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Writing and evaluating expressions

Turning words into symbols, naming the parts, and substituting a value.

Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.

1. What you will learn

In this lesson you write an expression from a description, name its parts (terms, coefficients, variables and constants) and evaluate it for given values. Translating words into symbols decides whether the rest goes well: 'five less than a number' is $n - 5$ and not $5 - n$, and that order catches almost everybody once. Evaluating uses the order of operations you already know, and formulas for area, volume and cost turn out to be expressions with a job.

2. What you already know

You can add, subtract, multiply and divide whole numbers, decimals and fractions. You know the order of operations: parentheses first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right. You have also seen a letter stand in for a missing number, as in $5 + \square = 12$. This lesson puts those pieces together. A letter becomes a number you do not know yet, or a number that can change, and a string of numbers, letters and operation signs becomes something you can write, read aloud and work out.

3. Words this lesson uses

4. A sentence written in math

An expression is a short way of writing a calculation. Instead of saying 'take a number, multiply it by 3, then add 7', you write

$$3x + 7.$$

The letter $x$ is a variable. It holds a place for a number. Writing $3x$ with no sign in between means 3 times $x$; mathematicians drop the $\times$ sign because it looks too much like the letter x.

There are two things to do with an expression. You can write one, turning words into symbols. And you can evaluate one: once you know what number the letter stands for, you put that number in and calculate. If $x = 5$, then $3x + 7$ becomes $3 \times 5 + 7 = 22$. If $x = 10$, the same expression gives 37. The expression stays the same while its value changes with the letter. That is its power: one short line describes a whole family of calculations.

An expression has no equals sign. $3x + 7$ is an expression; $3x + 7 = 22$ is an equation, a sentence claiming two things are equal. In this lesson you write and evaluate expressions. Solving equations comes later.

5. Turning words into symbols

Most word phrases match one operation. Learn these and translate one phrase at a time.

Watch the order. For adding and multiplying, order does not change the answer: $x + 8$ and $8 + x$ are the same. For subtracting and dividing it does. '8 less than $x$' means start with $x$ and take 8 away, so it is $x - 8$. The words say 8 first, but the symbols put it second. A quick way to be sure: try a number. 8 less than 20 is 12, and $20 - 8 = 12$, while $8 - 20$ is not 12.

The phrase 'the difference of $x$ and 8' keeps the order of the words: $x - 8$. 'The quotient of $x$ and 8' is $x \div 8$. Only 'less than' and 'subtracted from' flip the order.

When the words group things, use parentheses. 'Three times the sum of a number and 4' is $3(n + 4)$, because the sum is found first and then tripled. Without the parentheses, $3n + 4$ triples only the number.

6. Reading an expression and naming its parts

Take the expression $5a + 2b + 9$. Split it at the plus signs and you get its terms: $5a$, $2b$ and 9. Three terms.

In the term $5a$, the 5 is the coefficient: it multiplies the variable $a$. In $2b$ the coefficient is 2. The 9 has no letter, so it is a constant: whatever $a$ and $b$ are, it stays 9. A letter on its own, like the $a$ in $a + 6$, has coefficient 1, because $a$ is the same as $1 \times a$.

Being able to name the parts lets you read an expression aloud and talk about it. You can say '$5a + 2b + 9$ is the sum of three terms' or 'the product $5a$ is the first term'. An expression with parentheses can be one term made of factors: $4(x + 1)$ is the product of two factors, 4 and the quantity $x + 1$. Reading it as 'four times the sum of $x$ and 1' says exactly how it is built.

7. Evaluating: substitute, then follow the order of operations

To evaluate, replace every copy of the letter with its value. Write the value in parentheses at first. It keeps you from gluing digits together: if $x = 4$, then $3x$ is $3(4) = 12$, not 34.

Then follow the order of operations exactly as for any calculation:

1. Parentheses.
2. Exponents, such as $x^2$, which means $x \times x$.
3. Multiplication and division, from left to right.
4. Addition and subtraction, from left to right.

For example, evaluate $2x^2 - x$ when $x = 3$. Substitute: $2(3)^2 - 3$. The exponent comes first: $3^2 = 9$. Then multiply: $2 \times 9 = 18$. Then subtract: $18 - 3 = 15$. Squaring before multiplying matters. $(2 \times 3)^2$ would be 36, a different number, because the exponent in $2x^2$ belongs to the $x$ alone.

If the same letter appears twice, it gets the same value both times. Different letters may have different values, and each is substituted separately.

8. Formulas are expressions with a job

A formula is an expression that computes something useful, written with a letter for the result. You have met some already. The area of a rectangle is $A = lw$. The perimeter of a square is $P = 4s$. The volume of a cube is $V = s^3$.

