Roughly how much?

“One Thread” series, article 2

Working with teachers and tutors, we keep hearing the same question: how do you explain something so that every child understands? We have no ready-made recipes, but we do have observations and techniques we would like to share. We will try to share our experience regularly, one small example from school maths at a time, and to follow where it leads — from primary school all the way into adult life.

A customer buys 38 packs of exercise books at 49 som each. The shop assistant taps it into a calculator and says: “That’s 18,620.” Someone in the queue immediately feels that this cannot be right, while someone else reaches for their wallet. The calculator made no mistake: it honestly multiplied what it was given. The assistant simply pressed an extra zero and multiplied 380 by 49.

People often say that mental arithmetic is out of date: the phone is always at hand and never makes mistakes. But a phone answers precisely only the question that was typed into it. Whether it is the right question, and whether the answer looks like the truth, can only be judged by a person who knows roughly how much it should come to before pressing any buttons. Today’s conversation is about that “roughly”.

Primary school: keep the bundles, drop the sticks

In the previous issue, when we looked at column subtraction, we used bundles of sticks: a ten is a bundle, a hundred is a bundle of bundles. Estimation starts exactly here. Thirty-eight is almost four bundles. Forty-nine is almost five. To round a number to the nearest ten means to keep the bundles and stop counting the separate sticks.

Four tens times five tens is twenty hundreds, that is, two thousand. And note: a second-grader who understands that ten times ten makes a hundred gets this answer in a second, without any multiplication table.

Here we rearranged and regrouped the factors: we worked out 4 × 5 separately and 10 × 10 separately. This is allowed by the commutative and associative laws. The numbers in a problem come in the order the author wrote them, not in the order that is convenient to calculate. Whoever knows that they may be rearranged calculates in a convenient order.

We rounded both numbers up, so the real answer is a little less than two thousand. And if we round both down — three tens times four tens — we get one thousand two hundred, and the real answer is certainly more. Now we know the main thing: the purchase costs between one thousand two hundred and two thousand som, closer to two thousand. Eighteen thousand is impossible however you look at it.

This is estimation from below and from above — the same work with bundles, and it can be taught at the same time as multiplication. A child who is used to saying “it will be about…” later checks their own answer, without a teacher.

Fifth and sixth grade: where to put the decimal point

A fifth-grader is given an example: 0.38 × 4.9. They multiply in a column, as with whole numbers, and get 1862. Now the decimal point has to be placed, and the rule comes into play: count how many digits there are after the point in both factors — two and one, three in all — and count off the same number from the right. The result is 1.862. The rule is correct. But if the child slips by one digit, the exercise book will show 18.62 or 0.1862. Each of these answers looks just as plausible as the correct one if you do not know roughly how much it should be.

And knowing that is not hard. 0.38 is almost 0.4, and 4.9 is almost 5. Four tenths times five is two. So the answer is about two, and of the three options only 1.862 fits. The decimal point fell into place before we even remembered the rule.

Seventh to ninth grade: how many places

In the seventh or eighth grade standard form appears: 1862 is written as 1.862 · 10³. In fact this is the most honest way to write an estimate: the power of ten tells how many places the number has, that is, its order of magnitude. Our purchase of “about two thousand” is exactly 2 · 10³.

And here is what a pupil needs to understand: for many questions the order of magnitude matters more than all the other digits. Try it on yourself. How much is a million seconds? A million and a billion sound alike: both are “a lot”, both have zeros. But a million seconds is just over eleven days. And a billion seconds is almost thirty-two years. Three extra places — and “next week” turns into “half a lifetime”.

A person who has a feel for the order of magnitude will not confuse a million with a billion in the news. This does not need a separate lesson: it is enough for the question “how many places are there here?” to be asked in every problem with very large or very small numbers.

Estimate plus correction

Now the most interesting part. Back to the exercise books: 38 packs at 49 som. We estimated “about two thousand” by rounding both numbers. And what if we round only one? Which one is up to us: rearranging the factors does not change the product (that is the commutative law), so we choose the more convenient one. 49 is 50 minus one. Thirty-eight packs at 50 som is 1900. But each pack costs one som less, so we have overpaid by 38 som. Let’s take them back: 1900 − 38 = 1862. This is no longer an estimate but an exact answer, worked out in the head.

Behind this move stands the distributive law: 38 × (50 − 1) = 38 × 50 − 38 × 1. In second grade it is written on the board, and the children dutifully memorise it without understanding what it is for. Here you can see what it is for: it lets you split a hard multiplication into an easy one and a correction.

