Counting on: An addition strategy where you start with the larger number and count up by the amount of the smaller number.
Note: This strategy builds number sense and is particularly useful when one number is small.
- Identify the larger number (8)
- Start counting from the larger number
- Count up by the amount of the smaller number (4)
- Stop when you've counted the full amount
- The number you land on is the sum
8 is larger than 4, so start counting from 8
Count 4 numbers starting from 8: 9, 10, 11, 12
8 + 4 = 12
12
• Efficiency: Start counting from the larger number
• Sequential counting: Count each number in sequence
• Completion: Count the full amount of the second number
• Practice Tip: Counting on builds addition fluency and mental math skills
- 7 + 3: Start at 7, count 8, 9, 10 → 7 + 3 = 10
- 9 + 2: Start at 9, count 10, 11 → 9 + 2 = 11
- 6 + 5: Start at 6, count 7, 8, 9, 10, 11 → 6 + 5 = 11
- Always start counting from the larger number for efficiency
- Use your fingers or manipulatives if needed
- Practice with small numbers first, then increase complexity
Q: Why do I start counting from the larger number?
A: Starting from the larger number is more efficient because you count fewer numbers. For 8+4, you count 4 numbers instead of 8.
Q: When should I use counting on?
A: Use counting on when one of the numbers is small (≤5) or when you're still building addition fluency.
Counting back: A subtraction strategy where you start with the larger number and count backward by the amount of the smaller number.
Note: This strategy is effective when the number being subtracted is small and builds foundational subtraction skills.
- Start with the larger number (13)
- Count backward by the amount of the smaller number (5)
- Count each number as you move backward
- Stop after counting back the full amount
- The number you land on is the difference
Begin at 13
Count back 5 numbers: 12, 11, 10, 9, 8
13 - 5 = 8
8
• Starting point: Begin with the minuend (first number)
• Backward counting: Move in reverse numerical order
• Step completion: Count the exact number of steps specified
• Practice Tip: Counting back strengthens subtraction fluency
- 11 - 3: Start at 11, count back 10, 9, 8 → 11 - 3 = 8
- 10 - 4: Start at 10, count back 9, 8, 7, 6 → 10 - 4 = 6
- 9 - 2: Start at 9, count back 8, 7 → 9 - 2 = 7
- Use counting back when the number being subtracted is small (≤5)
- Count slowly and carefully to avoid mistakes
- Verify by counting forward: 8 + 5 = 13 ✓
Q: When should I use counting back?
A: Use counting back when the number being subtracted is small (≤5) and you're still developing subtraction fluency.
Q: What if I count back too far?
A: Double-check by counting more carefully. For 13-5, count exactly 5 steps: 12, 11, 10, 9, 8.
Number line strategy: Using a visual number line to model addition or subtraction by making jumps in the appropriate direction.
Note: Number lines provide a concrete visual representation of counting strategies and help students understand the concept of adding or subtracting.
- Draw a number line with appropriate range (0 to 15)
- Mark the starting number (7) on the line
- Make forward jumps equal to the number being added (6 jumps)
- Count each jump to ensure accuracy
- Identify the number where you land
Create a line with numbers from 0 to 15
Place a mark at 7 (the first number)
Jump 6 spaces to the right: 8, 9, 10, 11, 12, 13
After 6 jumps from 7, you land on 13
13
• Direction: Addition moves right on the number line
• Distance: The number of jumps equals the second number
• Starting point: Begin at the first number in the expression
• Practice Tip: Number lines help visualize the counting process
- 6 + 4: Start at 6, jump 4 spaces right → lands on 10
- 8 + 3: Start at 8, jump 3 spaces right → lands on 11
- 5 + 7: Start at 5, jump 7 spaces right → lands on 12
- Always start at the first number
- Jump forward (right) for addition
- Jump backward (left) for subtraction
- Count each jump carefully to avoid mistakes
Q: How do I know which direction to jump on the number line?
A: For addition, jump forward (right). For subtraction, jump backward (left).
Q: What if I don't have a number line drawn?
A: You can draw a simple one or visualize it in your head. The important part is to visualize the movement.
Counting on: An addition strategy where you start with the larger number and count up by the amount of the smaller number.
Note: This strategy builds number sense and is particularly useful when one number is small.