Evaluating a formula is the same skill. For a cube with edges of 5 centimeters, $V = 5^3 = 5 \times 5 \times 5 = 125$ cubic centimeters. The letters are usually chosen to remind you of what they mean: $l$ for length, $w$ for width, $s$ for side. Before substituting, say what each letter stands for, so the right number goes in the right place.

Units follow the formula. A length in feet times a width in feet gives square feet. Write the unit on your answer, because 125 on its own does not say whether you measured a box or a swimming pool.

9. The method, step by step, and how to check it

To write an expression from words:

1. Choose the variable. Decide what number is unknown or changing and give it a letter.
2. Translate one phrase at a time, using the table of operation words.
3. Watch the order for 'less than' and 'subtracted from', and use parentheses when the words say 'the sum of' or 'the difference of' before another operation.
4. Check with a number. Pick an easy value, work the words out in your head, and evaluate your expression. They must agree.

To evaluate an expression:

1. Rewrite any hidden multiplication so you can see it.
2. Substitute each value, in parentheses.
3. Calculate in the order of operations, one step per line.
4. Check that the size makes sense. If a cost grows with the number of people, more people should give a bigger answer. If an answer looks far too big or small, look for a digit glued to a coefficient or an addition done before a multiplication.

Writing one operation per line is slower for a minute and faster in the end, because a mistake shows up on the line where it happened.

10. In the world: scoring a football game

In American football, a touchdown is worth 6 points, a field goal 3 points, and the kick after a touchdown, the extra point, 1 point. If a team scores $t$ touchdowns, $f$ field goals and $e$ extra points, its score is

$$6t + 3f + e.$$

Each coefficient is the value of one kind of score, and each variable counts how many times it happened. Suppose a team scores 4 touchdowns, makes 3 of the 4 extra-point kicks, and adds 2 field goals. Evaluate: $6(4) + 3(2) + 3 = 24 + 6 + 3 = 33$ points.

The expression also answers 'what if' questions. Can a team score exactly 20 points with touchdowns and field goals only? Look at $6t + 3f$: 6 and 3 are both multiples of 3, so the total is always a multiple of 3, and 20 is not. The team needs extra points too: $6(2) + 3(2) + 2 = 12 + 6 + 2 = 20$ works, with 2 touchdowns, 2 field goals and 2 extra points. Sports statisticians use expressions like this to compare how teams earn their points.

11. In the world: calories on a food label

Food energy is measured in Calories. Each gram of protein gives about 4 Calories, each gram of carbohydrate about 4, and each gram of fat about 9. So a food with $p$ grams of protein, $c$ grams of carbohydrate and $f$ grams of fat has about

$$4p + 4c + 9f$$

Calories. A granola bar with 3 grams of protein, 20 grams of carbohydrate and 5 grams of fat has about $4(3) + 4(20) + 9(5) = 12 + 80 + 45 = 137$ Calories. Many food companies in the United States use these same factors when they print the Nutrition Facts label, so the expression explains where the number on the wrapper comes from. Notice how the coefficient 9 makes fat count for more than twice as much as the same weight of protein.

12. Mistakes to watch for

Flipping 'less than'. '6 less than $n$' is $n - 6$, not $6 - n$. Test with a number: 6 less than 10 is 4.

Gluing digits. If $x = 4$, $3x$ is 12, not 34. Put the value in parentheses: $3(4)$.

Adding before multiplying. In $2 + 5x$ with $x = 3$, multiply first: $2 + 15 = 17$, not $7 \times 3 = 21$.

Squaring the coefficient too. $2x^2$ with $x = 3$ is $2 \times 9 = 18$, not $6^2 = 36$. The exponent belongs only to the letter next to it.

Thinking a lone letter has no coefficient. $x$ means $1 \cdot x$. In $x + 3x$ there are four $x$'s, not three.

Forgetting parentheses for a group. 'Twice the sum of $a$ and 5' is $2(a + 5)$. Writing $2a + 5$ doubles only the $a$.

13. Writing an expression from words

1. Write an expression for 'nine less than four times a number'. Choose a letter for the number.

   $\text{the number} = y$

   Any letter works; the letter just holds the place of the unknown number.

2. Translate 'four times a number'.

   $4y$

   Times means multiply, and a coefficient is written in front of its letter.

3. Translate 'nine less than' that amount.

   $4y - 9$

   Nine less than something means start with it and take nine away.

4. Test the expression with $y = 5$.

   $4(5) - 9 = 20 - 9 = 11$

   An easy value lets you compare the symbols with the words.

5. Work the words out for 5 and compare.

   $4 \times 5 = 20, \text{ and nine less than } 20 \text{ is } 11$

   The words and the expression agree, so the translation is right; $9 - 4y$ would have given $-11$.

