Class 7 Mathematics Chapter 4: Expressions using Letter-Numbers

One sentence, many cases

Meera is four years older than her brother Arun. When Arun is 7, Meera is 11. When Arun is 12, Meera is 16.

Two numbers answered, and there are hundreds more. Writing them all out is impossible, so we need a way to write the one rule instead.

A letter stands for a number

Let x be Arun's age in years. Then Meera's age is x + 4.

The letter does not stand for Meera, or for age, or for a word. It stands for a number we have not yet fixed.

Why that is useful

The single phrase x + 4 covers every case at once. Put 7 in place of x and it gives 11. Put 12 in place of x and it gives 16.

One line of writing now does the work of an endless list.

Substituting a value

Replacing the letter by a number is called substituting. Nothing else in the phrase changes.

Substituting x = 7 into x + 4 gives 7 + 4, which is 11. The phrase was the rule; the substitution asks the rule one question.

A letter-number is still a number

Because x stands for a number, x + 4 is an arithmetic expression like any other. It has terms, and it can be read aloud.

Everything learnt about expressions still applies. Only the fact that one of the numbers is unnamed has changed.

Reading a relation out of a situation

A tricycle has 3 wheels. How many wheels on n tricycles?

Each tricycle brings 3, so n tricycles bring 3 lots of 3n. The expression is 3n, and the letter counts the tricycles.

Say what the letter means, every time

3n means nothing until we say what n counts. If n were the number of wheels, 3n would be wrong.

So every expression comes with a sentence: let n be the number of tricycles. Without it the expression cannot be checked.

The same letter, the same number

Within one expression, one letter always stands for the same value. If x is 7 in one place, it is 7 everywhere in that expression.

Different letters, though, may stand for the same number or for different ones. Nothing forces them apart.

A letter can carry a condition

Any odd number can be written 2k + 1, where k is a whole number.

Put k = 0 and the expression gives 1. Put k = 3 and it gives 7. Every odd number appears exactly once as k runs through the whole numbers.

Why that one is worth knowing

2k + 1 does not merely describe odd numbers. It explains them: an odd number is a double with one left over.

An expression can record a fact and show the reason for it in the same line.

Choosing the letter

Letters are a free choice. x + 4 and a + 4 say the same thing if x and a count the same quantity.

Pick a letter that helps you remember. Using n for a number of things, or t for a count of tricycles, costs nothing and saves mistakes.

A phrase, not an answer

x + 4 is not waiting to be finished. It is already the complete answer to "how old is Meera?"

An expression with a letter in it looks unfinished because there is no single number at the end. There is no single number because the question had no single case.

Two expressions can describe one situation

Meera's age is x + 4. Arun's age, written in terms of Meera's age y, is y - 4.

Both record the same four-year gap. Which one you write depends on which age you are given.

Checking an expression against the words

Read your expression back as a sentence. Does x + 4 say that Meera is four years older? Yes, because adding four makes it larger.

If you had written x - 4, reading it back would have claimed Meera is younger. Reading back is the cheapest check there is.

A common slip

Asked for "four more than a number", some write 4x. That is four times the number, not four more than it.

More means added. Times means multiplied. The words choose the operation, and nothing else does.

What carries forward

A letter in an expression stands for a number that has not been fixed, and one expression then covers every case of a situation at once.

Always state what the letter counts, and always read the expression back against the words. Those two habits prevent most of the errors in this chapter.

Scenario:

A family keeps a chart showing that Meera is four years older than her brother Arun. A visitor asks how old Meera is and nobody can answer, because Arun birthday has not happened yet this year. The eldest child writes x + 4 on the chart and says that is the answer. A younger cousin objects that x + 4 is not a number and so cannot be an answer at all. The eldest child then shows that when Arun is 7 the chart gives 11, and when he is 12 it gives 16. A second cousin, asked to write four more than a number, writes 4x instead.

Analysis:

The disagreement is about what counts as an answer, and the second cousin error is about which operation the word more names.

Activity:

Write three relations from your own family in letters, defining each letter first, then test each one with two values.

Title:

Write the rule, then test it

Safety:

No safety concerns. Notebook and pencil only.

What You Notice:

One expression answers every case, and the two forms of the same relation are both correct. Reading an expression back as a sentence catches a wrong operation faster than substituting ever does.

The expression 2k + 1 generates every odd number exactly once as k runs through the whole numbers, so it is a complete list in one line.

The forms x + 4 and 4x are equal only when x is one and a third, so no whole number at all makes the two agree.

The choice of letter is free: x + 4 and a + 4 are the same rule as long as both letters count the same thing.

More means add; times means multiply.

Define the letter, then read the line back.

The sign that goes missing

Three times m could be written 3 x m. In algebra it is written 3m instead.

The multiplication sign is simply left out. Nothing else has changed, and the two forms mean exactly the same thing.

