Add and Subtract Using Likes

Understanding "Likes" in Mathematics

In mathematics, "likes" refer to values that occupy the same place value position. When we add or subtract two-digit numbers, we can make the process much easier by grouping and working with likes: tens with tens, and ones with ones. This fundamental concept is the foundation for all multi-digit arithmetic and helps develop strong number sense.

What Are Likes?

Think of likes as things that are similar and belong together:

• Tens are likes: The tens digits from different numbers can be added or subtracted together
• Ones are likes: The ones digits from different numbers can be added or subtracted together
• Rule: We combine or subtract likes separately, then put our answers together

For example, in 23 + 15: - The tens (2 and 1) are likes—they can be worked with together - The ones (3 and 5) are likes—they can be worked with together - We calculate: tens (2 + 1 = 3) and ones (3 + 5 = 8) to get 38

Why "Likes" Matter

Understanding likes helps you: - Break complex problems into simple parts: Big problems become a series of small, easy problems - Understand place value deeply: Each position has its own value and can be worked with independently - Develop mental math skills: You can solve problems in your head by working with likes - Build foundation for algebra: The concept of "like terms" in algebra comes directly from this idea

Place Value Foundation

To work with likes effectively, you need to understand place value—the idea that a digit's position determines its value.

The Two-Digit Number Structure

Every two-digit number has two parts: - Tens place: The left digit shows how many tens (groups of 10) - Ones place: The right digit shows how many individual ones

Example: The number 47 - 4 in the tens place = 4 tens = 40 - 7 in the ones place = 7 ones = 7 - Total: 40 + 7 = 47

Visualizing with Base-Ten Blocks

Base-ten blocks make place value concrete: - Tens rods: Long pieces representing 10 - Unit cubes: Small cubes representing 1

For 35: - 3 tens rods (representing 30) - 5 unit cubes (representing 5) - Together they show 35

Expanded Form

Writing numbers in expanded form helps you see the likes clearly: - 23 = 20 + 3 - 15 = 10 + 5

Now you can see exactly what you're adding: - Tens: 20 + 10 = 30 - Ones: 3 + 5 = 8 - Total: 30 + 8 = 38

Adding Using Likes

When you add two-digit numbers, treating tens and ones separately makes the process manageable.

The Step-by-Step Process

Problem: 32 + 45

Step 1: Identify the likes - Tens: 3 and 4 - Ones: 2 and 5

Step 2: Add the tens - 3 tens + 4 tens = 7 tens - Or: 30 + 40 = 70

Step 3: Add the ones - 2 ones + 5 ones = 7 ones - Or: 2 + 5 = 7

Step 4: Combine the results - 7 tens + 7 ones = 77 - Or: 70 + 7 = 77

More Examples

Example 1: 41 + 27 - Tens: 4 + 2 = 6 (or 40 + 20 = 60) - Ones: 1 + 7 = 8 - Answer: 60 + 8 = 68

Example 2: 53 + 24 - Tens: 5 + 2 = 7 (or 50 + 20 = 70) - Ones: 3 + 4 = 7 - Answer: 70 + 7 = 77

Example 3: 62 + 13 - Tens: 6 + 1 = 7 (or 60 + 10 = 70) - Ones: 2 + 3 = 5 - Answer: 70 + 5 = 75

Why This Method Works

This method works because of the associative and commutative properties of addition: - You can add numbers in any order - You can group numbers in any way - So (20 + 3) + (10 + 5) = (20 + 10) + (3 + 5)

Subtracting Using Likes

The same principle applies to subtraction: subtract tens from tens, ones from ones.

The Step-by-Step Process

Problem: 68 - 23

Step 1: Identify the likes - First number: 6 tens, 8 ones - Second number: 2 tens, 3 ones

Step 2: Subtract the tens - 6 tens - 2 tens = 4 tens - Or: 60 - 20 = 40

Step 3: Subtract the ones - 8 ones - 3 ones = 5 ones - Or: 8 - 5 = 5

Step 4: Combine the results - 4 tens + 5 ones = 45 - Or: 40 + 5 = 45

More Examples

Example 1: 89 - 45 - Tens: 8 - 4 = 4 (or 80 - 40 = 40) - Ones: 9 - 5 = 4 - Answer: 40 + 4 = 44

Example 2: 76 - 32 - Tens: 7 - 3 = 4 (or 70 - 30 = 40) - Ones: 6 - 2 = 4 - Answer: 40 + 4 = 44

Example 3: 57 - 26 - Tens: 5 - 2 = 3 (or 50 - 20 = 30) - Ones: 7 - 6 = 1 - Answer: 30 + 1 = 31

Important Note About Regrouping

The problems we're focusing on here are ones where: - In subtraction, the ones digit on top is larger than the ones digit on bottom - In addition, the ones don't add up to more than 9

When ones add to more than 9, or when subtracting requires "borrowing," we need additional strategies. For now, we're building the foundation with simpler problems.

Mental Math Strategies

Working with likes is perfect for mental math because you can solve problems in your head without writing them down.

Strategy 1: Say It Out Loud

For 34 + 52: - Say: "30 plus 50 is 80" - Then: "4 plus 2 is 6" - Finally: "80 plus 6 is 86"

Verbalizing helps organize your thinking!

