Chapter 2 · Lesson 4 of 6 · Grade 6 · The Number System · MAT-06-NS-004
Add, Subtract, Multiply, and Divide Multi-Digit Decimals
Represent and solve add, subtract, multiply, and divide multi-digit decimals problems using precise mathematical notation, models, and reasoning.
Math V4 · Course chapter
Chapter 2: Division, Decimals, Factors, and Multiples
Extending arithmetic fluency with structure
Essential question: How can number structure make complicated calculations more understandable, efficient, and verifiable?
Where this lesson fits
Lesson 4 of 6. Each lesson builds on earlier chapter representations and ideas rather than resetting the topic.
Chapter destination
Design and cost a batch-production plan that uses fractional quantities, decimal operations, common factors, and common multiples.
Learning objectives
What you should be able to do
- Represent and solve add, subtract, multiply, and divide multi-digit decimals problems using precise mathematical notation, models, and reasoning.
- Explain why a method for add, subtract, multiply, and divide multi-digit decimals 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 2: Division, Decimals, Factors, and Multiples
Extending arithmetic fluency with structure. This is Lesson 4 of 6 in Chapter 2. It builds on earlier chapter ideas instead of restarting the topic from scratch.
Connection to the course
Cumulative knowledge used here includes Chapter 1 ratio reasoning, fraction multiplication, place value, whole-number factors. As you work, connect today's idea to earlier lessons in the chapter and keep those earlier representations available for comparison.
Mathematical meaning
Add, Subtract, Multiply, and Divide Multi-Digit Decimals extends arithmetic to signed, fractional, decimal, or coordinate quantities. Operations must preserve both magnitude and direction.
Represent the relationship
The Math Box uses base ten blocks 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 with an inverse operation, equivalent representation, estimate, or second method so both the calculation and reasoning are auditable.
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.
Base-Ten Lab
Change hundreds, tens, and ones and watch the total rebuild from place value.
- Model one example of add, subtract, multiply, and divide multi digit decimals 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.
Add 12.50 + 2.10.
12.50 + 2.10 = 14.60Align equal place values before adding.
Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Subtract 2.26 from 12.87.
12.87 − 2.26 = 10.61The representation should show the same mathematical relationship as the calculation. Align equal place values and regroup when needed.
Explain why a valid method works, then solve: Multiply 2.42 × 4.
2.42 × 4 = 9.68A complete explanation names the relationship or property being preserved. Multiply the units, then preserve the decimal place-value scale.
Estimate or predict first, then calculate and decide whether the result is reasonable: Divide 7.75 ÷ 5.
7.75 ÷ 5 = 1.55The estimate is a reasonableness check, not a replacement for the exact result. Division partitions the decimal quantity into equal groups; multiplication verifies the quotient.
Solve and verify the result with a second method, inverse operation, or equivalent representation: Add 13.98 + 2.74.
13.98 + 2.74 = 16.72Verification should independently support the result. Align equal place values before adding.
Interpret the answer in context after solving. What does the result mean here? Subtract 2.90 from 14.35.
14.35 − 2.90 = 11.45State the result with its meaning, units, direction, or comparison—not only a number. Align equal place values and regroup when needed.
Error analysis: a student says, "Applying sign rules without checking direction or magnitude." Explain the mistake, then solve this related task correctly: Multiply 3.06 × 2.
Use a number line, context, or inverse operation. Correct solution: 3.06 × 2 = 6.12Error analysis requires identifying the broken idea, not merely replacing the final answer. Multiply the units, then preserve the decimal place-value scale.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Divide 7.80 ÷ 4. Case B: Add 18.42 + 2.42.
Case A: 7.80 ÷ 4 = 1.95 Case B: 18.42 + 2.42 = 20.84Compare the structure, representation, units, and result rather than only the numbers. Division partitions the decimal quantity into equal groups; multiplication verifies the quotient.
Add 15.46 + 2.26.
