Class 7 Mathematics Chapter 2: Arithmetic Expressions

A phrase, not a sentence

In language, a phrase is not yet a statement. "The tall boy" names something without claiming anything.

Numbers work the same way. Write 9 + 6 and you have named a quantity without yet saying what it comes to. A phrase like that is called an arithmetic expression.

Every expression has a value

An expression names one number, and that number is its value.

The value of 9 + 6 is 15. The value of 25 - 7 is 18. The value of 8 x 4 is 32. The value of 36 divided by 6 is 6. Finding the value is called evaluating the expression.

The equality sign joins the two

We write the expression, then the sign, then its value.

So 9 + 6 = 15. The sign does not mean "the answer is". It means that what stands on the left and what stands on the right are the same number, written two ways.

Reading an expression aloud

Each operation has more than one name, and both are worth knowing.

9 + 6 can be read as "nine plus six" or as "the sum of nine and six". 8 x 4 is "eight times four" or "the product of eight and four". 25 - 7 is "twenty-five minus seven" or "the difference of twenty-five and seven".

Writing an expression from a situation

This is the step that does real work, and it comes before any arithmetic.

A tiffin costs 18 rupees and is bought on six days. The expression is 6 x 18. Notice what has been done: the situation has been turned into a phrase, and no multiplying has happened yet.

Only then evaluate

With the phrase written, the arithmetic is the easy part.

6 x 18 = 108, so the six tiffins cost 108 rupees. Writing the expression first and evaluating second keeps the two jobs separate, and it is much easier to check.

A rough value first

A quick estimate is worth making before any careful working. The phrase 48 x 11 must be a little more than 48 x 10, which is 480. So a final answer near 528 looks right, and one near 100 does not.

An estimate never replaces the working. It tells you whether the working you have already done can be trusted.

Many phrases, one value

A number can be named in a great many ways.

Twenty-four is the value of 20 + 4. It is also the value of 30 - 6, and of 6 x 4, and of 48 divided by 2. Four different phrases, one number.

Why that matters

If several expressions share a value, the expression carries information the value does not.

The value 108 tells you a total. The expression 6 x 18 tells you it was six lots of eighteen. Keeping the expression tells the reader where the number came from.

Comparing two expressions

Expressions can be compared with the same three signs used for numbers.

Those signs are =, < and >. To compare 7 x 5 with 30 + 6, evaluate each: 35 and 36. Since 35 is less than 36, we write 7 x 5 < 30 + 6.

Comparing without full evaluation

Often you can decide without finishing either sum.

Compare 9 x 8 with 9 x 7. Both are nine lots of something, and eight is more than seven, so the first must be larger. No multiplication was needed at all.

A longer expression

An expression may name more than two numbers.

A library holds 40 books, and seven more shelves arrive holding nine books each. The expression for the total is 40 + 7 x 9. Its value is 103, because seven nines are 63 and 40 added to 63 is 103.

The order is not obvious

Look at 40 + 7 x 9 again and notice something uncomfortable.

Read left to right and you would add 40 and 7 first, giving 47, then multiply by 9 to get 423. That is a different answer. So an expression with two different operations needs a rule, and that rule is the next topic.

What an expression is not

An expression is not an equation, and the difference is worth holding.

9 + 6 is an expression; it names a number. 9 + 6 = 15 is an equation; it claims that two things are equal. An expression can be evaluated. An equation can be true or false.

Checking your own work

The cheapest check is to read the expression back against the situation.

For the tiffins, 6 x 18 should be read as "six lots of eighteen rupees". If the reading matches the situation, the expression is right, whatever the arithmetic turns out to be.

What carries forward

Two habits from this topic last the whole chapter.

Write the expression before evaluating it, and keep it after evaluating it. The value answers the question; the expression explains the answer.

