Advanced Mental Addition and Subtraction for Year 6 Students

Challenging exercises covering advanced mental math strategies: expanded counting on, advanced making tens, complex decomposing, compensation method, and number line strategies for Year 6 learners.

Year 6 Exercises: Problems 1 to 3
1 Advanced Counting On Strategy
Problem 1: 78 + 9
Solve mentally using the advanced counting on strategy.
Show your thinking process and explain why you chose this method.
Definition:

Advanced Counting On Strategy: Start with the larger number and count up by the smaller number. In Year 6, this method is extended to include counting by groups (like 2s, 5s, or 10s) when appropriate and for larger numbers. Students also learn to visualize the counting process mentally and apply it efficiently.

Step-by-step method:
  1. Identify the larger number (78) and the smaller number (9)
  2. Start counting from the larger number: 78
  3. Count up by the smaller number: 79, 80, 81, 82, 83, 84, 85, 86, 87
  4. The final number is the answer: 87
Start
78
Count Up
+9
Result
87
Step 1: Identify numbers

78 is the larger number, 9 is the smaller number

Step 2: Start with larger number

Start counting from 78

Step 3: Count up by smaller number

78 → 79 → 80 → 81 → 82 → 83 → 84 → 85 → 86 → 87 (count 9 numbers)

Step 4: Record the final number

The last number counted is the answer: 87

78 + 9 = 87
Final answer:

78 + 9 = 87

Applied rules:

• Counting On: Start with the larger number to minimize counting

• Sequential Counting: Add one number at a time

• Small Numbers: Best used when adding 1-9

2 Advanced Making Tens Strategy
Problem 2: 97 + 6
Solve mentally using the advanced making tens strategy.
Show how you break apart the numbers and explain the process.
Definition:

Advanced Making Tens Strategy: Break apart one number to make a ten with the other number, then add the remaining amount. In Year 6, this includes making multiples of 10 beyond just the immediate next ten, and students also consider making hundreds when appropriate. This strategy builds on number bonds knowledge.

Original
97 + 6
Make Ten
97 + 3 + 3
Simplify
100 + 3 = 103
Step 1: Identify how many needed to make a ten

From 97, we need 3 more to reach 100 (the next hundred)

Step 2: Break apart the second number

Break 6 into 3 + 3 (so we can make 100 with 97)

Step 3: Add to make the ten

97 + 3 = 100

Step 4: Add the remaining amount

100 + 3 = 103

97 + 6 = 103
Final answer:

97 + 6 = 103

Applied rules:

• Ten Complements: Know how many more to make the next ten

• Decomposition: Break apart numbers strategically

• Easy Addition: Adding multiples of 10 is easier

3 Complex Decomposing Numbers Strategy
Problem 3: 69 + 26
Solve mentally using the complex decomposing numbers strategy.
Show how you separate tens and ones, and handle regrouping.
Definition:

Complex Decomposing Numbers: Break apart both numbers into tens and ones, add the tens together, add the ones together, then combine the results. When ones sum to 10 or more, regroup into tens. This is the most flexible strategy for any addition problem and builds on place value understanding.

Original
69 + 26
Decompose
60+9 + 20+6
Result
80+15 = 95
Step 1: Decompose each number

69 = 60 + 9 (6 tens and 9 ones)

26 = 20 + 6 (2 tens and 6 ones)

Step 2: Add the tens together

60 + 20 = 80

Step 3: Add the ones together

9 + 6 = 15

Step 4: Combine the sums

80 + 15 = 95

69 + 26 = 95
Final answer:

69 + 26 = 95

Applied rules:

• Place Value: Separate tens and ones

• Group Like Terms: Add tens with tens, ones with ones

• Regrouping: If ones sum to 10 or more, add to tens

Advanced Mental Math Strategies for Year 6
Counting On: Start with larger number
Strategy 1
Making Tens
97 + 6 = 97 + 3 + 3 = 100 + 3
Break apart to make 10
Decomposing
69 + 26 = 60+9 + 20+6 = 80+15
Separate tens and ones
Compensation
89 + 8 = 90 + 7 = 97
Round and adjust
Key definitions:

Mental Math: Calculating in your head without paper or tools

Decomposing: Breaking numbers into parts (tens and ones)

Compensation: Adjusting numbers to make calculations easier

Place Value: Understanding that digits represent different values based on their position

Regrouping: Converting 10 ones into 1 ten or vice versa

Number Bonds: Pairs of numbers that add up to another number (like 7 and 3 bond to make 10)

