Writing Equations with Variables/Letters for Unknown Values - Third Grade Mathematics

In earlier grades, when a number was missing in a math problem, you might have seen an empty box or a question mark. In third grade, you take your first magnificent step into the language of algebra by using letters to represent unknown quantities! A letter that stands in for an unknown number is called a variable. In this chapter, you will learn how to choose variables, write algebraic equations from word stories, and solve for the unknown value.

What Is a Variable?

A variable is a letter or symbol that represents an unknown number or a quantity that can change.

Earlier Grades:      3 + [ ? ] = 10
Third Grade Algebra: 3 + n = 10

Any letter of the alphabet can be used as a variable! Often, mathematicians pick a letter that matches the object in the story:
- Use 'b' for books

- Use 'c' for cookies

- Use 'm' for money or marbles

- Or use standard algebra letters like 'x', 'y', or 'n' (for number).


A variable is like a wrapped birthday present: inside the package is an exact number waiting to be unwrapped!

The Anatomy of an Equation

An equation is a mathematical statement showing that two expressions are equal in value, connected by an equal sign (=).

Left Expression          Equal Sign          Right Expression
    5 x n                    =                      30

Think of an equation like a balanced scale: both sides must carry the exact same weight or value!

               5 x n             30
            =============^=============
                  Balanced Scale!
            Since 5 x 6 = 30, n MUST be 6!

Translating Words into Equations with Variables

Translating English words into mathematical equations is like translating between languages. Look for keywords that represent operations and relationships:

+-----------------------+-------------------------+-------------------------+
| Operation             | English Phrases         | Algebraic Translation   |
+-----------------------+-------------------------+-------------------------+
| Addition (+)          | sum, more than, total   | n + 8 = 15              |
| Subtraction (-)       | difference, fewer than  | 20 - x = 12             |
| Multiplication (x)    | product, times, groups  | 4 x b = 24              |
| Division (÷)          | split equally, quotient | 18 ÷ p = 6              |
+-----------------------+-------------------------+-------------------------+

Example 1: Finding an Unknown Addend

Story: Maya had 14 stickers. Her friend gave her some more stickers, s. Now Maya has 25 stickers. Equation: 14 + s = 25 Solving for s: Think: 14 + ? = 25 -> 25 - 14 = 11. So, s = 11 stickers!

Example 2: Finding an Unknown Factor

Story: Caleb arranged 32 toy soldiers into 4 equal rows with r soldiers in each row. Equation: 4 x r = 32 Solving for r: Think: 4 times what number equals 32? 32 ÷ 4 = 8. So, r = 8 soldiers in each row!

Example 3: Two-Step Equations with Letters

Story: Jordan bought 3 packs of baseball cards with c cards in each pack. His brother gave him 5 more cards. Now Jordan has 29 cards in all. Equation: (3 x c) + 5 = 29 Solving for c:
- Step 1: Subtract the 5 cards: 3 x c = 29 - 5 = 24.

- Step 2: Divide by 3: c = 24 ÷ 3 = 8.
Each pack had 8 cards!

Chapter Practice Exercises

  1. Write an equation using the letter n for the unknown, then solve: a. A number plus 16 equals 30. b. A number decreased by 9 equals 15. c. 6 times a number equals 48. d. 35 divided by a number equals 7.
  2. Choose a letter for the unknown, write an equation, and solve for each word story: a. Liam had 42 pennies. He put them equally into 6 jars with p pennies in each jar. b. Sophia saved d dollars. Her grandmother gave her 20 dollars. Now Sophia has 65 dollars. c. A box has 4 rows of chocolates with c chocolates in each row. There are 28 chocolates in total.
  3. Solve for the variable in each equation: a. 8 x y = 56 b. 45 ÷ m = 9 c. k - 18 = 32 d. 27 + w = 60
  4. Write an equation with a letter for the unknown for this two-step problem: "A baker baked 4 trays of cookies with b cookies on each tray. He sold 10 cookies and has 22 cookies left."
  5. What is the value of b in exercise 4? Show your step-by-step thinking.

Solutions and Step-by-Step Answers

  1. Equations and solutions: a. n + 16 = 30 -> n = 30 - 16 = 14. b. n - 9 = 15 -> n = 15 + 9 = 24. c. 6 x n = 48 -> n = 48 ÷ 6 = 8. d. 35 ÷ n = 7 -> n = 35 ÷ 7 = 5.
  2. Word stories: a. Equation: 6 x p = 42 (or 42 ÷ 6 = p). p = 7 pennies in each jar. b. Equation: d + 20 = 65. d = 65 - 20 = 45 dollars saved. c. Equation: 4 x c = 28. c = 28 ÷ 4 = 7 chocolates in each row.
  3. Variable solutions: a. 8 x y = 56 -> y = 56 ÷ 8 = 7. b. 45 ÷ m = 9 -> m = 45 ÷ 9 = 5. c. k - 18 = 32 -> k = 32 + 18 = 50. d. 27 + w = 60 -> w = 60 - 27 = 33.
  4. Two-step equation: (4 x b) - 10 = 22.
  5. Solving for b:
  6. Step 1: Add 10 back: 4 x b = 22 + 10 = 32.
  7. Step 2: Divide by 4: b = 32 ÷ 4 = 8 cookies per tray.