Fluency for Addition and Subtraction Within 20: Comprehensive Guide and Exercises

Master addition and subtraction within 20: mental math strategies, number bonds, regrouping, and problem-solving methods through these 5 detailed exercises.

Comprehensive Summary: Fluency for Addition and Subtraction Within 20

Key Concepts and Definitions

Fluency: The ability to perform mathematical operations quickly, accurately, and flexibly. In the context of addition and subtraction within 20, fluency means being able to solve problems mentally without relying on finger counting or other physical aids.
Number Bonds
Pairs of numbers that combine to make a specific total. For example, number bonds for 10: 1+9, 2+8, 3+7, 4+6, 5+5.
Regrouping Strategy
Breaking apart numbers to make calculations easier, often by making tens. For example, 8 + 7 = 8 + (2 + 5) = (8 + 2) + 5 = 15.
Fact Families
Sets of related addition and subtraction facts that use the same three numbers. Example: 7, 8, 15 → 7+8=15, 8+7=15, 15-7=8, 15-8=7.
Landmark Numbers
Numbers that are easy to work with, especially multiples of 10. Using landmark numbers makes mental math more efficient.

Core Rules and Principles

  • Commutative Property: a + b = b + a (order doesn't matter in addition)
  • Associative Property: (a + b) + c = a + (b + c) (grouping doesn't change the sum)
  • Inverse Operations: Addition and subtraction are opposites (a + b = c ⟺ c - b = a)
  • Identity Property: a + 0 = a (adding zero doesn't change the number)
  • Compensation Strategy: Adjusting one number to make the calculation easier (e.g., 19 + 8 = 20 + 8 - 1)

Step-by-Step Methods and Procedures

Mental Math Strategies:
  1. Counting On: Starting from the larger number and counting forward
  2. Counting Back: Starting from the larger number and counting backward
  3. Using Doubles: Leveraging known doubles facts (e.g., 7 + 7 = 14)
  4. Near Doubles: Using doubles and adjusting (e.g., 7 + 8 = 7 + 7 + 1 = 15)
  5. Compensation: Adjusting one number to make the calculation easier
  6. Decomposition: Breaking apart numbers to make calculations simpler

Examples and Applications

Simple Example: 6 + 7

Using near doubles: 6 + 7 = 6 + 6 + 1 = 12 + 1 = 13

Intermediate Example: 15 - 8

Using decomposition: 15 - 8 = 15 - (5 + 3) = (15 - 5) - 3 = 10 - 3 = 7

Advanced Example: 14 + 8

Using regrouping: 14 + 8 = 14 + (6 + 2) = (14 + 6) + 2 = 20 + 2 = 22

Tips, Tricks, and Memory Aids

Tip 1: Practice number bonds for 10 regularly: (1+9), (2+8), (3+7), (4+6), (5+5).
Tip 2: Use doubles facts as anchor points for near-doubles (e.g., 6 + 7 = 6 + 6 + 1).
Tip 3: When subtracting, think "How many more?" rather than just taking away.
Tip 4: Always verify your answer by using the inverse operation.
Tip 5: Look for ways to make tens when adding or subtracting.
Tip 6: Use compensation: adjust one number to make the calculation easier (e.g., 19 + 8 = 20 + 8 - 1).
Solution: Exercises 1 to 3
1 Addition with Regrouping
Exercise 1
Solve: 16 + 7 using regrouping strategy
Problem Statement: Find the sum of 16 and 7 by breaking apart numbers to make ten.
Definition:

Regrouping Strategy: Breaking apart numbers to make ten, then adding the remainder. Also known as "making ten" or "bridging through ten".

Addition Method:
  1. Identify the larger number (16) and see how much is needed to reach the next ten
  2. Break apart the smaller number accordingly
  3. Add to make ten, then add the remaining amount
  4. Combine to find the final sum
Expression
16 + 7
Break Apart
16 + (4 + 3)
Make Ten
(16 + 4) + 3 = 20 + 3
Final Sum
23
Step 1: Analyze the problem

We have 16 + 7. The larger number is 16, which needs 4 more to reach 20.