Scenario:

A school library keeps 40 books on its reference shelf. During a stocktake seven new shelves arrive, each holding nine books. Two pupils are asked to write down the total. Asha writes the expression 40 + 7 x 9 and then works it out. Bikram writes the same expression but works through it from left to right, as he would read a sentence. They get different numbers and each is sure the other has made an arithmetic slip. The librarian points out that neither has made a slip in the arithmetic itself.

Analysis:

Both pupils evaluated correctly according to the order they used, which is why neither can find a mistake in their own working. Asha multiplied first, getting 63 and then 103. Bikram added first, getting 47 and then 423. The disagreement is not about arithmetic but about which operation comes first, and nothing in the written expression settles it. That is exactly why an order rule is needed, and it is the subject of the next topic.

Activity:

Write one situation of your own that needs two different operations, then ask someone else to evaluate your expression and compare answers.

Title:

One number, many phrases

Safety:

No safety concerns. Paper and pencil only.

What You Notice:

One number can be named in a great many ways. The value never tells you which phrase produced it, which is why a worked answer should show the expression and not only the result.

Reading 40 + 7 x 9 from left to right gives 423 instead of the correct 103.

The equality sign was first printed in 1557, to save writing "is equal to".

A value never reveals its own expression, which is why working is shown.

Phrase first, arithmetic second.

Sum for plus, product for times, difference for minus.

A comma changes a sentence

Read this: "The cat watched the boy eating fish." The boy is eating.

Now add one comma: "The cat watched the boy, eating fish." Now the cat is eating. Nothing else changed. Punctuation decided the meaning.

Brackets are the punctuation of arithmetic

An expression can be ambiguous in exactly the same way.

Brackets remove the ambiguity. They say plainly which part to work out first, and everything inside them is settled before anything outside is touched.

Inside the brackets first

That is the whole rule, and there is nothing more to it.

Take 60 + (8 x 7). Work inside: eight sevens are 56. Then the addition: 60 + 56 = 116. The bracket was an instruction about order, not about value.

Brackets can change the answer

This is why they are worth writing.

Compare (15 + 5) x 3 with 15 + 5 x 3. The first is 20 x 3, which is 60. The second works out the product first, giving 15 + 15, which is 30. Same digits, same operations, different answers.

And they can change a division too

The effect is not special to multiplication.

Compare 84 divided by (6 + 1) with 84 divided by 6, plus 1. The first is 84 divided by 7, which is 12. The second is 14 + 1, which is 15.

Subtracting a bracket

Brackets work the same way when a whole group is taken away.

A shopper pays 200 rupees for items costing 45 and 38. The change is 200 - (45 + 38). Inside first: 45 + 38 = 83. Then 200 - 83 = 117 rupees.

Losing the bracket loses the answer

Suppose that bracket is dropped. The line then reads 200 - 45 + 38, which comes to 193.

Taking away the 45 is right. But the 38 is now being added rather than taken away. The gap between 193 and 117 is 76, which is exactly twice the 38. Here a bracket is not decoration; it carries the whole meaning.

But what if there are no brackets?

Most expressions you meet are written without them.

So 70 + 6 x 9 has no punctuation at all. It still has one correct value, and the idea that settles it is the idea of a term.

Terms: the parts separated by plus

An expression splits into terms at every plus sign.

In 12 + 7 the terms are 12 and 7. In 30 + 5 + 9 the terms are 30, 5 and 9. Each term is a piece that will be added in at the end.

Subtraction splits into terms too

A minus sign looks like a problem for this, but it is not.

Taking 23 away does the same work as putting in a term of -23. So 90 - 23 is 90 + (-23), and its terms are 90 and -23. Every expression can be read as a sum of terms.

A term may be a product

A term is not always a single number.

In 70 + 6 x 9 the terms are 70 and 6 x 9. The second term is a product, and it is worked out as a whole before being added in.

So the order falls out

The terms give the rule without anyone having to memorise a list.

Work out each term on its own, then add the terms. For 70 + 6 x 9, the second term is 54, and 70 + 54 = 124.

Two products, added

The same reading handles longer expressions.

