M.2 Evaluate equations with mixed operations: true or false
What is an equation?
An equation is a mathematical sentence that states two expressions are equal. The equal sign (=) acts like a balance. Whatever is on the left side must have the same value as whatever is on the right side. In fourth grade, you will work with equations that contain more than one operation, such as addition, subtraction, multiplication, and division.
- 5 + 3 = 8
- 12 − 4 = 8
- 3 × 4 = 12
- 16 ÷ 2 = 8
An equation is always a true statement when both sides have the same value. If the values are different, the number sentence is false.
Which operation do I perform first? The rules
When a mathematical expression has more than one operation, you must follow a set of rules called the order of operations. These rules tell you which operation to do first so that your answer is correct.
If there are parentheses, always calculate inside them first.
-
Expression: (3 + 4) × 2
First operation: 3 + 4 = 7
Then: 7 × 2 = 14 -
Expression: (10 − 6) ÷ 2
First operation: 10 − 6 = 4
Then: 4 ÷ 2 = 2
After parentheses, do multiplication and division. Work from left to right.
-
Expression: 8 + 2 × 3
First operation: 2 × 3 = 6
Then: 8 + 6 = 14 -
Expression: 15 − 6 ÷ 2
First operation: 6 ÷ 2 = 3
Then: 15 − 3 = 12 -
Expression: 20 ÷ 5 × 2
First operation: 20 ÷ 5 = 4
Then: 4 × 2 = 8
Finally, do addition and subtraction. Work from left to right.
-
Expression: 10 − 3 + 2
First operation: 10 − 3 = 7
Then: 7 + 2 = 9 -
Expression: 5 + 6 − 4
First operation: 5 + 6 = 11
Then: 11 − 4 = 7
A helpful way to remember this order is: Please Excuse My Dear Aunt Sally.
Parentheses → Multiply & Divide (left to right) → Add & Subtract (left to right).
Evaluating equations with mixed operations
To evaluate an equation means to find out whether it is true or false. When an equation has mixed operations, you must apply the order of operations to simplify each side before comparing them.
- 3 + 4 × 2 = 11 → First multiply: 4 × 2 = 8. Then add: 3 + 8 = 11. The equation is true.
- 10 − 6 ÷ 2 = 7 → First divide: 6 ÷ 2 = 3. Then subtract: 10 − 3 = 7. The equation is true.
- 5 × 3 − 4 = 10 → First multiply: 5 × 3 = 15. Then subtract: 15 − 4 = 11. Since 11 does not equal 10, the equation is false.
Always perform multiplication and division before addition and subtraction. If you do not follow this order, you may get the wrong answer and decide an equation is true when it is actually false.
Using all four operations in one equation
Some equations use all four operations: addition, subtraction, multiplication, and division. To evaluate these equations correctly, you must apply the order of operations step by step. Work from left to right, but always complete multiplication and division before moving to addition and subtraction.
- 8 + 12 ÷ 4 − 2 × 3 = 5 → First, divide: 12 ÷ 4 = 3. Multiply: 2 × 3 = 6. The equation becomes 8 + 3 − 6. Then add: 8 + 3 = 11. Subtract: 11 − 6 = 5. The equation is true.
- 20 − 4 × 3 + 8 ÷ 2 = 12 → Multiply: 4 × 3 = 12. Divide: 8 ÷ 2 = 4. The equation becomes 20 − 12 + 4. Subtract: 20 − 12 = 8. Add: 8 + 4 = 12. The equation is true.
- 7 + 5 × 2 − 16 ÷ 4 = 10 → Multiply: 5 × 2 = 10. Divide: 16 ÷ 4 = 4. The equation becomes 7 + 10 − 4. Add: 7 + 10 = 17. Subtract: 17 − 4 = 13. Since 13 does not equal 10, the equation is false.
When an equation has multiple operations, rewrite it step by step. Crossing out completed steps can help you keep track and avoid mistakes.
True or false: understanding the equal sign
The equal sign does not mean “the answer is coming.” Instead, it means that the value on the left is exactly the same as the value on the right. When evaluating an equation as true or false, you are checking whether this balance is correct. Sometimes the equal sign appears in different positions, but the meaning stays the same.
- 15 = 10 + 5 → This is true because 15 equals 15.
- 24 ÷ 3 = 2 × 4 → Left side: 24 ÷ 3 = 8. Right side: 2 × 4 = 8. The equation is true.
- 18 − 6 = 4 × 3 → Left side: 18 − 6 = 12. Right side: 4 × 3 = 12. The equation is true.
- 7 + 8 = 20 − 6 → Left side: 7 + 8 = 15. Right side: 20 − 6 = 14. Since 15 does not equal 14, the equation is false.
