Mathematics
Grade 9
15 min
Which even or odd number comes before or after?
Which even or odd number comes before or after?
Tutorial Preview
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Introduction & Learning Objectives
Learning Objectives
Represent any even number algebraically as `2n` and any odd number as `2n + 1` for some integer `n`.
Derive the algebraic expression for the even or odd number that comes before or after a given algebraic representation.
Set up and solve linear and simple quadratic equations involving consecutive even or odd integers.
Use algebraic representations to prove basic properties of even and odd numbers (e.g., the sum of two odd numbers is even).
Analyze the parity (evenness or oddness) of polynomial function outputs for generalized even or odd integer inputs.
Distinguish between representing consecutive integers (`x, x+1, x+2`) and consecutive even/odd integers (`x, x+2, x+4`).
Ever noticed how house numbers on one side of a street are all even or all odd? 🤔 H...
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Key Concepts & Vocabulary
TermDefinitionExample
Even NumberAn integer that is divisible by 2. Algebraically, it is any number that can be expressed in the form `2n`, where `n` is an integer.If `n = -4`, the even number is `2(-4) = -8`. If `n = 5`, the even number is `2(5) = 10`.
Odd NumberAn integer that is not divisible by 2. Algebraically, it is any number that can be expressed in the form `2n + 1`, where `n` is an integer.If `n = 3`, the odd number is `2(3) + 1 = 7`. If `n = -1`, the odd number is `2(-1) + 1 = -1`.
ParityThe property of an integer of being either even or odd. Two integers have the same parity if they are both even or both odd.The numbers 12 and -4 have the same parity (both even). The numbers 9 and 11 have the same parity (both odd).
Consecutive Even IntegersEven integers that follow each other...
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Core Formulas
Finding the Next/Previous Even Number
Given an even number `E = 2n`:
Next Even: `E + 2 = 2n + 2`
Previous Even: `E - 2 = 2n - 2`
Use this rule when you need to find an even number that comes immediately before or after another even number. The key is to add or subtract 2, not 1.
Finding the Next/Previous Odd Number
Given an odd number `O = 2n + 1`:
Next Odd: `O + 2 = (2n + 1) + 2 = 2n + 3`
Previous Odd: `O - 2 = (2n + 1) - 2 = 2n - 1`
Use this rule to find an odd number immediately before or after another odd number. Just like with even numbers, the difference is 2.
Finding the Next/Previous Integer (Parity Change)
Given any integer `k`:
Next Integer: `k + 1`
Previous Integer: `k - 1`
Use this when you need the very next or previous integer, regardless of parit...
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Challenging
Which statement provides a valid algebraic proof that the product of any two odd integers is always odd?
A.3 * 5 = 15. Since 3 and 5 are odd and 15 is odd, the statement is true.
B.Let the integers be x and x+2. Then x(x+2) = x^2+2x. This is always even.
C.Let the integers be 2n+1 and 2m+1. Their product is 4nm + 2n + 2m + 1 = 2(2nm+n+m) + 1, which is in the form 2k+1.
D.Let the integers be 2n+1 and 2m+1. Their product is (2n+1)(2m+1). Since it contains +1, the result must be odd.
Challenging
Let `f(x) = x^2 + bx + c`, where `b` and `c` are integers. If `f(x)` is always an even integer whenever `x` is an even integer, what must be true about the coefficient `c`?
A.c must be odd.
B.c must be even.
C.c can be any integer.
D.c must be zero.
Challenging
The average of four consecutive odd integers is 24. What is the product of the smallest and largest of these integers?
A.597
B.567
C.621
D.525
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