# 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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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]

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]

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]

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]

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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Complete Index of Starters

#### Roman Numerals Jigsaw

This is a wonderful activity for someone who does not know Roman numerals. By completing the activity an understanding of the symbols develops and a great sense of achievement is enjoyed.

### Suggestions

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#### Plus

A number arranging puzzle with seven levels of challenge. So far this activity has been accessed 18494 times and 1400 people have earned a Transum Trophy for completing it.

### Exam Style Questions

Still looking for an Advanced Starter? Try an Exam Style Question. There are hundreds to choose from.

Yes past paper questions can make ideal Lesson Starters. Have them projected onto your whiteboard and your students can start working as soon as they sit down. The (I)GCSE Higher/Extended and the IB questions have fully worked solutions that you can reveal line by line when you go through the questions with the class. All this saves you valuable time at the beginning of your lesson.

Here are some questions I like as Starters to get you going:

Two circles just fit side by side in a larger circle. What fraction of the larger circle do they cover? more...

One is added to the product of two consecutive positive even numbers. Show that the result is a square number. more...

The diagrams below show a sequence of patterns made from red and yellow tiles. more...

This Venn diagram shows the relationship between the sets of numbers. more...

Twitter and the media were buzzing with comments about a particular question about Hanna's sweets. more...

Calculate the area of the front face of the dart board. more...

A restaurant offers 6 starters, 14 main courses and 5 different desserts. more...

Find the length of the elastic band that fits around three crayons. more...

The graph shows the height of water in a container over a period time. more...

This draft emoji design includes a circle and a kite. more...

Minerva's Revenge is the longest rollercoaster in the Funville Theme Park. more...

Windthrup bought a car for £9500 which depreciated by 6% in the first year. more...

Match the equation with the letter of its graph. more...

### Maths Riddles:

Yes, mathematical riddles make great lesson starters too.

Animals at the Oasis: Calculate the number of animals gathered at the oasis according to the clues.  [Topic(s): Arithmetic]

Audible Riddles: Listen to these audible mathematical riddles and use your ingenuity to figure out the answers.  [Topic(s): Audible]

Bus Driver: A riddle about a bus driver on quite a busy route.  [Topic(s): Calculation]

Calendar Clue: Find the murderer from the clue written on the calendar  [Topic(s): Calendar]

Calendar Riddle: Work out the date of my birthday from the clues in rhyme.  [Topic(s): Number, Factors, Multiples, Square]

Company or Crowd?: If two's company and three's a crowd, what's four and five?  [Topic(s): Riddle]

Cube Ages: Calculate the mean age of the two fathers and two sons with the given clues.  [Topic(s): Cube Numbers, Average, Mean]

Doctor's Son: The doctor's son's age does not add up!  [Topic(s): Inverse]

Does not add up!: Why would eight and five add up to one?  [Topic(s): Addition]

Fractions of Animals: What is three sevenths of a chicken, two thirds of a cat and 50% of a goat?  [Topic(s): Fractions]

Fried Eggs: How many eggs remain after the events that took place in the kitchen?  [Topic(s): Riddle]

Growing Fast: How can it be that Percy will be 16 next year when he was only 13 the day before yesterday?  [Topic(s): Age, Time, Riddle]

Moses' Roses: How can ten roses be planted in five rows of four?  [Topic(s): Roses, Rows]

Rejuvenate: A girl was 16 years old in 2010 but just 8 years old in 2018  [Topic(s): Calendar]

Rope Ladder: How many rungs of a rope ladder remain above the water line after the tide has risen two metres?  [Topic(s): Ship]

Rosie's Cube: How many of the small cubes does Rosie need to complete the big cube?  [Topic(s): Volume 3D Shape]

Sharing Apples: Share five apples between five children while leaving one in the basket.  [Topic(s): Division]

Sibling Count: How many boys and how many girls are there in the family?  [Topic(s): Fractions]

Smart Animals: Who is most likely to be able to work out the square root of 121?  [Topic(s): Square root]

Sums With Hundreds: When is eighteen hundred minus fifty the same as seventeen hundred plus ten?  [Topic(s): Hundreds]

The Blue Arrows: What is the smallest number of jets that could have been in the formation according to the sighting information?  [Topic(s): Planes, Jets]

Tricky Takeaway: Take away some of the letters in a word to get a surprising result.  [Topic(s): Letters, Subtract]

Two Coins: What two coins add up to 70p (one of them is not a 50p coin)  [Topic(s): Coins, Money]

Wandering Bear: A bear walks south, east then north and finds it is back where it started. What colour is the bear?  [Topic(s): Bear]

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