Strategy 2: Visualize Place Value

For 67 - 43: - Picture: 6 tens rods and 7 unit cubes - Remove: 4 tens rods and 3 unit cubes - Left with: 2 tens rods and 4 unit cubes = 24

Strategy 3: Use Your Fingers

For addition (42 + 35): - Hold up 4 fingers on one hand (4 tens) - Hold up 3 fingers on the other hand (3 tens) - Count total fingers: 7 (represents 70) - Now do the ones: 2 + 5 = 7 - Answer: 77

Strategy 4: Break It Down

For 58 + 31: - Think: "58 plus 30 is 88 (adding the tens)" - Then: "88 plus 1 is 89 (adding the ones)"

Real-World Applications

Adding and subtracting likes appears constantly in everyday situations:

Money

Problem: "I have $45 and earn $32 more. How much do I have?" - Tens: $40 + $30 = $70 - Ones: $5 + $2 = $7 - Total: $77

Time

Problem: "A movie is 68 minutes, commercials are 21 minutes. What's the difference?" - Tens: 60 - 20 = 40 - Ones: 8 - 1 = 7 - Difference: 47 minutes

Measurement

Problem: "One board is 47 inches, another is 32 inches. Total length?" - Tens: 40 + 30 = 70 - Ones: 7 + 2 = 9 - Total: 79 inches

Collections

Problem: "I have 56 trading cards, trade away 24. How many left?" - Tens: 50 - 20 = 30 - Ones: 6 - 4 = 2 - Left: 32 cards

Ages

Problem: "Grandma is 73, Grandpa is 75. How much older is Grandpa?" - Tens: 70 - 70 = 0 - Ones: 5 - 3 = 2 - Difference: 2 years

Practice Activities

Make learning interactive and fun:

Activity 1: Place Value Dice Game

Materials: Two dice, paper

How to play: 1. Roll two dice to create your first two-digit number (first roll = tens, second roll = ones) 2. Roll again to create your second two-digit number 3. Add them using likes 4. Check with a calculator 5. Try 10 rounds!

Activity 2: Likes Sorting Cards

Materials: Index cards

Create cards with: - Different two-digit numbers - Sort them into groups where addition/subtraction would be easy (no regrouping needed) - Practice adding/subtracting each pair

Activity 3: Base-Ten Block Building

Materials: Base-ten blocks (or drawings)

Activity: 1. Build a number with blocks (like 45: 4 tens rods, 5 unit cubes) 2. Build another number (like 32: 3 tens rods, 2 unit cubes) 3. Push the tens rods together and count them 4. Push the unit cubes together and count them 5. Write your equation!

Activity 4: Mental Math Challenge

Materials: Timer, paper for recording

Challenge: 1. Partner calls out a problem (like "34 + 42") 2. Solve it mentally using likes 3. Say the answer out loud 4. How many can you get right in 2 minutes?

Activity 5: Real-World Scavenger Hunt

Activity: - Find two-digit numbers around your home - Addresses, prices, page numbers, dates - Create addition and subtraction problems using pairs - Solve them using likes!

Building Fluency

Fluency means solving these problems quickly and accurately:

Progressive Practice

Week 1: Focus on adding tens (numbers with 0 in ones place) - 20 + 30, 40 + 10, etc.

Week 2: Add problems with small ones digits (1-3) - 21 + 32, 41 + 13, etc.

Week 3: Add problems with medium ones digits (4-6) - 24 + 35, 44 + 26, etc.

Week 4: Mix all types of addition

Week 5: Introduce subtraction using the same progression

Daily Practice Routine

Morning (5 minutes): - 5 addition problems - Work mentally, then check written

Afternoon (5 minutes): - 5 subtraction problems - Work mentally, then check written

Evening Review: - Review any mistakes - Understand what went wrong

Multiple Representations

Practice the same problem in different ways: - Symbolic: 45 + 32 = 77 - Verbal: "Forty-five plus thirty-two equals seventy-seven" - Expanded: (40 + 30) + (5 + 2) = 70 + 7 = 77 - Visual: Draw base-ten blocks

Common Challenges and Solutions

Challenge: "I forget which digits to add together"

Solution: Write numbers vertically, lining up place values:

  45
+ 32
----

Now tens are stacked with tens, ones with ones!

Challenge: "I mix up the order"

Solution: Always work right to left (ones first, then tens). This habit will help later with more complex problems.

Challenge: "I get confused when the answer has different digits"

Solution: Double-check by using expanded form: - 32 + 45 = (30 + 40) + (2 + 5) = 70 + 7 = 77

Challenge: "I can't do it in my head"

Solution: Start by writing it out in expanded form until the pattern becomes automatic. Mental math develops with practice!

Connecting to Other Math Concepts

Three-Digit Addition

The same principle extends to hundreds: - 245 + 132 = (200 + 100) + (40 + 30) + (5 + 2) = 377

Decimals (Future Learning)

Working with likes prepares you for: - 2.5 + 3.2 = (2 + 3) + (0.5 + 0.2) = 5.7

Algebra

"Like terms" in algebra use the same idea: - 3x + 5x = 8x (combining like terms)

Assessment Checkpoints

You've mastered this skill when you can: - ✓ Quickly identify tens and ones in any two-digit number - ✓ Add tens separately from ones mentally - ✓ Subtract tens separately from ones mentally - ✓ Explain why we add/subtract likes - ✓ Solve problems without regrouping fluently - ✓ Apply this strategy to real-world problems

Looking Ahead

This foundation prepares you for: - Addition with regrouping: When ones add to more than 9 - Subtraction with borrowing: When you need to regroup tens - Three-digit arithmetic: Same principles with hundreds, tens, ones - Multi-step problems: Using this skill in word problems - Algebraic thinking: Understanding like terms

Conclusion

Adding and subtracting using likes is a fundamental skill that makes multi-digit arithmetic manageable and logical. By understanding that we can work with tens and ones separately, you gain both confidence and speed in calculation. This isn't just a trick—it's a deep understanding of how our number system works through place value. Practice regularly, use visual tools when needed, and celebrate your growing ability to solve problems both on paper and in your head. You're building mathematical foundations that will support increasingly complex mathematics as you grow!