# Advanced Maths Lesson Starters

Here is a list of advanced Maths lesson starters suitable for students studying for A Level, IB, Advanced placement etc.

In addition, many of the 'Starter of the Day' activities are easily developed for the more able student.

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### Advanced Starters:

Algebraic Product: Finding the value of the expression is easier than you think!  [Topic(s): Algebra]

Angle Thinking: Find the range of possible angles, x, for which tan x > cos x > sin x  [Topic(s): Trigonometry Inequalities]

Average Cycling Speed: Work out the average speed of two journeys. The obvious answer is not the correct answer.  [Topic(s): Average speed]

Back To The Factory: Find all the numbers below 1000 which have exactly 20 factors  [Topic(s): Factors, Possibilities, Prime, Number]

Barmy BIDMAS: A misleading way of stating the answer to a simple calculation.  [Topic(s): BIDMAS Factorial]

Bertrand's Box Paradox: Bertrand's box paradox is a paradox of elementary probability theory, first posed by Joseph Bertrand in 1889  [Topic(s): Conditional probability]

Best Dice: Which of the unusual dice would you choose to give you the best chance of winning the prize?  [Topic(s): Probability]

Biased Coin: Use a biased coin to obtain a fair result  [Topic(s): Probability]

Car Inequalities: Solve three simultaneous inequalities to find how many cars I own.  [Topic(s): Inequality, Inequalities, natural numbers]

Charging Rhinos: Find the easy way to solve this kinematics problem involving a fly and two rhinos.  [Topic(s): Speed Distance Time]

Cheryl's Birthday: Use a process of elimination to work out the correct date from the clues given.  [Topic(s): Logic]

Coordinate Distance: Find k given that(-2,k) is 13 units away from (10,9)  [Topic(s): Coordinate Geometry, Quadratics, Coordinates]

Cuboid: Find the dimensions of a cuboid matching the description given  [Topic(s): Measures]

Curved Angles: How many right angles are there in a semicircle?  [Topic(s): Angles]

Difference Cipher: Find the mathematical word from the cipher  [Topic(s): code, cipher, number]

Divisible by 11: Can you prove that a three digit number whose first and third digits add up to the value of the second digit must be divisible by eleven?  [Topic(s): Algebra, Place Value]

Double or Half?: At ten percent change per day is doubling achieved faster than halving?  [Topic(s): Percentages]

Exceeds by 99: Find the number whose double exceeds its half by exactly 99.  [Topic(s): Algebra]

Fence Optimisation: Find the length of a rectangle enclosing the largest possible area.  [Topic(s): Differentiation Calculus Optimisation]

Fermi Problem: The classic Fermi problem using standard estimation techniques  [Topic(s): Fermi, Estimate, Standard Form]

Find The Radius: Find the radius of the circle from the small amount of information provided.  [Topic(s): Geometry Trigonometry Algebra]

Four Fraction Division: Explain why the answer to a series of fraction divisions is a whole number.  [Topic(s): Fractions]

GDC Challenge: Produce the given graph on a graphic display calculator  [Topic(s): Graph]

Geometry Snack: Find the value of the marked angle in this diagram from the book Geometry Snacks  [Topic(s): Angles, Geometry]

Grandmother: How far would grandma have travelled after a suitably large number of days given her walking regime?  [Topic(s): Sequences and series]

Hands Together: The hands of a clock are together at midnight. At what time are they next together?  [Topic(s): Angles, Speed]

HCF and LCM given: If given the HCF, LCM and the smaller of two numbers can you find the other?  [Topic(s): HCF, LCM, Factor, Multiple]

How Many Left Handers?: Work out the number of members if the probability of left-handed members being randomly selected is given.  [Topic(s): Probability]

Hundred and Fifty Percent: Divide 110 into two parts so that the larger part is 150% of the smaller part.  [Topic(s): Percentages Ratio]

Key Eleven: Prove that a four digit number constructed in a certain way will be a multiple of eleven.  [Topic(s): Number, Proof, Algebra]

Log Perfection: Determine whether the given statements containing logarithms are true or false  [Topic(s): Logarithms, Logs]

Logarithm Equation: Solve an equation containing logarithms with different bases  [Topic(s): Logarithms, Logs]

Maximum Product: Two numbers add up to 10. What's the largest possible product they could have?  [Topic(s): Optimise Differentiate Calculus]

Multiple of sum of digits: What number is six times the sum of its digits?  [Topic(s): Number]

Nine Digit Numbers: How many different nine digit numbers are their that contain each of the digits from one to nine?  [Topic(s): Number]

Other Child's Gender: What is the probability that the other child is also a boy?  [Topic(s): Probability]

Paper Ratio: Calculate the ratio of the sides of an A4 sheet of paper without any measuring.  [Topic(s): Ratio]

Paper Surprising Perimeter: Find the perimeter of a folded sheet of A4 paper as described in this short video.  [Topic(s): Perimeter, Surds, Ratio]

