Estimating a quantity nobody can count: guess, model, assume, approximate

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

Multiplying a two- or three-digit number by a two-digit number
Reading and comparing quantities in hundreds, thousands, lakhs and crores
Unit conversion between grams and kilograms, and between kilometres and hours at a stated speed
Scientific notation, and why the standard form is 1 ≤ x < 10 — reporting a quantity to as many digits as you actually know
The Class 7 chapter on large numbers, which the chapter names as the place this work began

What they should be able to do

Write a relation among named quantities before attempting any arithmetic
Make a stated, defensible assumption for a quantity nobody has looked up
Produce an instinctive guess before calculating, and compare it afterwards
Compute an estimate from a relation and a set of assumptions
Report an estimate at an accuracy the assumptions support
Rework the same relation with a different unit substituted, and say what changes
Run a relation backwards, solving for a quantity that is not the last one named
Judge two estimates that disagree, and decide whether the disagreement is in the model or only in the assumptions

Where it usually goes wrong

"There is a right answer and my number is wrong." Two students who assume different coin masses will get different counts and both may be right. What can be wrong is the relation — dividing where you should multiply, or forgetting a conversion. Mark the model, not the arithmetic.
"Guessing first is a waste of time." The guess is what tells you afterwards whether the answer is plausible. A student who computes 300 coins for 45 kg and never guessed has nothing to notice with.
"An estimate should be given to the last digit." Reporting 3,142,857 coins from an assumed coin mass claims an accuracy the assumption cannot support. Round to the digits you have earned.
"Assume means make something up." It means state a value, out loud, that you could defend and that someone else could disagree with. An unstated assumption is the actual error.
"Money problems and distance problems are different techniques." They are the same four steps. The chapter deliberately runs jaggery, coins, notebooks, meals and walking through one routine.
"Working backwards needs a different method." The relation is the same; you solve for a different slot in it. The pādayātra question is the same product as the coin question, read in reverse.
"If I cannot look it up, I cannot do it." The coin question has no number in it at all. That is not a defect of the question, it is the question.

Questions to check understanding

Given a situation, write the relation among quantities before computing
State the assumptions needed for a described estimate and give a defensible value for each
Compute an estimate and report it to a sensible number of digits
Given someone else's estimate and assumptions, judge whether it is reasonable
Re-estimate the same quantity with one assumption changed, and say by what factor the answer moves
Work backwards from a distance and a rate to a duration
Choose between two proposed relations for one situation and justify the choice
Explain why two correct estimates can differ — the reasoning-style question this section exists for

Examples worth working on the board

The opening situation (Part I p.33). Nanjundappa wants to give away jaggery matching Roxie's weight and wheat matching Estu's, and wants to know the cost.
The balance illustration (Part I p.33, artwork). Two beam balances side by side. In the left one a girl sits cross-legged in one pan with stacked blocks of jaggery in the other; in the right one a boy sits opposite sacks of wheat. Verified on the printed page. It carries no numbers and is the figure that makes the relation obvious without stating it.
The two relations, as the chapter writes them (Part I p.33). The rupee worth of the jaggery is Roxie's weight in kilograms multiplied by the price of one kilogram of jaggery. The rupee worth of the wheat is Estu's weight in kilograms multiplied by the price of one kilogram of wheat.
The two ages, given as constraints (Part I p.33): Roxie is 13 and Estu is 11. These are the only facts supplied, and they are there to make a weight assumption defensible rather than arbitrary.
The chapter's own assumptions and results (Part I p.33). Taking Roxie at 45 kg and jaggery at ₹70 a kilogram gives 45 × 70 = ₹3150. Taking Estu at 50 kg and wheat at ₹50 a kilogram gives 50 × 50 = ₹2500. These are the book's numbers, worked on the page.
The culture box (Part I p.33, tinted panel). The practice of giving goods equal in weight to a person is old, is still followed in parts of southern India, stands as a token of gratitude and of bhakti, and also supports the community.
The four-step routine (Part I pp.33–34). Step 1 is guessing: name an answer instinctively, with no calculation. Step 2 has three parts — describe the relations among the quantities needed; make reasonable assumptions and approximations where information is missing; then compute, and check the guess against the result.
Roxie's coin question (Part I p.33): if 1-rupee coins were used instead of jaggery, how many would it take to match her weight? A speech bubble on Part I p.34 supplies the missing move — weigh a coin and find out. The chapter offers the guess in brackets: hundreds, thousands, lakhs, crores, or more.
The owl panel on guessing (Part I p.34, tinted): early guesses may be far off and that is fine; the skill improves with practice, and guessing and estimating build a feel for quantities.
Estu's variation (Part I p.34): what if 5-rupee coins or 10-rupee notes are used instead? Guess first, then work it out. Note as a check: notes and coins have very different masses, so this is not a rescaling of the previous answer — the relation must be rebuilt.
Two adult intentions (Part I p.34). Estu wants to give away notebooks worth his weight every year; Roxie wants to do annadāna worth her weight every year. The question, flagged Math Talk, is how many people each of these yearly gifts might reach — a different relation again, since it needs a quantity per person.
The pādayātra (Part I p.34). Roxie and Estu overhear that a group walked about 400 km to arrive early that morning. How long ago did they set out? This is the relation run backwards: distance and walking speed are known or assumable, and the time is the unknown.
The pilgrimage box (Part I p.35, tinted panel with a long line of walkers drawn beneath it). Walking long distances as a spiritual practice is observed across religions in the country under different names. Six pilgrimages are named, among them Pandharpur Wari and the Kānwar Yatra.
Circumnavigation (Part I p.35): how many times could a person walk right round the world in a lifetime, walking without stopping? The chapter supplies one number only — take the distance round the Earth as 40,000 km. Everything else is the student's to assume.
The Note to the Teacher (Part I p.34, in a dashed red box). It grants that what a student assumes may vary a great deal, so the computed answers vary too, and says that is fine. What matters is modelling the situation properly, and there may be more than one good way to do that. Accuracy of assumed quantities improves with practice.

Figures to have open

An empty relation frame — a product with named, initially blank slots, each filling from either a given or a stated assumption, colour-coded by which. This is the spine of the whole topic and the chapter has no equivalent; not in the book.
A beam balance with a person in one pan, and the contents of the other pan swapping between jaggery, coins, notes and notebooks while the relation underneath updates. Built from the chapter's own artwork idea (Part I p.33), redrawn.
A guess-versus-answer strip, logarithmically spaced so a guess that is ten times out looks ten times out. Not in the book, and it pairs naturally with the power line of Reading a power line: multiplication as movement along a scale.
No photograph is needed. The chapter's walkers illustration on Part I p.35 is decorative.

Where this sits in the book

NCERT Ganita Prakash Class 8, Part I, printed Chapter 2, "Power Play", §2.5 "Did You Ever Wonder?", Part I pp.33–35, up to the unnumbered subheading "Linear Growth vs. Exponential Growth".
The Note to the Teacher sits at the foot of Part I p.34.
The chapter names the Class 7 chapter on large numbers as where this thread started (Part I p.33).
The chapter's SUMMARY (Part I p.46) restates the routine — guess, model, assume and approximate, then calculate.
The same routine is applied to a specific pair of growth processes in Additive growth versus multiplicative growth, and to quantities beyond counting in Why the nearest power of ten is the only handle on a quantity too big to picture.