Chapter 4 · Lesson 4 of 10 · Grade 6 · Expressions & Equations · MAT-06-EE-004
Apply the Order of Operations
Represent and solve apply the order of operations problems using precise mathematical notation, models, and reasoning.
Math V4 · Course chapter
Chapter 4: Expressions, Equations, and Inequalities
Using symbols to describe relationships
Essential question: How can a verbal or numerical relationship be expressed symbolically, transformed without changing its meaning, and solved?
Where this lesson fits
Lesson 4 of 10. Each lesson builds on earlier chapter representations and ideas rather than resetting the topic.
Chapter destination
Model a real pricing or distance situation with variables, expressions, equations, inequalities, tables, and graphs.
Learning objectives
What you should be able to do
- Represent and solve apply the order of operations problems using precise mathematical notation, models, and reasoning.
- Explain why a method for apply the order of operations works, interpret the result in context, and verify it independently.
Prerequisite check
Make sure the foundation is ready.
The new lesson depends on this prerequisite as active knowledge.
Learn
Build the idea from meaning, not memorization.
Chapter 4: Expressions, Equations, and Inequalities
Using symbols to describe relationships. This is Lesson 4 of 10 in Chapter 4. It builds on earlier chapter ideas instead of restarting the topic from scratch.
Connection to the course
Cumulative knowledge used here includes Chapter 1 rates, Chapter 2 operation fluency, Chapter 3 signed quantities. As you work, connect today's idea to earlier lessons in the chapter and keep those earlier representations available for comparison.
Mathematical meaning
Apply the Order of Operations uses symbols to describe relationships. Equality and inequality are relationships that must be preserved on both sides.
Represent the relationship
The Math Box uses equation balance to expose the structure. Change one input, predict the result, then connect the visual change to an equation, table, graph, number line, or geometric model.
Calculate, interpret, and verify
Verify algebra by substituting the proposed solution into the original statement; equivalent expressions must agree for the same input.
Math Box · Interactive lesson
Touch the math. Change it. See what stays true.
Use the interactive model before and after the worked examples. Change the inputs, make a prediction, then use the model to test whether your reasoning holds.
Equation Balance
Adjust both sides. Equality means both expressions have the same value.
Balanced: both sides have equal value.
- Model one example of apply the order of operations in the Math Box.
- Change one input, predict the effect, and test the prediction.
- Connect the model to another representation used earlier in this chapter.
- State the relationship or invariant that explains what stayed mathematically consistent.
Worked examples
Twenty different ways to see the concept work.
Evaluate (2 + 3) × 2.
(2 + 3) × 2 = 5 × 2 = 10Grouping symbols are evaluated before multiplication.
Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Evaluate (3 + 5) × 3.
(3 + 5) × 3 = 8 × 3 = 24The representation should show the same mathematical relationship as the calculation. Grouping symbols are evaluated before multiplication.
Explain why a valid method works, then solve: Evaluate (4 + 7) × 4.
(4 + 7) × 4 = 11 × 4 = 44A complete explanation names the relationship or property being preserved. Grouping symbols are evaluated before multiplication.
Estimate or predict first, then calculate and decide whether the result is reasonable: Evaluate (5 + 9) × 5.
(5 + 9) × 5 = 14 × 5 = 70The estimate is a reasonableness check, not a replacement for the exact result. Grouping symbols are evaluated before multiplication.
Solve and verify the result with a second method, inverse operation, or equivalent representation: Evaluate (6 + 3) × 6.
(6 + 3) × 6 = 9 × 6 = 54Verification should independently support the result. Grouping symbols are evaluated before multiplication.
Interpret the answer in context after solving. What does the result mean here? Evaluate (7 + 5) × 7.
(7 + 5) × 7 = 12 × 7 = 84State the result with its meaning, units, direction, or comparison—not only a number. Grouping symbols are evaluated before multiplication.
Error analysis: a student says, "Changing only one side of an equation." Explain the mistake, then solve this related task correctly: Evaluate (8 + 7) × 8.
Use equivalent operations on both sides. Correct solution: (8 + 7) × 8 = 15 × 8 = 120Error analysis requires identifying the broken idea, not merely replacing the final answer. Grouping symbols are evaluated before multiplication.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Evaluate (2 + 9) × 9. Case B: Evaluate (4 + 3) × 9.
Case A: (2 + 9) × 9 = 11 × 9 = 99 Case B: (4 + 3) × 9 = 7 × 9 = 63Compare the structure, representation, units, and result rather than only the numbers. Grouping symbols are evaluated before multiplication.
