Solved Exercises on Strategies Based on Place Value and Properties of Operations in Grade 2

Master strategies based on place value and properties of operations: commutative, associative, and distributive properties combined with place value for efficient addition and subtraction through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Commutative Property with Place Value
Exercise 1
Show that 34 + 27 = 27 + 34 using strategies based on place value and properties of operations
Definition:

Commutative Property: Changing the order of addends does not change the sum (a + b = b + a).

Commutative Property Method:
  1. Decompose both numbers into tens and ones
  2. Add using place value strategy
  3. Reverse the order and add again
  4. Verify the sums are equal
First: 34 + 27
30+4 + 20+7 = 50+11=61
Second: 27 + 34
20+7 + 30+4 = 50+11=61
Result
61 = 61
Step 1: Decompose 34 and 27 into tens and ones

34 = 30 + 4 and 27 = 20 + 7

Step 2: Add using place value strategy

(30 + 20) + (4 + 7) = 50 + 11 = 61

Step 3: Reverse the order and decompose again

27 = 20 + 7 and 34 = 30 + 4

Step 4: Add in reversed order using place value

(20 + 30) + (7 + 4) = 50 + 11 = 61

34 + 27 = 27 + 34 = 61
Final answer:

34 + 27 = 27 + 34 = 61

Applied rules:

• Commutative Property: Order of addends doesn't affect the sum

• Place Value Addition: Adding digits with the same place value

• Associative Property: Grouping doesn't affect the sum

2 Associative Property with Place Value
Exercise 2
Show that (23 + 15) + 12 = 23 + (15 + 12) using strategies based on place value and properties of operations
Definition:

Associative Property: Grouping of addends does not change the sum ((a + b) + c = a + (b + c)).

Left side: (23 + 15) + 12
38 + 12 = 50
Right side: 23 + (15 + 12)
23 + 27 = 50
Result
50 = 50
Step 1: Calculate the left side (23 + 15) + 12

First: 23 + 15 = (20 + 3) + (10 + 5) = 30 + 8 = 38

Step 2: Add 38 + 12

38 + 12 = (30 + 8) + (10 + 2) = 40 + 10 = 50

Step 3: Calculate the right side 23 + (15 + 12)

First: 15 + 12 = (10 + 5) + (10 + 2) = 20 + 7 = 27

Step 4: Add 23 + 27

23 + 27 = (20 + 3) + (20 + 7) = 40 + 10 = 50

(23 + 15) + 12 = 23 + (15 + 12) = 50
Final answer:

(23 + 15) + 12 = 23 + (15 + 12) = 50

Applied rules:

• Associative Property: Grouping of addends doesn't affect the sum

• Place Value Addition: Adding digits with the same place value

• Regrouping: When ones sum to 10+, convert to tens and ones

3 Properties with Subtraction
Exercise 3
Solve 78 - 34 using strategies based on place value and properties of operations
Definition:

Subtraction with Properties: Using place value and number relationships to simplify subtraction operations.

Problem
78 - 34
Decompose
(70+8)-(30+4)
Subtract by place
40+4=44
Step 1: Decompose both numbers into tens and ones

78 = 70 + 8 and 34 = 30 + 4

Step 2: Subtract the tens

70 - 30 = 40

Step 3: Subtract the ones

8 - 4 = 4

Step 4: Combine the differences

40 + 4 = 44

78 - 34 = 44
Final answer:

78 - 34 = 44

Applied rules:

• Place Value Subtraction: Subtract digits with the same place value

• Number Composition: Reassemble the final difference from its components

• Regrouping: Apply when the top digit is smaller than the bottom digit

Rules and methods, laws,...
34 + 27 = 27 + 34 (commutative property)
Commutative Property
Property 1
Commutative
a + b = b + a
Property 2
Associative
(a + b) + c = a + (b + c)
Property 3
Place Value
Working with same place values
Key definitions:

Commutative Property: Changing the order of addends does not change the sum (a + b = b + a)

Associative Property: Grouping of addends does not change the sum ((a + b) + c = a + (b + c))

Place Value: The value of a digit based on its position in a number (ones, tens, hundreds)

Properties of Operations: Rules that describe how numbers behave when added or subtracted

Complete methodology:
  1. Identify the operation: Determine if addition or subtraction
  2. Recognize applicable properties: Identify which property can be used
  3. Apply place value strategy: Decompose numbers into tens and ones
  4. Perform operations: Add or subtract digits in the same place
  5. Combine results: Put the tens and ones back together
Tip 1: Use the commutative property to rearrange numbers for easier computation.
Tip 2: Use the associative property to group numbers that add to round numbers.
Tip 3: Always decompose numbers to see tens and ones clearly.
Tip 4: Check your answer by using the inverse operation.
Common errors: Forgetting that subtraction is not commutative, misapplying properties to subtraction, adding tens and ones together before completing operations.
Key strategies: Practice identifying which properties apply, visualize with manipulatives, check answers with inverse operations.
Properties of operations:

• Addition is commutative: a + b = b + a

• Addition is associative: (a + b) + c = a + (b + c)

• Subtraction is NOT commutative: a - b ≠ b - a

• Subtraction is NOT associative: (a - b) - c ≠ a - (b - c)

• Place value helps decompose numbers for easier computation

Solution: Exercises 4 to 5
4 Mixed Operations with Properties
Exercise 4
Solve using strategies based on place value and properties of operations: 45 + 28 - 17
Definition:

Mixed Operations with Properties: Perform operations sequentially using place value methods and properties of operations.

