Solved Exercises on Counting Integer Solutions in Grade 1

Master counting integer solutions: addition and subtraction strategies with visual methods through these 3 detailed exercises.

Solution: Exercises 1 to 3
1 Counting integer solutions
Exercise 1
Find all pairs of whole numbers (a, b) where a + b = 5 and both a and b are greater than 0. How many solutions are there?
Definition:

Integer solutions: Whole number answers that satisfy an equation

Solution method:
  1. List all possible values for the first number
  2. Calculate what the second number must be
  3. Check if both numbers are positive
  4. Count the valid pairs
1
+
4
=
5
(1, 4)
1 + 4 = 5
(2, 3)
2 + 3 = 5
(3, 2)
3 + 2 = 5
(4, 1)
4 + 1 = 5
Step 1: List possible values for a

a can be 1, 2, 3, or 4 (not 0 or 5 because b must also be positive)

Step 2: Calculate corresponding b values

When a = 1, b = 5 - 1 = 4

When a = 2, b = 5 - 2 = 3

When a = 3, b = 5 - 3 = 2

When a = 4, b = 5 - 4 = 1

Step 3: Verify both numbers are positive

All pairs (1,4), (2,3), (3,2), (4,1) have positive values

Solutions: (1,4), (2,3), (3,2), (4,1)
Final answer:

There are 4 integer solutions where a + b = 5 and both are positive.

Applied rules:

• Systematic listing: Go through values in order

• Constraint checking: Verify both numbers meet requirements

• Verification: Check that each pair satisfies the equation

2 Subtraction solutions
Exercise 2
Find all pairs of whole numbers (x, y) where x - y = 3 and both x and y are positive numbers less than 10. How many solutions are there?
Definition:

Subtraction solutions: Pairs where the first number minus the second equals the target

4
-
1
=
3
4
→
3
2
1
1
Step 1: List possible values for y

y can be 1, 2, 3, 4, 5, 6 (since x = y + 3 must be less than 10)

Step 2: Calculate corresponding x values

When y = 1, x = 1 + 3 = 4

When y = 2, x = 2 + 3 = 5

When y = 3, x = 3 + 3 = 6

When y = 4, x = 4 + 3 = 7

When y = 5, x = 5 + 3 = 8

When y = 6, x = 6 + 3 = 9

Step 3: Verify both numbers are positive and less than 10

All pairs satisfy the conditions

Solutions: (4,1), (5,2), (6,3), (7,4), (8,5), (9,6)
Final answer:

There are 6 integer solutions where x - y = 3 and both are positive numbers less than 10.

Applied rules:

• Reorganization: Rewrite subtraction as addition: x - y = 3 becomes x = y + 3

• Bounded constraints: Consider upper and lower limits

• Systematic enumeration: List possibilities in order

3 Sum with constraints
Exercise 3
Find all pairs of whole numbers (m, n) where m + n = 7 and m is greater than n. How many solutions are there?
Definition:

Constrained solutions: Pairs that satisfy both the equation and additional conditions

4
+
3
=
7
4
→
5
6
7
7
Step 1: List all pairs where m + n = 7

(1,6), (2,5), (3,4), (4,3), (5,2), (6,1)

Step 2: Apply the constraint m > n

(1,6): 1 > 6? No

(2,5): 2 > 5? No

(3,4): 3 > 4? No

(4,3): 4 > 3? Yes

(5,2): 5 > 2? Yes

(6,1): 6 > 1? Yes

Step 3: Count valid solutions

(4,3), (5,2), (6,1) satisfy both conditions

Solutions: (4,3), (5,2), (6,1)
Final answer:

There are 3 integer solutions where m + n = 7 and m is greater than n.

Applied rules:

• Two-step approach: First find all solutions, then apply constraints

• Verification: Check each potential solution against all conditions

• Order matters: The constraint m > n eliminates half the solutions

Detailed Summary: Counting Integer Solutions

Definitions of Key Concepts

Integer Solutions: Whole number answers that satisfy an equation. In grade 1, these are positive counting numbers (1, 2, 3, etc.).

Equation: A mathematical statement showing that two expressions are equal

Variable: A letter that represents an unknown number

Constraint: A condition that limits what values the variables can have

Core Rules and Principles

Rule 1: Always check that your solution makes the equation true.

Rule 2: Verify that all constraints are satisfied.

Rule 3: Systematically list possibilities to ensure you don't miss any solutions.

