PrepShorts · Study sheet · Class 8 Mathematics · Chapter 6, We Distribute, Yet Things Multiply
Chapter 6 · We Distribute, Yet Things Multiply
(a + b)² and (a − b)²: why there is a middle term at all
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Squaring a sum is not squaring its parts, and the amount you lose by is not an abstraction. It is two rectangles you can point at.
The idea
Squaring a sum is not squaring its parts, and the reason is something you can point at: the square built on a + b holds two smaller squares and two rectangles, and those two rectangles are congruent. That congruence is the entire origin of the doubled middle term — not a rule, a pair of identical pieces. The subtraction version is then not a second thing to learn. It is the same identity with the second letter replaced by its negative, and the sign rules alone decide that the middle term flips while the last term does not. The chapter also builds it a second way, by starting from the bigger square and removing too much, which is why the small square comes back with a plus.
What you should be able to do
- Compute the square of a two-digit number by splitting it into a round part and a small part and adding four areas
- Expand
(a + b)²by the distributive property and account for the middle term as two congruent rectangles - State Identity 1A and use it on numbers and on expressions
- Decide when
(a + b)²exceedsa² + b²and when it does not, in terms of the signs ofaandb - Choose a split of a given number that makes both squares easy, and show that different splits give the same total
- Expand the square of a two-term expression such as
(6x + 5)², both by distributivity and by the identity, and compare the two routes - Build the square of a difference geometrically, by removing two overlapping strips from a larger square and restoring the doubly-removed corner
- Derive Identity 1B from Identity 1A by substituting a negative, and say why only the middle term changes sign
- Apply both identities to squares of numbers just above or just below a round number
Words to know
| Term | Definition in one line | First introduced |
|---|---|---|
| sidelength | the length of a side of a square or rectangle, written by the chapter as one word | printed in this chapter (Part I, §6.2, p.145) |
| area | the amount of surface a region covers | printed in this chapter (Part I, §6.2, p.145) |
| square of the sum | the result of adding two numbers and then squaring | printed as part of a subheading in this chapter (Part I, §6.2, p.145) |
| Identity 1A | the chapter's label for the expansion of the square of a sum | printed in this chapter (Part I, §6.2, p.145) |
| Identity 1B | the chapter's label for the expansion of the square of a difference | printed in this chapter (Part I, §6.2, p.147) |
| like terms | terms built from exactly the same letters to the same powers | printed in this chapter (Part I, §6.1, p.141) |
| distributive property | the rule that multiplying a sum is the same as multiplying each part and adding | printed in this chapter (Part I, §6.1, p.137) |
| exponent notation | writing a repeated product of one letter as that letter with a raised count | printed in this chapter (Part I, §6.1, p.141) |
| middle term | the doubled cross term standing between the two squares | an added phrasing; the chapter writes the term out and never labels it |
| overcorrection | removing an area twice and adding one copy back | an added word for the move the chapter narrates as a question |
Where people slip up
- "The square of a sum is the sum of the squares." The single most durable error in school algebra, and the chapter's figure is the antidote: the two rectangles are drawn, tinted and labelled, and they are plainly not nothing. Run it numerically too — 60 squared plus 5 squared is not 65 squared, and the shortfall is exactly the two 300s.
- "The middle term is
ab." It is two copies ofab, because there are two rectangles and they are congruent. Get a student to point at both. - "The square of a difference is
a² − b²." Two separate errors live here: the missing middle term, and the belief that the last term should be negative. Squaring a negative gives a positive, which is why only the cross term flips. The next topic in this chapter is wherea² − b²actually comes from, and confusing the two is the reason to teach them in this order. - "The square of a sum always beats the sum of the squares." Not when the two numbers have opposite signs, and not when either is zero. The chapter asks this as a Math Talk question for exactly that reason.
- "There is a right way to split a number before squaring." 65 can be cut at 60 and 5, at 30 and 35, or at 52 and 13. The total is fixed; only the difficulty changes. The chapter sets all three.
- "You must remember both identities." 1B is 1A with the second letter negated, and the chapter shows the substitution explicitly. One identity plus the sign rules is enough — and the owl box says that even that is optional if you can expand.
- "The square of
6xis6x²." The printed working takes it to36x²in a visible step. Do not skip it. - "The 55-square argument is faulty because it removes the corner twice." It does remove it twice, and that is the interesting part. The chapter narrates the double removal as a problem and then repairs it, which is a better lesson than a clean derivation would have been.
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Worked answers to this chapter’s exercises · this video explains Figure it Out · 6.4 Q2, Figure it Out · 6.4 Q5
Transcript1,302 words
Sixty times sixty is three thousand six hundred. Five times five is twenty-five. You already know both of those, and neither one cost you any effort. So here is the question. Can those two easy squares give you sixty-five squared? Add them. Three thousand six hundred and twenty-five. Sixty-five squared is four thousand two hundred and twenty-five. So the answer is no, and it is no by six hundred. This whole video is about where that six hundred went, and it is somewhere you can point at.
Draw a square with sixty-five down every side. Cut the top edge at sixty, and cut the left edge at sixty too. Two cuts. The square falls into four regions, and not one of them is optional. The big one is sixty by sixty, which is three thousand six hundred. The small one is five by five, which is twenty-five. And there are two more, each of them sixty by five, each of them three hundred.
