Class 8 Mathematics Chapter 6: We Distribute, Yet Things Multiply

A question about nudging

Take the product 23 times 27, which is 621. By how much does it grow if 23 becomes 24? What if 27 becomes 28 instead? And what if both are raised by one? There is a pattern, and finding it is the point of this topic.

The easiest of the three

Raise 27 to 28. Multiplying by 28 rather than 27 means one more lot of 23, so the product grows by 23. You can see it from the definition of multiplication alone, and the answer 644 confirms it.

Saying it with a property

There is a better way to see it. Multiplying a number by a sum is the same as multiplying by each part and adding. In letters, a times b plus c equals ab plus ac. This is the distributive property, and it does the rest of this chapter.

Seeing it as a rectangle

Picture a dots rows wide. Put b columns of dots beside c columns of dots. The whole block holds a times b plus c dots. Counted in two pieces it holds ab dots and ac dots. Both counts describe one block, so they agree.

Back to the nudge

Now 23 times 28 is 23 times the quantity 27 plus 1, which the property turns into 23 times 27, plus 23. The growth of 23 is read straight off. In general, a times b plus 1 is ab plus a.

Raising both factors

What if 23 and 27 both go up by one? That is 24 times 28, and the question is what it adds to 621. Treat 24 as a single thing and distribute over the second bracket: 24 times 27, plus 24 times 1.

Working it through

Each of those two pieces opens up again. So 24 times 28 is 23 times 27, plus 27, plus 23, plus 1. The growth is 51, and 23 plus 27 plus 1 is 51 exactly. In letters, raising both a and b by one adds a plus b plus 1.

One up and one down

Here is the interesting case. Raise 23 to 24 and lower 27 to 26. Do the two nudges cancel? Many people expect so. The answer is 624, which is 3 more than 621, so they do not.

Why they do not cancel

Expanding gives 23 times 27, plus 27, minus 23, minus 1. The gain of 27 comes from the extra lot of the second number, and the loss of 23 plus 1 from the lot taken away. Those are different sizes, so a gap of 3 is left.

And it can go either way

Swap the roles and the product falls instead. Raise 27 to 28 while lowering 23 to 22 and the product drops. Whether the change is a gain or a loss turns on which of the two numbers was the larger.

Chasing the general case

Three cases have each needed their own working. That is a sign to look for one rule covering them all. Suppose a moves by m and b moves by n. The new product is a plus m, times b plus n.

Identity 1

Expanding gives ab plus mb plus an plus mn. So the product moves by mb plus an plus mn. Call that Identity 1. Everything earlier is a case of it, with m and n set to 1, or to 1 and minus 1.

Negatives need no new rule

Lowering a number by 3 is the same as raising it by minus 3. So a fall is already covered: set n to minus 3 and the sign rules for multiplying integers look after the rest. Checked at m equal to minus 2 and n equal to 3, the identity holds.

The pattern in the expansion

Look at the four pieces. The first bracket holds a and m; the second holds b and n. The four products are a times b, m times b, a times n, m times n. Each term of one bracket has met each term of the other.

That rule is the whole method

This is worth holding on to, because it is how every bracket in the chapter is opened. Multiply each term by each term and add the results, watching the signs. Nothing else is needed, however many terms there are.

What makes it an identity

Two expressions are called equal when they give the same value for every number their letters stand for. A statement of that equality is an identity. So a times b plus 8 equals ab plus 8a is an identity, and so is Identity 1.

Negative numbers are included

The distributive property holds for negative integers as well as positive ones. So every expression in this topic keeps working when a letter stands for a negative number. Try a equal to minus 5 and b equal to 8, or both negative, and the identities still hold.

Scenario:

A stationery supplier packs pencils into boxes. The order is for 23 boxes with 27 pencils in each. A clerk works out the total as 621. The buyer then rings twice. First he asks for one more box and one more pencil in every box. Then he changes his mind and asks instead for one more box with one FEWER pencil in each. The clerk says the second change cancels itself out and the total stays at 621. The supervisor disagrees and wants both new totals worked out properly.

Analysis:

The first change is the double-rise case and the second is the one-up-one-down case, and only one of them behaves the way intuition expects.

Title:

Build the four products

Safety:

Take care with the ruler edge and keep pens off the table.

What You Notice:

The four areas come to ab, mb, an and mn, and they add to the whole. That picture is Identity 1, and it shows why each term must meet each term.

Raising one factor of 23 times 27 and lowering the other leaves the product 3 bigger, not the same. The two nudges do not cancel, because the numbers they act on are different sizes.

A pair where raising one by 2 and lowering the other by 4 leaves the product untouched does exist: 1 times 6, 2 times 8, 3 times 10 and so on without end.

A minus is a plus in disguise — One identity handles rises and falls together, because taking away 3 is the same as adding minus 3. The sign rules then look after the rest.

