Solved Exercises on Place-Value Strategies in Grade 2

Master place-value strategies: decomposing numbers, regrouping, expanded form, and using place value for efficient addition and subtraction through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Expanded Form Strategy
Exercise 1
Write 84 in expanded form and add 17 using place-value strategy
Definition:

Expanded Form: Writing a number as the sum of its place values (e.g., 84 = 80 + 4).

Expanded Form Method:
  1. Write each number in expanded form
  2. Add the tens together
  3. Add the ones together
  4. Combine the sums
  5. Regroup if necessary
Problem
84 + 17
Expanded form
80+4 + 10+7
Add by place
90+11=101
Step 1: Write each number in expanded form

84 = 80 + 4 and 17 = 10 + 7

Step 2: Add the tens together

80 + 10 = 90

Step 3: Add the ones together

4 + 7 = 11

Step 4: Combine the sums

90 + 11 = 101

84 + 17 = 101
Final answer:

84 + 17 = 101

Applied rules:

• Expanded Form: Expressing numbers as the sum of their place values

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

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

2 Regrouping Strategy
Exercise 2
Solve using place-value regrouping: 58 + 37
Definition:

Regrouping Strategy: Converting 10 ones to 1 ten when adding, or 1 ten to 10 ones when subtracting.

Problem
58 + 37
Decompose
(50+8)+(30+7)
Add by place
80+15=95
Step 1: Decompose both numbers into tens and ones

58 = 50 + 8 and 37 = 30 + 7

Step 2: Add the tens together

50 + 30 = 80

Step 3: Add the ones together

8 + 7 = 15

Step 4: Regroup when ones sum to 10+

15 ones = 10 ones + 5 ones = 1 ten + 5 ones

Step 5: Add the regrouped ten to the tens sum

80 + 10 = 90, then 90 + 5 = 95

58 + 37 = 95
Final answer:

58 + 37 = 95

Applied rules:

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

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

• Number Composition: Reassemble the final sum from its components

3 Subtraction with Regrouping
Exercise 3
Solve using place-value strategy: 72 - 38
Definition:

Subtraction with Regrouping: Converting 1 ten to 10 ones when the top ones digit is smaller than the bottom ones digit.

Problem
72 - 38
Decompose
(70+2)-(30+8)
Regroup
70+2=60+12
Step 1: Decompose both numbers into tens and ones

72 = 70 + 2 and 38 = 30 + 8

Step 2: Check if ones can be subtracted

2 - 8 cannot be done, so we need to regroup

Step 3: Regroup from tens to ones

Take 1 ten from 70 (leaving 60) and add to 2 ones: 60 + 12

Step 4: Subtract tens and ones separately

60 - 30 = 30 and 12 - 8 = 4

Step 5: Combine the results

30 + 4 = 34

72 - 38 = 34
Final answer:

72 - 38 = 34

Applied rules:

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

• Regrouping: When the top digit is smaller, borrow from the next higher place

• Number Composition: Reassemble the final difference from its components

Rules and methods, laws,...
84 = 80 + 4 (expanded form)
Expanded Form Strategy
Strategy 1
Expanded Form
Writing numbers as tens + ones
Strategy 2
Regrouping
Converting between place values
Strategy 3
Place Value
Working with same place values
Key definitions:

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

Expanded Form: Writing a number as the sum of its place values (e.g., 84 = 80 + 4)

Regrouping: Converting from one place value to another (borrowing in subtraction, carrying in addition)

Decomposing: Breaking a number into its component parts (tens and ones)

Complete methodology:
  1. Identify place values: Determine the tens and ones in each number
  2. Separate by place: Work with tens and ones separately
  3. Perform operations: Add or subtract digits in the same place
  4. Regroup when necessary: Borrow or carry between places
  5. Combine results: Put the tens and ones back together
Tip 1: Always write numbers in expanded form to see the tens and ones clearly.
Tip 2: In subtraction, if the top ones digit is smaller than the bottom ones digit, you must regroup.
Tip 3: Remember: 1 ten = 10 ones (for regrouping purposes).
Tip 4: Always check your answer by using the inverse operation.
Common errors: Forgetting to regroup when necessary, adding tens and ones together before completing operations, misidentifying place values.
Key strategies: Practice decomposing numbers, visualize with base-10 blocks, check answers with inverse operations.
Place-value rules:

• In 2-digit numbers, the left digit is the tens place, right digit is the ones place

• When adding, add tens to tens and ones to ones

• When subtracting, subtract tens from tens and ones from ones

• Regroup when the top digit is smaller than the bottom digit in subtraction

• 1 ten = 10 ones (for regrouping purposes)

Solution: Exercises 4 to 5
4 Mixed Operations Strategy
Exercise 4
Solve using place-value strategy: 46 + 29 - 18
Definition:

Mixed Operations Strategy: Perform operations sequentially using place-value methods.