- Identify the larger number (9)
- Start counting from the larger number
- Count up by the amount of the smaller number (5)
- Stop when you've counted the full amount
- The number you land on is the sum
9 is larger than 5, so start counting from 9
Count 5 numbers starting from 9: 10, 11, 12, 13, 14
9 + 5 = 14
14
• Efficiency: Start counting from the larger number
• Sequential counting: Count each number in sequence
• Completion: Count the full amount of the second number
• Practice Tip: Counting on builds addition fluency and mental math skills
- 8 + 4: Start at 8, count 9, 10, 11, 12 → 8 + 4 = 12
- 7 + 6: Start at 7, count 8, 9, 10, 11, 12, 13 → 7 + 6 = 13
- 6 + 3: Start at 6, count 7, 8, 9 → 6 + 3 = 9
- Always start counting from the larger number for efficiency
- Use your fingers or manipulatives if needed
- Practice with small numbers first, then increase complexity
Q: Why do I start counting from the larger number?
A: Starting from the larger number is more efficient because you count fewer numbers. For 9+5, you count 5 numbers instead of 9.
Q: When should I use counting on?
A: Use counting on when one of the numbers is small (≤5) or when you're still building addition fluency.
Counting back: A subtraction strategy where you start with the larger number and count backward by the amount of the smaller number.
Note: This strategy is effective when the number being subtracted is small and builds foundational subtraction skills.
- Start with the larger number (15)
- Count backward by the amount of the smaller number (7)
- Count each number as you move backward
- Stop after counting back the full amount
- The number you land on is the difference
Begin at 15
Count back 7 numbers: 14, 13, 12, 11, 10, 9, 8
15 - 7 = 8
8
• Starting point: Begin with the minuend (first number)
• Backward counting: Move in reverse numerical order
• Step completion: Count the exact number of steps specified
• Practice Tip: Counting back strengthens subtraction fluency
- 12 - 4: Start at 12, count back 11, 10, 9, 8 → 12 - 4 = 8
- 11 - 3: Start at 11, count back 10, 9, 8 → 11 - 3 = 8
- 10 - 6: Start at 10, count back 9, 8, 7, 6, 5, 4 → 10 - 6 = 4
- Use counting back when the number being subtracted is small (≤5)
- Count slowly and carefully to avoid mistakes
- Verify by counting forward: 8 + 7 = 15 ✓
Q: When should I use counting back?
A: Use counting back when the number being subtracted is small (≤5) and you're still developing subtraction fluency.
Q: What if I count back too far?
A: Double-check by counting more carefully. For 15-7, count exactly 7 steps: 14, 13, 12, 11, 10, 9, 8.
Visual strategy: Using pictures, diagrams, or manipulatives to model and solve mathematical problems.
Note: Visual representations help students understand abstract concepts by making them concrete and tangible.
- Represent the first number with objects (6 circles)
- Represent the second number with objects (8 circles)
- Count all objects together
- Record the sum
Draw 6 circles: ●●●●●●
Draw 8 circles: ●●●●●●●●
Count all circles: 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14
There are 14 circles in total
14
• Concrete modeling: Use visual representations to understand abstract concepts
• Addition concept: Combining two groups to find the total
• Counting principle: Each object represents one unit
• Practice Tip: Visual strategies help build number sense and understanding
- 5 + 7: Draw 5 circles + 7 circles = 12 circles
- 4 + 9: Draw 4 circles + 9 circles = 13 circles
- 7 + 6: Draw 7 circles + 6 circles = 13 circles
- Use simple shapes like circles or squares to represent numbers
- Organize your visual models in rows to make counting easier
- Count slowly and carefully to avoid mistakes
Q: Can I use visual strategies for all addition facts?
A: Yes! Visual strategies work well for all addition facts and help build understanding of the concept.
Q: What if I can't draw well?
A: Simple dots, tally marks, or even using fingers work perfectly fine for visual representations.
Missing number problems: Equations where one operand is unknown, requiring inverse operations to find the value.
Note: These problems reinforce the relationship between addition and subtraction, helping students understand that addition and subtraction are inverse operations.
- Rewrite the equation as: ? + 6 = 11
- Use inverse operation: ? = 11 - 6
- Solve the subtraction problem
- Verify by adding the result to the known number
? + 6 = 11
To find the missing addend, subtract: ? = 11 - 6
11 - 6 = 5 (using counting back: 11, 10, 9, 8, 7, 6)
5 + 6 = 11 ✓
5
• Inverse operations: Addition and subtraction are opposites
• Equation solving: Isolate the unknown variable
• Verification: Always check your answer by substitution
• Practice Tip: Missing number problems strengthen operational understanding
- ? + 4 = 9 → ? = 5
- ? + 3 = 8 → ? = 5
- ? + 5 = 10 → ? = 5
- Missing number problems always use inverse operations
- For addition problems: ? + A = B means ? = B - A
- For subtraction problems: A - ? = C means ? = A - C
- Always verify by substituting the found number back into the original
Q: How do I know which operation to use for missing number problems?
