Fluency and Strategies Within 20: Detailed Exercises and Solutions

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

Solution: Exercises 1 to 3
1 Addition with Regrouping
Exercise 1
Solve: 17 + 6 using regrouping strategy
Problem Statement: Find the sum of 17 and 6 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 (17) 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
17 + 6
Break Apart
17 + (3 + 3)
Make Ten
(17 + 3) + 3 = 20 + 3
Final Sum
23
Step 1: Analyze the problem

We have 17 + 6. The larger number is 17, which needs 3 more to reach 20.

Step 2: Break apart the smaller number

Split 6 into 3 + 3, where 3 goes to make 20 with 17.

Step 3: Make ten and add remainder

17 + 3 = 20, then add the remaining 3: 20 + 3 = 23

Step 4: Verify the calculation

Counting: 17, 18, 19, 20, 21, 22, 23. Yes, 17 + 6 = 23

17 + 6 = 23
Final answer:

17 + 6 = 23

Applied rules:

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

• Associative Property: Grouping doesn't change the sum (17 + 6 = 17 + (3 + 3) = (17 + 3) + 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 17, it's 3 (since 17 + 3 = 20).
2 Subtraction with Decomposition
Exercise 2
Solve: 15 - 8 using decomposition strategy
Problem Statement: Find the difference between 15 and 8 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
15 - 8
Break Apart
15 - (5 + 3)
Go Through Ten
(15 - 5) - 3 = 10 - 3
Final Difference
7
Step 1: Analyze the problem

We have 15 - 8. Start with 15 and determine how to subtract 8 efficiently.

Step 2: Break apart the subtrahend

Split 8 into 5 + 3, where 5 is subtracted first to reach 10.

Step 3: Subtract to reach landmark number

15 - 5 = 10, then subtract the remaining 3: 10 - 3 = 7

Step 4: Verify the calculation

Counting backwards: 15, 14, 13, 12, 11, 10, 9, 8. That's 7 steps backward from 15 to 8.

15 - 8 = 7
Final answer:

15 - 8 = 7

Applied rules:

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

• Subtraction Property: Subtracting a sum equals subtracting each addend (15 - 8 = 15 - (5 + 3) = (15 - 5) - 3)

• 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 15 - 8: subtract 5 to get to 10, then subtract 3 more.
3 Number Bonds and Fact Families
Exercise 3
Complete the fact family for 7, 9, and 16
Problem Statement: Write all four related addition and subtraction facts for the numbers 7, 9, and 16.
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
7, 9, 16
Addition Facts
7 + 9 = 16
9 + 7 = 16
Subtraction Facts
16 - 7 = 9
16 - 9 = 7
Step 1: Verify the relationship

Check that 7 + 9 = 16 (which is true: 7 + 9 = 16)

Step 2: Write the addition facts

Using the commutative property: 7 + 9 = 16 and 9 + 7 = 16

Step 3: Write the subtraction facts

From the total (16), subtract each part: 16 - 7 = 9 and 16 - 9 = 7

Step 4: Verify all relationships

All four facts are consistent: 7 + 9 = 16, 9 + 7 = 16, 16 - 7 = 9, 16 - 9 = 7

7 + 9 = 16
9 + 7 = 16
16 - 7 = 9
16 - 9 = 7
Final answer:

Fact family: 7 + 9 = 16, 9 + 7 = 16, 16 - 7 = 9, 16 - 9 = 7

Applied rules:

• Commutative Property: Order doesn't matter in addition (7 + 9 = 9 + 7)

• 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."
Key Concepts and Strategies
Making Ten: 8 + 5 = 8 + (2 + 3) = (8 + 2) + 3 = 10 + 3 = 13
Regrouping Strategy
Strategy 1
Making Ten
Add or subtract to reach a multiple of 10 first
Strategy 2
Decomposition
Break apart numbers to simplify calculations
Strategy 3
Fact Families
Use relationships between addition and subtraction
Key Definitions:

Fluency: The ability to perform calculations accurately, efficiently, and flexibly

Number Bonds: Pairs of numbers that combine to make a specific total

Landmark Numbers: Numbers that are easy to work with, especially multiples of 10

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
Tip 1: Practice number bonds for 10: (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.
Solution: Exercises 4 to 5
4 Mixed Operations Word Problem
Exercise 4
Sarah has 14 stickers. She gives 6 to her friend and then finds 5 more. How many stickers does she 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
14 stickers
Give Away
14 - 6 = 8
Find More
8 + 5 = 13
Step 1: Identify the starting amount

Sarah starts with 14 stickers.