Estimation: Finding approximate answers to check reasonableness

Fact Families: Related addition and subtraction facts that help with mental math

Number Properties: Commutative, associative, and distributive properties that aid mental computation

Complete methodology:
  1. Look at the numbers: Identify which strategy might work best
  2. Choose the method: Counting on, making tens, decomposing, or compensation
  3. Apply the strategy: Follow the steps of the chosen technique
  4. Handle regrouping: Convert 10 ones to 1 ten if needed
  5. Combine results: Put the pieces back together
  6. Verify: Check your answer by using another method
Tip 1: Always start with the larger number when counting on.
Tip 2: Practice complements to 10 and 20 (like 7 needs 3 to make 10, 17 needs 3 to make 20).
Tip 3: Decompose both numbers to add tens and ones separately.
Tip 4: Round difficult numbers to make them easier to work with.
Tip 5: Use number bonds to help with making tens strategy.
Tip 6: Look for doubles facts to help with addition.
Tip 7: Estimate first to check if your answer is reasonable.
Common errors: Forgetting to count the starting number, miscounting, adding instead of subtracting when compensating, forgetting to regroup.
Practice areas: Number bonds to 10 and 20, basic addition facts, place value understanding, regrouping.
Memorization tips: Remember that addition is commutative (order doesn't matter).
Year 6 Exercises: Problems 4 to 5
4 Advanced Compensation Method
Problem 4: 89 + 8
Solve mentally using the advanced compensation method.
Show how you round and adjust, including multiple steps if needed.
Definition:

Advanced Compensation Method: Round one number to make it easier to work with, then adjust the answer by the amount you changed. In Year 6, this may involve multiple adjustments or more complex rounding, and students also learn to consider compensation for subtraction. This strategy develops flexibility in mathematical thinking.

Original
89 + 8
Round Up
90 + 7
Result
97
Step 1: Look for numbers near tens

89 is close to 90 (just 1 away)

Step 2: Round one number up

Add 1 to 89 to make it 90

Step 3: Adjust the other number

Subtract 1 from 8 to keep the total the same: 8 - 1 = 7

Step 4: Add the rounded numbers

90 + 7 = 97

89 + 8 = 97
Final answer:

89 + 8 = 97

Applied rules:

• Compensation: When you add to one number, subtract from the other

• Rounding: Make numbers easier by rounding to nearest ten

• Balancing: Keep the equation balanced by adjusting both sides

5 Advanced Mental Subtraction Strategy
Problem 5: 93 - 7
Solve mentally using the advanced compensation strategy.
Show how you make subtraction easier with multiple approaches.
Definition:

Advanced Mental Subtraction: Use compensation, counting back, or other strategies to subtract without paper. In Year 6, students learn multiple approaches for different situations and begin to consider which strategy is most efficient. They also understand the inverse relationship between addition and subtraction.

Original
93 - 7
Compensate
93 - 10 + 3
Result
86
Step 1: Look for numbers near tens

7 is close to 10 (3 away)

Step 2: Subtract the round number

93 - 10 = 83

Step 3: Add back the difference

We subtracted 3 too many, so add 3 back: 83 + 3 = 86

Step 4: Verify the result

86 + 7 = 93 ✓

93 - 7 = 86
Final answer:

93 - 7 = 86

Applied rules:

• Compensation for Subtraction: Subtract more than needed, then add back

• Verification: Check by adding the answer to the subtracted number

• Mental Flexibility: Choose the easiest strategy for the numbers

Year 6 Mental Math: Advanced Learning Objectives and Progression
Counting On: 78 + 9 = 78 → 79 → 80 → 81 → 82 → 83 → 84 → 85 → 86 → 87
Strategy Overview
Key definitions:

Mental Math: Performing calculations in your head without paper or tools

Number Bonds: Pairs of numbers that add up to another number (like 7 and 3 bond to make 10)

Place Value: Understanding that digits represent different values based on their position

Commutative Property: Order doesn't matter in addition (a + b = b + a)

Associative Property: Grouping doesn't matter in addition ((a + b) + c = a + (b + c))

Regrouping: Converting 10 ones into 1 ten or vice versa

Estimation: Finding approximate answers to check reasonableness

Fact Families: Related addition and subtraction facts that help with mental math

Number Properties: Commutative, associative, and distributive properties that aid mental computation