Three thousand six hundred, and three hundred, and three hundred, and twenty-five. Four thousand two hundred and twenty-five. Squaring the parts found you two of the four pieces and walked past the other two. Now make the same two cuts with letters instead of numbers. A square, a plus b down each side, cut once across and once down. Top left is a by a. Bottom right is b by b.
The top right piece is b across and a down. The bottom left piece is a across and b down. Different corners of the figure, but built from the same two sidelengths, which makes them the same rectangle turned round. And they are the only pair in this figure that matches. Add the four areas up. A squared, and a b, and a b, and b squared. Two of those four are the same thing, so they collect into two a b.
The square of a plus b is a squared, plus two a b, plus b squared. That two is not a rule anybody decided. It is a count of pieces. There are two rectangles because there are two ways to put a against b, and both of them are inside the square. Change the figure to a real rectangle, a and b across but a and u down, and nothing matches any more.
Four products still, but four terms in the answer, and no doubling anywhere. The doubling belonged to the square. So the square of a sum is the sum of the squares, plus that middle term. Is it always the bigger of the two? Everything hangs on the middle term, and the middle term carries the sign of a times b. Take every pair of whole numbers from minus six to six. A hundred and sixty-nine pairs.
In seventy-two of them the square of the sum is the bigger one. In seventy-two it is the smaller one. In the remaining twenty-five they are exactly equal. Bigger when the two numbers share a sign, smaller when they do not, and equal when either one of them is zero. Nothing forced that cut at sixty and five. Cut sixty-five at thirty and thirty-five instead. Nine hundred, plus one thousand and fifty twice, plus one thousand two hundred and twenty-five. Four thousand two hundred and twenty-five.
Cut it at fifty-two and thirteen. Two thousand seven hundred and four, plus six hundred and seventy-six twice, plus one hundred and sixty-nine. The same total again. Every one of the sixty-six ways to cut sixty-five into two whole parts lands on the same number. The total was never in question. Only how easy the four pieces are, and that is your choice to make. Now square something with a letter in it. Six x plus five.
The long way is four products, exactly as before. Six x times six x is thirty-six x squared. Thirty-six, not six, because the six gets squared along with the x. Then six x times five, twice over, which is sixty x. Then five times five, which is twenty-five. The short way says first thing squared, twice the product, second thing squared, and stops. Thirty-six x squared, plus sixty x, plus twenty-five. The same three terms, four lines earlier.
And if the identity will not come to you, the long way still gets there. It is a shortcut, not a gate. Now turn it round and build a smaller square inside a bigger one. Here is a square of side sixty, and tucked into its top left corner, a square of side fifty-five. What is left over is two strips, each of them five wide. Take away the strip down the right hand side. Three thousand six hundred less three hundred is three thousand three hundred.
Take away the strip along the bottom. Another three hundred gone, which leaves three thousand. But fifty-five squared is three thousand and twenty-five. Two strips came off, both of them honestly, and the answer is twenty-five short. Look at where the two strips cross each other. That little five by five corner is in both of them. So it went once with the first strip, and it went again with the second.
It has been taken away twice, and it was only ever there once. Put one copy back. Three thousand and twenty-five, which is fifty-five squared. Not two copies and not none. One, because it was removed exactly one time too many. The argument was never wrong. It overshot, and then it said so. In letters the four pieces are a squared, minus a b, minus a b, plus b squared.
So the square of a minus b is a squared, minus two a b, plus b squared. Now notice which sign changed and which one did not. Take the first identity and put minus b wherever b stood. The middle term holds one b, so it flips. The last term holds two, and a negative times a negative is positive, so it does not. One identity and the sign rules cover both cases. There is no second thing to memorise.
It also tells you why the square of a difference is not a squared minus b squared. That answer is only right when b is zero, or when a and b are equal. Now put it to work, and let the split do the thinking. A hundred and four is a hundred plus four. Ten thousand, plus four hundred twice, plus sixteen. Ten thousand eight hundred and sixteen. Thirty-seven is forty less three. Sixteen hundred, less two hundred and forty, plus nine. One thousand three hundred and sixty-nine.
Ninety-nine is a hundred less one. Nine thousand eight hundred and one. Fifty-eight is sixty less two. Three thousand three hundred and sixty-four. None of that is a different method. It is the same four pieces every time. The skill is choosing where to cut, so that both squares are ones you already know. One last thing this buys you, and it is not about arithmetic at all. Every even number is two k. Square it and you get four k squared, so every even square is a whole number of fours.
Every odd number is two k plus one. Square it and you get four k squared, plus four k, plus one. Which is four, times k, times k plus one, and then one more. But of k and k plus one, one of them is always even. So that four is really an eight. Every odd square is a multiple of eight with exactly one left over. Not sixteen, though. Three squared is nine, and nine is not sixteen and one.
Two easy squares never did give you the third one. What they were missing was a pair of rectangles, and you can see both of them.
Where this fits
Taken from the notes each video was made from, not from the reading order — these are the ideas this one rests on and the ones that later rest on it.
Builds on
- Multiplying two two-term expressions, and where the four terms come fromClass 8 · Ch 6, We Distribute, Yet Things Multiply
Comes up again in
- Why the product of a sum and the matching difference is a² − b², and the patterns that followClass 8 · Ch 6, We Distribute, Yet Things Multiply
- Many different-looking expressions for one growing patternClass 8 · Ch 6, We Distribute, Yet Things Multiply
Either side of this one
- Fast mental multiplication, powered by distributionClass 8 · Ch 6, We Distribute, Yet Things Multiply