Chapter 5 · Expressions
Interpreting and Writing Expressions in Context
Learn what each part of an expression means in a real situation. Then write your own expressions by defining a variable and building a part for each quantity.
Your class wants to order T-shirts. A print shop charges a one-time setup fee of 20 dollars, plus 7 dollars for each shirt. How can you describe the cost of any order, large or small?
You can use an expression: a mathematical phrase made of numbers, variables and operations such as addition and multiplication, with no equals sign. A variable is a letter that stands for a number that can change.
Let be the number of shirts in the order, so can be , , and so on. Then the total cost in dollars is , where means times .
The situation that an expression describes is its context. In this lesson you will interpret expressions, which means saying what each part stands for in the context. Then you will write your own.
The parts of an expression have names.
- Terms are the parts that you add or subtract. The terms of are and .
- A coefficient is the number that multiplies a variable. In the term , the coefficient is .
- A constant is a term that is only a number. It has no variable, so its value stays the same. Here the constant is .
A name tells you what kind of part you have. The context tells you what the part means. To find that meaning, you need three things: the quantities (amounts you can count or measure) named in the situation, the definition of the variable, and the operations in the expression. The table below gives the meaning and the unit, such as dollars, of each part of .
| Part | Name | Meaning in the shirt order | Unit |
|---|---|---|---|
variable | number of shirts in the order | shirts | |
coefficient | price of one shirt | dollars per shirt | |
term | cost of all the shirts | dollars | |
constant | setup fee, paid once | dollars | |
whole expression | total cost of the order | dollars |
Check the table with a number. Suppose the order has shirts, so . Each of the shirts costs dollars, so the shirts cost dollars. (The dot means times.) The class pays the setup fee once, so the order costs dollars.
Notice the units. Dollars per shirt times a number of shirts gives dollars. Dollars plus dollars gives dollars. That is why you can add the two terms.
Key concept
Each part stands for a quantity
When an expression describes a situation, each part stands for a quantity in that situation, such as a price in dollars or a time in hours.
- To interpret an expression, start from the definition of the variable. Then say what each coefficient, each term and each expression in parentheses measures.
- To write an expression, define the variable first. Then build one part for each quantity and join the parts with the operations the situation describes.
A check you can use when the parts are amounts with units: parts that you add or subtract must have the same unit, as the two dollar amounts in do.
Worked example
Read each part of an expression
A candle's height in centimeters is after it has burned for hours. What do , and mean? How tall is the candle after hours? For which values of does the expression make sense?
Start with the constant 18. It has no variable, so it stays the same while the candle burns. To see what it measures, replace h with 0, the moment before any burning. The value is 18, so 18 is the starting height of the candle: 18 centimeters.
Read the term 2h next. The expression subtracts it from a height in centimeters, so it must be in centimeters too. It is the length of candle that has burned away after h hours. Taking it away from the starting height leaves the height that remains.
Now read the 2 that multiplies h. When h grows by 1, the term 2h grows by 2: it is 2 when h = 1, 4 when h = 2 and 6 when h = 3. So 2 is the number of centimeters that burn away in each hour. Centimeters per hour times hours gives centimeters, which fits the term 2h.
Notice the sign. Subtracting 2h gives the same result as adding negative 2h, so you can read the coefficient of h as negative 2. The negative sign tells you that the height goes down as h goes up: 2 centimeters lower for each hour.
Find the height after 4 hours. Replace h with 4. Multiply first and then subtract, because the order of operations puts multiplication before subtraction.
Find the values of h that fit the situation. Time burned cannot be negative, so h is at least 0. The candle loses 2 centimeters each hour, so its 18 centimeters last 9 hours, because 18 divided by 2 is 9. The expression gives 0 when h = 9. At h = 10 it gives negative 2, and a height cannot be negative. So the expression makes sense only for h from 0 to 9.
Answer
The is the starting height in centimeters, is the length burned away after hours, and the is the centimeters burned each hour. After hours the candle is centimeters tall. The expression makes sense for from to .
Writing an expression is interpreting in reverse. You start with the situation and build one part for each quantity.