A: Use the inverse operation. For addition problems, use subtraction. For subtraction problems, use addition.
Q: What if the missing number is being subtracted?
A: For A - ? = C, solve as ? = A - C. For ? - B = C, solve as ? = C + B.
Multiple step problems: Mathematical expressions that require more than one operation to solve, following the order of operations.
Note: When addition and subtraction appear together, solve from left to right in the order they appear.
- Perform the first operation (8 + 3)
- Take the result and perform the next operation (11 - 5)
- Continue for any additional operations
- State the final answer
8 + 3 = 11
11 - 5 = 6
The result is 6
6
• Left to right: Perform operations in the order they appear
• Sequential processing: Complete each operation before moving to the next
• Tracking: Keep track of intermediate results
• Practice Tip: Verify by solving in reverse or estimating
- 7 + 5 - 4: First 7+5=12, then 12-4=8
- 9 - 3 + 2: First 9-3=6, then 6+2=8
- 6 + 3 - 5: First 6+3=9, then 9-5=4
- Always work from left to right
- Keep track of your intermediate results
- Verify by adding back: 6 + 5 - 3 = 8, so 8 + 3 - 5 = 6 ✓
Q: Can I solve multiple operations in any order?
A: No! For addition and subtraction, always work from left to right in the order they appear.
Q: What if I have addition and subtraction mixed?
A: Still work from left to right: perform the operations in the order they appear.
Real world application: Mathematical problems that occur in everyday situations, requiring computation to solve practical questions.
Note: Real world problems help students connect mathematical concepts to their daily lives and develop problem-solving skills.
- Identify the quantities in the problem (12 and 4)
- Determine that "sold" indicates subtraction is needed
- Set up the subtraction problem (12 - 4)
- Solve using appropriate strategy (counting back)
- State the answer in the context of the original problem
The store started with 12 balloons
They sold 4 balloons
"Sold" indicates we need to subtract
12 - 4 = 8 (counting back: 12, 11, 10, 9, 8)
The store has 8 balloons left
8 balloons
• Context identification: Recognize mathematical operations in real-world scenarios
• Problem translation: Convert verbal descriptions to mathematical expressions
• Application: Use mathematical skills to solve practical problems
• Practice Tip: Real-world problems help understand the relevance of math skills
- Cookies: 10 - 3 = 7 cookies left
- Books: 15 - 6 = 9 books remaining
- Apples: 13 - 5 = 8 apples left
- Real world problems often use words like "left," "remaining," "sold," or "used"
- Always include units (balloons, dollars, etc.) in your final answer
- Estimate first to check if your answer is reasonable
Q: How do I know if my answer makes sense in real-world problems?
A: Estimate first and check if your exact answer is close to your estimate. For example, 12 - 4 ≈ 8, which is reasonable.
Q: What if the problem has extra information?
A: Focus only on the numbers and operations needed to answer the specific question asked.
Challenge problem: A complex problem that combines multiple operations and requires careful attention to detail and sequential reasoning.
Note: Challenge problems help students synthesize multiple skills and build confidence in tackling difficult mathematical situations.
- Identify the initial quantity (14 students)
- Process each event in chronological order
- Apply the appropriate operation for each event
- Track the running total after each operation
- State the final answer in the context of the problem
There were 14 students in class at the beginning
3 students went to the office: 14 - 3 = 11 students
2 new students joined: 11 + 2 = 13 students
4 students went home: 13 - 4 = 9 students
There are 9 students in class at the end of the day
9 students
• Sequential processing: Handle events in the order they occur
• Operation identification: "Went" means subtraction, "joined" means addition
• Running total: Keep track of the current count after each event
• Practice Tip: Challenge problems build confidence and fluency
- Birds: 12 - 2 + 3 - 4 = 9 birds
- Cars: 15 - 4 + 2 - 3 = 10 cars
- Books: 18 - 6 + 4 - 2 = 14 books
- Process events in the order they happen chronologically
- Keep track of your running total after each event
- Always verify your answer makes sense in the context
Q: What if there are many changes in a problem?
A: Handle them one at a time in the order they occur, keeping track of your current total after each change.
Q: Can I do multiple operations at once?
A: It's safer to do them one at a time to avoid mistakes, especially when learning.