Step 2: Break apart the smaller number

Split 7 into 4 + 3, where 4 goes to make 20 with 16.

Step 3: Make ten and add remainder

16 + 4 = 20, then add the remaining 3: 20 + 3 = 23

Step 4: Verify the calculation

Counting: 16, 17, 18, 19, 20, 21, 22, 23. Yes, 16 + 7 = 23

16 + 7 = 23
Final answer:

16 + 7 = 23

Applied rules:

• Regrouping: Break apart one number to make ten with another

• Associative Property: Grouping doesn't change the sum (16 + 7 = 16 + (4 + 3) = (16 + 4) + 3)

• Mental Math: Use known facts (like making tens) to solve problems efficiently

Tip: Always identify what number added to the larger number makes the next ten. For 16, it's 4 (since 16 + 4 = 20).
2 Subtraction with Decomposition
Exercise 2
Solve: 14 - 6 using decomposition strategy
Problem Statement: Find the difference between 14 and 6 by decomposing the subtrahend.
Definition:

Decomposition Strategy: Breaking apart the number being subtracted to make the calculation easier by going through a landmark number (usually ten).

Expression
14 - 6
Break Apart
14 - (4 + 2)
Go Through Ten
(14 - 4) - 2 = 10 - 2
Final Difference
8
Step 1: Analyze the problem

We have 14 - 6. Start with 14 and determine how to subtract 6 efficiently.

Step 2: Break apart the subtrahend

Split 6 into 4 + 2, where 4 is subtracted first to reach 10.

Step 3: Subtract to reach landmark number

14 - 4 = 10, then subtract the remaining 2: 10 - 2 = 8

Step 4: Verify the calculation

Counting backwards: 14, 13, 12, 11, 10, 9, 8. That's 6 steps backward from 14 to 8.

14 - 6 = 8
Final answer:

14 - 6 = 8

Applied rules:

• Decomposition: Break apart the number being subtracted to make tens

• Subtraction Property: Subtracting a sum equals subtracting each addend (14 - 6 = 14 - (4 + 2) = (14 - 4) - 2)

• Mental Math: Use known facts (like counting to tens) to solve problems efficiently

Tip: Think about how many you need to subtract to get to 10, then subtract the rest. For 14 - 6: subtract 4 to get to 10, then subtract 2 more.
3 Number Bonds and Fact Families
Exercise 3
Complete the fact family for 8, 5, and 13
Problem Statement: Write all four related addition and subtraction facts for the numbers 8, 5, and 13.
Definition:

Fact Family: A set of related addition and subtraction facts that use the same three numbers. It demonstrates the inverse relationship between addition and subtraction.

Numbers
8, 5, 13
Addition Facts
8 + 5 = 13
5 + 8 = 13
Subtraction Facts
13 - 8 = 5
13 - 5 = 8
Step 1: Verify the relationship

Check that 8 + 5 = 13 (which is true: 8 + 5 = 13)

Step 2: Write the addition facts

Using the commutative property: 8 + 5 = 13 and 5 + 8 = 13

Step 3: Write the subtraction facts

From the total (13), subtract each part: 13 - 8 = 5 and 13 - 5 = 8

Step 4: Verify all relationships

All four facts are consistent: 8 + 5 = 13, 5 + 8 = 13, 13 - 8 = 5, 13 - 5 = 8

8 + 5 = 13
5 + 8 = 13
13 - 8 = 5
13 - 5 = 8
Final answer:

Fact family: 8 + 5 = 13, 5 + 8 = 13, 13 - 8 = 5, 13 - 5 = 8

Applied rules:

• Commutative Property: Order doesn't matter in addition (8 + 5 = 5 + 8)

• Inverse Operations: Addition and subtraction are opposite operations

• Part-Whole Relationship: Two parts combine to make a whole, and the whole minus a part gives the other part

Tip: Remember the "part-whole" concept: the largest number is the "whole," and the other two are "parts."
Solution: Exercises 4 to 5
4 Mixed Operations Word Problem
Exercise 4
Tom has 12 marbles. He finds 5 more marbles and then gives 3 to his friend. How many marbles does he have now?
Problem Statement: Solve a two-step word problem involving addition and subtraction within 20.
Definition:

Word Problem: A mathematical problem presented in a narrative format requiring identification of operations and numbers.

