Expressions, equations, and inequalities
Algebra I · 9th grade
Solving a formula for one letter
By the end, Hannah will be able to rearrange a formula so one chosen letter stands alone by undoing the same operations on both sides.
About 40 minutes to teach
Teach the idea
A formula is just an equation people use so often it got a famous name. It comes apart the same way any other equation does. Decide which letter you want standing alone, then undo everything that was done to it, and do that same undo on both sides.
Start with the area of a rectangle: A = lw. The width w has been multiplied by l. Divide both sides by l to free w.
A = lw
A ÷ l = w
So w = A ÷ l.
Check with numbers you already trust. If A is 48 and l is 8:
48 ÷ 8 = 6
The width is 6, which matches what the original formula would give.
Now a longer one, the perimeter of a rectangle: P = 2l + 2w. Here w was multiplied by 2 and then 2l was added. Undo the addition first. Subtract 2l from both sides.
P − 2l = 2w
Then divide both sides by 2.
(P − 2l) ÷ 2 = w
So w = (P − 2l) ÷ 2.
The brackets are not decoration. They tell you the whole of P minus 2l is being halved, not just the 2l piece.
Test it. Let P = 30 and l = 9:
2 × 9 = 18
30 − 18 = 12
12 ÷ 2 = 6
Width is 6 again. If you skip the brackets or undo only part of a side, the numbers stop matching, and that is how you catch the mistake.
Rearranging once is worth the time whenever you will use the same formula many times. After that, every new set of numbers is just one quick line of arithmetic instead of a full solve.
Do it together
1. Take paper and a pencil. Write the volume formula for a rectangular box: V = lwh. Ask Hannah to solve it for h so height stands alone. Guide Hannah to divide both sides by l and by w (or by the product lw) until the result is h = V ÷ (lw). Then give Hannah V = 96, l = 6, and w = 4, and have Hannah find the height.
2. Write the distance formula d = rt. This time ask Hannah to solve it for r. Hannah should divide both sides by t and land on r = d ÷ t. Check the rearranged form together with d = 200 and t = 5.
3. Write y = mx + b. Ask Hannah to solve it for b. Have Hannah subtract mx from both sides so b = y − mx. Test it with y = 25, m = 3, and x = 4, and confirm the value of b.
4. Ask Hannah why anyone bothers to rearrange a formula at all if the numbers could just be plugged into the original each time. Listen for the idea that solving for the letter you keep needing once turns every later question into one short arithmetic step. Then move on to the activity below.
Activity
Together: Rearrange the one you keep using. Measure a few rectangular things around the house, a book, a tray, a picture frame, and write down each perimeter and each width. Ask Hannah how to get the length from those two numbers, and let the first one be done the long way, by putting the numbers into the perimeter formula and solving. After the second, ask whether there is a way to stop solving from scratch, and rearrange the formula together into l = (P − 2w) ÷ 2. Use it on the rest, then measure one length to check. If Hannah writes less: Hannah reads out the measurements and says which letter is wanted on its own, and you keep the written record.
Tip If Hannah undoes only part of a side or drops the brackets, have Hannah plug the same known numbers into both the original formula and the new version. When the answers disagree, the missing step usually jumps straight out. If Hannah is racing ahead, hand Hannah V = lwh again and ask Hannah to solve for l instead of h with a fresh set of numbers.
On paper
Worksheet
Make the Letter You Want the Subject
The worksheet, as it prints
Algebra I · Worksheet
Make the Letter You Want the Subject
Rearrange each formula so the letter asked for stands alone, then test the rearranged form on the numbers given.
Name: Hannah
Score: ____ / 6
1. The distance travelled at a steady speed is given by d = rt. Solve it for t, then use your answer with d = 147 and r = 21.
2. The equation of a line is given by y = mx + b. Solve it for m, then use your answer with y = 63, b = 14 and x = 7.
3. The perimeter of a rectangle is given by P = 2l + 2w. Solve it for w, then use your answer with P = 22 and l = 5.
4. The area of a rectangle is given by A = lw. Solve it for w, then use your answer with A = 30 and l = 10.
5. Theo rearranged P = 2l + 2w into w = P − 2l ÷ 2, with no brackets. Test that with P = 66 and l = 21, and say what the brackets are for.
6. What would happen to the width of a rectangle with perimeter 66 centimeters if its length grew by 1 centimeter and the perimeter stayed the same?
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