Why it is dropped

The letter x is a common name for an unfixed number, and the multiplication sign looks almost the same. Writing 3 x x invites a misreading.

Leaving the sign out removes the clash, and it makes long expressions far quicker to read.

It is dropped only for multiplication

Addition keeps its sign, so three more than m stays 3 + m. Subtraction keeps its sign too.

So 3m and 3 + m are different expressions. The missing sign always means multiply, never add.

The number goes in front

We write 3m and not m3. The number is called the coefficient, and it is written first by convention.

In 7y the coefficient is 7. In y the coefficient is 1, even though no number is written, because one y is still y.

Terms, as before

An expression still splits into terms at its plus signs. In 5p + 3p the terms are 5p and 3p.

Everything learnt about terms carries over. A term may now hold a letter as well as numbers.

Like terms

Two terms are like terms when they carry exactly the same letter part. So 5p and 3p are like terms, and 4a and 2a are like terms.

But 3a and 3b are not. Their coefficients match while their letters do not, and it is the letter that has to match.

Why like terms can be collected

Five p plus three p is eight p, in the same way that five apples and three apples make eight apples.

The letter says what is being counted. If the thing being counted is the same, the counts can simply be added.

Collecting a subtraction

Seven y take away two y leaves five y. So 7y - 2y simplifies to 5y.

Again the letter did not change. Only the count in front of it did.

Unlike terms cannot be collected

3a + 3b stays as it is. There is no way to write it more shortly, because the two terms count different things.

Three apples and three bananas are six pieces of fruit, but they are not six apples and not six bananas. The expression is already as simple as it gets.

A longer expression

Take 5p + 8q + 3p. The terms are 5p, 8q and 3p.

The 5p and the 3p are like terms and collect to 8p. The 8q has no partner at all, so the simplest form is 8p + 8q.

Collecting works term by term. A term with no partner is carried along unchanged rather than forced to join something.

A mixed expression

Take 4a + 6 + 2a. The terms are 4a, 6 and 2a.

The 4a and the 2a are like terms and collect to 6a. The 6 is a plain number and has nothing to join, so the answer is 6a + 6.

Why the 6 stays separate

A plain number is counting nothing in particular, so it is not like a term carrying a letter.

Writing 6a + 6 as 12a would be adding a count of a to a count of ones. That is the commonest error in the whole chapter.

Simplifying means collecting

To simplify an expression, gather the like terms and leave the rest alone. The value never changes; only the writing gets shorter.

Substituting a = 5 into 4a + 6 + 2a gives 36, and into 6a + 6 also gives 36. A simplification that changed the value would be wrong.

Multiplying into a bracket

Consider 8(q + 2). The bracket is one quantity, and eight of it are wanted.

Eight lots of q is 8q, and eight lots of 2 is 16. So 8(q + 2) is 8q + 16. Every term inside gets multiplied, not just the first.

Why every term inside

Eight lots of a bundle is eight lots of each thing in the bundle. Missing one of them would leave part of the bundle uncounted.

Substituting q = 5 gives 8 times 7, which is 56. And 8q + 16 gives 40 + 16, which is 56 as well.

Checking a simplification

Substitute any value into both forms. If they disagree, the simplification is wrong, and the value you chose has found the error for you.

One value is usually enough to catch a slip, though it cannot prove a simplification right on its own.

What carries forward

A missing sign means multiply. The coefficient goes in front, and a letter with no number has a coefficient of 1.

Like terms carry exactly the same letter part and may be collected by adding their coefficients. Unlike terms, and plain numbers, stay where they are.

Scenario:

A stall sells pens and notebooks. A helper writes the takings as 5p for the pens and 8q for the notebooks, having defined p as the number of pens sold and q as the number of notebooks. Asked for a single expression, the helper writes 5p + 8q and then wonders whether it can be tidied into 13pq. Later the same helper is given 4a + 6 + 2a to simplify and writes 12a, and is given 8(q + 2) to expand and writes 8q + 2. The stall owner checks all three by putting numbers in.

Analysis:

All three errors come from collecting things that count different quantities, or from stopping a multiplication partway through a bracket.

Activity:

Price three items at your own stall with three letters, write the takings as one expression, and say which terms could collect and which could not.

Title:

Collect, and then test

Safety:

No safety concerns. Counters, notebook and pencil only.

What You Notice:

Counters of one colour collect and counters of two colours do not, which is exactly the like-terms rule. The substitution shows that 6a + 6 and 12a are different expressions, giving 36 and 60.

A letter standing alone has a coefficient of one, and writing that one in would be correct but is simply never done.

Collecting like terms is the same act as counting apples with apples; the letter is only a label for what is being counted.

A simplification can be disproved by one substitution but never proved by one, because two different expressions can agree at a single value.

No sign means times, never plus.

