Complete Mental Math Practice: Addition and Subtraction Strategies

Comprehensive practice covering all mental math strategies: counting on, making tens, decomposing numbers, compensation method, and number line strategies.

Practice Sessions: Problems 1 to 3
1 Counting On Practice
Problem 1: 34 + 7
Solve mentally using the counting on strategy.
Show your thinking process step by step.
Definition:

Counting On Strategy: Start with the larger number and count up by the smaller number. This is the most efficient method when adding small numbers (1, 2, 3, 4, or 5) and helps develop number sense.

Step-by-step method:
  1. Identify the larger number (34) and the smaller number (7)
  2. Start counting from the larger number: 34
  3. Count up by the smaller number: 35, 36, 37, 38, 39, 40, 41
  4. The final number is the answer: 41
Start
34
Count Up
+7
Result
41
Step 1: Identify numbers

34 is the larger number, 7 is the smaller number

Step 2: Start with larger number

Start counting from 34

Step 3: Count up by smaller number

34 → 35 → 36 → 37 → 38 → 39 → 40 → 41 (count 7 numbers)

Step 4: Record the final number

The last number counted is the answer: 41

34 + 7 = 41
Final answer:

34 + 7 = 41

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 1-5

2 Making Tens Practice
Problem 2: 48 + 6
Solve mentally using the making tens strategy.
Show how you break apart the numbers and explain the process.
Definition:

Making Tens Strategy: Break apart one number to make a ten with the other number, then add the remaining amount. This is especially useful when one number is close to a multiple of 10.

Original
48 + 6
Make Ten
48 + 2 + 4
Simplify
50 + 4 = 54
Step 1: Identify how many needed to make a ten

From 48, we need 2 more to reach 50 (the next ten)

Step 2: Break apart the second number

Break 6 into 2 + 4 (so we can make 10 with 48)

Step 3: Add to make the ten

48 + 2 = 50

Step 4: Add the remaining amount

50 + 4 = 54

48 + 6 = 54
Final answer:

48 + 6 = 54

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 Decomposing Numbers Practice
Problem 3: 53 + 28
Solve mentally using the decomposing numbers strategy.
Show how you separate tens and ones.
Definition:

Decomposing Numbers: Break apart both numbers into tens and ones, add the tens together, add the ones together, then combine the results. This strategy works well for any two-digit addition.

Original
53 + 28
Decompose
50+3 + 20+8
Result
70+11 = 81
Step 1: Decompose each number

53 = 50 + 3 (5 tens and 3 ones)

28 = 20 + 8 (2 tens and 8 ones)

Step 2: Add the tens together

50 + 20 = 70

Step 3: Add the ones together

3 + 8 = 11

Step 4: Combine the sums

70 + 11 = 81

53 + 28 = 81
Final answer:

53 + 28 = 81

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

Mental Math Practice Overview
Counting On: Start with larger number
Strategy 1
Making Tens
48 + 6 = 48 + 2 + 4 = 50 + 4
Break apart to make 10
Decomposing
53 + 28 = 50+3 + 20+8 = 70+11
Separate tens and ones
Compensation
67 + 9 = 70 + 6 = 76
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)

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. Combine results: Put the pieces back together
  5. 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 (like 7 needs 3 to make 10).
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: Practice regularly to build speed and accuracy.
Common errors: Forgetting to count the starting number, miscounting, adding instead of subtracting when compensating.
Practice areas: Number bonds to 10, basic addition facts, place value understanding.
Memorization tips: Remember that addition is commutative (order doesn't matter).
Practice Sessions: Problems 4 to 5
4 Compensation Method Practice
Problem 4: 67 + 8
Solve mentally using the compensation method.
Show how you round and adjust the numbers.
Definition:

Compensation Method: Round one number to make it easier to work with, then adjust the answer by the amount you changed. This is helpful when one number is close to a round number.

Original
67 + 8
Round Up
70 + 5
Result
75
Step 1: Look for numbers near tens

67 is close to 70 (3 away)

Step 2: Round one number up

Add 3 to 67 to make it 70

Step 3: Adjust the other number

Subtract 3 from 8 to keep the total the same: 8 - 3 = 5

Step 4: Add the rounded numbers

70 + 5 = 75

67 + 8 = 75
Final answer:

67 + 8 = 75

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 Mental Subtraction Practice
Problem 5: 72 - 6
Solve mentally using the counting back strategy.
Show how you make subtraction easier mentally.
Definition:

Mental Subtraction: Use counting back or compensation to subtract without paper. Students learn multiple approaches for different situations.

Original
72 - 6
Count Back
72 - 6 = 66
Result
66
Step 1: Start with the larger number

Begin with 72

Step 2: Count back by the smaller number

Count back 6 numbers: 71, 70, 69, 68, 67, 66

Step 3: Record the final number

The number you land on is the answer: 66

Step 4: Verify the result

66 + 6 = 72 ✓

72 - 6 = 66
Final answer:

72 - 6 = 66

Applied rules:

• Counting Back: Move backwards on the number line

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

• Mental Flexibility: Choose the easiest strategy for the numbers

Mental Math Practice Guidelines
Counting On: 34 + 7 = 34 → 35 → 36 → 37 → 38 → 39 → 40 → 41
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

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. Verify the result: Check your answer by using another method or reversing the operation
  5. Practice regularly: Repetition builds speed and accuracy
Tip 1: Practice number bonds to 10 daily (pairs that make 10).
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.
Common errors: Forgetting to count the starting number, miscounting, confusing addition and subtraction steps.
Practice areas: Number bonds to 10, basic addition facts, place value understanding.
Memorization tips: Remember that addition is commutative (order doesn't matter).
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

Mental Math: Strategy Effectiveness
Comparing Mental Math Approaches for 54 + 19
Different strategies for solving: 54 + 19
Counting On: 54 → 55 → 56... (19 counts)
Making Tens: 54 + 6 + 13 = 60 + 13 = 73
Decomposing: 50+4 + 10+9 = 60+13 = 73
Compensation: 55 + 18 = 73

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

Mental Math 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 48 + 7, you take 2 from 7 to make 48 into 50, leaving 5. So it becomes 50 + 5 = 55.
  • Decomposing: You break apart BOTH numbers into tens and ones. Example: For 53 + 28, you break 53 into 50+3 and 28 into 20+8, then add 50+20=70 and 3+8=11, finally 70+11=81.

When to use which:

  • Use Making Tens when one number is close to a multiple of 10 (like 48, 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 8 + 34.

  • If you start from 8 and count on 34: 9, 10, 11, 12... (34 counts!) - This is hard to keep track of!
  • If you start from 34 and count on 8: 35, 36, 37, 38, 39, 40, 41, 42 - 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: 67 + 8

  • I add 3 to 67 to make it 70
  • So I must subtract 3 from 8 to keep the total the same
  • Now I have 70 + 5 = 75

Example with subtraction: 72 - 6

  • I can count back 6 from 72: 71, 70, 69, 68, 67, 66
  • So 72 - 6 = 66

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