You can evaluate an equation by simplifying both sides independently. If the simplified values match, the equation is true. If they do not match, it is false.
Common mistakes when evaluating mixed operations
When working with equations that have mixed operations, fourth-grade students often make a few common mistakes. Recognizing these mistakes can help you avoid them and evaluate equations correctly.
- Mistake: 4 + 3 × 5 = 35 (adding before multiplying). Correction: Multiply first: 3 × 5 = 15, then add: 4 + 15 = 19. The correct equation is 4 + 3 × 5 = 19.
- Mistake: 20 − 10 ÷ 2 = 5 (subtracting before dividing). Correction: Divide first: 10 ÷ 2 = 5, then subtract: 20 − 5 = 15. The correct equation is 20 − 10 ÷ 2 = 15.
- Mistake: 12 ÷ 3 × 2 = 2 (multiplying before dividing). Correction: Work from left to right for multiplication and division: 12 ÷ 3 = 4, then 4 × 2 = 8. The correct equation is 12 ÷ 3 × 2 = 8.
- Mistake: Forgetting to simplify both sides of the equal sign. Correction: Always simplify the left side and the right side separately before comparing them.
When multiplication and division appear together, work from left to right. Do not assume that multiplication always comes before division. The same rule applies to addition and subtraction: work from left to right after completing all multiplication and division.
Strategies for evaluating equations
Using a clear strategy will help you evaluate equations quickly and accurately. Follow these steps every time you see an equation with mixed operations.
- Step 1: Look at the equation. Identify all operations on both sides of the equal sign.
- Step 2: On each side, perform all multiplication and division from left to right.
- Step 3: On each side, perform all addition and subtraction from left to right.
- Step 4: Compare the simplified values on both sides. If they are equal, the equation is true. If they are not equal, the equation is false.
- Equation: 9 + 6 ÷ 3 − 2 × 4 = 3
- Left side: First, divide: 6 ÷ 3 = 2. Multiply: 2 × 4 = 8. The left side becomes 9 + 2 − 8.
- Left side: Next, add: 9 + 2 = 11. Subtract: 11 − 8 = 3.
- Right side: The right side is already 3.
- Compare: 3 = 3. The equation is true.
Writing down each step on paper helps prevent mental math errors. Use scratch paper to show your work clearly.
Equations with parentheses
Sometimes an equation includes parentheses ( ). Parentheses group parts of an expression together. When parentheses appear, you must calculate what is inside them first, before following the rest of the order of operations. This changes how you evaluate the equation.
- (4 + 3) × 2 = 14 → First, add inside parentheses: 4 + 3 = 7. Then multiply: 7 × 2 = 14. The equation is true.
- 24 ÷ (8 − 2) = 4 → First, subtract inside parentheses: 8 − 2 = 6. Then divide: 24 ÷ 6 = 4. The equation is true.
- (10 − 6) × (2 + 1) = 8 → First, solve each parentheses: 10 − 6 = 4, and 2 + 1 = 3. Then multiply: 4 × 3 = 12. Since 12 does not equal 8, the equation is false.
Parentheses always come first in the order of operations. If there are parentheses inside other parentheses, start with the innermost set first.
Real-world meaning of true and false equations
Understanding whether an equation is true or false helps you solve real-world problems. When you check if a calculation is correct, you are evaluating an equation. This skill is used in shopping, cooking, building projects, and many other everyday activities.
- You have $20. You buy 3 notebooks for $4 each and a pen for $2. The cashier says your total is $14. Is this true? Evaluate: 3 × 4 + 2 = 12 + 2 = 14. The equation is true.
- A recipe calls for 2 cups of flour per batch. You make 4 batches, but you only have 10 cups of flour. The recipe says you need 8 cups. Evaluate: 2 × 4 = 8. The equation is true, so you have enough flour.
- You are saving money. You save $5 each week for 6 weeks. Your goal is $35. Evaluate: 5 × 6 = 30. Since 30 does not equal 35, the equation is false. You need to save more or adjust your goal.
Evaluating equations helps you check your work and make sure your calculations are accurate. It is a valuable skill for both school and daily life.
Common Core alignment:
CCSS.MATH.CONTENT.4.OA.A.1 – Interpret a multiplication equation as a comparison.
CCSS.MATH.CONTENT.4.OA.A.2 – Multiply or divide to solve word problems involving multiplicative comparison.
Focus standard: CCSS.MATH.CONTENT.4.OA.A.3 – Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which remainders must be interpreted. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.
Notes for teachers
This lesson is aligned with CCSS.MATH.CONTENT.4.OA.A.3 and supports the foundational skill of evaluating equations with mixed operations. Students learn to apply the order of operations to determine whether an equation is true or false. All content is 100% free, use it for whole-class instruction, independent study and practice, or homework.