PrepShorts · Study sheet · Class 8 Mathematics · Chapter 2, Power Play
Chapter 2 · Power Play
Reading a power line: multiplication as movement along a scale
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Stand all the powers of one number in an evenly spaced column and multiplication stops being an operation. It becomes a move.
The idea
Set the successive powers of one number at equal spacing along a line and multiplication stops being an operation and becomes a movement. Multiplying by the base is one step; multiplying by its square is two steps; dividing is the same steps the other way. That is why so many unlike-looking expressions land on the identical rung — they are different routes to the same place — and why "how many times larger" is settled by counting rungs rather than by dividing the values. The exponents are the addresses; the rules of the previous section are just the arithmetic of walking.
What you should be able to do
- Build a line of the powers of a chosen base, with the exponents in order
- Locate a given power on that line and read off its value
- Interpret multiplying by the base as a one-rung move, and by a power of the base as a move of that many rungs
- Show that several different expressions land on the same rung, and say why
- Compute "how many times as large" for two powers of a shared base by subtracting exponents
- Extend the line below the exponent 0 and place the reciprocal values correctly
- Answer a mixed product-and-quotient question by walking a printed line rather than by computing
- Explain why the rungs are equally spaced although the values are not
Words to know
| Term | Definition in one line | First introduced |
|---|---|---|
| power line | a line carrying the successive powers of one base at equal spacing | printed in this chapter, as a subheading and again at Part I p.30 |
| base | the number whose powers the line carries | printed in bold in this chapter (Part I p.22) |
| exponent | the label on each rung; also the address of a position on the line | printed in bold in this chapter (Part I p.22) |
| times larger | the chapter's phrasing for a comparison by division | printed in this chapter (Part I p.30) |
| rung | one labelled position on the line | an added term; not printed in this chapter, which draws the positions and names none of them |
| scale | a line whose rungs are read as positions rather than as counts | an added term; not printed in this chapter |
| ratio | the quotient of two values, as against their difference | an added term; not printed in this chapter, which performs the divisions without naming the quantity |
Where people slip up
- "The line is a number line, so the values are equally spaced." They are not, and cannot be — 1 and 4 would be a whisker apart while 16384 and 65536 would need half a kilometre. What is equally spaced is the exponent. Say this explicitly the first time the line appears, or students will try to read intermediate values off it.
- "Sixteen times larger means sixteen more." The chapter's phrasing invites this. Sixteen times as large is a multiplication; sixteen larger is an addition. Put 1024 + 16 beside 1024 × 16 once.
- "Going down the line means subtracting." Going down divides. The whole point of the figure is that a move is a multiplication or a division, never an addition or subtraction — even though the exponents add and subtract.
- "4⁰ is a special case that interrupts the line." It sits at even spacing like every other rung, between 4¹ and 4⁻¹, with the value 1. The chapter draws it that way on purpose, right after proving why.
- "The rungs below 4⁰ are a different kind of thing." They obey the same step rule. One rung down from 1 divides by 4 and gives a quarter, exactly as one rung down from 4 gives 1.
- "Different expressions landing on one rung is a coincidence." It is the three rules of the previous section, drawn. Trace each of the three left-hand expressions as a route along the line and the bracket explains itself.
- "You need the values to answer the questions." You need the exponents. On the 7-line the answer to 2,401 × 49 is found by counting rungs, and the value is then read off — the reverse of the order students expect.
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Worked answers to this chapter’s exercises
Transcript1,341 words
Take one number — four — and write its powers in a column. Four to the eighth at the top. Then the seventh, the sixth, the fifth, and on down. Four to the first, four to the zero, and keep going below that: four to the minus one, four to the minus two. Eleven rungs. Now the important part, and it is a decision, not a fact. Space them evenly.
Every rung is the same distance from its neighbour, all the way down. Hold on to that, because by the end of this it will look like the only sensible thing anybody could have done. Now write what each one is worth, on the right. Sixty-five thousand five hundred and thirty-six at the top. Then sixteen thousand three hundred and eighty-four. Four thousand and ninety-six. One thousand and twenty-four. Two hundred and fifty-six, sixty-four, sixteen, four.
One. Then a quarter, then one sixteenth. Look at those two columns together. On the left, evenly spaced. On the right, nothing like evenly spaced. Here is what the picture buys you. Put a token on any rung and move it up one. Sixteen becomes sixty-four. Sixty-four becomes two hundred and fifty-six. Every single time, one rung up is multiplying by four. And one rung down is dividing by four.
Two hundred and fifty-six becomes sixty-four, sixty-four becomes sixteen. So on this line, multiplying is not an operation you carry out. It is a move. That is the whole idea, and everything else today is a consequence of it. If one rung is a four, what is two rungs? Four twice. Sixteen. So a jump of two rungs upward is multiplying by sixteen, and two rungs down is dividing by sixteen.
Watch it happen. From four thousand and ninety-six, two rungs up, and you land on sixty-five thousand five hundred and thirty-six. That is times sixteen, and nobody multiplied anything. Going the other way: from sixty-four, two rungs down, and you land on four. Divide by sixteen. The size of the multiplier is decided entirely by how far you travel. Now something that looks like a coincidence and is not. Here are three expressions.