Step-by-Step Methods

  1. Read carefully: Identify the equation and all constraints
  2. Isolate variables: Solve for one variable in terms of the other
  3. List possibilities: Go through potential values systematically
  4. Check constraints: Verify each potential solution meets all conditions
  5. Count solutions: Tally the valid solutions

Examples - Simple to Advanced

Simple Example: a + b = 3, a > 0, b > 0

Possible values: (1,2), (2,1)

Count: 2 solutions

Medium Example: x + y = 6, x ≥ 2, y ≥ 1

Possible values: (2,4), (3,3), (4,2), (5,1)

Count: 4 solutions

Advanced Example: m + n = 8, m > n, m ≤ 6

All pairs: (1,7), (2,6), (3,5), (4,4), (5,3), (6,2), (7,1)

With m > n: (5,3), (6,2)

With m ≤ 6: (5,3), (6,2)

Count: 2 solutions

Tips, Tricks, and Common Pitfalls

Tip 1: Start with the smallest possible value and work upward systematically.
Tip 2: Use visual aids like number lines to understand relationships.
Tip 3: Make a table to organize possible values and check them off as you go.
Pitfall 1: Don't forget to check that your solutions satisfy all constraints.
Pitfall 2: Be careful not to include zero or negative numbers when the problem requires positive integers.

Key Notes for Memorization

  • Always verify your solutions by substituting back into the original equation
  • Constraints reduce the number of possible solutions
  • Systematic listing prevents missing solutions
  • For a + b = N, there are typically N-1 solutions when both must be positive
  • Order matters when constraints like m > n are applied
  • Check boundary conditions (minimum and maximum values)
Visual Counting Strategies
Counting Integer Solutions
Systematic Enumeration Methods
Table Method
a b a + b Valid?
1 4 5 Yes
2 3 5 Yes
3 2 5 Yes
4 1 5 Yes
Number Line Method
1
2
3
4
5
6
7
8
9
10

For x - y = 3: If y = 2, then x = 5 (move 3 steps right)

Verification Method
4
+
3
=
7

Check: Does 4 + 3 = 7? Yes ✓

Check: Is 4 > 3? Yes ✓

Therefore (4,3) is valid

Constraint Application
Filtering Valid Solutions
Solution Filtering

Starting set: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1)

Apply constraint m > n:

Keep: (4,3), (5,2), (6,1)

Discard: (1,6), (2,5), (3,4)

Final count: 3 solutions

Counting Laws, Methods, Rules, and Definitions
Counting Integer Solutions
Fundamental Counting Principles
Key definitions:

Integer: A whole number (positive, negative, or zero)

Solution: A value that makes an equation true

Positive integer: A whole number greater than zero

Complete counting methodology:
  1. Identify the equation: Understand what relationship must be satisfied
  2. Identify constraints: Note any limitations on the variables
  3. Systematic enumeration: List possible values in an organized way
  4. Verification: Check each possibility against all conditions
  5. Counting: Tally the valid solutions
Tip 1: Always write down the equation and constraints clearly before starting.
Tip 2: Use tables to organize your work and avoid missing solutions.
Tip 3: Check your final count by verifying that you haven't skipped any values.
Tip 4: Remember that order matters when there are inequality constraints.
Counting laws: Each valid solution is counted once, constraints reduce the solution set, systematic approach ensures completeness.
Best practices: List solutions in order, verify each solution, double-check boundary cases.
Counting principles to remember:

• Completeness: Include all possible solutions

• Uniqueness: Count each solution only once

• Verification: Check that each solution satisfies all conditions

• Systematic approach: Use ordered methods to avoid omissions

• Constraint awareness: Apply all given restrictions

• Accuracy: Verify the final count

Questions & Answers

Question: My child is struggling with finding all the solutions when there are constraints. How can I help them be more systematic?

Answer: Help your child develop a systematic approach:

  • Make a plan: Decide to start with the smallest possible value and work up
  • Use a table: Create columns for each variable and check each possibility
  • Check off: Mark each value as you consider it to avoid repeats
  • Visual aids: Use number lines or counters to represent the relationships

For example, with a + b = 5: Start with a = 1, find b = 4. Then a = 2, find b = 3. Continue until a = 4, b = 1. This ensures no solutions are missed!

The key is developing a consistent method they can rely on.

Question: Sometimes I count too many solutions. How do I know when to stop?

Answer: Set clear boundaries before you start counting:

  • Minimum values: What's the smallest number allowed?
  • Maximum values: What's the largest number allowed?
  • Logical limits: If a + b = 5 and both must be positive, neither can be 5 or more

Example: For a + b = 5 with positive integers:

a can be 1, 2, 3, or 4 (not 0 or 5)

Once you reach the maximum possible value, you're done!

Also, check if continuing would violate the original equation.

Question: How can I help my students understand that constraints reduce the number of solutions?

Answer: Use concrete examples to demonstrate how constraints filter solutions:

  • Start broad: Show all possible solutions without constraints
  • Apply filters: Visually cross out solutions that don't meet criteria
  • Count remaining: Show how the count decreases
  • Real-world context: Use scenarios like "How many ways to share 5 cookies?" vs "How many ways if I get more than you?"

For example, with a + b = 6: All solutions are (1,5), (2,4), (3,3), (4,2), (5,1) = 5 solutions

With constraint a > b: Only (4,2), (5,1) = 2 solutions

Students can literally see solutions being eliminated.