Evaluate (3 + 3) × 10.
(3 + 3) × 10 = 6 × 10 = 60Grouping symbols are evaluated before multiplication.
Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Evaluate (4 + 5) × 2.
(4 + 5) × 2 = 9 × 2 = 18The representation should show the same mathematical relationship as the calculation. Grouping symbols are evaluated before multiplication.
Explain why a valid method works, then solve: Evaluate (5 + 7) × 3.
(5 + 7) × 3 = 12 × 3 = 36A complete explanation names the relationship or property being preserved. Grouping symbols are evaluated before multiplication.
Estimate or predict first, then calculate and decide whether the result is reasonable: Evaluate (6 + 9) × 4.
(6 + 9) × 4 = 15 × 4 = 60The estimate is a reasonableness check, not a replacement for the exact result. Grouping symbols are evaluated before multiplication.
Solve and verify the result with a second method, inverse operation, or equivalent representation: Evaluate (7 + 3) × 5.
(7 + 3) × 5 = 10 × 5 = 50Verification should independently support the result. Grouping symbols are evaluated before multiplication.
Interpret the answer in context after solving. What does the result mean here? Evaluate (8 + 5) × 6.
(8 + 5) × 6 = 13 × 6 = 78State the result with its meaning, units, direction, or comparison—not only a number. Grouping symbols are evaluated before multiplication.
Error analysis: a student says, "Changing only one side of an equation." Explain the mistake, then solve this related task correctly: Evaluate (2 + 7) × 7.
Use equivalent operations on both sides. Correct solution: (2 + 7) × 7 = 9 × 7 = 63Error analysis requires identifying the broken idea, not merely replacing the final answer. Grouping symbols are evaluated before multiplication.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Evaluate (3 + 9) × 8. Case B: Evaluate (5 + 3) × 8.
Case A: (3 + 9) × 8 = 12 × 8 = 96 Case B: (5 + 3) × 8 = 8 × 8 = 64Compare the structure, representation, units, and result rather than only the numbers. Grouping symbols are evaluated before multiplication.
Evaluate (4 + 3) × 9.
(4 + 3) × 9 = 7 × 9 = 63Grouping symbols are evaluated before multiplication.
Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Evaluate (5 + 5) × 10.
(5 + 5) × 10 = 10 × 10 = 100The representation should show the same mathematical relationship as the calculation. Grouping symbols are evaluated before multiplication.
Explain why a valid method works, then solve: Evaluate (6 + 7) × 2.
(6 + 7) × 2 = 13 × 2 = 26A complete explanation names the relationship or property being preserved. Grouping symbols are evaluated before multiplication.
Estimate or predict first, then calculate and decide whether the result is reasonable: Evaluate (7 + 9) × 3.
(7 + 9) × 3 = 16 × 3 = 48The estimate is a reasonableness check, not a replacement for the exact result. Grouping symbols are evaluated before multiplication.
Practice
Guided practice with hints
01Solve and verify the result with a second method, inverse operation, or equivalent representation: Evaluate (8 + 3) × 4.
Hint: Choose a representation before computing.
Answer: (8 + 3) × 4 = 11 × 4 = 44
02Interpret the answer in context after solving. What does the result mean here? Evaluate (2 + 5) × 5.
Hint: Explain what relationship or property makes your method valid.
Answer: (2 + 5) × 5 = 7 × 5 = 35
03Error analysis: a student says, "Changing only one side of an equation." Explain the mistake, then solve this related task correctly: Evaluate (3 + 7) × 6.
Hint: Choose a representation before computing.
Answer: Use equivalent operations on both sides. Correct solution: (3 + 7) × 6 = 10 × 6 = 60
04Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Evaluate (4 + 9) × 7. Case B: Evaluate (6 + 3) × 7.
Hint: Explain what relationship or property makes your method valid.
Answer: Case A: (4 + 9) × 7 = 13 × 7 = 91 Case B: (6 + 3) × 7 = 9 × 7 = 63
05Evaluate (5 + 3) × 8.
Hint: Choose a representation before computing.
Answer: (5 + 3) × 8 = 8 × 8 = 64
06Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Evaluate (6 + 5) × 9.
Hint: Explain what relationship or property makes your method valid.
Answer: (6 + 5) × 9 = 11 × 9 = 99
Practice
Independent practice
01Explain why a valid method works, then solve: Evaluate (7 + 7) × 10.
Answer: (7 + 7) × 10 = 14 × 10 = 140
02Estimate or predict first, then calculate and decide whether the result is reasonable: Evaluate (8 + 9) × 2.