Four to the eighth, divided by four cubed. Four cubed, times four squared. Four to the seventh, times four to the minus two. They look nothing alike, and each one needs a different rule. But watch where each of them lands. Eight minus three is five. Three plus two is five. Seven plus minus two is five. All three arrive at the same rung, because all three are the same journey described three ways.
Write those same three in plain numbers and the mystery goes. Sixty-five thousand five hundred and thirty-six divided by sixty-four. Sixty-four times sixteen. Sixteen thousand three hundred and eighty-four, times one sixteenth. One thousand and twenty-four, every time. And notice what did not happen: none of those three needed working out. Every number in them is already sitting on the line, so each one is a start rung, a number of steps, and a finish rung.
You read the answer off. You do not compute it. The rungs below the zero one worry people, so look at them properly. Nothing changes. One rung down from four is one. One rung down from one is a quarter. One rung down from a quarter is one sixteenth. Same step, same spacing, same rule — divide by four each time. The rung holding one is not a wall and it is not a special case.
It sits between four and a quarter at exactly the same distance from each, and it has to, because the step from four to one is a divide by four and so is the step from one to a quarter. Here is a question that gets asked badly. Is sixteen thousand three hundred and eighty-four sixteen times as large as one thousand and twenty-four? You could divide. Or you could count.
They are two rungs apart, and two rungs is sixteen. Yes. And now the trap in the wording, which is worth stopping on. Sixteen times as large is not sixteen larger. One thousand and twenty-four plus sixteen is one thousand and forty. One thousand and twenty-four times sixteen is sixteen thousand three hundred and eighty-four. Those are not close. Try one that crosses the middle. How many times as large is four squared as four to the minus two?
You do not need the values at all. Count the rungs. From minus two up to two is four steps. Four steps is four multiplied by itself four times: two hundred and fifty-six. Check it if you like — sixteen divided by one sixteenth is two hundred and fifty-six — but the checking is not the method. The method was counting, and it did not care that the journey passed through the zero rung on the way.
None of this was about the number four. Build the same thing for seven. Seven to the seventh at the top: eight hundred and twenty-three thousand, five hundred and forty-three. Then a hundred and seventeen thousand, six hundred and forty-nine. Sixteen thousand eight hundred and seven. Two thousand four hundred and one, three hundred and forty-three, forty-nine, seven, one. And below: one seventh, one over forty-nine, one over three hundred and forty-three, one over two thousand four hundred and one.
Twelve rungs, evenly spaced, exactly as before. Only the step size changed. One rung is now a seven. So here are eight questions, and the point is how little arithmetic they take. Two thousand four hundred and one, times forty-nine. Those are rungs four and two, so the answer is rung six: a hundred and seventeen thousand, six hundred and forty-nine. Forty-nine cubed. Forty-nine is rung two, and cubing means three of those, so rung six again — the same answer, by a different road.
Three hundred and forty-three times two thousand four hundred and one: rungs three and four, so rung seven. Sixteen thousand eight hundred and seven divided by forty-nine: five take away two, rung three, three hundred and forty-three. Seven divided by three hundred and forty-three: one take away three, rung minus two, one over forty-nine. And sixteen thousand eight hundred and seven divided by eight hundred and twenty-three thousand five hundred and forty-three is five take away seven — rung minus two again, the same answer.
A hundred and seventeen thousand six hundred and forty-nine, times one over three hundred and forty-three: six take away three, rung three. And the last one is the interesting one. One over three hundred and forty-three, times itself. Minus three and minus three is minus six — and there is no rung minus six, because the drawing stops at minus four. The answer is one over a hundred and seventeen thousand, six hundred and forty-nine, and to see it you have to extend the line by two more rungs.
Last thing. Why is it fair to space these evenly when the values so obviously are not? Look at the gaps on the value side. From four thousand and ninety-six up to sixteen thousand three hundred and eighty-four is a jump of twelve thousand two hundred and eighty-eight. From one down to a quarter is a jump of three quarters. Ten gaps, ten different numbers, not one of them repeated.
Now look at the ratios instead. Every neighbouring pair is in the ratio four. Ten gaps, one number. That is the whole answer: the thing that is constant here is the multiplier, so the thing that gets the even spacing is the exponent. Draw it by value instead and the picture dies — with the top rung at the top of a board ten units tall, everything from one sixteenth up to sixty-four is crushed into less than one hundredth of a single unit, and the first two rungs, one and four, sit less than half a thousandth apart.
Even spacing is not an approximation you are putting up with. It is what makes the picture say something true.
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
- The laws of exponents, and where each one comes fromClass 8 · Ch 2, Power Play
- Zero and negative exponents: extending the rule rather than inventing a meaningClass 8 · Ch 2, Power Play
Comes up again in
- Why the nearest power of ten is the only handle on a quantity too big to pictureClass 8 · Ch 2, Power Play
Either side of this one
- Powers of 10, and place value written out for whole numbers and decimalsClass 8 · Ch 2, Power Play