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  • Number
    • Arithmetic >
      • The Four Operations >
        • The Four Operations (QQI)
        • The Four Operations (10QQI)
        • The Four Operations (QQI Count Down)
        • The Four Operations (QQI Relay)
        • The Four Operations (QQI BINGO)
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        • The Four Operations (Video)
        • Timestables Square (QQI)
        • Grid Multiplication (QQI)
      • Missing Numbers >
        • Missing Numbers (QQI)
        • Missing Numbers (10QQI)
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        • Missing Numbers (QQI Relay)
        • Missing Numbers (QQI BINGO)
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      • Order of Operations >
        • Order of Operations (QQI)
        • Order of Operations (10QQI)
        • Order of Operations (QQI Count Down)
        • Order of Operations (QQI Relay)
        • Order of Operations (QQI BINGO)
        • Order of Operations (QQI Worksheets)
      • Powers of Ten >
        • Powers of Ten (QQI)
        • Powers of Ten (10QQI)
        • Powers of Ten (QQI Count Down)
        • Powers of Ten (QQI Relay)
        • Powers of Ten (QQI BINGO)
        • Powers of Ten (QQI Worksheets)
      • Decimal Operations >
        • Decimal Operations (QQI)
        • Decimal Operations (10QQI)
        • Decimal Operations (QQI Count Down)
        • Decimal Operations (QQI Relay)
        • Decimal Operations (QQI BINGO)
        • Decimal Operations (QQI Worksheets)
      • Rounding >
        • Rounding (QQI)
        • Rounding (10QQI)
        • Rounding (QQI Count Down)
        • Rounding (QQI Relay)
        • Rounding (QQI BINGO)
        • Rounding (QQI Worksheets)
      • Products and Sums (QQI)
      • Products and Sums (10QQI)
    • Fractions >
      • Cancelling Fractions >
        • Cancelling Fractions (QQI)
        • Cancelling Fractions (10QQI)
        • Cancelling Fractions (QQI Count Down)
        • Cancelling Fractions (QQI Relay)
        • Cancelling Fractions (QQI BINGO)
        • Cancelling Fractions (QQI Worksheets)
      • Mixed Numbers and Improper Fractions >
        • Mixed Numbers and Improper Fractions (QQI)
        • Mixed Numbers and Improper Fractions (10QQI)
        • Mixed Numbers and Improper Fractions (QQI Count Down)
        • Mixed Numbers and Improper Fractions (QQI Relay)
        • Mixed Numbers and Improper Fractions (QQI BINGO)
        • Mixed Numbers and Improper Fractions (QQI Worksheets)
      • Fractions of Amounts >
        • Fractions of Amounts (QQI)
        • Fractions of Amounts (10QQI)
        • Fractions of Amounts (QQI Count Down)
        • Fractions of Amounts (QQI Relay)
        • Fractions of Amounts (QQI BINGO)
        • Fractions of Amounts (QQI Worksheets)
      • Fraction Arithmetic >
        • Fraction Arithmetic (QQI)
        • Fraction Arithmetic (10QQI)
        • Fraction Arithmetic (QQI Count Down)
        • Fraction Arithmetic (QQI Relay)
        • Fraction Arithmetic (QQI BINGO)
        • Fraction Arithmetic (QQI Worksheets)
    • FDP >
      • Fraction Decimal Conversions Drill
    • Percentages >
      • Percentages of Amounts >
        • Percentages of Amounts (QQI)
        • Percentages of Amounts (10QQI)
        • Percentages of Amounts (QQI Count Down)
        • Percentages of Amounts (QQI Relay)
        • Percentages of Amounts (QQI BINGO)
        • Percentages of Amounts (QQI Worksheets)
        • Percentage of Amounts (Video)
      • Writing Numbers as a Percentage >
        • Writing Numbers as a Percentage (QQI)
        • Writing Numbers as a Percentage (10QQI)
        • Writing Numbers as a Percentage (QQI Count Down)
        • Writing Numbers as a Percentage (QQI Relay)
        • Writing Numbers as a Percentage (QQI BINGO)
        • Writing Numbers as a Percentage (QQI Worksheets)
        • Writing Numbers as a Percentage (Video)
      • Percentage Change >
        • Percentage Change (QQI)
        • Percentage Change (10QQI)
        • Percentage Change (QQI Count Down)
        • Percentage Change (QQI Relay)
        • Percentage Change (QQI Worksheets)
        • Percentage Change (Video)
      • Increase and Decrease by a Percentage >
        • Increase and Decrease by a Percentage (QQI)
        • Increase and Decrease by a Percentage (10QQI)
        • Increase and Decrease by a Percentage (QQI Count Down)
        • Increase and Decrease by a Percentage (QQI Relay)
        • Increase and Decrease by a Percentage (QQI BINGO)
        • Increase and Decrease by a Percentage (QQI Worksheets)
        • Increase and Decrease by a Percentage (Video)
      • Compound Interest and Simple Interest >
        • Compound Interest and Simple Interest (QQI)
        • Compound Interest and Simple Interest (10QQI)
        • Compound Interest and Simple Interest (QQI Count Down)
        • Compound Interest and Simple Interest (QQI Relay)