Complete methodology:
  1. Analyze the numbers: Look for patterns, round numbers, or easy combinations
  2. Choose the strategy: Counting on, making tens, decomposing, or compensation
  3. Apply the method: Follow the specific steps for the chosen technique
  4. Handle regrouping: Convert 10 ones to 1 ten if needed
  5. Verify the result: Check your answer by using another method or reversing the operation
  6. Practice regularly: Repetition builds speed and accuracy
Tip 1: Practice number bonds to 10 and 20 daily (pairs that make 10 and 20).
Tip 2: When adding, start with the larger number to minimize counting.
Tip 3: Round difficult numbers to the nearest ten for easier calculations.
Tip 4: Always verify your mental math by checking with a different method.
Tip 5: Practice regularly to build speed and accuracy.
Tip 6: Use visual aids like number lines or drawings when first learning.
Tip 7: Look for doubles and near-doubles to help with addition.
Common errors: Forgetting to compensate properly, miscounting, confusing addition and subtraction steps, forgetting to regroup.
Key memorizations: Number bonds to 10 and 20, doubles facts, place value understanding, basic addition/subtraction facts.
Advanced techniques: Using known facts to derive unknown facts, looking for patterns, recognizing when to switch strategies mid-problem.
Mental Math Strategies Summary:

• Counting On: Start with larger number, count up by smaller number

• Making Tens: Break numbers to form a ten, then add remainder

• Decomposing: Separate into tens and ones, add separately

• Compensation: Round numbers to make calculation easier, adjust result

• Number Line: Visualize jumps forward or backward on a number line

• Regrouping: Convert 10 ones to 1 ten when necessary

Year 6 Mental Math: Advanced Strategy Effectiveness
Comparing Mental Math Approaches for 85 + 18
Different strategies for solving: 85 + 18
Counting On: 85 → 86 → 87... (18 counts)
Making Tens: 85 + 5 + 13 = 90 + 13 = 103
Decomposing: 80+5 + 10+8 = 90+13 = 103
Compensation: 87 + 16 = 103

Analysis: Different strategies offer varying levels of efficiency depending on the numbers involved.

  • Counting On: Simple but lengthy for larger additions
  • Making Tens: Efficient when one number is close to a ten
  • Decomposing: Systematic approach for any two-digit numbers
  • Compensation: Great for numbers near round figures

Year 6 Questions & Answers

Question: I don't understand when to use making tens versus decomposing numbers. Can you explain the difference?

Answer: Great question! The key difference is in how you manipulate the numbers:

  • Making Tens: You break apart ONE number to combine with the other to make a ten. Example: For 97 + 6, you take 3 from 6 to make 97 into 100, leaving 3. So it becomes 100 + 3 = 103.
  • Decomposing: You break apart BOTH numbers into tens and ones. Example: For 69 + 26, you break 69 into 60+9 and 26 into 20+6, then add 60+20=80 and 9+6=15, finally 80+15=95.

When to use which:

  • Use Making Tens when one number is close to a multiple of 10 (like 97, 59, 87)
  • Use Decomposing when neither number is particularly close to a round number

Both strategies aim to make the math easier by working with friendlier numbers!

Question: Why is it better to start counting on from the bigger number? Doesn't it matter which one I start with?

Answer: Yes, it definitely matters! Starting from the bigger number makes mental math much easier because you have fewer numbers to count.

Example: Let's say you want to solve 9 + 85.

  • If you start from 9 and count on 85: 10, 11, 12, 13... (85 counts!) - This is hard to keep track of!
  • If you start from 85 and count on 9: 86, 87, 88, 89, 90, 91, 92, 93, 94 - Much easier!

Starting with the larger number minimizes the amount of counting you need to do, which reduces the chance of making mistakes and helps you find the answer faster. It's all about efficiency in mental math!

Think of it like taking the shortest path - you want to do the least amount of work to get to your answer.

Question: My child struggles with the compensation method. How can I help them understand when to add versus subtract during compensation?

Answer: The compensation method can be tricky! Here's a simple rule to remember:

  • If you ADD to one number: SUBTRACT the same amount from the other number
  • If you SUBTRACT from one number: ADD the same amount to the other number

Example with addition: 89 + 8

  • I add 1 to 89 to make it 90
  • So I must subtract 1 from 8 to keep the total the same
  • Now I have 90 + 7 = 97

Example with subtraction: 93 - 7

  • I add 3 to 7 to make it 10 (easier to subtract)
  • So I must add 3 to 93 as well to keep the difference the same
  • Now I have 96 - 10 = 86

The key concept is keeping the equation balanced - whatever you do to one side, you must do to maintain equality.