More than two terms

The distributive property has been used on brackets holding two terms. Nothing restricts it to two. Take 3a times the bracket a minus b plus 5. Multiply each of the three terms inside by 3a and add the results.

Doing it carefully

Three a times a is 3 times a squared. Three a times minus b is minus 3ab. Three a times 5 is 15a. So the whole thing comes to 3a squared, minus 3ab, plus 15a. Notice the middle sign: a minus inside the bracket stays a minus.

Can it be tidied further?

Look at 3a squared and 15a. Both contain the letter a, so they look as if they might join. They may not. The first carries a squared and the second carries a, so they are different kinds of thing.

Like terms

Terms carrying exactly the same letters are called like terms, and only like terms may be joined. So 4ab and 7ab join to make 11ab, while 3a squared and 15a stay apart. The numbers in front make no difference to this.

A famous expansion

Try a plus b, times a plus b. Each term meets each term, which gives a times a, then b times a, then a times b, then b times b. That is a squared, then ba, then ab, then b squared.

Collecting it

Now ba and ab are the same thing, because the order of multiplication does not matter. So they are like terms and they join: ab plus ab is 2ab. The tidy answer is a squared plus 2ab plus b squared.

A useful habit

Expand first, then collect. Doing it the other way round, or trying to do both at once, is where most mistakes come from. The chapter sets out twelve worked expansions to be checked, and eight of them are wrong.

A quick way to multiply by 11

The same property gives a shortcut. Since 11 is 10 plus 1, any number times 11 is that number shifted one place to the left, added to itself. Written out, the digits end up added in neighbouring pairs.

Watching it happen

Take 4132 times 11. Write the first digit 4. Then 4 plus 1 is 5. Then 1 plus 3 is 4. Then 3 plus 2 is 5. Then the last digit 2. The answer reads 45452, and no multiplication was done at all.

When the sums spill over

Try 3874 times 11. The pairs give 3, then 11, then 15, then 11, then 4. Each sum past nine carries its ten into the column to its left, and the answer comes out as 42614. The method still works; it just needs the carries.

And the same trick for 101

A hundred and one is 100 plus 1, so a number times 101 is itself shifted two places, added to itself. For 47 that gives 4747. With 1001 the number simply repeats: 236 times 1001 is 236236.

Checking somebody else’s work

Reading a wrong expansion is a good way to learn. Here is one. Someone writes minus 3p times the bracket minus 5p plus 2q, and gets minus 3p plus 5p minus 2q. The outside term has been added instead of multiplied, and the answer is 15p squared minus 6pq.

A sign that went missing

Another writes 2 times the bracket x minus 1, plus 3 times the bracket x plus 4, and gets 2x minus 1 plus 3x plus 4. Only the first term inside each bracket was multiplied. The correct answer is 5x plus 10, not 5x plus 3.

The square that lost its middle

A third writes 5m plus 6n, all squared, and gets 25m squared plus 36n squared. The two squares are right and the middle term is missing. Each term must meet each term, which also gives 2 times 5m times 6n, or 60mn.

Terms joined that should not have been

Someone writes 5w squared plus 6w and makes it 11w squared. The letter is the same but the powers differ, so these are unlike terms and nothing can be done with them. The expression was already as tidy as it gets.

And one that is simply right

Not every item is wrong. Minus q plus 2, all squared, does come to q squared minus 4q plus 4. So does x plus 2, times x plus 5, coming to x squared plus 7x plus 10. Checking means deciding, not assuming a mistake.

What the mistakes have in common

Look back at the wrong ones. One added instead of multiplying. One reached only the first term in a bracket. One dropped a middle term. One joined unlike terms. Every single one is a failure of either each term meets each term, or same letters or stay apart.

Scenario:

A teacher hands out a sheet of twelve worked expansions and asks the class to find the mistakes. Two pupils compare their marking. Asha has marked every item wrong, reasoning that the exercise would not have been set otherwise. Bilal has marked four of them right. Three items are in dispute. The first is minus q plus 2, all squared, given as q squared minus 4q plus 4. The second is 5w squared plus 6w, given as 11w squared. The third is 2 times the bracket x minus 1, plus 3 times the bracket x plus 4, given as 5x plus 3.

Analysis:

Each of the three can be settled by substituting a number, and the two rules of the topic say which rule each error breaks.

Title:

Multiply by eleven in your head

Safety:

No equipment is involved, so no safety step is needed.

What You Notice:

The first five need no working at all. The last three need carries, which is the only extra step the method ever asks for.

Multiplying 4132 by 11 needs no working at all. Write 4, then 4 plus 1, then 1 plus 3, then 3 plus 2, then 2, and the answer 45452 is already there.

A number times 1001 simply repeats itself: 236 times 1001 is 236236, because 1001 is 1000 plus 1.

Same letters, or stay apart — Two terms may be joined only when their letters match exactly. Matching numbers in front of them counts for nothing at all.