Answer: (8 + 9) × 2 = 17 × 2 = 34
03Solve and verify the result with a second method, inverse operation, or equivalent representation: Evaluate (2 + 3) × 3.
Answer: (2 + 3) × 3 = 5 × 3 = 15
04Interpret the answer in context after solving. What does the result mean here? Evaluate (3 + 5) × 4.
Answer: (3 + 5) × 4 = 8 × 4 = 32
05Error analysis: a student says, "Changing only one side of an equation." Explain the mistake, then solve this related task correctly: Evaluate (4 + 7) × 5.
Answer: Use equivalent operations on both sides. Correct solution: (4 + 7) × 5 = 11 × 5 = 55
06Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Evaluate (5 + 9) × 6. Case B: Evaluate (7 + 3) × 6.
Answer: Case A: (5 + 9) × 6 = 14 × 6 = 84 Case B: (7 + 3) × 6 = 10 × 6 = 60
07Evaluate (6 + 3) × 7.
Answer: (6 + 3) × 7 = 9 × 7 = 63
08Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Evaluate (7 + 5) × 8.
Answer: (7 + 5) × 8 = 12 × 8 = 96
Common mistakes
Learn to catch the error, not just the answer.
Use equivalent operations on both sides.
Only terms with the same variable part are like terms.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Translate cost, distance, rate, and measurement situations into symbolic rules.
- Use equations and inequalities to determine unknown quantities, thresholds, and feasible ranges.
Challenge problems
Explain why a valid method works, then solve: Evaluate (8 + 7) × 9.
(8 + 7) × 9 = 15 × 9 = 135
Estimate or predict first, then calculate and decide whether the result is reasonable: Evaluate (2 + 9) × 10.
(2 + 9) × 10 = 11 × 10 = 110
Solve and verify the result with a second method, inverse operation, or equivalent representation: Evaluate (3 + 3) × 2.
(3 + 3) × 2 = 6 × 2 = 12
Interpret the answer in context after solving. What does the result mean here? Evaluate (4 + 5) × 3.
(4 + 5) × 3 = 9 × 3 = 27
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 6 Chapter 4 Reasoning Lab
Solve apply the order of operations problems accurately, choose an appropriate representation, and justify each result with a check.
Practice
Mastery check
01Error analysis: a student says, "Changing only one side of an equation." Explain the mistake, then solve this related task correctly: Evaluate (5 + 7) × 4.
Answer: Use equivalent operations on both sides. Correct solution: (5 + 7) × 4 = 12 × 4 = 48
02Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Evaluate (6 + 9) × 5. Case B: Evaluate (8 + 3) × 5.
Answer: Case A: (6 + 9) × 5 = 15 × 5 = 75 Case B: (8 + 3) × 5 = 11 × 5 = 55
03Evaluate (7 + 3) × 6.
Answer: (7 + 3) × 6 = 10 × 6 = 60
04Represent before calculating. Use words, an expression or equation, a table, and when useful a graph for this task, then solve: Evaluate (8 + 5) × 7.
Answer: (8 + 5) × 7 = 13 × 7 = 91
05Explain why a valid method works, then solve: Evaluate (2 + 7) × 8.
Answer: (2 + 7) × 8 = 9 × 8 = 72
06Estimate or predict first, then calculate and decide whether the result is reasonable: Evaluate (3 + 9) × 9.
Answer: (3 + 9) × 9 = 12 × 9 = 108
07Solve and verify the result with a second method, inverse operation, or equivalent representation: Evaluate (4 + 3) × 10.
Answer: (4 + 3) × 10 = 7 × 10 = 70
08Interpret the answer in context after solving. What does the result mean here? Evaluate (5 + 5) × 2.
Answer: (5 + 5) × 2 = 10 × 2 = 20
Terminology
Words to know
- variable
- A symbol representing a number that may vary or be unknown.
- coefficient
- A numerical factor multiplying a variable.
- expression
- A combination of numbers, variables, and operations.
- solution
- A value that makes an equation or inequality true.
Continue learning
Connected concepts
Curriculum references
Standards and instructional references
Grade 6 scope, sequence, standards, and public task structure
Reference for coherent sequencing, representations, dependency-aware progression, and reasoning-rich task types.Eureka Math² Grade 6 program structure
Reference for module/topic coherence, recap, mixed-practice, and cumulative-learning patterns.enVision Mathematics Grade 6 instructional model
Reference for problem-based entry points, visual learning, modeling, and application patterns.Common Core State Standards for Mathematics — Grade 6
Grade-level content and mathematical-practice expectations.