15.46 + 2.26 = 17.72Align equal place values before adding.
Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Subtract 2.42 from 15.83.
15.83 − 2.42 = 13.41The representation should show the same mathematical relationship as the calculation. Align equal place values and regroup when needed.
Explain why a valid method works, then solve: Multiply 2.58 × 6.
2.58 × 6 = 15.48A complete explanation names the relationship or property being preserved. Multiply the units, then preserve the decimal place-value scale.
Estimate or predict first, then calculate and decide whether the result is reasonable: Divide 7.05 ÷ 3.
7.05 ÷ 3 = 2.35The estimate is a reasonableness check, not a replacement for the exact result. Division partitions the decimal quantity into equal groups; multiplication verifies the quotient.
Solve and verify the result with a second method, inverse operation, or equivalent representation: Add 16.94 + 2.90.
16.94 + 2.90 = 19.84Verification should independently support the result. Align equal place values before adding.
Interpret the answer in context after solving. What does the result mean here? Subtract 3.06 from 17.31.
17.31 − 3.06 = 14.25State the result with its meaning, units, direction, or comparison—not only a number. Align equal place values and regroup when needed.
Error analysis: a student says, "Applying sign rules without checking direction or magnitude." Explain the mistake, then solve this related task correctly: Multiply 2.10 × 4.
Use a number line, context, or inverse operation. Correct solution: 2.10 × 4 = 8.40Error analysis requires identifying the broken idea, not merely replacing the final answer. Multiply the units, then preserve the decimal place-value scale.
Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Divide 5.50 ÷ 2. Case B: Add 21.38 + 2.58.
Case A: 5.50 ÷ 2 = 2.75 Case B: 21.38 + 2.58 = 23.96Compare the structure, representation, units, and result rather than only the numbers. Division partitions the decimal quantity into equal groups; multiplication verifies the quotient.
Add 18.42 + 2.42.
18.42 + 2.42 = 20.84Align equal place values before adding.
Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Subtract 2.58 from 18.79.
18.79 − 2.58 = 16.21The representation should show the same mathematical relationship as the calculation. Align equal place values and regroup when needed.
Explain why a valid method works, then solve: Multiply 2.74 × 2.
2.74 × 2 = 5.48A complete explanation names the relationship or property being preserved. Multiply the units, then preserve the decimal place-value scale.
Estimate or predict first, then calculate and decide whether the result is reasonable: Divide 18.90 ÷ 6.
18.90 ÷ 6 = 3.15The estimate is a reasonableness check, not a replacement for the exact result. Division partitions the decimal quantity into equal groups; multiplication verifies the quotient.
Practice
Guided practice with hints
01Solve and verify the result with a second method, inverse operation, or equivalent representation: Add 19.90 + 3.06.
Hint: Choose a representation before computing.
Answer: 19.90 + 3.06 = 22.96
02Interpret the answer in context after solving. What does the result mean here? Subtract 2.10 from 20.27.
Hint: Explain what relationship or property makes your method valid.
Answer: 20.27 − 2.10 = 18.17
03Error analysis: a student says, "Applying sign rules without checking direction or magnitude." Explain the mistake, then solve this related task correctly: Multiply 2.26 × 6.
Hint: Choose a representation before computing.
Answer: Use a number line, context, or inverse operation. Correct solution: 2.26 × 6 = 13.56
04Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Divide 17.75 ÷ 5. Case B: Add 24.34 + 2.74.
Hint: Explain what relationship or property makes your method valid.
Answer: Case A: 17.75 ÷ 5 = 3.55 Case B: 24.34 + 2.74 = 27.08
05Add 21.38 + 2.58.
Hint: Choose a representation before computing.
Answer: 21.38 + 2.58 = 23.96
06Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Subtract 2.74 from 21.75.
Hint: Explain what relationship or property makes your method valid.
Answer: 21.75 − 2.74 = 19.01
Practice
Independent practice
01Explain why a valid method works, then solve: Multiply 2.90 × 4.
