Odds & Evens
Introduction
In today's lesson, you will learn how to generalise about odd, even and consecutive numbers. You will need to be familiar with collecting like terms, multiplying algebraic terms and factorising algebraic expressions (click on the hyperlinks for videos of how to do them). Afterwards, you may wish to practise Collecting Like Terms, Algebraic Notations and Factorising (click on the hyperlinks for some self-marked practice).
In general, how do you express the following algebraically:
Even?
Odd?
Even + Even?
Even + Odd?
Odd + Odd?
Consecutive Evens?
Consecutive Odds?
To find out more, take a look at the powerpoint below:
Can you describe what the link between the pictures and the expressions is?
Try to use these expressions to prove/show different relationships and solve problems.
odds evens.pptxFurther Practice
Some of the essential skills introduced in this lesson are "substitution", "collecting like terms", "forming expressions", "multiplying algebraic terms" and "factorising". The relevant skills can be found on DrFrostMaths, CorbettMaths, MyiMaths and Eedi. Watch any video and/or go through any online lesson as you see fit.
Transum
Substitution:
Substitution Examples - not sure how substitution works, look at these examples.
Substitution - try this self-marked exercise. There are altogether 7 levels.
Connecting Rules - if you are given the values of x and y which of these equations is correct?
DiceGebra - a game for two players evaluating algebraic equations and inequalities.
Simplifying Algebra:
Algebra Matching - try this matching game with a friend become familiar with equivalent expressions.
Algebraic Notations - try this self-marked exercise. There are altogether 2 levels.
Collecting Like Terms - try this self-marked exercise. There are altogether 4 levels.
Algebragons - find the missing expressions in these partly completed algebraic arithmagon puzzles. There are altogether 5 levels.
Algebraic Perimeters - try this self-marked exercise. There are altogether 4 levels.
Brackets - try this self-marked exercise. There are altogether 10 levels, not all are appropriate.
Factorising - try this self-marked exercise. There are altogether 10 levels, only the first four are appropriate.
Changing the Subject - try this self-marked exercise. There are altogether 8 levels, not all are appropriate.
Writing Expressions - listen to the voice saying the algebraic expression then write it in its simplest form.
Superfluous -find a strategy to figure out the values of the letters used in these calculations.
For more goodies on algebra on transum, click on the hyperlink.
Extension
Don Steward
The task above is based on Don Steward's post of odd and even. For more of his goodies, take a look at: pre-algebra, algebra, algebra misconceptions, expressions, expressions forming, expressions tricks, expressions both ways.
Pyramids
In year 7, you learnt how to find the biggest and smallest top number by manipulating the bottom numbers. You also found rules to calculate the top number without the middle ones. Now in year 8 we revisit this topic but ask harder questions where you are required to manipulate algebraic expressions:
Look at the yellow pyramids on the right:
Work out the top expression in terms of the variables on the bottom row.
What pattern do you see? How might this pattern continue?
Test your conjecture with size 6, size 7 pyramids.
Explain in terms of the variables on the bottom row how you would make the top expression as large or as small as possible.
2. Now take some blank pyramids (you may want to start with a size 3 and work up to a size 5) and answer the following questions:
If all the numbers on the bottom row of a pyramid go up in a regular amount (arithmetic sequence), what is the relationship with the top number?
If all the numbers on the bottom row of a pyramid are multiplied by a common multiplier (geometric sequence), what is the relationship with the left-hand diagonal and the top number?
If all the numbers on the bottom row of a pyramid are consecutive triangular numbers, establish that the top number will always be the sum of two squares.
3. Take a look at the many pyramids below that involve manipulating algebraic expressions:
Pyramids - Task Sheet.pdfWalls (SMP Interact).pdfPyramids - Collecting Like Terms.pdf