Composing and Decomposing Numbers in Tens - Grade 1

Master composing and decomposing numbers in tens: understanding tens and ones, place value concepts, and number relationships through these 5 detailed exercises.

Solution: Exercises 1 to 3
1 Composing Numbers
Exercise 1
Compose the number: 3 tens and 4 ones = ?
Definition:

Composing numbers: Combining tens and ones to make a whole number. 1 ten = 10 ones.

Composition method:
  1. Count the number of tens and multiply by 10
  2. Add the number of ones
  3. Combine to form the total number
10
10
10

3 tens = 3 × 10 = 30

→
1
1
1
1

4 ones = 4

Tens
3 × 10 = 30
Ones
4
Total
30 + 4 = 34
Step 1: Count the tens

We have 3 tens, so 3 × 10 = 30

Step 2: Count the ones

We have 4 ones, so 4 × 1 = 4

Step 3: Add together

30 + 4 = 34

3 tens and 4 ones = 34
Final answer:

3 tens and 4 ones = 34

Applied rules:

• Tens rule: Each ten represents 10 ones

• Place value: Tens digit is worth 10 times its face value

• Composition: Combine tens and ones to form the number

2 Decomposing Numbers
Exercise 2
Decompose the number: 57 = ? tens and ? ones
Definition:

Decomposing numbers: Breaking a number into its tens and ones parts.

10
10
10
10
10

5 tens = 50

→
1
1
1
1
1
1
1

7 ones = 7

Number
57
Tens
5
Ones
7
Step 1: Identify the tens digit

In 57, the tens digit is 5, so we have 5 tens

Step 2: Identify the ones digit

In 57, the ones digit is 7, so we have 7 ones

Step 3: Verify the decomposition

5 tens (50) + 7 ones (7) = 50 + 7 = 57 ✓

57 = 5 tens and 7 ones
Final answer:

57 = 5 tens and 7 ones

Applied rules:

• Digit position: Left digit is tens, right digit is ones

• Place value: Tens digit × 10 gives tens value

• Verification: Add tens and ones to check the original number

3 Number Relationships
Exercise 3
Show that 24 can be composed in different ways:
24 = 2 tens + 4 ones OR 1 ten + 14 ones
Definition:

Regrouping: Converting tens to ones or ones to tens while keeping the total value the same.

10
10

2 tens = 20

→
1
1
1
1

4 ones = 4

10

1 ten = 10

→
1
1
1
1
1
1
1
1
1
1
1
1
1
1

14 ones = 14

Standard
24 = 2×10 + 4×1 = 20 + 4
Regrouped
24 = 1×10 + 14×1 = 10 + 14
Step 1: Standard decomposition

24 = 2 tens and 4 ones (20 + 4 = 24)

Step 2: Regroup one ten to ones

Take 1 ten (10) and convert to 10 ones

Now we have 1 ten and 4 + 10 = 14 ones

Step 3: Verify the regrouped form

1 ten (10) + 14 ones (14) = 10 + 14 = 24 ✓

24 = 2 tens + 4 ones = 1 ten + 14 ones
Final answer:

24 can be shown as 2 tens + 4 ones OR 1 ten + 14 ones

Applied rules:

• Regrouping: 1 ten = 10 ones

• Equivalence: Different compositions can represent the same number

• Flexibility: Understanding multiple representations helps with addition/subtraction

Rules and methods, laws,...
1 ten = 10 ones
Place Value Relationship
Composition
T × 10 + O
Tens × 10 + Ones
Decomposition
N → T + O
Number → Tens + Ones
Regrouping
1T = 10O
1 Ten = 10 Ones
Key definitions:

Tens: Groups of 10 units (10, 20, 30, ...)

Ones: Individual units (1, 2, 3, ...)

Place value: The value of a digit depends on its position

Composing: Combining tens and ones to make a number

Decomposing: Breaking a number into tens and ones

Complete methodology:
  1. Identify digits: Recognize tens and ones positions
  2. Apply place value: Tens digit × 10, ones digit × 1
  3. Compose or decompose: Combine or break apart accordingly
  4. Verify: Check that your answer makes sense
Tip 1: Remember that the left digit is always tens and the right digit is always ones in two-digit numbers.
Tip 2: Draw groups of 10 to visualize tens, and individual dots for ones.
Tip 3: Always check your work by adding tens and ones to see if you get the original number.
Tip 4: Practice with concrete objects like blocks or sticks to build understanding.
Key concept: In the number system, each position has a specific value (place value).
Memory aid: "Tens are groups of 10, ones are single units."
Essential rules to remember:

• 1 ten = 10 ones (always)

• In a two-digit number, the first digit is tens, second is ones

• To compose: multiply tens by 10 and add ones

• To decompose: separate the tens and ones digits

• You can regroup: 1 ten = 10 ones (and vice versa)

Solution: Exercises 4 to 5
4 Problem Solving
Exercise 4
Sarah has 43 stickers. She wants to put them in packs of 10.
How many full packs can she make and how many stickers will be left over?
Definition:

Application: Using place value to solve real-world problems involving grouping.

10
10
10
10

4 tens = 40 stickers in full packs

→
1
1
1

3 ones = 3 stickers left over

Total
43
Packs
4
Leftover
3
Step 1: Decompose the number

43 = 4 tens and 3 ones

Step 2: Interpret the result

4 tens = 4 packs of 10 stickers each = 40 stickers

3 ones = 3 individual stickers left over

Step 3: Verify the answer

4 packs × 10 stickers + 3 leftover = 40 + 3 = 43 ✓

4 full packs and 3 stickers left over
Final answer:

Sarah can make 4 full packs and will have 3 stickers left over.