Using what you already know

A square of side 60 has area 3600, and a square of side 5 has area 25. Can those two facts help with a square of side 65? Not by adding them, since 3625 is not the answer. But the picture shows what is missing.

Cutting the square up

Draw a square of side 65 and mark off 60 along each edge. The square falls into four pieces: a square of side 60, a square of side 5, and two rectangles measuring 60 by 5.

Adding the pieces

The four areas are 3600, 25, 300 and 300. Together they come to 4225, which is 65 squared exactly. So the two squares you knew were not enough on their own; the two rectangles make up the rest.

The same thing in letters

Replace 60 by a and 5 by b. The square of side a plus b holds a square of side a, a square of side b, and two rectangles a by b. So a plus b, all squared, equals a squared plus 2ab plus b squared.

And from the algebra too

The picture is not the proof. Opening the brackets gives a times a, b times a, a times b, b times b. The two middle pieces are like terms and join into 2ab, which is the same answer. Call this the square of a sum.

Now a difference

What about a square of side 55? Draw it inside the square of side 60. Start from 3600 and take away the two strips measuring 60 by 5, which is 300 twice over.

The piece removed twice

Look carefully at the corner. The two strips overlap there, on a small square of side 5, so its area has been taken away twice. One copy must be put back. That gives 3600 less 300, less 300, plus 25, which is 3025.

The second identity

In letters, a minus b, all squared, equals a squared minus 2ab plus b squared. Only the middle sign has changed from the first identity. The added-back square is why the last term is a plus and not a minus.

A shorter route to it

There is a neater way to see it. Taking b away is the same as adding minus b. Put minus b into the first identity and the middle term becomes 2 times a times minus b, which is minus 2ab, while the last term squares to plus b squared.

A pattern in numbers

Here is a table to explain. Nine squared less 1 squared is 80, and so is 10 times 8. Eight squared less 6 squared is 28, and so is 14 times 2. Seven squared less 2 squared is 45, and so is 9 times 5. Something is going on.

Reading the pattern

In each row the answer is the sum of the two numbers, times their difference. Written in letters, the guess is that a squared minus b squared equals a plus b, times a minus b. A guess is all it is so far.

Checking the guess

Expand the right-hand side. Each term meets each term, giving a squared, minus ab, plus ba, minus b squared. The two middle pieces are like terms with opposite signs, so they cancel completely. What is left is a squared minus b squared.

The third identity

So the pattern is no accident. A sum times a difference always gives the difference of the two squares. This is the third identity, and it is the one that saves the most work in arithmetic.

Putting them to work

Squaring 104 by hand is tiresome. Write it as 100 plus 4 and the first identity gives 10000, plus 800, plus 16, which is 10816. Write 37 as 40 minus 3 and the second gives 1600, minus 240, plus 9, which is 1369.

And the third is the quickest

Try 98 times 102. Those are 100 minus 2 and 100 plus 2, so the answer is 100 squared less 2 squared, which is 9996. In the same way 45 times 55 is 50 squared less 5 squared, or 2475.

A pattern that needs two identities

One more. Twice the sum of two squares seems always to be a sum of two squares: twice 61 is 122, which is 121 plus 1. Why? Add the first two identities together. The middle terms cancel and what remains is 2a squared plus 2b squared.

Reading that result

So a plus b, all squared, plus a minus b, all squared, comes to twice the quantity a squared plus b squared. For 6 and 5 that reads 121 plus 1 equals twice 61, and the pattern is explained rather than merely noticed.

Many routes, one answer

A last word. A count or an area can often be built up in several different ways, and the expressions look nothing alike until they are expanded. When they agree, the identity is what shows that the two routes had to meet.

Scenario:

A tiler is quoting for two jobs and his calculator has died. The first job is a square terrace of side 104 metres, and he needs its area. The second is a rectangular courtyard measuring 98 metres by 102 metres, and he needs that area too. His apprentice suggests that since 98 and 102 average 100, the courtyard must be exactly 10000 square metres, the same as a square of side 100. The tiler is not sure, and he also wants to know whether the terrace is simply 10000 plus 16.

Analysis:

Both jobs are an identity away from being done in one line, and the apprentice has guessed at one of them.

Title:

Race the calculator

Safety:

No equipment is involved, so no safety step is needed.

What You Notice:

The identity wins on anything of the form 100 minus k, times 100 plus k, because the answer is 10000 less k squared and needs no working at all.

The product 98 times 102 can be done in one step. It is 100 squared less 2 squared, which is 9996, and no long multiplication is needed.

Twice the sum of two squares is always the sum of two other squares: for 6 and 5, twice 61 is 122, which is 11 squared plus 1 squared.

Two squares and two rectangles — A square of side a plus b holds a square of side a, a square of side b and two rectangles a by b. That picture is the whole identity.