Step 2: Identify the first operation

She gives 6 to her friend: 14 - 6. Use decomposition strategy: 14 - 6 = 14 - (4 + 2) = (14 - 4) - 2 = 10 - 2 = 8

Step 3: Identify the second operation

She finds 5 more: 8 + 5. Use regrouping strategy: 8 + 5 = 8 + (2 + 3) = (8 + 2) + 3 = 10 + 3 = 13

Step 4: Verify the solution

14 - 6 + 5 = 8 + 5 = 13. Check: Does this make sense? Yes, she lost some but gained others, ending with fewer than she started.

Sarah has 13 stickers now
Final answer:

Sarah has 13 stickers now.

Applied rules:

• Order of Operations: Perform operations from left to right

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

• Mental Math Strategies: Use decomposition and regrouping for efficient computation

Tip: Circle key numbers and underline action words ("gives" means subtract, "finds" means add).
5 Equation Solving
Exercise 5
Solve for the missing number: 13 - ? = 8
Problem Statement: Find the unknown subtrahend in a subtraction equation.
Definition:

Unknown Subtrahend: In an equation of the form minuend - ? = difference, the unknown number being subtracted.

Equation
13 - ? = 8
Rewrite
? = 13 - 8
Solve
? = 5
Step 1: Understand the equation

We need to find what number subtracted from 13 gives 8.

Step 2: Use inverse operations

To find the missing subtrahend, subtract the difference from the minuend: ? = 13 - 8

Step 3: Solve using decomposition

13 - 8: Break apart 8 into 3 + 5. So 13 - 8 = 13 - (3 + 5) = (13 - 3) - 5 = 10 - 5 = 5

Step 4: Verify the solution

Check: 13 - 5 = 8 ✓. This confirms our answer is correct.

13 - 5 = 8
Final answer:

The missing number is 5.

Applied rules:

• Inverse Operations: Addition and subtraction undo each other

• Equation Solving: To find a missing subtrahend, subtract the difference from the minuend

• Verification: Always substitute back to verify the solution

Tip: Remember: Minuend - Subtrahend = Difference, so Subtrahend = Minuend - Difference.
Comprehensive Guide: Laws, Methods, and Strategies
a + b = c ⟺ c - a = b and c - b = a
Inverse Relationship
Key definitions:

Addition: Combining two or more quantities to find a total

Subtraction: Finding the difference between two quantities or finding a missing part

Regrouping: Breaking apart numbers to make calculations easier, often by making tens

Complete methodology:
  1. Understand the problem: Identify what operation is needed and the numbers involved
  2. Choose the strategy: Select the most efficient method (counting on, making ten, decomposition, etc.)
  3. Execute the calculation: Apply the chosen strategy step by step
  4. Verify the result: Check using inverse operations or estimation
Tip 1: Always look for ways to make tens when adding or subtracting.
Tip 2: Use known facts like doubles and near-doubles to build new knowledge.
Tip 3: Practice number bonds regularly to build automaticity.
Tip 4: Use manipulatives or drawings when first learning a strategy.
Common errors: Forgetting to regroup properly, misapplying inverse operations, not checking answers.
Fluency building: Practice daily with varied problems, focus on accuracy first then speed.
Properties to know:

• Commutative Property: a + b = b + a

• Associative Property: (a + b) + c = a + (b + c)

• Identity Property: a + 0 = a

• Inverse Operations: a + b = c ⟺ c - b = a

Visual Learning: Number Relationships
Exercise 6: Number Bond Visualization
Understanding number bonds to 20 through visual representation:
13 = 10 + 3
17 = 10 + 7
19 = 10 + 9

Analysis: Visualizing how numbers break down into tens and ones helps with mental math strategies.

  • Breaking into tens makes addition/subtraction easier
  • Recognizing patterns builds fluency
  • Visual models support conceptual understanding

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.