Visualizing Parity
Dot pairing for odd and even sums.
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Adding two odd numbers combines their leftover items into a complete pair, making the sum even.
Students will derive rules for parity addition and express parity patterns algebraically.
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Introduce the sum of 5 odd numbers puzzle.
Kishor is working on a puzzle with 5 empty boxes. He has a set of number cards containing only odd numbers (e.g., 3, 5, 7, 9, 11, 13). His goal is to place exactly one card in each box so that the 5 numbers add up to exactly 30.
Dot pairing for odd and even sums.
Adding two odd numbers combines their leftover items into a complete pair, making the sum even.
Worked example showing how to add multiple parities.
Problem. Find the parity of the sum of 2 even numbers and 2 odd numbers.
(For example: )
Faded practice for adding multiple odds and evens.
Let's find the parity of the sum of two odd numbers and three even numbers by grouping them step-by-step. First, we add the two odd numbers together. We know from pairing dots that results in an number. Next, we group the three even numbers. Adding any amount of even numbers together, like , always yields an number. Now we just add our two intermediate results. We are adding an even number to an even number. Therefore, the final parity of the entire sum must be .
Introduce 2n and 2n-1.
Consider the counting sequence of even numbers:
Notice that every even number is exactly double its position in the sequence. To find the even number at any position , we simply multiply the position by . Thus, the even number is written algebraically as .
Algebraic expressions for parity.
Substitute into the odd number formula:
Stepwise practice identifying parity of an algebraic expression.
State the algebraic expression we are evaluating.
Show your substitution and calculation for n = 3.
Is the result for n = 3 even or odd?
Show your substitution and calculation for n = 8.
Is the result for n = 8 even or odd?