Problem
45 + 28 - 17
First: 45 + 28
70 + 13 = 83
Then: 83 - 17
70 + 13 - 17 = 66
Step 1: First perform 45 + 28 using place value

45 = 40 + 5, 28 = 20 + 8, so (40 + 20) + (5 + 8) = 60 + 13 = 73

Step 2: Now perform 73 - 17 using place value

73 = 70 + 3, 17 = 10 + 7

Step 3: Since 3 < 7, regroup 70 + 3 to 60 + 13

Now we have (60 + 13) - (10 + 7)

Step 4: Subtract tens and ones separately

60 - 10 = 50, and 13 - 7 = 6

Step 5: Combine the results

50 + 6 = 56

45 + 28 - 17 = 56
Final answer:

45 + 28 - 17 = 56

Applied rules:

• Order of Operations: Perform operations from left to right

• Place Value Strategy: Use decomposition for each operation

• Regrouping: Apply when needed in each step

5 Properties Application Strategy
Exercise 5
Use properties of operations to solve efficiently: 36 + 18 + 14
Definition:

Properties Application Strategy: Use properties of operations to group numbers in a way that makes computation easier.

Original: 36 + 18 + 14
Reorder: 36 + 14 + 18
Using commutative property
50 + 18 = 68
Final result
68
Step 1: Recognize that 36 + 14 is easier to compute than 36 + 18

Both 36 and 14 have 6 + 4 = 10 in the ones place

Step 2: Use the commutative property to reorder

36 + 18 + 14 = 36 + 14 + 18

Step 3: Compute 36 + 14 using place value

36 + 14 = (30 + 6) + (10 + 4) = 40 + 10 = 50

Step 4: Add the remaining number

50 + 18 = (50 + 10) + 8 = 60 + 8 = 68

36 + 18 + 14 = 68
Final answer:

36 + 18 + 14 = 68

Applied rules:

• Commutative Property: Reordering addends to make computation easier

• Place Value Addition: Adding digits with the same place value

• Strategic Grouping: Choosing which numbers to add first for efficiency

Strategies Based on Place Value and Properties of Operations Concepts
34 + 27 = 27 + 34 = 61 (commutative property)
Commutative Property Application
Key definitions:

Commutative Property: Changing the order of addends does not change the sum (a + b = b + a)

Associative Property: Grouping of addends does not change the sum ((a + b) + c = a + (b + c))

Place Value: The value of a digit based on its position in a number (ones, tens, hundreds)

Properties of Operations: Rules that describe how numbers behave when added or subtracted

Complete methodology:
  1. Identify the operation: Determine if addition or subtraction
  2. Recognize applicable properties: Identify which property can be used
  3. Apply place value strategy: Decompose numbers into tens and ones
  4. Perform operations: Add or subtract digits in the same place
  5. Combine results: Put the final answer together
Tip 1: Use the commutative property to rearrange numbers for easier computation (e.g., put numbers that add to 10 together)
Tip 2: Use the associative property to group numbers that add to round numbers (like 20, 30, etc.)
Tip 3: Remember that subtraction is NOT commutative (a - b ≠ b - a)
Tip 4: Practice recognizing number combinations that make 10 or other round numbers
Common errors: Forgetting that subtraction is not commutative, misapplying properties to subtraction, adding tens and ones together before completing operations.
Success habits: Regular practice with identifying properties, visualizing with manipulatives, checking answers.
Properties of operations with place value:

• Addition is commutative: a + b = b + a (can be used with place value)

• Addition is associative: (a + b) + c = a + (b + c) (can be used with place value)

• Subtraction is NOT commutative: a - b ≠ b - a

• Subtraction is NOT associative: (a - b) - c ≠ a - (b - c)

• Place value helps decompose numbers for easier application of properties

Strategies Based on Place Value and Properties Workflow

🔢
Step-by-Step Process
1
Identify operation
2
Recognize properties
3
Apply place value
4
Perform operations
5
Combine results
Properties Visualization
Understanding properties of operations helps us rearrange numbers to make calculations easier while maintaining accuracy.
Master Properties and Place Value for Efficient Math!

Questions & Answers

Question: Why do we need to learn about properties of operations? Isn't it enough to just know how to add and subtract?

Answer: Learning properties of operations makes math easier and faster! Here's why:

  • The commutative property lets you add numbers in any order: 36 + 18 is the same as 18 + 36, so pick whichever is easier for you!
  • The associative property lets you group numbers differently: (25 + 17) + 3 is the same as 25 + (17 + 3) = 25 + 20 = 45, which is easier!
  • These properties help you recognize patterns and shortcuts in math problems.

Think of properties like helpful rules that make math more flexible and easier to solve!

Question: How can I help my child practice strategies based on place value and properties of operations at home?

Answer: Here are effective ways to practice strategies based on place value and properties of operations at home:

  • Use manipulatives: Base-10 blocks to show place value and grouping
  • Play number games: Have your child solve problems in different orders to see that the answer stays the same
  • Real-world connections: Use money to show that $3.40 + $2.70 = $2.70 + $3.40
  • Pattern spotting: Look for numbers that add to 10 or other round numbers

Start with simple examples like 5 + 3 = 3 + 5, then move to larger numbers. Practice decomposing numbers regularly to strengthen place value understanding.

Make it visual and hands-on - the more concrete experiences your child has, the better they'll understand abstract concepts.

Question: Do the same properties work for subtraction? Can I change the order when subtracting?

Answer: Great question! No, the commutative property does NOT work for subtraction. Here's why:

  • With addition: 5 + 3 = 3 + 5 = 8 (same answer)
  • With subtraction: 5 - 3 = 2, but 3 - 5 = -2 (different answers!)

Also, the associative property does NOT work for subtraction:

  • (10 - 5) - 2 = 5 - 2 = 3
  • 10 - (5 - 2) = 10 - 3 = 7
  • So (10 - 5) - 2 ≠ 10 - (5 - 2)

That's why we focus on addition properties when talking about commutative and associative properties!