute mentally, then something like 99 x 99.
Now that you have learned a little bit about squares and the products of various numbers, why don’t you test yourself with the following problems:
- 65 x 75
- 80 x 120
- 37 x 43
- If the average of the length and width of a rectangle (that is not a square) is 0.5 cm greater than the side length of a square, will the rectangle always have a greater area?
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