Chapter 5 · Numerical expressions
Read, Write and Compare Numerical Expressions
Read a numerical expression as a set of instructions, write expressions from words, and compare two expressions without working them out.
A numerical expression is a calculation written with numbers and signs for operations. The operations are adding (), subtracting (), multiplying () and dividing (). An expression may also have parentheses, the curved marks ( ). It has no equals sign. For example, is a numerical expression.
You already know how to evaluate an expression. That means to work out its value, the one number it equals. Today you will do something different. You will read an expression like a set of instructions, write an expression from words, and compare two expressions. Most of the time, you will not calculate at all.
Key concept
An expression is a set of instructions
A numerical expression records a calculation. Its numbers, signs and parentheses tell you what to do and in what order. So you can say it in words, and write it from words, without finding its value. When two expressions share the same part, you can often compare them without calculating. Just look at what is done to that part.
To read an expression in words, ask two questions. What is done first? What is done to that result?
To find the first step, use the order of operations. These are the rules you already know for which step comes first: parentheses first, then multiplying and dividing from left to right, then adding and subtracting from left to right.
Four words name the result of a step:
- A sum is the result of adding. is the sum of 9 and 6.
- A difference is the result of subtracting. is the difference between 15 and 4.
- A product is the result of multiplying. is the product of 7 and 8.
- A quotient is the result of dividing. is the quotient of 36 and 4.
| Expression | Done first | Done next | In words |
|---|---|---|---|
Add 9 and 6. | Multiply the sum by 2. | 2 times the sum of 9 and 6 | |
Add 12 and 8. | Subtract the sum from 50. | 50 minus the sum of 12 and 8 | |
Multiply 7 by 8. | Add 3 to the product. | 3 more than the product of 7 and 8 | |
Subtract 4 from 40. | Divide the difference by 6. | the difference between 40 and 4, divided by 6 |
Worked example
Read an expression in words
Say in words what tells you to do. Do not evaluate it.
Ask what is done first. Parentheses come first in the order of operations, and they hold 26 + 14. So the first step is to add 26 and 14.
Ask what is done to that result. After the parentheses you see ÷ 5, so the sum is divided by 5.
Put the two steps together in the order they are done, because the order is part of the instruction: first add, then divide. You did not need to find any value.
Answer
The expression says to add 26 and 14, then divide the sum by 5.
Now go the other way, from words to symbols. Start with one operation.
For adding and multiplying, you may write the two numbers in either order. “Add 8 and 7” can be or . “Multiply 6 by 9” can be or . The value is the same either way.
Subtracting and dividing are different. Taking 5 away from 12 is not the same as taking 12 away from 5. And is 5, but is less than 1. So for these two operations, the order of the numbers matters.
- In subtraction, the number you start with goes first. “Subtract 5 from 12” starts with 12, so it is . “8 less than 30” starts with 30, so it is .
- In division, the number being divided goes first. “Divide 20 by 4” is . “How many 6s are in 48?” divides 48, so it is .
Some descriptions have two steps, such as “add 8 and 7, then multiply by 2”. The words tell you which step comes first. Your expression must say the same thing. Parentheses are how you mark a step that must be done first.
Write the first step. Then write what is done to its result. Then ask: would the order of operations do my first step first anyway?
- If no, put parentheses around the first step. “Add 8 and 7, then multiply by 2” is . Without parentheses, would do first, because multiplying comes before adding.
- If yes, you do not need parentheses. “Multiply 4 by 5, then add 3” is , because multiplying already comes before adding. Writing is also correct.
| Words | Expression | Parentheses needed? |
|---|---|---|
Add 10 and 6, then divide by 4 | Yes. Without them, would be done first. | |
Divide 36 by 4, then subtract 2 | No. Dividing already comes before subtracting. | |
Subtract the sum of 3 and 4 from 20 | Yes. Without them, would be done first. | |
Divide 60 by the sum of 2 and 4 | Yes. Without them, would be done first. | |
Add 5 to the product of 6 and 9 | No. Multiplying already comes before adding. |
Worked example
Write an expression with two steps
Write one expression for these words: subtract 6 from 20, then multiply by 3.
Write the first step. The words say “from 20”, so 20 is the starting number and it goes first.
Write the second step. The whole difference is multiplied by 3, so the difference must be found first. Put parentheses around it to keep it together.
Check that the parentheses are needed. Without them, 20 − 6 × 3 would do 6 × 3 first, because multiplying comes before subtracting. That reads “multiply 6 by 3, then subtract the product from 20”. Those are not the words you were given, so the parentheses must stay.
Read your own expression back in words: subtract 6 from 20, then multiply the difference by 3. It matches the words, so the expression is right.
Answer
The expression is . You can also write , because the order of the two parts you multiply does not change the product.
Sometimes you can compare two expressions without calculating. Look for a part that is the same in both.
Look at and . Both hold the same sum. Think of that sum as one amount. You do not need to know what it is.
The first expression is that amount, once. The second says to multiply that amount by 3, and multiplying by 3 gives three copies of it. So is three times as large as . You never have to add those big numbers. Below, you can see the three copies.
Worked example
Compare without calculating
Without calculating, compare each of these with : first , then .
Find the part that is the same. All three expressions hold the sum 7,264 + 518. Treat it as one amount. You do not need its value, because each expression uses the very same amount.
Read the first expression. It multiplies that sum by 4, which makes 4 copies of it. So it is four times as large as the sum.
Read the second expression. It divides that sum by 2, which splits it into 2 equal parts and keeps one part. So it is half as large as the sum.
Put the three in order. The sum is greater than 0, so half of it is smaller than the sum, and four times it is larger than the sum.
Answer
is four times as large as , so it is the largest. is half as large as , so it is the smallest.
The big idea: a numerical expression is a set of instructions. You can read it in words and write it from words without finding its value. When two expressions share a part, you can often compare them by looking at what is done to that part.
When you write an expression, make two checks. For subtracting and dividing, put the numbers in the right order. Use parentheses when the order of operations would not do your first step first.
Next, you will write one expression that records all the steps of a word problem and say what each part stands for. You will also find and fix mistakes in written expressions.