Unit 8: Problem Solving · Lesson 7
Problem Solving - Lesson 7
What this lesson is for
- Students will understand and practice Problem Solving - Lesson 7
- Students will solve problems related to this topic
Train up a child in the way he should go: and when he is old, he will not depart from it.
Proverbs 22:6
Number Theory Problems: Consecutive Integers
Number theory problems — involving integers, their relationships, and divisibility — have fascinated mathematicians since antiquity. They sharpen algebraic reasoning and logical precision.
Consecutive Integer Vocabulary:
- Consecutive integers: n, n+1, n+2, ...
- Consecutive even integers: n, n+2, n+4, ... (where n is even)
- Consecutive odd integers: n, n+2, n+4, ... (where n is odd)
Note: Both consecutive even and odd integers are represented as n, n+2, n+4, etc. The distinction is what value n starts at.
Example 1: The sum of three consecutive integers is 75. Find them. n + (n+1) + (n+2) = 75 3n + 3 = 75 3n = 72 n = 24 The integers are 24, 25, 26.
Example 2: The product of two consecutive even integers is 168. Find them. n(n+2) = 168 n² + 2n − 168 = 0 Discriminant: 4 + 672 = 676 = 26² n = (−2 + 26)/2 = 12 The integers are 12 and 14. (12 × 14 = 168 ✓)
Example 3: The larger of two consecutive odd integers is 3 more than twice the smaller. Find them. Let n = smaller (odd). Larger = n + 2. n + 2 = 2n + 3 −n = 1 n = −1 The integers are −1 and 1.
Number Relationship Problems: Example: One number is 4 more than three times another. Their sum is 36. Find both numbers. Let x = smaller. Other = 3x + 4. x + (3x + 4) = 36 4x = 32 x = 8; other = 28.
Practice:
- The sum of four consecutive integers is 66. Find them.
- The sum of the squares of two consecutive positive integers is 85. Find the integers.
- Three consecutive odd integers have a sum of 33. Find them.
- One number is 5 less than twice another. Their sum is 40. Find both.
Guided practice
The sum of three consecutive integers is 63. Find all three integers.
Show the steps and the answer
Let the integers be n, n+1, n+2. Sum: n + (n+1) + (n+2) = 63. Simplify: 3n + 3 = 63. Solve: 3n = 60, so n = 20. The integers are 20, 21, 22.
Answer: 20, 21, 22
The product of two consecutive even integers is 48. Find them.
Show the steps and the answer
Let consecutive even integers be n and n+2. Equation: n(n+2) = 48. Expand: n² + 2n - 48 = 0. Factor: (n+8)(n-6) = 0. So n = -8 or n = 6. This gives -8, -6 or 6, 8.
Answer: −8 and −6 (or 6 and 8)
Three consecutive odd integers have a sum of 57. Find them.
Show the steps and the answer
Let odd integers be n, n+2, n+4. Sum: n + (n+2) + (n+4) = 57. Simplify: 3n + 6 = 57. Solve: 3n = 51, so n = 17. The integers are 17, 19, 21.
Answer: 17, 19, 21
Homework
Complete the practice problems for Problem Solving - Lesson 7.
Note for the parent
Today's lesson covers Problem Solving - Lesson 7. Estimated time: 55 minutes. Encourage independent work but be available for questions.
The first three of this lesson's 10 questions
Three consecutive integers can be represented as n, n+1, and n+2. If their sum is 99, the equation is:
Show the answer
Answer: 3n + 3 = 99
Solving 3n + 3 = 99 for consecutive integers summing to 99:
Show the answer
Answer: n = 33, giving 33, 34, 35
Consecutive EVEN integers differ by:
Show the answer
Answer: 2
Another 7 questions follow in the lesson itself, each one graded the moment it is answered.