Words into expressions

Words such as "more than", "less than", "twice" and "divided by" each mean an operation. Write the operations in the order the words say them, then check with a number.

"Twice as many", "5 less than", "shared between 4": every maths sentence about a number can be written with a letter.

A letter can stand for any number, so a sentence about "a number" can be written as an expression. Each word that tells you what to do to the number, such as more than, less than, twice or divided by, becomes an operation. The skill is writing the operations in the order the words give them.

Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.

PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.

5 less than n

Try it with a real number: what is 5 less than 12? Which expression gives that?

n

is a number. Which expression means "5 less than

n

5 - n

\frac{n}{5}

5n

n - 5

TryTry it on the board and compare the result with your prediction.

What do the words give?

The board does what the words say to $n$ , and shows the result $T$ . Try other values of $n$ . Is $5 - n$ ever the same as what the board shows?

Try it yourselfYour edits are saved in this browser.

Set $n = 12$ . The board gives "5 less than $n$ ". What does it show? Does that match your prediction?

Use the board. This ticks when you get there.

InvestigateWork out why it happens, one step at a time.

From words to an expression

Here is how to turn words into an expression, one step at a time.

Worked exampleBefore each step, say what you think comes next.

$\text{3 more than twice } n$ Break the words into the parts they name.
$\text{twice } n$ Start with the part said first: twice a number is $2 \times n$ .
$2n$ A number next to a letter means multiply, so write $2n$ .
$\text{3 more than } 2n$ "More than" means add 3 to that.
$2n + 3$ The expression.
$n = 4: \; 2 \times 4 + 3 = 11$ Check with a number: twice 4 is 8, and 3 more is 11.

Read Ana's working one line at a time. What is 7 less than 10? What does her expression give?

Write each expression, then try a number to check it.

Write an expression for "3 more than $x$ ".

A number

n

is divided by 4. Which expression is that?

\frac{n}{4}

\frac{4}{n}

4n

Ana writes "7 less than $n$ ". Click the line where she goes wrong.

ModifyChange the problem on purpose and see what happens.

Other words

The board does what the words say, and shows the answer. Write your expression, then check it with a number.

Try it yourselfYour edits are saved in this browser.

Write an expression for "half of $n$ ".

Write an expression for "5 more than 3 times $n$ ".

Write an expression for "2 less than 4 times $n$ ".

Write an expression for "the product of $a$ and $b$ ".

PractiseA few on your own, to check you have it.

Say it, then write it

Ben is $n$ years old. His sister is 6 years older. Write an expression for his sister's age.

Eggs come in boxes of 12. Write an expression for the number of eggs in $b$ boxes.

A pizza has 8 slices. $n$ slices are eaten. Write an expression for the slices left.

Review: find $3n + 5$ when $n = 4$ .

MakeMake your own problem using what you have learned. Expect a few tries; that is normal.

Find the rule

Say your expression in words, the way you did in this lesson.

Stretch: write a story that your expression could describe.

Try it yourselfYour edits are saved in this browser.

Need a hint?

Try n = 0 and n = 1. How much does T go up each time n goes up by 1?

Another hint

At n = 0 the board shows −1.

The mystery board follows a rule in $n$ . Move $n$ and watch $T$ . Write an expression for $T$ .

For teachers

Students turn words such as more than, less than, twice and divided by into expressions, writing the operations in the order the words say them and checking with a number (AC9M7A02, node 1). About 45 minutes.

Open the teacher guide: answers, solutions, slides and a worksheet.

Students will be able to

I can say what operation each word in a sentence about a number means.
I can write an expression from words, with the operations in the right order.
I can check my expression by trying a number I can work out in words.

Before class

Do the Run and Modify boards yourself first. They hide the formula on purpose: the board does what the words say and shows the result, so students check their expression against it.

The new move is turning each word into an operation. The most common error is "less than" (Ana's mistake): the numbers are written in the order they are spoken, 7 − n, when the number you start from comes first, n − 7. Always check with a real number: 7 less than 10 is 3.

The pizza question in Practise is the opposite on purpose: "n slices are eaten from 8" starts from 8, so 8 − n is correct. Ask students what is being taken away from what.

Running the lesson

Predict: take a show of hands. Ask a student who chose 5 − n to work out 5 less than 12, then try 5 − 12.
Run: ask students to say what the board is doing to n before they write anything.
Investigate: model the worked example live. Say the words in steps and write each step as you say it.
Modify: at each task, ask 'what is the first thing the words do to n?' before writing.
Make: ask for the expression in words as well as in symbols.

When students get stuck

A student writes 7 − n for '7 less than n' (Ana's mistake). Ask: What is 7 less than 10? What does 7 − 10 give? Fix: 'Less than' takes the amount away from the number: start from n, then subtract: n − 7.
A student writes 5n for '5 more than n'. Ask: If n is 10, what is 5 more than n? What is 5n? Fix: 'More than' means add. Multiply is for 'times' or 'lots of'.

Stretch and support

Stretch: Write expressions for: 'the next whole number after n', '3 less than half of n', 'the cost of n pens at 2 dollars each and one ruler at 5 dollars'.

Support: Give a table of words and their operation (more than: +, less than: −, times, twice, half, divided by) to use as a key.

Where it leads

Next: when brackets are needed. 'Add 3 to n, then double it' is not 2n + 3. After that, expressions describe whole situations, which leads into Linear relations (y = mx + c) and into Equations.

What each step is for

5 less than n: Committing to an expression for 'less than' exposes writing the numbers in the order they are spoken: 5 − n.
what do the words give?: Comparing the board's numbers with the words fixes the meaning of 'less than' before any expression is written.
from words to an expression: Each word names an operation; an expression is those operations written in the order the words say them, and a number check catches a swapped order.
other words: One change at a time: dividing, adding after multiplying, subtracting after multiplying, and two letters.
say it, then write it: Turning everyday sentences into expressions, where the order of a subtraction depends on the story; the review is substituting.
find the rule: Reading an expression off a mystery board reverses the lesson: from numbers back to words and an expression.

Now you can

Say what operation each word in a sentence about a number means.
Write an expression from words, with the operations in the right order.
Check my expression by trying a number I can work out in words.

Practise this skillNew questions, new numbers each timeGet 3 right in a row to mark it practised.Next lesson5. When brackets are neededBrackets keep a part together so it is worked out first. If a sentence says to do something to a whole result, such as "add 3, then double it", the expression needs brackets.

Source: content/maths/algebra/words-to-expressions/lesson.md