- Define the variable. Say which number it stands for and give its unit: "let be the number of weeks," not just " is weeks."
- Look for an amount that you count once. If you count it one time whatever the value of the variable, it is a constant.
- Look for an amount that repeats. When the situation has the same amount for each one of something, such as dollars for each week, multiply that amount by how many there are.
- Join the parts. Add amounts that the situation puts together. Subtract an amount that the situation takes away.
- Test a value. Choose a value the situation allows. Work out the amount straight from the situation, then from your expression. If the results differ and your arithmetic is right, the expression is wrong. If they agree, that supports your expression, but only your reasoning about each part shows that it works for every allowed value.
Worked example
Write an expression from a situation
You have dollars in a savings jar. At the end of every week you add dollars. Write an expression for the number of dollars in the jar after weeks. Then find the amount after weeks.
Define the variable so that every part has a clear meaning. Let w be the number of weeks that have passed. You add money once at the end of each week, so w is a whole number: 0, 1, 2 and so on. The expression will stand for the number of dollars in the jar.
Look for an amount that you count once. The 45 dollars is in the jar at the start, and you count it one time however many weeks pass. It is the constant.
Look for an amount that repeats. You add the same 8 dollars for each week, so after w weeks you have added 8 times w dollars. The 8 is the coefficient of w.
Join the parts. The situation puts the dollars you add together with the dollars you started with, so add the two terms. Both terms are in dollars, so the total is in dollars.
Test the value w = 4. Straight from the situation, you start with 45 dollars and add 8 dollars four times, which makes 77 dollars. The expression also gives 77. The results agree, which supports the expression.
Answer
The expression gives the number of dollars in the jar after weeks, where is a whole number that is or greater. After weeks the jar holds dollars.
Sometimes the amount that repeats is built from two amounts, such as a price with a discount taken off. Then you multiply the whole amount, and parentheses hold it together as one part. When you multiply numbers or expressions, each one is called a factor.
Equivalent expressions have the same value for every value of the variable. You already know that the distributive property rewrites as the equivalent expression , because it multiplies each term inside the parentheses by . In a context, two equivalent forms describe the same amount, and their parts can tell you about different quantities.
Worked example
One situation, two forms
Six friends buy tickets to a trampoline park. The full price of one ticket is dollars, and each friend uses a coupon for dollars off. Write an expression for the total cost in dollars for all six friends. Then say what the parts of the equivalent expression mean.
Define the variable. Let p be the full price in dollars of one ticket. Each coupon takes 2 dollars off, so the situation makes sense when p is greater than 2. The expression will stand for the total cost in dollars for all 6 friends.
Build the cost for one friend. The coupon takes 2 dollars away from the full price, so subtract 2 from p. Both amounts are in dollars, so the result is in dollars.
Look for what repeats. Each of the 6 friends pays that same amount, so multiply it by 6. The parentheses keep the amount together as one factor, so the 6 multiplies both the full price and the coupon.
Rewrite the expression with the distributive property: the 6 multiplies each term inside the parentheses. This is true for every value of p, so the two forms are equivalent.
Interpret the new parts. The term 6p is 6 tickets at the full price of p dollars each, so it is the total cost before any coupons. The 12 comes from 6 times 2, so it is the total that the 6 coupons take off. The form subtracts it because the coupons lower the cost. Both parts are in dollars.
Test the value p = 12. Straight from the situation, each friend pays 12 minus 2, which is 10 dollars, so 6 friends pay 60 dollars. Both forms give 60 as well.
Answer
The total cost in dollars is , where is the cost for one friend. In the equivalent form , the term is the cost of tickets at full price and is the total number of dollars the coupons take off.
Takeaway. When an expression describes a situation, each variable, coefficient, term and expression in parentheses stands for a quantity in that situation. To interpret, start from the definition of the variable and say what each part measures. To write, define the variable, build a part for each quantity, and join the parts with the operations the situation describes. Equivalent forms have the same value, and their parts can stand for different quantities.
Next, you will rewrite expressions for percent increase and percent decrease, and you will use an equivalent form of an expression to solve a problem.