Parallel Graphs: Determine from their equations which of the straight line graphs are parallel and perpendicular.  [Topic(s): Graphs, Coordinate Geometry]

Penny Bags: Can you place 63 pennies in bags in such a way that you can give away any amount of money (from 1p to 63p) by giving a selection of these prepacked bags?  [Topic(s): Puzzle, Binary]

Perennial Rivals: Which football team will be first to win four games?  [Topic(s): Probability]

Permutable Functions: Find pairs of functions that are commutative under composition.  [Topic(s): functions]

Piece of String: Find where a piece of string should be cut to form a circle and a square of equal areas.  [Topic(s): Equations, Problem Solving]

Pizza Slice: A problem which can be solved by considereing the areas of a triangle and a sector of a circle.  [Topic(s): Mensuration, Area, Trigonometry]

Product of Indices: Find the product of the unknown indices that feature in two equations  [Topic(s): Indices, Logarithms]

Quad Midpoints: What shape is created when the midpoints of the sides of a quadrilateral are joined together?  [Topic(s): Geometry, Proof, Vectors]

Restrained Flea: How many different places could the flea find itself after 8 foot-long jumps either north, south, east or west?  [Topic(s): Loci, Combinations]

Reverse Connection: Find a general rule for the difference between a two digit number and that same number with the digits reversed.  [Topic(s): Algebra, Place Value]

Rice on a Chess Board: How many grains of rice are on a chess board if each square has twice the number of grains as the previous square.  [Topic(s): Sequences and series]

Rich or Poor?: An interesting outcome of percentage increase and decrease  [Topic(s): Percentages]

Road Connections: Design roads to connect four houses that are on the corners of a square, side of length one mile, to minimise the total length of the roads.  [Topic(s): Calculus Trigonometry]

Same Series Sum: Find an arithmetic series and a geometric series that have the same sum of the first five terms.  [Topic(s): Sequences and series]

Same Three Digits: Find expressions which contain indices and evaluate to numbers containing the same digits.  [Topic(s): Puzzle, Indices, Problem Solving]

Seventeen Camels: Explain the mathematics of the classic ninteenth century fraction sharing story.  [Topic(s): Fractions]

Simplify: Simplify an algebraic fraction  [Topic(s): Indices]

Single Fraction: Simplify an expression involving fractions, exponents and a quare root.  [Topic(s): Fractions, Indices, Roots]

Speed Circles: Find the diameters of the circles in the corners of the square.  [Topic(s): Mensuration Circles]

Sphere Hole: Find the volume of the remaining part of a sphere after a 10cm cylindrical hole has been drilled through it.  [Topic(s): Volume, Mensuration]

Square in Rectangle: Find the area of a square drawn under the diagonal of a rectangle  [Topic(s): Algebra, area, similar triangles]

Square Root of 1%: What is the square root of one percent?  [Topic(s): Root, Percentage]

Tan 22.5: Find an exact value for tan 22.50 without using a calculator.  [Topic(s): Trigonometry]

Test Scores: Explore the misconception that when adding fractions you add both the numerators and the denominators  [Topic(s): Fractions]

Three Right Triangles: Calculate the lengths of the unlabelled sides of these right-angled triangles.  [Topic(s): Similarity]

Transum Tonic: What is the largest number of bottles that it is not possible to buy if they come in packs of 6, 9, and 20?  [Topic(s): Multiples]

Tri-Junction: A real life situation that can be analysed with the use of a tree diagram.  [Topic(s): Tree diagrams, probability]

Triangle or Quadrilateral: Can a quadrilateral have a straight angle?  [Topic(s): Shape Angle]

Trig Blot: Find all the measurements of the given triangle.  [Topic(s): Trigonometry]

Two Equals One: What is wrong with the algebraic reasoning that shows that 2 = 1 ?  [Topic(s): Equations Algebra]

Two real numbers: The sum of the reciprocals of two real numbers is -1, and the sum of their cubes is 4. What are they?  [Topic(s): Simultaneous Equations]

Unfinished Game: Share the prize in a fair ratio according to the probability of each player willing.  [Topic(s): Ratio, Probability]

Unlucky Seven Eleven: Follow the instructions to multiply a chosen number then explain the result you get.  [Topic(s): Number, Place Value]

Vowel Code: How many ways can you create a code for the vowels by assigning to each vowel a different vowel?  [Topic(s): Codes, Permutations]

Weather Reports: Which five different integers multiply together to give 12?  [Topic(s): Prime Factors, Negative numbers]

What Question?: Write down all the possible questions that could have been asked if this was the diagram provided in a mathematics textbook.  [Topic(s): Trigonometry]

X Divided by 2Y: Why do different calculators not agree on the order of operations?  [Topic(s): BIDMAS Substitution Calculator]

Zero Even: Prove that zero is an even number.  [Topic(s): Proof]

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