14. Evaluating with the order of operations

1. Evaluate $3x^2 + 2(x - 1)$ when $x = 4$. Substitute 4 for every $x$.

   $3(4)^2 + 2(4 - 1)$

   Both copies of $x$ get the same value.

2. Work out the parentheses.

   $4 - 1 = 3$

   Parentheses come first in the order of operations.

3. Work out the exponent.

   $4^2 = 4 \times 4 = 16$

   Exponents come next, and the square belongs to the 4 alone.

4. Multiply in the first term.

   $3 \times 16 = 48$

   Multiplication comes before addition.

5. Multiply in the second term.

   $2 \times 3 = 6$

   The 2 multiplies the value of the parentheses.

6. Add the two terms.

   $48 + 6 = 54$

   Addition is last, once each term is a single number.

15. Writing and evaluating from a real situation

1. Maya has 45 dollars saved and adds 8 dollars every week. Write an expression for her savings after some weeks. Choose the variable.

   $w = \text{number of weeks}$

   The number of weeks is what changes, so it gets the letter.

2. Write the part that grows.

   $8w$

   8 dollars each week for $w$ weeks is 8 times $w$.

3. Add the part that stays the same.

   $45 + 8w$

   The 45 dollars she started with is a constant; it is there from week 0.

4. Evaluate for 6 weeks: substitute.

   $45 + 8(6)$

   The question asks about 6 weeks, so $w = 6$.

5. Multiply the weekly amount by the number of weeks.

   $8 \times 6 = 48$

   Multiplication comes before addition.

6. Add the starting amount.

   $45 + 48 = 93 \text{ dollars}$

   The savings are the starting amount plus what was added.

7. Check by counting week by week.

   $45 \to 53 \to 61 \to 69 \to 77 \to 85 \to 93$

   Six jumps of 8 dollars land on 93, the same as the expression.

16. Your turn: evaluate $5m - 2n$ when $m = 6$ and $n = 7$

1. Substitute both values.

   $5(6) - 2(7)$

   Each letter gets its own value.

2. Multiply in each term.

   $30 - 14$

   Both multiplications come before the subtraction.

3. Your turn: work this step out. Its working is at the end of the packet.

   Subtract the second product from the first.

17. Guided practice

In the expression $9w + 2t + 24$, match each part to its name.

18. Guided practice

A box is 7 inches long, 4 inches wide and 5 inches tall. The area of cardboard needed to cover it is given by the formula $S = 2lw + 2lh + 2wh$, in square inches. Complete the solution.

1. Substitute the three measurements.

   $S = 2(7)(4) + 2(7)(5) + 2(4)(5)$

   Each letter is replaced by its own number, and parentheses keep the numbers apart.

2. Work out the first product: top and bottom.

   $2 \times 7 \times 4 =$ _top

   Multiplication comes before addition, so each term is found on its own first.

3. Work out the second product: front and back.

   $2 \times 7 \times 5 =$ _front

   Two faces of the box are length by height.

4. Work out the third product: the two ends.

   $2 \times 4 \times 5 =$ _ends

   The last two faces are width by height.

5. Add the three products.

   $S =$ _total square inches

   The six faces together are all the cardboard the box needs.

19. Guided practice

Which words describe the expression $5(n + 15)$?

• 5 times the sum of a number and 15
• 15 more than 5 times a number
• the sum of 5 and a number, plus 15
• 5 more than the product of a number and 15

20. Practice

Evaluate $39 + 3k$ when $k = 4$.

When $k = 4$, the expression equals _answer.

21. Practice

Write an expression for: the quotient of a number $n$ and 3, decreased by 8.

Answer:

22. Practice

A class trip to a science museum costs 311 dollars for the bus plus 17 dollars for each student's ticket. The total cost in dollars for $s$ students is $311 + 17s$. What do the tickets cost, and what is the total cost, when 48 students go?

The tickets cost _tickets dollars, and the whole trip costs _total dollars.

23. Somewhere new

Weather reports in most countries give temperatures in degrees Celsius. To change a Celsius temperature $C$ into degrees Fahrenheit, use the formula $F = \dfrac{9C}{5} + 32$. A city reports 20 degrees Celsius. What is that in degrees Fahrenheit?

Answer:

24. Lesson test

Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.

25. Test question

A class trip to a science museum costs 258 dollars for the bus plus 9 dollars for each student's ticket. The total cost in dollars for $s$ students is $258 + 9s$. What do the tickets cost, and what is the total cost, when 54 students go?

The tickets cost _tickets dollars, and the whole trip costs _total dollars.

26. What you can do now

You can write an expression from words and evaluate it. Without looking: write 'five less than a number', and say why it is not $5 - n$.

Working for the steps left to you

16. Your turn: evaluate $5m - 2n$ when $m = 6$ and $n = 7$, step 3