Look at how this move works. The first step is to bring the number to a round one, to the next place: take 50 instead of 49. If you stop here, you get an estimate. The second step is the correction: give back what we added. Do both steps — and you have the exact answer. Estimation and mental calculation are not two different skills but one, simply taken to the end or not.

A whole family of tricks that people love to show in short videos rests on this same move:

• To multiply by 9 means to multiply by 10 and subtract the number itself. 47 × 9 = 470 − 47 = 423. Nine is ten minus one, and the distributive law lets you remove the extra part separately.
• To multiply by 5 means to multiply by 10 and halve. 64 × 5 = 640 : 2 = 320. Five is half of ten, and the associative law lets you multiply by 10 first and then divide by 2.

Each time we do the same thing: we move to a round number, that is, to the next place, where multiplying comes down to shifting digits, and then we correct. Subtraction “with a margin” from the previous issue works in exactly the same way, only there it is the associative law of addition at work: subtracting 17 is the same as subtracting 20 and adding 3.

A child meets these laws in primary school as rules with wordings, and in seventh grade — again, in algebra: expanding brackets rests on the same distributive law. If the laws were a tool the child used for mental calculation, algebra will turn out to be a way of writing down what they have long been able to do. It is just as important to understand that not everything can be rearranged: seven minus three is not the same as three minus seven, and in life you put on your socks first and only then your shoes. The commutative and associative laws are a special property of addition and multiplication, and place-value notation turns this property into a tool.

A trick shown as a magic trick is forgotten within a month. A trick understood to the end is one the child will start inventing on their own — and that is a sure sign that place value has been understood.

Upper grades and life: interest that grows on interest

A problem: a deposit grows by 10% a year. How many times will it grow in 25 years? The most common answer is “by 250%”, that is, three and a half times: 10% multiplied by 25 years. But this is wrong. Each year the 10% is added not to the original sum but to the sum that has already grown.

Let’s follow this without a calculator. To add 10% means to add a tenth, and a tenth of a number is the same number shifted one place to the right. We had one — we add 0.1 and get 1.1. We add a tenth of 1.1, that is 0.11, and get 1.21. A tenth of 1.21 is 0.121, the same number shifted one place to the right. Adding: 1.21 + 0.121 = 1.331, or, rounded, 1.33. And so at every step: shift the number one place to the right and add it to itself. Then 1.33, then 1.46, 1.61, 1.77, 1.95. Seven steps — and the sum has almost doubled. So in fourteen years it will grow almost fourfold, in twenty-one — more than sevenfold. The remaining four years add almost another half, and it comes to about eleven. There was a thousand — now there are almost eleven thousand, not three and a half. The difference is not in the percentages but in the place value: the sum has gained an extra digit.

Now the same arithmetic from the other side. A loan at 24% a year: every year the debt grows by almost a quarter. Add almost a quarter to one — 1.24; almost another quarter of that — about 1.54; once more — about 1.91. In three years the debt has almost doubled. A person who can estimate like this sees all of it before signing the contract. To someone who cannot, only time will show it.

Tomorrow in class

Everything said comes down to one simple habit, and it can be introduced in any grade — from the second to the eleventh. Before a pupil starts calculating, ask them to name the answer roughly. Not to guess, but to estimate: round to a convenient place and say “it will be about two thousand” or “it will be a little less than two”. Then let them calculate exactly — on paper, in their head or on a calculator. And then compare one with the other.

This third step — comparing — is the most important one. If the exact answer and the estimate are far apart, one of them contains a mistake, and the pupil looks for it on their own, without a hint from the teacher. If they are close, the pupil gets something no answer key at the back of the textbook can give: the confidence that they understand what they are doing.

This does not need a new textbook: it only needs the question “roughly how much?” to sound as familiar in class as “how much is it?”.

What the thread rests on

• Grades 2–3. To round means to keep the place you need and drop the lower ones.
• Grade 2. Factors can be rearranged and regrouped — the commutative and associative laws.
• Grades 2–4. A multiplication can be split into an easy part and a correction — the distributive law.
• Grade 5. A product of decimals has as many digits after the point as both factors together, and an estimate places the point before the rule does.
• Grades 7–8. The power in standard form is the number of places.
• Grade 10. With compound interest, interest is added to interest, and the sum grows not linearly but exponentially.

You can discuss the article, share your own techniques and suggest a question for the next issues in the “One Thread” Telegram channel @youraizone.