Evaluate Formulas and Find Mistakes | Grade 6 Math | LevelKite

A formula is an equation that shows how one quantity depends on other quantities. Each letter in it is a variable: it stands for a number. For example, in $P = 2l + 2w$ , the variable $P$ is the perimeter of a rectangle, $l$ is its length and $w$ is its width.

In this lesson, the quantity you want is alone on the left side of each formula. To find it, you evaluate the expression on the right side, which means you find its value. First you substitute: you replace each variable with its value. Then you calculate.

You already know how to substitute and how to use the order of operations. Here you use those skills on formulas. Then you use them to find mistakes in worked solutions.

The formulas used in this lesson

Formula	What it finds	What the variables stand for	Unit of the result
$P = 2l + 2w$	$P$ , the perimeter of a rectangle	$l$ is the length and $w$ is the width, both in the same unit	That same length unit, such as meters
$A = 6s^2$	$A$ , the surface area of a cube	$s$ is the side length, the length of one edge	Square units, such as square centimeters when $s$ is in centimeters
$V = s^3$	$V$ , the volume of a cube	$s$ is the side length, the length of one edge	Cubic units, such as cubic centimeters when $s$ is in centimeters
$F = 1.8C + 32$	$F$ , a temperature in degrees Fahrenheit	$C$ is the same temperature in degrees Celsius	Degrees Fahrenheit

Key concept

Substitute, then keep every line equal

To evaluate a formula, replace each variable with its value in parentheses. Then follow the order of operations: grouping symbols such as parentheses, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right.

Each correct step swaps part of the expression for something equal to it, so every line of the work keeps the same value. If a line has a different value from the line above it, that line contains a mistake.

Worked example

Perimeter of a rectangle

A rectangle is $12$ meters long and $7.5$ meters wide. Use the formula $P = 2l + 2w$ to find its perimeter.

Match each measurement with its variable. The length is 12 meters and the width is 7.5 meters. Both are in meters, so the perimeter will be in meters too.
$l = 12, \quad w = 7.5$
Write the formula, then put each value in parentheses in place of its variable. A number written next to a variable means multiply, so the 2 next to the length becomes 2 times 12, not the number 212.
$P = 2l + 2w = 2(12) + 2(7.5)$
Multiply before you add. Each pair of parentheses holds just one number, so no addition is grouped to go first, and multiplication comes before addition. Since 2 times 12 is 24 and 2 times 7.5 is 15, the line becomes 24 plus 15.
$P = 24 + 15$
Add the two products. The sum is the perimeter.
$P = 39$

Answer

The perimeter of the rectangle is $39$ meters.

Worked example

Surface area and volume of a cube

A cube has a side length of $5$ centimeters. Use $A = 6s^2$ to find its surface area and $V = s^3$ to find its volume.

Start with the surface area. A cube has 6 square faces, and the area of each face is the side length times the side length. Put 5 in parentheses in place of the variable.
$A = 6s^2 = 6(5)^2$
Evaluate the exponent before you multiply. The exponent 2 is on the 5 only, and no grouping symbols join the 6 to the 5, so the power comes before the multiplication. Five squared means 5 times 5, which is 25.
$A = 6(25)$
Multiply. Six faces of 25 square centimeters each make 150 square centimeters.
$A = 150$
Now find the volume. Put 5 in parentheses in place of the variable in the volume formula.
$V = s^3 = (5)^3$
Evaluate the power. The exponent 3 tells you to use 5 as a factor three times. Since 5 times 5 is 25 and 25 times 5 is 125, the volume is 125 cubic centimeters.
$V = 5 \cdot 5 \cdot 5 = 125$

Answer

The cube has a surface area of $150$ square centimeters and a volume of $125$ cubic centimeters.

Now think about checking work that someone else wrote. In correct work, each step replaces part of the expression with a number equal to it. For example, $25$ replaces $5^2$ . So in correct work, the expression after the equals sign has the same value on every line. Call the value of that expression the value of the line.

This gives you a way to find a mistake:

Check that the first line matches the formula and that each variable has its own value in its place. If it does not, the mistake is in the substitution.
Find the value of each line.
Look for the first line whose value is different from the line above it. That line contains a mistake.

The table uses these checks on the mistaken cube work. Notice that the last line is a correct step. It only carries the wrong value forward.

Checking the mistaken surface area work line by line

Line	What was written	Value of the line	Same value as the line above?
1	$A = 6(5)^2$	$6(25) = 150$	This is the first line. It matches $A = 6s^2$ with $s = 5$ .
2	$A = 30^2$	$900$	No. The value changed from $150$ to $900$ , so the mistake is in this line.
3	$A = 900$	$900$	Yes. $30^2$ is $900$ , so this step is correct.

After you find the line, name the mistake. Ask three questions:

Substitution: Did the writer replace each variable with its own value and keep every multiplication? Writing $2l$ as $212$ when $l = 12$ is a substitution mistake.
Order of operations: Did the writer do each step in the correct order? Writing $6(5)^2$ as $30^2$ is an order mistake.
Arithmetic: Is each calculation correct? Writing $5^3$ as $15$ is an arithmetic mistake, because $5^3$ means $5 \cdot 5 \cdot 5$ , which is $125$ .

Worked example

Find the mistake in a temperature conversion

The formula $F = 1.8C + 32$ changes a temperature of $C$ degrees Celsius into $F$ degrees Fahrenheit. Maya used it with $C = 20$ . She wrote Line 1: $F = 1.8(20) + 32$ . Line 2: $F = 1.8(52)$ . Line 3: $F = 93.6$ . Which line has the first mistake? Name the mistake and correct the work.

Check the first line against the formula. The formula multiplies the Celsius temperature by 1.8 and then adds 32. Line 1 puts 20 in place of the variable and keeps both operations, so Line 1 is correct.
$F = 1.8(20) + 32$
Find the value of Line 1. No grouping symbols surround a sum here, so multiplication comes before addition. Multiply 1.8 by 20 to get 36. Then add 32 to get 68.
$1.8(20) + 32 = 36 + 32 = 68$
Find the value of Line 2 and compare. The product of 1.8 and 52 is 93.6, which is not 68. The value changed, so Line 2 contains a mistake.
$1.8(52) = 93.6 \neq 68$
Name the mistake. Maya added 20 and 32 first. But the 20 is multiplied by 1.8, and no grouping symbols surround the sum of 20 and 32, so the multiplication has to come first. This is an order of operations mistake.
Correct the work from Line 2 on. Keep the product 36 and then add 32. Line 3 was a correct step from Line 2, but it carried the wrong value forward.
$F = 36 + 32 = 68$

Answer

The first mistake is in Line 2, where Maya added before multiplying. The correct value is $F = 68$ , so $20$ degrees Celsius is $68$ degrees Fahrenheit.

Takeaway. A formula shows how quantities are related. To evaluate one, replace each variable with its value, follow the order of operations, and state the answer with its unit. To check someone's work, find the value of each line and compare it with the line above. In correct work the value never changes. So a line where the value changes is a line that contains a mistake.