Take 3 x 11 + 2 x 13. The terms are 3 x 11 and 2 x 13. They come to 33 and 26, and adding them gives 59.

Why multiplication seems to come first

People often say multiplication is done before addition. That is true, but it is not the reason.

The reason is that a multiplication sits INSIDE a term, while an addition joins one term to the next. You finish a term before you add it on.

Say all of before a bracket

A bracket is easy to miss on a small screen, so reading it aloud helps. Say all of twenty plus four, times five. Those two words, all of, hold the group together while you read on.

A reader who says twenty, plus four times five has already lost the bracket. One reading gives 120 and the other gives 40.

Brackets beat terms

When both are present, the brackets win.

In (15 + 5) x 3 the bracket makes 15 + 5 a single quantity, so the expression is one term, namely 20 x 3. Brackets are allowed to override the term reading, which is exactly what they are for.

Writing brackets you do not need

Sometimes a bracket changes nothing, and that is still useful.

In 70 + (6 x 9) the bracket is unnecessary, because the term reading already gives that order. But writing it makes the intention obvious to a reader, and it costs nothing.

Scenario:

A small shop has two tills. At the first, a customer buys items costing 45 and 38 rupees and hands over 200. The assistant writes 200 - 45 + 38 on the slip and gives back 193 rupees. At the second till the same purchase is rung up, and that assistant writes 200 - (45 + 38) and gives back 117 rupees. The customer notices the difference of 76 rupees and asks which slip is right. Both assistants insist they subtracted and added correctly, and neither has made a slip in the arithmetic itself.

Analysis:

The second slip is right and the first has lost a bracket. Taking away a whole group means the group must be bracketed, so the correct expression is 200 - (45 + 38), which gives 117. The first assistant subtracted 45 and then added 38, which is 200 - 45 + 38, or 193. That expression describes something else entirely: paying for one item and being given a second sum. The arithmetic was right in both cases; only one expression matched the situation.

Activity:

Take a till receipt from home with at least two items and a cash payment. Write the change as one bracketed expression, then evaluate it.

Title:

Does the bracket matter?

Safety:

No safety concerns. Paper and pencil only.

What You Notice:

Sometimes the bracket changes the value and sometimes it changes nothing. When it changes nothing, the term reading was already giving that order.

One comma can change a sentence, and one bracket can change an expression.

Brackets affect division too: 84 divided by (6 + 1) is 12, not 15.

A bracket that changes nothing is still worth writing for clarity.

Inside the brackets first, always.

Finish the term, then add it on.

Terms are added, so they can move

Once an expression is read as a sum of terms, something useful follows.

Addition does not care about order. So the terms may be written in any order at all, and the value will not change.

Swapping two terms

Take 11 - 7. Read as a sum, its terms are 11 and -7.

Swap them and you get -7 + 11. Both come to 4. The minus travelled with its own number, which is the part that matters.

A lift goes up, then down

A lift rises nine floors and then descends five.

The expression is 9 - 5, with terms 9 and -5. Writing it as -5 + 9 describes going down first and then up, and it ends at the same floor. The order of the journey changed; the destination did not.

Regrouping to make life easier

Terms can also be gathered in any grouping you like.

Take 46 + 29 + 54. Added in order it is awkward. But 46 and 54 make exactly 100, so grouping them first gives 100 + 29, which is 129. The answer was always 129; the regrouping only made it easy.

Scanning a long line first

With a long line of terms it pays to scan before adding anything. A pair that makes a round hundred turns awkward work into easy work.

In 28 + 59 + 72 the 28 and the 72 make exactly 100, and 100 + 59 is 159. Adding left to right reaches the same 159 by a harder road.

Terms that cancel

Sometimes regrouping shows that two terms destroy each other.

In 17 + 38 - 17 the terms are 17, 38 and -17. The 17 and the -17 come to nothing, so the value is simply 38. No arithmetic on 55 was ever needed.

