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Level Six > Number and Algebra

# AO1: Form and solve linear equations and inequations, quadratic and simple exponential equations, and simultaneous equations with two unknowns.

Children
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## I'm So Sorry I Ate Chocolate

This unit explores the magnitudes of sides and angles of a triangle and leads to the discovery and proof of the Sine Rule. This Rule is then used to solve triangles, some of which arise in practical situations. Finally we compare the use of the Sine Rule and the Cosine Rule.

Note that this unit may contain more material than you need for some classes. You may want to choose only some of the sessions if you have not planned to allot five sessions on the Sine Rule and its relation to the Cosine Rule.

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## The Why and How of General Terms

This unit looks at two general equations associated with a sequence or pattern. These are (i) recurrence relations as general ways to see what happens between consecutive terms of a sequence; and (ii) the general term of a sequence as a function of the number of that term. Recurrence relations make it easy to generate a table of values for the sequence. A table can lead to finding the general term. But we also see how to use a geometric approach as an alternative way to finding this general term. The work in this unit is valid for any arithmetic progression, that is, any sequence where the difference between any pair of consecutive terms is the same.

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## The Power of Algebra

Here we look for patterns in number problems and use algebra as an efficient means of solving general problems.

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## Fibonacci II

In this unit we take a deeper look into Fibonacci sequences. Here we see situations other than rabbits that produce these numbers; a related set of numbers – the Lucas numbers; and the use of quadratic equations to find a general term of a sequence of numbers that is generated by a recurrence relation similar to Fibonacci’s. We also tie the unit into What Happens on Average? (Algebra, Level 5) and see what happens as the number of terms approaches infinity.

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## Pigs, Goats and Sheep

Use algebraic skills to solve a problem involving three variables.

Devise and use problem solving strategies to explore situations mathematically (be systematic, use algebra).

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## Square Milk Bottle Crates

Find a rule for a quadratic pattern

Justify or explain the rule for a quadratic pattern

Devise and use problem solving strategies to explore situations mathematically (be systematic, draw a diagram, use a table, use algebra).

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## Rectangular Milk Bottle Crate

Find and use patterns to solve a problem (algebraically or by using a table)

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Solve a problem in a number of ways, including using algebraic expressions.

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## Pizza Toppings 2

Make predictions using a rule

Use a systematic method to solve a problem

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## Diophantus I

Solve a problem in a number of ways, including using algebraic expressions

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## Diophantus II

solve a problem in a number of ways, including using algebraic expressions

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## Cosine Rule

This unit explores the magnitudes of sides and angles of a triangle and leads to the discovery and proof of the Cosine Rule. This Rule is then used to solve triangles, some of which arise in practical situations. We note that the Cosine Rule is a generalisation of Pythagoras’ Theorem.

Note that this unit may contain more material than you need for some classes. You may want to choose from parts of this unit if you have not allotted 5 sessions on the Cosine Rule to your class.

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## What is s?

Use guessing to make conjectures

Solve a problem using algebraic expressions

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## What is t?

Use guessing to make conjectures

Solve a problem using algebraic expressions

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## What is s and t?

Use guessing to make conjectures

Solve a problem using algebraic expressions

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## Peter's Third String

Determine the maximum area of a rectangle with a given perimeter

Consider how the area of a quadrilateral changes as its shape changes

Interpret a relationship from a graph

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## Karen's Second Tiles

Determine the maximum area of a regular polygon with a given perimeter

Appreciate the concept of limit as it applies to the area of regular n-gons and circle that both have the same perimeter

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## Polygonal Strings

This is a problem from the number and algebra strand.

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## The Old Chicken Run Problem

use algebraic equations to determine the maximum area of a rectangle with a given partial perimeter.

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Use algebra to simplify expressions

Use geometry to assist their algebra.

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## Weighing Time

Solve simultaneous equations

Devise and use problem solving strategies to explore situations mathematically (be systematic, use equipment, draw a diagram, use smaller cases, use algebra).

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## Difference Magic Squares

This is a problem from the number and algebra strand.

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## Egyptian Fractions

This unit has a brief look at what is known about Egyptian Fractions. These are unit fractions – fractions whose numerator is one. We look at how fractions can be represented in terms of Egyptian Fractions. Finally we suggest a web site that students might like to use as resource material for a small project.
During this unit, students gain considerable practice in adding and subtracting fractions. They also look for patterns and will need some algebra to handle these.

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## Babylonian Mathematics I

This unit uses a Babylonian clay tablet and the mathematics found on it as a catalyst to investigate a variety of mathematical ideas.  This same catalyst is also used for the unit: Babylonian Mathematics 2.  Areas under enlargement are discussed in the present unit, and lying behind the various activities in both these units is the idea of incommensurability, which means, roughly, ‘things which cannot be measured no matter how accurate the ruler’.

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## Babylonian Mathematics 2

This unit uses a Babylonian clay tablet and the mathematics found on it as a catalyst to investigate a variety of mathematical ideas.  This same catalyst is also used for the unit: Babylonian Mathematics 1.  Areas under enlargement are discussed in the present unit, and lying behind the various activities in both these units is the idea of incommensurability, which means, roughly, ‘things which cannot be measured no matter how accurate the ruler’

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## Discovery

Here we look for patterns in some number problems and see how far we can extend two basic ideas.  The main point of this unit is to produce conjectures although we will spend a little time on proofs.