PrepShorts · Study sheet · Class 8 Mathematics · Chapter 6, We Distribute, Yet Things Multiply
Chapter 6 · We Distribute, Yet Things Multiply
Many different-looking expressions for one growing pattern
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A growing pattern does not come with a formula. It comes with a value at every step and any number of honest ways of counting to it.
The idea
A growing pattern does not come with a formula. It comes with a value at every step, and any number of honest ways of counting to that value — each of which produces a different-looking expression. So the interesting question is not "what is the formula" but "are these two the same formula", and the picture cannot answer it: geometry records how you counted, and checking a step or two only records a coincidence. Only expanding both expressions to a common form settles it. That is what the chapter is really teaching in this section, and it is why the same section sets four ways of counting circles, three ways of measuring one shaded region, and then asks for more.
What you should be able to do
- Continue a drawn pattern by one step, and count the units in the drawn steps
- Decompose one pattern in several ways, and write an expression for the step number from each decomposition
- Expand each expression and show that all of them reduce to a single common form
- Explain why agreement at one or two steps does not establish that two expressions are equal, and why expansion does
- Use a derived expression to find the count at a step too large to draw
- Write an expression for a shaded area by subtraction and again by direct measurement, and verify the two agree
- Evaluate competing expressions at given values and confirm they give one number
- Read a figure whose parts are described in letters, and identify the dimensions of each part
- Produce a second method for a pattern or an area, deliberately unlike the first
Words to know
| Term | Definition in one line | First introduced |
|---|---|---|
| Step | the chapter's name for one stage of a growing pattern, numbered from one | printed in this chapter (Part I, §6.4, p.150) |
| basic unit | the repeated shape a pattern is built from | printed in this chapter (Part I, §6.4, p.156) |
| algebraic expression | a combination of letters and numbers built with the operations | printed in this chapter (Part I, §6.3, p.150; §6.4 uses it on Part I p.153) |
| equivalent | said of two expressions that take the same value for every substitution | printed in this chapter (Part I, §6.4, p.154) |
| simplify | to rewrite an expression with fewer or tidier terms | printed in this chapter (Part I, §6.4, p.152) |
| identity | an equality of two expressions that holds for every substitution | printed in this chapter (Part I, §6.1, p.139) |
| sidelength | the length of a side of a square or rectangle, written by the chapter as one word | printed in this chapter (Part I, §6.4, p.153) |
| like terms | terms built from exactly the same letters to the same powers | printed in this chapter (Part I, §6.1, p.141) |
| method | the chapter's word for one particular way of decomposing a pattern | printed in this chapter (Part I, §6.4, p.151) |
| decomposition | the way a figure has been cut up before counting | an added word; the chapter shows the cuts by colour and framing and gives them no name |
| common form | the simplified expression that two different-looking expressions both reduce to | an added phrasing |
Where people slip up
- "Every pattern has one formula, and my job is to find it." It has one value at each step. The chapter prints four expressions for one pattern and invites a fifth, and the point of section 6 is that this is not a problem to be resolved but the normal state of affairs.
- "Different-looking expressions are different expressions." The four printed ones look nothing like each other and are the same. The only way to know is to expand.
- "If two expressions agree at Step 1, they agree." All four agree at Step 1 by construction — they were all read off the same drawing. Agreement at a step is guaranteed and therefore proves nothing about any other step. Make this explicit; it is the transferable idea in the section.
- "My method is wrong because it is not the book's." The chapter prints four and asks whether the reader's matches any of them or is different, with no suggestion that being different is a fault.
- "You cannot subtract to find an area." Two of the printed methods in this section do exactly that, and one of them subtracts four rectangles from a square. Subtraction is a way of counting.
- "The shaded area depends on which student you ask." Three named methods, three different expressions, one number when the letters are replaced. That is the whole design of the Fig. 1 item.
- "Drawing the next figure answers the question." It answers one question. Step 10 is drawable with patience and Step 100 is not, which is what forces the expression.
- "You can read the expression off the picture and stop." The picture tells you how you counted. Whether your count and someone else's always agree is an algebraic question, and expanding is how it is answered.
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Worked answers to this chapter’s exercises · this video explains Figure it Out · 6.4 Q10, Figure it Out · 6.4 Q11
Transcript1,391 words
Here is a pattern of circles, and it grows. Step one holds three. Step two holds eight. Step three holds fifteen. Draw the next one and you get twenty-four. That was easy enough. Now: how many at step ten? You could draw it. You would rather not. How many at step one hundred? Now you cannot, and neither can I. So the question turns into this. Is there an expression, a rule in the step number, that hands you the count at any step at all?
The picture will not answer that. A picture holds a value at every step, and it holds no formula whatsoever. So count it yourself. Here is step three again, and here is one way to see it. It is a square, four across and four down, with a single circle missing from one corner. Four squared, take away one. Fifteen. Step one is the same shape. A two by two square, less one. Three.