Problem
46 + 29 - 18
First: 46 + 29
70 + 15 = 85
Then: 85 - 18
70 + 15 - 18 = 67
Step 1: First perform 46 + 29 using place value

46 = 40 + 6, 29 = 20 + 9, so (40 + 20) + (6 + 9) = 60 + 15 = 75

Step 2: Now perform 75 - 18 using place value

75 = 70 + 5, 18 = 10 + 8

Step 3: Since 5 < 8, regroup 70 + 5 to 60 + 15

Now we have (60 + 15) - (10 + 8)

Step 4: Subtract tens and ones separately

60 - 10 = 50, and 15 - 8 = 7

Step 5: Combine the results

50 + 7 = 57

46 + 29 - 18 = 57
Final answer:

46 + 29 - 18 = 57

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 Comparison Strategy
Exercise 5
Which is greater: 63 - 27 or 58 - 23? Use place-value strategy.
Definition:

Comparison Strategy: Solve each expression using place-value methods, then compare the results.

First: 63 - 27
60+3 - 20+7 = 40-4 = 36
Second: 58 - 23
50+8 - 20+3 = 30+5 = 35
Compare
36 > 35
Step 1: Solve 63 - 27 using place value

63 = 60 + 3, 27 = 20 + 7

Step 2: Since 3 < 7, regroup 60 + 3 to 50 + 13

Now we have (50 + 13) - (20 + 7)

Step 3: Subtract tens and ones separately

50 - 20 = 30, and 13 - 7 = 6

Step 4: Combine the results

30 + 6 = 36

Step 5: Solve 58 - 23 using place value

58 = 50 + 8, 23 = 20 + 3, so (50 - 20) + (8 - 3) = 30 + 5 = 35

Step 6: Compare the results

36 > 35, so 63 - 27 is greater

63 - 27 = 36, 58 - 23 = 35, so 63 - 27 > 58 - 23
Final answer:

63 - 27 is greater than 58 - 23

Applied rules:

• Place Value Subtraction: Apply decomposition method to both expressions

• Comparison: Once both expressions are simplified, compare the results

• Regrouping: Apply when needed in each expression

Place-Value Strategies Concepts and Techniques
84 = 80 + 4 (expanded form strategy)
Place-Value Decomposition
Key definitions:

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

Expanded Form: Writing a number as the sum of its place values (e.g., 84 = 80 + 4)

Regrouping: Converting between place values (1 ten = 10 ones)

Decomposing: Breaking a number into its component parts (tens and ones)

Complete methodology:
  1. Identify place values: Look at each digit and determine its value
  2. Separate by place: Group tens and ones separately
  3. Perform operations: Work with tens and ones separately
  4. Regroup when necessary: Convert between places when needed
  5. Combine results: Put the final answer together
Tip 1: Think of tens as bundles of 10 ones - this helps visualize regrouping
Tip 2: Use base-10 blocks or drawings to physically represent the numbers
Tip 3: In subtraction, always check if you need to regroup BEFORE subtracting
Tip 4: Practice with number lines to see how place value affects the magnitude of numbers
Common errors: Forgetting to regroup when necessary, adding tens and ones together before completing operations, misidentifying place values.
Success habits: Regular practice with decomposing numbers, visualizing with manipulatives, checking answers.
Place-value strategies:

• Decompose numbers into tens and ones before operating

• Add/subtract digits with the same place value

• Regroup when necessary (10 ones = 1 ten)

• Check your answer by using the inverse operation

• Use visual models to support understanding

Place-Value Strategy Workflow

🔢
Step-by-Step Process
1
Identify place values
2
Decompose numbers
3
Operate by place
4
Regroup if needed
5
Combine results
Place-Value Visualization
Understanding place value helps us see that 84 is actually 8 tens and 4 ones, making operations easier to manage.
Master Place-Value Strategies for Strong Math Foundation!

Questions & Answers

Question: I still don't understand why we need to regroup. Can you explain it in a simpler way?

Answer: Sure! Think of regrouping like trading. Let's say you have 72 cents and want to give someone 38 cents:

  • You have 7 dimes and 2 pennies (72 cents)
  • But you need to give 8 pennies and you only have 2 pennies
  • So you trade 1 dime for 10 pennies (regrouping)
  • Now you have 6 dimes and 12 pennies
  • You can give 8 pennies and keep 4 pennies
  • Then give 3 dimes and keep 3 dimes
  • You're left with 3 dimes and 4 pennies = 34 cents

Regrouping is just trading between place values to make the math easier!

Question: How can I help my child understand place-value strategies at home?

Answer: Here are effective ways to practice place-value strategies at home:

  • Use manipulatives: Base-10 blocks, straws bundled in tens, or groups of 10 beans
  • Play number games: Have your child build numbers with place-value cards
  • Real-world connections: Use money to show tens and ones (dimes and pennies)
  • Drawings: Have your child draw sticks for tens and dots for ones

Start with simple numbers like 12, 23, 34 and practice decomposing them: 12 = 10 + 2. Once comfortable, move to larger numbers and operations.

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

Question: Why do we have to write numbers in expanded form? It seems like extra work!

Answer: Writing numbers in expanded form might seem like extra work, but it's actually a shortcut! Here's why:

  • It makes the math clearer: Instead of trying to add 58 + 37 all at once, you add 50 + 30 = 80, then 8 + 7 = 15, then 80 + 15 = 95
  • It prevents mistakes: You won't accidentally add 8 tens to 7 ones
  • It builds number sense: You understand that 58 really means 5 tens and 8 ones
  • It prepares you for bigger numbers: This same strategy works for hundreds and thousands later

Think of it like organizing your school supplies - it's easier to find what you need if everything has its place!