Start
12 marbles
Find More
12 + 5 = 17
Give Away
17 - 3 = 14
Step 1: Identify the starting amount

Tom starts with 12 marbles.

Step 2: Identify the first operation

He finds 5 more: 12 + 5. Use regrouping strategy: 12 + 5 = 12 + (5) = 17

Step 3: Identify the second operation

He gives 3 to his friend: 17 - 3. Use counting back: 17 - 3 = 14

Step 4: Verify the solution

12 + 5 - 3 = 17 - 3 = 14. Check: Does this make sense? Yes, he gained some and lost some, ending with more than he started.

Tom has 14 marbles now
Final answer:

Tom has 14 marbles now.

Applied rules:

• Order of Operations: Perform operations from left to right

• Problem-Solving Steps: Understand, Plan, Solve, Check

• Mental Math Strategies: Use regrouping and counting strategies for efficient computation

Tip: Circle key numbers and underline action words ("finds" means add, "gives" means subtract).
5 Equation Solving
Exercise 5
Solve for the missing number: 11 + ? = 18
Problem Statement: Find the unknown addend in an addition equation.
Definition:

Unknown Addend: In an equation of the form a + ? = c, the unknown number being added.

Equation
11 + ? = 18
Rewrite
? = 18 - 11
Solve
? = 7
Step 1: Understand the equation

We need to find what number added to 11 gives 18.

Step 2: Use inverse operations

To find the missing addend, subtract the known addend from the sum: ? = 18 - 11

Step 3: Solve using decomposition

18 - 11: Break apart 11 into 1 + 10. So 18 - 11 = 18 - (1 + 10) = (18 - 10) - 1 = 8 - 1 = 7

Step 4: Verify the solution

Check: 11 + 7 = 18 ✓. This confirms our answer is correct.

11 + 7 = 18
Final answer:

The missing number is 7.

Applied rules:

• Inverse Operations: Addition and subtraction undo each other

• Equation Solving: To find a missing addend, subtract the known addend from the sum

• Verification: Always substitute back to verify the solution

Tip: Remember: Addend + Addend = Sum, so Missing Addend = Sum - Known Addend.

Questions & Answers

Question: My students struggle with the "making ten" strategy. They don't understand why we break apart numbers. Can you explain the benefit?

Answer: The "making ten" strategy is crucial because our number system is based on tens. When students make ten, they're using a familiar landmark number that's easy to work with mentally.

Benefits of making ten:

  • Tens are easier to add and subtract mentally
  • Students can build on known facts (like 10 + 3 = 13)
  • It develops number sense and flexibility in thinking
  • Prepares students for more complex addition with larger numbers

For example, with 8 + 6: Instead of counting on 6 from 8, students make 8 + 2 = 10, then add the remaining 4 to get 14. This is faster and more accurate than counting.

Question: My child gets confused when solving word problems. How can I help them identify whether to add or subtract?

Answer: Help your child look for clue words and visualize the situation:

  • Addition clues: "total," "altogether," "combined," "more," "joined," "in all"
  • Subtraction clues: "left," "remaining," "difference," "less," "take away," "gave away"

More importantly, encourage them to ask: "Am I putting things together or taking them apart?" This conceptual understanding is more important than just looking for keywords.

Draw pictures or use manipulatives to model the problem before attempting to solve it numerically.

Question: How can I help students develop fluency without just drilling basic facts?

Answer: Focus on building number sense through meaningful activities:

  • Use number bonds and fact families to show relationships
  • Practice with games and puzzles that require strategic thinking
  • Encourage students to explain their thinking and share strategies
  • Connect to real-world situations regularly

Fluency comes from understanding relationships between numbers, not just memorization. When students understand that 7 + 8 can be thought of as 7 + 7 + 1 = 14 + 1 = 15, they're developing flexible thinking that transfers to new problems.