Solved Exercises on Using Doubles and Near Doubles in Grade 2

Master using doubles and near doubles: doubles facts, near doubles strategy, addition patterns, mental math techniques through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Doubles facts
Exercise 1
Solve 6 + 6 using the doubles fact.
Definition:

Doubles facts: Adding the same number to itself (e.g., 6 + 6, 7 + 7).

Doubles method:
  1. Recognize that both addends are the same
  2. Use the doubles fact or multiply by 2
  3. Recall the memorized doubles fact
6
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Step 1: Recognize the doubles pattern

6 + 6 means the same number added to itself

Step 2: Apply the doubles rule

Any number added to itself is double that number

6 + 6 = 2 × 6 = 12

Step 3: Verify the result

Count: 6 + 6 = 12 ✓

Doubles fact: 6 + 6

Same number added to itself = double the number

6 + 6 = 12
Final answer:

6 + 6 = 12 using the doubles fact.

Applied rules:

• Doubles principle: n + n = 2n

• Memorization: Doubles facts should be memorized for fluency

• Pattern recognition: Same numbers added together

2 Near doubles strategy
Exercise 2
Solve 7 + 8 using the near doubles strategy.
Definition:

Near doubles: Adding numbers that are one apart (e.g., 7 + 8, 5 + 6).

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Step 1: Recognize near doubles pattern

7 + 8 are consecutive numbers (one apart)

Step 2: Use the doubles fact

7 + 7 = 14 (double 7)

Step 3: Adjust for the difference

Since 8 is 1 more than 7, add 1 more

14 + 1 = 15

Step 4: Write the final answer

7 + 8 = 15

Near doubles: 7 + 8

7 + 7 = 14, then add 1 more = 15

7 + 8 = 15
Final answer:

7 + 8 = 15 using the near doubles strategy.

Applied rules:

• Near doubles principle: n + (n+1) = (n + n) + 1

• Pattern building: Use known doubles to solve near doubles

• Adjustment strategy: Add or subtract the difference

3 Alternative near doubles
Exercise 3
Solve 8 + 7 using the near doubles strategy (alternative approach).
Definition:

Alternative approach: Using the larger number as the double and adjusting down.

8
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Step 1: Recognize near doubles pattern

8 + 7 are consecutive numbers (one apart)

Step 2: Use the larger number as the double

8 + 8 = 16 (double 8)

Step 3: Adjust for the difference

Since 7 is 1 less than 8, subtract 1

16 - 1 = 15

Step 4: Write the final answer

8 + 7 = 15

Alternative near doubles: 8 + 7

8 + 8 = 16, then subtract 1 = 15

8 + 7 = 15
Final answer:

8 + 7 = 15 using the alternative near doubles strategy.

Applied rules:

• Alternative principle: (n+1) + n = (n+1) + (n+1) - 1

• Flexibility: Either number can be used as the base double

• Adjustment direction: Add if smaller number, subtract if larger number

Doubles and Near Doubles Rules and Methods
6 + 6 = 12
Doubles Fact
Rule 1
Doubles: n + n = 2n
Add same number twice
Rule 2
Near Doubles: n + (n+1)
Use doubles + 1
Rule 3
Near Doubles: (n+1) + n
Use doubles - 1
Key definitions:

Doubles facts: Addition equations where both addends are the same (e.g., 5 + 5, 6 + 6)

Near doubles: Addition equations where addends are consecutive numbers (e.g., 5 + 6, 7 + 8)

Consecutive numbers: Numbers that follow each other in counting order (like 7 and 8)

Memorization: Learning facts automatically without calculation

Complete methodology:
  1. Identify doubles: Recognize when both addends are the same
  2. Identify near doubles: Recognize when addends are one apart
  3. Use doubles fact: Apply known doubles fact
  4. Adjust for difference: Add or subtract the difference
Tip 1: Practice doubles facts until they become automatic.
Tip 2: Use doubles to help solve near doubles problems.
Tip 3: Look for the pattern of numbers being one apart.
Tip 4: You can use either number as the base double.
Common errors: Forgetting doubles facts, confusing near doubles with other patterns, not adjusting properly for differences.
Memory aid: "Doubles are twins" and "Near doubles are neighbors."
Rules to remember:

• Step 1: Identify if it's a doubles or near doubles problem

• Step 2: Recall the relevant doubles fact

• Step 3: Adjust by adding or subtracting the difference

• Step 4: Practice regularly for fluency

Doubles Result Near Doubles Result
1 + 1 2 1 + 2 3
2 + 2 4 2 + 3 5
3 + 3 6 3 + 4 7
4 + 4 8 4 + 5 9
5 + 5 10 5 + 6 11
Solution: Exercises 4 to 5
4 Mental math application
Exercise 4
Solve 9 + 10 using the near doubles strategy mentally.
Definition:

Mental math: Solving problems in your head using strategies like doubles.

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Step 1: Recognize the near doubles pattern

9 + 10 are consecutive numbers (one apart)

Step 2: Think of the doubles fact

9 + 9 = 18 (I know this doubles fact)

Step 3: Adjust for the difference

10 is 1 more than 9, so add 1 to the doubles fact

18 + 1 = 19

Step 4: State the answer

9 + 10 = 19

Mental near doubles: 9 + 10

9 + 9 = 18, then add 1 more = 19

9 + 10 = 19
Final answer:

9 + 10 = 19 using mental math with near doubles strategy.

