Web Design by. Skip to main content. Special Factoring: Differences of Squares Diff. Purplemath When you learn to factor quadratics , there are three other formulas that they usually introduce at the same time.
Content Continues Below. Share This Page. Terms of Use Privacy Contact. Advertising Linking to PM Site licencing. Visit Our Profiles. Start the pattern, silently again, with students, and continue only far enough for them to get the idea of what this new exploration is. This time, you can offer the marker to students right away, since the stage has already been set by the earlier activities.
You might do three or four examples, perhaps ending like this and then encourage students to do it on their own, trying several more examples. When they are ready, sketch and hand the marker to someone. Of course, it gets followed by You might try one or two more, but then offer a new challenge: If this proves to be too much of a leap, silently offer this hint:. The same procedure — silent beginning, some exploration on their own, silent extensions — can be used for researching three and four steps away.
Perhaps they can find something in their work that would let them predict the rule for five steps away! Even if they do, of course, they should check that prediction by doing the research in the same way.
For five steps away, the description of the pattern might look like this: is always 25 more than. The rule for five steps away, by the way, makes it easy to square two-digit numbers that end in 5.
Take 35, for example. According to the rule, the product of numbers five steps away should be 25 less than the square of that middle number. Turning both columns and placing them at the bottom gives us an arrangement of 25 tiles same as it started that is now 7 rows tall , and mostly 3 columns wide, but again with a part that sticks out.
Each bottom row sticks out two, because the top of the figure is narrower by two columns than it started out. Learning a trick is not important. But learning to feel this comfortable with numbers, realizing how much one can do, is important.
Asher, a particularly bright and eager child who knew his multiplication facts as he had just turned eight and was entering third grade, delighted in multiplying numbers that were twelve steps away from a number he could easily square in his head.
The logic is the same, but it is worth seeing again. This is 47 units wide and 53 units tall or 47 columns and 53 rows, or 53 rows with 47 tiles in each row.
Here is one of many experiments that help to make multiplication of negative numbers feel natural. All of the explorations above involved either 1 starting with a number, squaring it, and comparing that result to the product of numbers that differed from it on both sides by some fixed quantity or 2 starting with two numbers whose difference was even, and comparing their product to the square of their mean. There is one other possibility: comparing the product of consecutive numbers numbers that differ by 1 with the product of their neighbors some fixed distance to the right or left.
If the neighbors are one step away, the situation looks like this,. If they are two steps away, it looks like this,. After exploring one step and two steps as above , we may explore three and four and more steps away, looking for a generalization. Two tempting but wrong ones are:. Of course, generalizing from only two cases is likely to lead to wrong conjectures, but it is quite natural to have hunches even before one has checked things out. Asher was not yet nine years old when he was exploring this multiplication pattern.
After seeing just the 2 and 6, he had a hunch, but a very different and much more sophisticated one than the ones given above. It is just icing on the cake to mention that Asher made this discovery and conjecture in a situation where there was nothing to write on or write with: no calculator, no paper, no pencils.
The numbers, the computations, and the results all had to be held mentally. He had met the original problem — difference of squares — the same way. Holding up three fingers and later, five he was comparing the square of the number at the middle finger with the square of the two numbers next to it:.
That is a nice bonus, if students get good at it, and the students love it because it makes them feel smart, but being able to multiply certain pairs of numbers mentally is not the goal. And learning a way of describing patterns is important.
If by any chance you spot an inappropriate image within your search results please use this form to let us know, and we'll take care of it shortly. Term » Definition. Word in Definition. Wiktionary 5. Freebase 1. How to pronounce difference of two squares? Alex US English. David US English. Mark US English. Daniel British. Libby British.
Mia British. The difference of two numbers is found by subtracting. The difference of two squares means one squared term subtract another squared term. The difference of two squares can be factorised into brackets using the method above for factorising quadratics.
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