## How did algebra develop and why do we use it?

### Key questions

1. 1

How do we recognise and work with different types of equations and inequalities?

2. 2

How can we manipulate algebraic expressions involving surds, indices, and fractions?

3. 3

How are equations formed and what do they mean?

4. 4

How can algebra be used to generalise our thinking?

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#### Introducing...

Resource type Title
Many ways problem Two-way algebra
Many ways problem Which is bigger?

#### Developing...

Resource type Title
Fluency exercise Equation Sudoku
Fluency exercise Multiple manipulations
Package of problems Index issues
Package of problems Nested surds
Problem requiring decisions Real-world problems?
Food for thought $n^5 - n$
Food for thought Comparing continued fractions
Food for thought Powerful fractions

#### Review questions

Title Ref
Can we factorise $(x+y)^6 - (x-y)^6$ completely? R9404
Can we find the dimensions of this rectangle? R6127
Can we find the radii of two circles given two clues? R7032
Can we fully factorise $x^4+4y^4$? R8219
Can we list all $p/q$ between $0$ and $1$ if $2 \le q \le 5$? R9354
Can we prove that, for a triangle, $4(s-b)(s-c) \le a^2$? R8056
Can we prove these bounds for this fraction involving surds? R7077
Can we prove these inequalities involving $a, b, c$ and $d$? R7865
Can we show this surd is less than $6$? R5701
Can we simplify these expressions involving indices? R8779
Can we simplify these algebraic fractions? R5364
Can we solve these equations in $x, y, z, a$ and $b$? R6155
Can we solve these simultaneous equations of degree 1 and 2? R6797
Can we solve these two equations involving $x,y,1/x$ and $1/y$? R6694
Given that $a, b > 0$, can we prove that $a/b + b/a \ge 2$? R7354
Given these power facts, can we show $z^6=cx$? R9160
How long was the walk to Glastonbury Tor? R6121
If these equations have an infinity of solutions, how are $a$ and $b$ related? R6327
If we know the area-sum and the perimeter-sum, what are the sides? R5505
When do these simultaneous equations have a positive solution? R8158
When is the inequality $x^4<8x^2+9$ satisfied? R9603
When is this product of powers an integer? R6983