Step two, three by three less one, is eight. Step four, five by five less one, is twenty-four. Every time the square is one bigger than the step, in both directions. So this way of counting says: one more than the step, squared, take away one. Here is a second way to see exactly the same drawing. There is a square in the middle whose side is the step number, and then two arms. One column down the right, one row along the bottom.
At step three the square holds nine, and the two arms hold six between them. Nine and six is fifteen. At step two, four in the square and four in the arms. Eight. At step one, one in the square and two in the arms. Three. At step four, sixteen in the square and eight in the arms. Twenty-four. So this way says: the step squared, plus twice the step.
A third way, and for this one you have to stop looking for a square. Take a rectangle that is the step number across and one more than the step down. Then add one more column beside it. Step three: three across, four down, and a column of three. Fifteen. Step one: one across, two down, and a column of one. Three. Step two: two across, three down, plus two. Eight.
Step four: four across, five down, plus four. Twenty-four. So this way says: the step, times one more than the step, plus the step. And a fourth, which is the tidiest of the four. Slide the circles round and they make a single rectangle with nothing left over at all. The step number across, and two more than the step down. Step three: three by five. Fifteen. Step one: one by three. Three.
Step two: two by four, eight. Step four: four by six, twenty-four. So this way says: the step, times two more than the step. One drawing. Four honest counts. Four expressions that look nothing like one another. Put them side by side and look at what you actually have. One more than the step, squared, less one. The step squared plus twice the step. The step times one more than the step, plus the step. And the step times two more than the step.
One of them has a bracket squared in it. One has no bracket at all. Two of them add something on the end and two do not. So here is the honest question, and you should feel the doubt in it. Are these one rule written four ways, or four different rules that have happened to agree so far? The picture cannot tell you. A picture is a record of how you counted, not of whether your count and somebody else's will always agree.
That is an algebra question, and there is exactly one way to answer it. Expand them. All four of them. One more than the step, squared, opens out to the step squared, plus twice the step, plus one. Then take one away, and the plus one and the minus one annihilate. The step squared plus twice the step. The second one is already there. It needs nothing done to it.
The step times one more than the step is the step squared plus the step. Add another step and you have the step squared plus twice the step. And the step times two more than the step opens straight out to the step squared plus twice the step, in a single move. Four routes, one destination, term for term. They were never four rules. They were one rule, counted four ways.
Now the part that matters, and it is not really about circles. All four agreed at step one. Of course they did. They were all read off the same drawing. Agreeing at a step you copied from is not evidence of anything at all. Watch how cheap agreement is. Take the straight line through the first two counts: five times the step, take away two. At step one that gives three, which is right. At step two it gives eight, which is right.
At step three it gives thirteen, and the truth is fifteen. It is out by two, and from there it is wrong forever. Two steps of perfect agreement bought you nothing. Expanding is what settles it, and nothing short of expanding does. Which is what an expression is for. Step ten. The step squared is a hundred, twice the step is twenty, and the count is a hundred and twenty.
You could have drawn that one. Slowly, and with a ruler. Step fifteen. Two hundred and twenty-five, plus thirty. Two hundred and fifty-five. Nobody is drawing that. And notice what has just happened between you and the picture. The drawing is what let you find the rule. The rule is now what goes where no drawing can follow. That is the whole trade. The same move works on an area, and there it is worth watching closely.
A square whose side is a plus b, with four identical rectangles packed around a hole in the middle. One way to measure that hole: take the whole square, and subtract the four rectangles. The other way: look at it. The hole is itself a square, and its side is a minus b. Measure it directly. Subtracting is a way of counting. It is not a lesser one, and it is not a trick.
Expand the two and they land on the same thing, which is what you would want, and is not what the drawing told you. Two accounts of one region. The algebra is what makes them agree. Now three people, one shaded region, and three methods that share nothing. The first takes a square of side a, and subtracts a rectangle a by b. The second takes the whole outer rectangle, a across and a plus two b down, and subtracts all three of the bars standing inside it.
The third refuses to subtract anything. It adds the two flanking pieces, each of them half the difference of a and b wide, and a tall. Three expressions, and not one of them looks like another. Expand all three and every single one is a squared minus a b. Put a equal to eight and b equal to three into any of them and you get forty. Not three answers. One.
So here are some to take away with you. A ring of tiles. Eight at step one, twelve at step two, sixteen at step three. What is it at step four, at step ten, and at any step at all? Twenty, and forty-four, if you want to mark yourself afterwards. A rectangle with a band down one side and along the bottom, and a region left in the corner. Nine wide, six tall, band three and a half.
The corner is thirteen and three quarters, the band around it is forty and a quarter, and together they are fifty-four. Get there two different ways. And a sequence that runs five, eleven, nineteen. At step ten it is a hundred and thirty-one. Find a method. Then go and find a different one, and expand them both. Because a second method is not a waste of your time. It is the only thing that will ever tell you the first one was right.
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
- (a + b)² and (a − b)²: why there is a middle term at allClass 8 · Ch 6, We Distribute, Yet Things Multiply
- 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
Either side of this one
- Why some resized images look right and others look stretchedClass 8 · Ch 7, Proportional Reasoning-1