Algebra Problems Extension
To view the problem statement, double click on the problem
title. This will open up the problem in a new window. From there, you can
download the full problemandsolution Word document.
Examples
Page Ref 
Problem
Title 
Objectives
Ref 
Description 
Key
Words 
114, 150, 158 
The
Sum of r ^{3} 
Formulae,
Proof, Index Laws 
1^{3}
+ 2^{3} + 3^{3} ... from a really simple diagram 
Formulae,
T_{n}, Predict, Verify 
115, 150 
Paper
Sizes 
Trial +
Improvement, Similarity 
A3, A4 paper etc 
Forming
equations 
120, 239 
Simpson's Rule and
the Volume of a Sphere 
Formulae 
This exercise leads the pupil through applications of
Simpson’s rule to the formula for the volume of a
sphere. It is intriguing that all these volumes are given exactly by the rule. 
Cubic
expression, Formulae, Like terms 
108, 120 
Continued
Square Roots 
Quadratic
Equations, Trial + Improvement 
Easy answers to
nastylooking
types 
Subject
of the formula, Iteration, Factorize 
105, 110, 120 
Large Perfect Squares 
Algebraic expressions,
Calculations 
1111² etc 
These are all based on the idea of squaring a bracket 
Partition,
Multiply out, Square, Verify 
119,
127 
A
Whole Diophantine 
Algebraic expressions 
Solve
some unusual Simultaneous Equations for Integer solutions only  adding
and factorising a little... 
Simultaneous
Equations, Linear Expressions 
121 
Target
Practice 
Algebraic expressions 
Prove
a simple result about rings on a circular target 
Multiply out, Square 
121, 186, 189 
How
Many Triangles 
Triangles,
Algebraic expressions 
How
many different triangles can you make with rods of 6 different lengths ? Good exercise in algebraic
manipulation. 
Formulae,
T_{n}, Predict, Verify 
121, 281, 285 
Domino
Match 
Probability,
Sample Space, Rearranging Algebra 
What's the
likelihood of any 2 dominoes matching each other ? Nice algebraic
cancelling ... 
Formulae,
Take out common factors 
127 
Simultaneous
Equations 
Simultaneous
Equations 
Symmetric, but nastylooking, Simultaneous
Equations yield beautifully simple answers if an
intelligent approach is taken to their 'adding and
subtracting'. 
Factorize,
Simultaneous Equations 
127, 139, 141 
Diophantine
Pairs 
Equations,
Linear Expressions 
Finding
all the Integer solutions to an Equation 
Change
Subject, Linear Expression 
91, 127, 189 
The
Magic of Pythagoras 
Forming
and Simultaneous Equations, Elimination 
This Problem is a variation on the Well in the
Courtyard problem... and has a lovely, surprising answer ! 
Simultaneous
Equations 
133,
141, 151, 159 
Peachy
Pyramids 
Use
ICT to explore nonlinear sequences 
Exploring
the beautiful square fruit pyramids that are seen on greengrocers' stalls 
Spreadsheet,
Sequence, Formula, Squares, ICT 
133,
141, 151, 159 
Plummy
Pyramids 
Use
ICT to explore nonlinear sequences 
Exploring
the morecomplex rectangular fruit pyramids that are seen on greengrocers'
stalls 
Spreadsheet,
Sequence, Formula, Squares, ICT 
129 
Absolute
Equations 
Simultaneous
Equations 
Introducing the x notation, and
solving simple equations using it 
Simultaneous
Equations, Graphical method 
57, 91, 133 
Power
Twins 
Index
Laws, Powers 
2 ^{4}
= 4 ^{2}  find some more examples ! 
Index
Laws, Power, Exponent 
133, 227, 234 
Overlapping
Squares 1 
Graphs, Trial +
Improvement 
This problem offers an
element of surprise in that the equation of the locus of possible solutions isn’t the
straight line that pupils may well expect... 
Graphs,
Subject of the formula, Reciprocal 
119, 171 
Swap
It and See 
Complex
expressions 
Why
is (0.43^2 + 0.57) equal to (0.57^2 + 0.43)? 
Brackets,
Quadratic graphs, 
143,
119 
Perfect
Cubes 
Complex
expressions 
Can
both (n + 3) and (n^2 + 3) be perfect cubes? 
Expand
Brackets, Tn, 
149, 161 
Toffee
Maths 
Function,
Linear sequence 
A
foldonitself process that shows up the nature of 'Chaos', using a simple
iterative function 
Rationals
+ Irrationals, Iterative Processes, Mappings, Spreadsheets 
150, 175 
Graphical
Convergence 
Functions,
Graphs, Proof 
A
mapping for x and y sends points to a convergent 'invariant point'  nice,
visual, algebraic exercise 
Sequence, Convergent, Predict, Verify 
108, 150 
Funny
Functions 
Functions,
Calculators, Proof 
Analyzing
the Calculator functions 'SQRT' and 'INT', looking at associativity etc 
Brackets,
Associative, Mappings 
109, 151, 155 
PC Plod... 
Generate nonlinear sequences 
How many
ways can a policeman patrol a grid of streets ? 
Sequence, Pascal's triangle, Notation 