Why the minus must travel

A warning before the next rule, because this is where errors come from.

When a term moves, its sign goes with it. The terms of 11 - 7 are 11 and -7, never 11 and 7. Leaving the minus behind turns -7 into 7 and changes the value by fourteen.

Taking away a whole group

Now suppose a bracket follows a minus sign.

A shopper has 300 rupees and buys items costing 80 and 25. The amount left is 300 - (80 + 25). Working inside first gives 105, and 300 - 105 leaves 195.

The same answer, taken one at a time

There is a second route to that answer, and it is often quicker.

Take the 80 away first: 300 - 80 = 220. Then take the 25 away: 220 - 25 = 195. What was done here is 300 - 80 - 25, and it agrees.

The first rule for removing brackets

Those two routes give the same value, which is a rule.

Subtracting a group is the same as subtracting each member of it. In symbols, a - (b + c) = a - b - c. The plus inside the bracket became a minus outside it.

Why that happens

It is worth seeing the reason rather than memorising the shape.

Taking away a bundle of two things is the same as taking away the first and then taking away the second. Nothing is left over and nothing is counted twice.

When the bracket holds a subtraction

The second case is the one people get wrong.

Consider 180 - (70 - 25). Inside first: 70 - 25 is 45, and 180 - 45 is 135. Now take the other route and watch the signs carefully.

The second rule for removing brackets

Taking 70 away gives 180 - 70 = 110. But too much has been taken away.

The group was only 45, not 70, so 25 must be given back: 110 + 25 = 135. So a - (b - c) = a - b + c. The minus inside became a plus outside.

A single rule covers both

Rather than remember two shapes, remember one action.

A minus in front of a bracket flips the sign of every term inside it. Pluses become minuses and minuses become pluses. Both rules above are that one action.

A longer bracket

The rule does not care how many terms are inside.

Take 500 - (120 + 60 + 95). Every inner sign flips, giving 500 - 120 - 60 - 95. That is 225, and evaluating inside the bracket first gives 500 - 275, which is also 225.

Checking a removed bracket

There is a cheap check, and it costs one extra line.

Evaluate the expression both ways: once with the bracket worked out first, and once with the bracket removed. The two must agree. If they differ, a sign was not flipped, and the gap will be exactly twice the term you missed.

What carries forward

Two freedoms and one caution.

The freedoms are that terms may be swapped and regrouped at will. The caution is that a sign belongs to its term, and a minus before a bracket reaches every term inside.

Scenario:

A water tank holds 180 litres at the start of the day. One pipe draws off 70 litres for the garden. Later a second pipe returns 25 litres of collected rainwater to the tank. The caretaker wants a single expression for what the tank holds at the end. He writes the net change as a bracketed group, 70 - 25, and subtracts it. His assistant says brackets are unnecessary and writes 180 - 70 - 25 instead. They get 135 and 85, and disagree by 50 litres.

Analysis:

The caretaker is right. The net loss to the tank is 70 - 25, which is 45 litres, so the tank holds 180 - 45, or 135 litres. The assistant subtracted the 25 instead of adding it, because he carried the outer minus on to the second term without flipping its sign. Removing the bracket correctly gives 180 - 70 + 25, which is 135 and agrees. The 50-litre gap is exactly twice the 25, which is the signature of a single sign error.

Activity:

Find a situation at home with one amount taken out and a smaller amount put back. Write it as a single bracketed expression and then remove the brackets correctly.

Title:

Flip every sign

Safety:

No safety concerns. Paper and pencil only.

What You Notice:

The two correct routes always agree. Flipping only the first sign leaves an error of exactly twice the term you failed to flip, which is a useful signature to recognise in your own work.

Missing one sign flip leaves an error of exactly twice the unflipped term.

In 17 + 38 - 17 two terms cancel, so no real arithmetic is needed at all.

Regrouping never changes a value; it only changes how hard the sum is.

The sign travels with its term.

Minus outside, flip every sign inside.