Same letter part, or it will not collect.

A pattern you can see growing

Lay out squares in a row, each new square sharing one side with the last. One square needs 4 matchsticks, two need 7, three need 10.

Counting is easy for three squares and tiresome for thirty. A rule would settle every case at once.

Count what changes

Each new square adds 3 matchsticks, not 4, because one side is already there.

So the pattern starts at 4 and goes up in threes: 4, 7, 10, 13. The constant step is the thing worth noticing.

Turn the step into an expression

For n squares there are n lots of 3 matchsticks, plus one extra stick to close the very first square.

So the count is 3n + 1. Putting n = 1 gives 4, n = 2 gives 7 and n = 3 gives 10, which are the numbers we counted.

Test it where counting is hard

Putting n = 10 into 3n + 1 gives 31. Putting n = 30 gives 91.

Nobody would want to count 91 matchsticks one at a time. This is what the expression was for.

A second pattern, with a different shape

Square tables are pushed into a row. One table seats 4 people, two seat 6, three seat 8.

Each extra table adds 2 seats, because the two joined sides are lost. The pattern is 4, 6, 8 and goes up in twos.

Reading its rule

For n tables there are n lots of 2 seats along the sides, plus one seat at each end of the row.

So the seats number 2n + 2. Putting n = 1 gives 4, and putting n = 5 gives 12 seats.

Where the constant comes from

In both rules the number added at the end came from the ends of the arrangement, not from the repeating part.

3n + 1 has one stick closing the first square. 2n + 2 has one seat at either end. The constant is always a thing you can point at.

Guessing is not justifying

A pattern starts 1, 2, 4. What comes next?

8 is a reasonable guess, if the rule is doubling. But 7 is equally reasonable, if the rule adds 1, then 2, then 3. The first few terms do not fix the rule.

So a rule needs a reason

To be sure of a pattern we must say WHY each step happens. In the matchstick row we could point at the shared side.

A rule read off the numbers alone is a guess. A rule read off the structure is a result.

Checking a rule properly

Test the rule on a case you have not used to build it. If the rule came from the first three terms, check the fourth.

Putting n = 4 into 3n + 1 gives 13, and four squares really do need 13 matchsticks. That is evidence, because the fourth term was not used to make the rule.

A relationship hiding in plain numbers

Add the odd numbers in order. One is 1. One and three are 4. One, three and five are 9.

Those totals are 1, 4 and 9, which are the square numbers. The sum of the first n odd numbers is n times n.

Why that one is true

Picture a square of dots growing. To turn a 2-by-2 square into a 3-by-3 you add a row, a column, and one corner dot.

That added strip always holds an odd number of dots. So every square number is a sum of odd numbers, built one strip at a time.

What an expression lets you do

Once a pattern has a rule, three things become possible. You can find any term without listing the ones before it.

You can work backwards from a term to its position. And you can compare two patterns without drawing either of them.

Working backwards

A matchstick row uses 61 sticks. Which term is it?

If 3n + 1 is 61, then 3n is 60, so n is 20. Twenty squares, found without building anything.

Comparing two patterns

The matchstick rule 3n + 1 grows by 3 each time; the table rule 2n + 2 grows by 2.

So the matchstick counts pull ahead and stay ahead. Comparing the coefficients answers that without computing a single term.

What carries forward

Find the constant step first, turn it into the term holding the letter, then find where the leftover constant comes from in the arrangement.

A rule guessed from a few terms is only a guess. Say why each step happens, then test the rule on a term you did not use to build it.

Scenario:

A class is arranging square tables in one long row for a school lunch. One table seats four children, two tables seat six, and three tables seat eight. The teacher asks how many children a row of twenty tables would seat, and one pupil begins drawing all twenty. A second pupil says the answer must be eighty, because one table seats four. Meanwhile a third pupil is building a matchstick row of squares and has counted 4, 7 and 10 sticks for the first three. She writes the rule 4n and is surprised when ten squares turn out to need 31 sticks rather than 40.

Analysis:

Both wrong rules come from treating the first term as the repeating amount, instead of finding the constant step and the end pieces separately.

Activity:

Build a matchstick row of six squares and a paper row of six tables, count both, and check each against its rule before and after you build them.

Title:

Build it, count it, rule it

Safety:

No safety concerns. Matchsticks are unlit throughout.

What You Notice:

The prediction for the fourth square is real evidence, because that term was not used to build the rule. The prediction for ten shows what the rule is actually for, since counting 31 sticks by hand is slow and error-prone.

The rules 3n + 1 and 4n agree at one square only, which is exactly why checking the first term can never expose that error.

Every square number is a sum of odd numbers, because growing a square adds a row, a column and one corner dot.

A rule lets you compare two patterns without computing a single term, by comparing the numbers in front of the letter.

Step first, constant second, reason always.

Test on a term you did not use.