Applied rules:

• Mental calculation: Use known facts to solve in your head

• Efficiency: Doubles facts make mental math faster

• Pattern recognition: Identify consecutive numbers quickly

5 Word problem application
Exercise 5
Sarah has 7 red balloons and 8 blue balloons. How many balloons does she have in total? Use the near doubles strategy.
Definition:

Word problem: A real-world scenario requiring the use of doubles and near doubles strategies.

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Step 1: Identify the numbers in the problem

Sarah has 7 red balloons and 8 blue balloons

Need to find: 7 + 8

Step 2: Recognize the near doubles pattern

7 and 8 are consecutive numbers (one apart)

Step 3: Use the doubles fact

7 + 7 = 14 (known doubles fact)

Step 4: Adjust for the difference

8 is 1 more than 7, so add 1 more to 14

14 + 1 = 15

Step 5: Answer in context

Sarah has 15 balloons in total

Word problem: 7 + 8

7 + 7 = 14, then add 1 more = 15

7 + 8 = 15
Final answer:

Sarah has 15 balloons in total using the near doubles strategy.

Applied rules:

• Problem translation: Convert word problems to mathematical expressions

• Strategy application: Use appropriate doubles/near doubles method

• Context answer: Include units and context in final answer

Doubles and Near Doubles Laws, Methods, and Definitions
n + n = 2n
Doubles Formula
Key definitions:

Doubles facts: Addition equations where both addends are identical (e.g., 4 + 4, 6 + 6, 8 + 8)

Near doubles: Addition equations where addends are consecutive integers (e.g., 4 + 5, 6 + 7, 8 + 9)

Consecutive numbers: Numbers that follow each other in the counting sequence (differ by 1)

Memorization: Learning basic facts automatically without computation

Mental math: Calculating in your head using known patterns and strategies

Pattern recognition: Identifying mathematical relationships and structures

Complete methodology:
  1. Pattern identification: Recognize doubles and near doubles situations
  2. Doubles recall: Retrieve memorized doubles facts
  3. Adjustment calculation: Add or subtract the difference for near doubles
  4. Verification: Check the result using alternative methods
  5. Application: Use in word problems and mental math
Tip 1: Practice doubles facts daily until they become automatic.
Tip 2: Use near doubles when you see numbers that are one apart.
Tip 3: You can use either number as the base double in near doubles.
Tip 4: Look for the "neighbor" pattern in consecutive numbers.
Common errors: Forgetting doubles facts, confusing near doubles with other patterns, incorrect adjustment direction.
Memory aids: "Doubles are twins" and "Near doubles are neighbors" help pattern recognition.
Essential laws and rules:

• Doubles law: n + n = 2n (adding the same number twice)

• Near doubles law: n + (n+1) = (n + n) + 1

• Alternative near doubles: (n+1) + n = (n+1) + (n+1) - 1

• Memorization principle: Doubles facts should be automatic

• Adjustment rule: Add if smaller number, subtract if larger number

8 + 9 = (8 + 8) + 1 = 16 + 1 = 17
Near Doubles Example
Visual representation: Using objects or drawings helps understand the doubling concept.
Real-world connection: Doubles appear in pairs like shoes, eyes, ears.

Questions & Answers

Question: Why do we need to memorize doubles facts? Can't I just count to find the answer?

Answer: You can count, but memorizing doubles facts is much faster:

  • Counting: For 6 + 6, you'd count 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
  • Memorized: 6 + 6 = 12 instantly!

Memorizing doubles facts:

  • Makes math faster and easier
  • Helps with near doubles problems
  • Built confidence in math abilities
  • Prepares you for harder math later

It's like knowing your address - you don't have to figure it out every time!

Question: What's a near double? How do I know if numbers are near doubles?

Answer: Near doubles are numbers that are neighbors - they are only 1 apart:

  • Examples: 5 + 6, 7 + 8, 9 + 10, 3 + 4
  • Not near doubles: 5 + 7 (2 apart), 3 + 8 (5 apart)

To identify near doubles:

  • Look for consecutive numbers (like 6 and 7)
  • Check if they differ by only 1
  • Think: "Are these neighbors on the number line?"

Near doubles are like having one more or one less than a doubles fact!

Question: My child gets confused about whether to add or subtract when using near doubles. How can I help?

Answer: The key is to think about which number is closer to the doubles fact:

  • For 7 + 8: Use 7 + 7 = 14, then add 1 more (because 8 is 1 more than 7)
  • For 8 + 7: Use 7 + 7 = 14, then add 1 more (because 8 is 1 more than 7)

Help your child remember:

  • Look at the smaller number for the doubles fact
  • Figure out how much more the larger number is
  • Add that amount to the doubles fact

Or use the larger number as the base and subtract the difference.

Question: Can I use either number as the base double in near doubles? Like for 8 + 9?

Answer: Yes, you can use either number as the base double! Both ways work:

  • Method 1: 8 + 8 = 16, then add 1 more (for the 9) = 17
  • Method 2: 9 + 9 = 18, then subtract 1 (because 8 is 1 less than 9) = 17

Both give the same answer: 8 + 9 = 17

Some people prefer using the smaller number as the base, others prefer the larger number. Choose whichever feels easier to you!

The important thing is to adjust in the right direction!

Question: Why do we learn doubles and near doubles? When will I use them?

Answer: Doubles and near doubles are super useful for many reasons:

  • Fast math: They help you solve problems quickly in your head
  • Harder problems: They help with bigger numbers later (like 25 + 26)
  • Real life: Shopping, cooking, sports scores often involve these patterns
  • Confidence: Knowing facts makes math feel easier and more fun

Think of them like shortcuts - they help you get to the answer faster!

The more you practice, the more automatic and helpful they become.