Reading a power line: multiplication as movement along a scale

These teaching notes are for members

What the video covers, what to say before it, where a class usually goes wrong, and what to set afterwards. An account is free, and it opens every chapter of every book.

Create a free account Sign in

What to assume they know

The laws of exponents, and where each one comes from — the product and power-of-a-power rules
Zero and negative exponents: extending the rule rather than inventing a meaning — the division rule, and zero and negative exponents
Reading positions on a marked line, including positions below a marked zero
The powers of 4 up to 65536 and of 7 up to 823543, or the willingness to build them by repeated multiplication
Unit fractions: 1/4, 1/16, 1/7, 1/49 and so on

What they 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

Where it usually goes wrong

"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.

Questions to check understanding

Place a given power on a partially labelled power line and supply its value
Fill missing values on a power line that runs below the exponent 0
Say how many times as large one power is than another, for a shared base
Evaluate a product or quotient of two listed values by locating them on the line
Write three different expressions that all name the same power
Given a move described in words ("divide by 64"), say how many rungs it is and in which direction
Explain why the rungs are equally spaced although the values are not — the reasoning-style question this figure is built for
Build the first few rungs of a power line for a new base, such as 3 or 10

Examples worth working on the board

The line of powers of 4 (Part I p.29). A vertical rule with eleven rungs. The exponent labels sit on the left, top to bottom: 4⁸, 4⁷, 4⁶, 4⁵, 4⁴, 4³, 4², 4¹, 4⁰, 4⁻¹, 4⁻². The values sit on the right, in the same order: 65536, 16384, 4096, 1024, 256, 64, 16, 4, 1, 1/4, 1/16. All eleven pairs are printed.
The four arrows (Part I p.29), read off the printed page and confirmed on the printed page:
A curved arrow labelled ÷ 4 runs from the 4⁸ rung down to the 4⁷ rung — one rung.
A curved arrow labelled × 16 runs from the 4⁶ rung up to the 4⁸ rung — two rungs.
A curved arrow labelled ÷ 16 runs from the 4³ rung down to the 4¹ rung — two rungs.
A curved arrow labelled × 4 runs from the 4¹ rung up to the 4² rung — one rung.
The left-hand bracket (Part I p.29). Three expressions are braced together and joined to the 4⁵ rung: 4⁸ ÷ 4³, then 4³ × 4², then 4⁷ × 4⁻². Each needs a different rule and all three arrive at the same place.
The right-hand bracket (Part I p.29). Three arithmetic statements are braced together and joined to the value 1024: 65536 ÷ 64, then 64 × 16, then 16384 multiplied by one sixteenth. These are the left bracket's three expressions with the values substituted in.
The worked comparison (Part I p.30). Is 16384, which is 4⁷, sixteen times larger than 1024, which is 4⁵? The chapter answers yes, because 4⁷ ÷ 4⁵ = 4².
The open comparison (Part I p.30): the chapter then poses the same comparison across the exponent 0, asking by what factor 4² exceeds 4⁻². Left for the student.
The line of powers of 7 (Part I p.30). Twelve rungs, exponents on the left and values on the right: 7⁷ with 823543; 7⁶ with 117649; 7⁵ with 16807; 7⁴ with 2401; 7³ with 343; 7² with 49; 7¹ with 7; 7⁰ with 1; 7⁻¹ with 1/7; 7⁻² with 1/49; 7⁻³ with 1/343; 7⁻⁴ with 1/2401.
The eight questions beside it (Part I p.30), each printed with an equals sign and a blank. In order: 2,401 × 49; then 49³; then 343 × 2,401; then 16,807 divided by 49; then 7 divided by 343; then 16,807 divided by 8,23,543; then 1,17,649 multiplied by 1/343; and finally 1/343 multiplied by 1/343. Every one is a walk along the printed line — use the eight items, not the answers.
A useful note on the printed values. Every number appearing in the eight questions is already a labelled value on the 7-line, so a student never has to multiply large numbers. That is the point of the figure.

Figures to have open

The power line itself, redrawn for base 4 with all eleven rungs, the values alongside, and the four arrows able to be shown moving. This is the chapter's own figure (Part I p.29) and it is the only figure this topic has; redraw it as a schematic rather than reproducing the printed art, and keep the arrow spans exactly as printed.
The line of powers of 7 (Part I p.30) with its twelve rungs, and the eight questions arranged so each can be traced on it.
A stepping token — a marker that slides along the line while the expression being evaluated builds underneath. An added device, and what turns a static figure into an argument.
A side-by-side of the two lines, 4 and 7, to show that the construction is about the exponents and not about the particular base. An added figure.
No photograph is needed; the chapter prints none on these two pages.

Where this sits in the book

NCERT Ganita Prakash Class 8, Part I, printed Chapter 2, "Power Play", §2.3 "The Other Side of Powers", the unnumbered subheading "Power Lines", Part I pp.29–30, running up to §2.4.
The rules being displayed were established at Part I pp.24–25 and Part I pp.27–29 (The laws of exponents, and where each one comes from and Zero and negative exponents: extending the rule rather than inventing a meaning).
The idea returns in a different guise at Part I pp.36–42, where the powers of 10 are used as a ladder for real quantities (Why the nearest power of ten is the only handle on a quantity too big to picture).
Chapter-end exercises that reward this figure: "Figure it Out" at Part I p.45, items 7 and 8.