Answer: 2.90 × 4 = 11.60
02Estimate or predict first, then calculate and decide whether the result is reasonable: Divide 15.80 ÷ 4.
Answer: 15.80 ÷ 4 = 3.95
03Solve and verify the result with a second method, inverse operation, or equivalent representation: Add 22.86 + 2.10.
Answer: 22.86 + 2.10 = 24.96
04Interpret the answer in context after solving. What does the result mean here? Subtract 2.26 from 23.23.
Answer: 23.23 − 2.26 = 20.97
05Error analysis: a student says, "Applying sign rules without checking direction or magnitude." Explain the mistake, then solve this related task correctly: Multiply 2.42 × 2.
Answer: Use a number line, context, or inverse operation. Correct solution: 2.42 × 2 = 4.84
06Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Divide 13.05 ÷ 3. Case B: Add 27.30 + 2.90.
Answer: Case A: 13.05 ÷ 3 = 4.35 Case B: 27.30 + 2.90 = 30.20
07Add 24.34 + 2.74.
Answer: 24.34 + 2.74 = 27.08
08Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Subtract 2.90 from 24.71.
Answer: 24.71 − 2.90 = 21.81
Common mistakes
Learn to catch the error, not just the answer.
Use a number line, context, or inverse operation.
For negatives, values closer to zero are greater.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Model temperature, elevation, debt, gains and losses, coordinates, and measurements beyond whole numbers.
- Compute accurately with fractions, decimals, and signed quantities while checking magnitude and sign.
Challenge problems
Explain why a valid method works, then solve: Multiply 3.06 × 6.
3.06 × 6 = 18.36
Estimate or predict first, then calculate and decide whether the result is reasonable: Divide 9.50 ÷ 2.
9.50 ÷ 2 = 4.75
Solve and verify the result with a second method, inverse operation, or equivalent representation: Add 25.82 + 2.26.
25.82 + 2.26 = 28.08
Interpret the answer in context after solving. What does the result mean here? Subtract 2.42 from 26.19.
26.19 − 2.42 = 23.77
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 6 Chapter 2 Reasoning Lab
Solve add, subtract, multiply, and divide multi-digit decimals problems accurately, choose an appropriate representation, and justify each result with a check.
Practice
Mastery check
01Error analysis: a student says, "Applying sign rules without checking direction or magnitude." Explain the mistake, then solve this related task correctly: Multiply 2.58 × 4.
Answer: Use a number line, context, or inverse operation. Correct solution: 2.58 × 4 = 10.32
02Compare two related cases and explain what changes and what stays mathematically consistent. Case A: Divide 30.90 ÷ 6. Case B: Add 30.26 + 3.06.
Answer: Case A: 30.90 ÷ 6 = 5.15 Case B: 30.26 + 3.06 = 33.32
03Add 27.30 + 2.90.
Answer: 27.30 + 2.90 = 30.20
04Represent before calculating. Use a number line, area/fraction model, place-value model, or operation equation for this task, then solve: Subtract 3.06 from 27.67.
Answer: 27.67 − 3.06 = 24.61
05Explain why a valid method works, then solve: Multiply 2.10 × 2.
Answer: 2.10 × 2 = 4.20
06Estimate or predict first, then calculate and decide whether the result is reasonable: Divide 27.75 ÷ 5.
Answer: 27.75 ÷ 5 = 5.55
07Solve and verify the result with a second method, inverse operation, or equivalent representation: Add 28.78 + 2.42.
Answer: 28.78 + 2.42 = 31.20
08Interpret the answer in context after solving. What does the result mean here? Subtract 2.58 from 29.15.
Answer: 29.15 − 2.58 = 26.57
Terminology
Words to know
- rational number
- A number expressible as a ratio of two integers with nonzero denominator.
- absolute value
- Distance from zero on a number line.
- opposite
- A number the same distance from zero on the other side.
- coordinate
- A number locating a point relative to an axis.
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.