Applied rules:

• Decomposition: Break the number into tens and ones

• Grouping: Each ten represents a group of 10 items

• Real-world connection: Apply mathematical concepts to practical situations

5 Number Puzzles
Exercise 5
I am a number between 20 and 30. My ones digit is 6.
What number am I? Show how I can be decomposed.
Definition:

Number sense: Using place value knowledge to identify and describe numbers.

10
10

2 tens = 20

→
1
1
1
1
1
1

6 ones = 6

Between 20-30
26
Tens
2
Ones
6
Step 1: Analyze the clues

• Between 20 and 30 means the tens digit is 2

• Ones digit is 6

• So the number is 26

Step 2: Decompose the number

26 = 2 tens and 6 ones

Step 3: Verify the answer

2 tens (20) + 6 ones (6) = 20 + 6 = 26 ✓

26 is between 20 and 30 ✓

Ones digit is 6 ✓

Step 4: Alternative representation

26 can also be shown as 1 ten and 16 ones

(1 ten = 10, 16 ones = 16, 10 + 16 = 26)

The number is 26 = 2 tens + 6 ones
Final answer:

The number is 26, which decomposes to 2 tens and 6 ones.

Applied rules:

• Logical reasoning: Use given clues to identify the number

• Place value: Understand how digits determine value

• Multiple representations: Same number can be shown in different ways

Place Value Laws, Methods, Rules, Definitions
N = T × 10 + O
Number Composition Formula
Key definitions:

Place value: The value of a digit depends on its position in the number.

Tens place: The second digit from the right, representing groups of 10.

Ones place: The rightmost digit, representing individual units.

Composing: Combining tens and ones to form a number.

Decomposing: Breaking a number into tens and ones.

Regrouping: Converting between tens and ones while maintaining the same total value.

Complete methodology:
  1. Identify positions: Locate tens and ones places in the number
  2. Determine values: Multiply tens digit by 10, keep ones digit as is
  3. Compose/decompose: Add for composition, separate for decomposition
  4. Verify: Check that your result equals the original number
Tip 1: Use base-10 blocks or drawings to physically represent tens and ones.
Tip 2: Remember: 10 ones can always be grouped into 1 ten.
Tip 3: Practice reading numbers aloud as "X tens and Y ones" (e.g., "34" = "3 tens and 4 ones").
Tip 4: When comparing numbers, compare the tens first, then the ones if needed.
Common error: Confusing tens and ones positions (e.g., thinking 23 has 2 ones and 3 tens).
Success strategy: Always start with the largest place value (tens) when working with two-digit numbers.
Essential formulas to remember:

• For any two-digit number XY: Number = X × 10 + Y × 1

• 1 ten = 10 ones (regrouping rule)

• Composing: T tens + O ones = (T × 10) + O

• Decomposing: N = T tens + O ones

• Verification: T × 10 + O should equal the original number

Visual Place Value Guide

📊
Tens and Ones Structure

Two-digit number format: [TENS][ONES]

Example 45: 4 tens (40) + 5 ones (5) = 45

10
10
10
10

4 tens = 40

1
1
1
1
1

5 ones = 5

Regrouping Examples

Same value, different representations:

32 = 3 tens + 2 ones OR 2 tens + 12 ones

56 = 5 tens + 6 ones OR 4 tens + 16 ones

Key Patterns

Numbers ending in 0: Only tens (20 = 2 tens + 0 ones)

Numbers ending in 0-9: Tens + corresponding ones

Smallest two-digit number: 10 (1 ten + 0 ones)

Largest two-digit number: 99 (9 tens + 9 ones)

Problem-Solving Workflow
1
Identify
2
Separate
3
Calculate
4
Verify
Master Place Value: Tens and Ones Are Your Foundation!

Questions & Answers

Question: I don't understand why 1 ten equals 10 ones. Why isn't it just 1? It seems confusing.

Answer: Great question! Think of it like this: A "ten" is a group of 10 individual items bundled together. For example:

  • If you have 1 bundle of 10 pencils, that bundle equals 10 individual pencils
  • If you have 1 ten-dollar bill, it's worth 10 one-dollar bills
  • If you have 1 box of 10 cookies, that box contains 10 individual cookies

So 1 ten doesn't mean the number 1—it means 1 GROUP of 10 ones. That's why 1 ten = 10 ones. It's like having 1 container that holds 10 things inside it.

Question: How can I help my child practice composing and decomposing numbers at home in a fun way?

Answer: Here are some engaging activities for home practice:

  • Counting objects: Use coins, buttons, or small toys. Group them into sets of 10.
  • Base-10 blocks: If available, these are perfect for visualizing tens and ones.
  • Number games: Play "I spy" with numbers, asking your child to identify tens and ones.
  • Cooking activities: Measure ingredients and count in groups of 10.
  • Card games: Use playing cards to create two-digit numbers and break them apart.

Make it a game! Children learn better when they're engaged and having fun.

Question: Sometimes I mix up the tens and ones. Like for 23, I say 2 ones and 3 tens. How can I remember which is which?

Answer: Here are some helpful memory tricks:

  • Position rule: The LEFT digit is always tens, the RIGHT digit is always ones
  • Size matters: Tens are bigger groups (10 each), so the left digit (which looks bigger in importance) is tens
  • Reading order: Read the number as "X tens and Y ones" (23 = "two tens and three ones")
  • Practice saying: "The first number tells me how many tens, the second tells me how many ones"

Remember: 23 means 2 tens (20) + 3 ones (3) = 23. The 2 comes first, so it's tens!