        • Compound Interest and Simple Interest (QQI BINGO)
        • Compound Interest and Simple Interest (QQI Worksheets)
        • Compound Interest and Simple Interest (Video)
      • Overall Percentage Change >
        • Overall Percentage Change (QQI)
        • Overall Percentage Change (10QQI)
        • Overall Percentage Change (QQI Count Down)
        • Overall Percentage Change (QQI Relay)
        • Overall Percentage Change (QQI BINGO)
        • Overall Percentage Change (QQI Worksheets)
      • Reverse Percentages >
        • Reverse Percentages (QQI)
        • Reverse Percentages (10QQI)
        • Reverse Percentages (QQI Count Down)
        • Reverse Percentages (QQI Relay)
        • Reverse Percentages (QQI BINGO)
        • Reverse Percentages (QQI Worksheets)
        • Reverse Percentages (Video)
      • Mixed Percentages >
        • Mixed Percentages (QQI)
        • Mixed Percentages (10QQI)
        • Mixed Percentages (QQI Count Down)
        • Mixed Percentages (QQI Relay)
        • Mixed Percentages (QQI BINGO)
        • Mixed Percentages (QQI Worksheets)
    • Factors and Multiples >
      • Number Properties (QQI)
      • Product of Primes >
        • Product of Primes (QQI)
        • Product of Primes (10QQI)
        • Product of Primes (QQI Count Down)
        • Product of Primes (QQI Relay)
        • Product of Primes (QQI BINGO)
        • Product of Primes (QQI Worksheets)
      • HCF and LCM >
        • HCF and LCM (QQI)
        • HCF and LCM (10QQI)
        • HCF and LCM (QQI Count Down)
        • HCF and LCM (QQI Relay)
        • HCF and LCM (QQI BINGO)
        • HCF and LCM (QQI Worksheets)
        • HCF and LCM (Video)
      • 100 Square Multiples (QQI)
      • 100 Square Types of Numbers (QQI)
    • Standard Form >
      • Standard Form Conversions >
        • Standard Form Conversions (QQI)
        • Standard Form Conversions (10QQI)
        • Standard Form Conversions (QQI Count Down)
        • Standard Form Conversions (QQI Relay)
        • Standard Form Conversions (QQI BINGO)
        • Standard Form Conversions 2 (QQI BINGO)
        • Standard Form Conversions (QQI Worksheets)
      • Standard Form Arithmetic >
        • Standard Form Arithmetic (QQI)
        • Standard Form Arithmetic (10QQI)
        • Standard Form Arithmetic (QQI Count Down)
        • Standard Form Arithmetic (QQI Relay)
        • Standard Form Arithmetic (QQI BINGO)
        • Standard Form Arithmetic (QQI Worksheets)
    • Ratio and Proportion >
      • Ratio (Video)
    • Surds >
      • Surds Activities >
        • Surds (QQI)
        • Surds (10QQI)
        • Surds (QQI Count Down)
        • Surds (QQI Relay)
        • Surds (QQI BINGO)
        • Surds (QQI Worksheets)
  • Algebra
    • Algebraic Manipulation >
      • Collecting Like Terms >
        • Collecting Like Terms (QQI)
        • Collecting Like Terms (10QQI)
        • Collecting Like Terms (QQI Count Down)
        • Collecting Like Terms (QQI Relay)
        • Collecting Like Terms (QQI BINGO)
        • Collecting Like Terms (QQI Worksheets)
      • Expanding Single Brackets >
        • Expanding Single Brackets (QQI)
        • Expanding Single Brackets (10QQI)
        • Expanding Single Brackets (QQI Count Down)
        • Expanding Single Brackets (QQI Relay)
        • Expanding Single Brackets (QQI BINGO)
        • Expanding Single Brackets (QQI Worksheets)
      • Factorising >
        • Factorising (QQI)
        • Factorising (10QQI)
        • Factorising (QQI Count Down)
        • Factorising (QQI Relay)
        • Factorising (QQI BINGO)
        • Factorising (QQI Worksheets)
      • Expanding Quadratic Brackets >
        • Expanding Quadratic Brackets (QQI)
        • Expanding Quadratic Brackets (10QQI)
        • Expanding Quadratic Brackets (QQI Count Down)
        • Expanding Quadratic Brackets (QQI Relay)
        • Expanding Quadratic Brackets (QQI BINGO)
        • Expanding Quadratic Brackets (QQI Worksheets)
      • Factorising Quadratics >
        • Factorising Quadratics (QQI)
        • Factorising Quadratics (10QQI)
        • Factorising Quadratics (QQI Count Down)
        • Factorising Quadratics (QQI Relay)
        • Factorising Quadratics (QQI BINGO)
        • Factorising Quadratics (QQI Worksheets)
        • Factorising Quadratic Expressions (Video)
        • Factorising Four Term Expressions (Video)
      • Indices >
        • Indices (QQI)
        • Indices (10QQI)
        • Indices (QQI Count Down)
        • Indices (QQI Relay)
        • Indices (QQI BINGO)
        • Indices (QQI Worksheets)
      • Completing the Square >
        • Completing the Square (QQI)
        • Completing the Square (10QQI)
        • Completing the Square (QQI Count Down)
        • Completing the Square (QQI Relay)
        • Completing the Square (QQI BINGO)
        • Completing the Square 2 (QQI BINGO)
        • Completing the Square (QQI Worksheets)
      • Algebraic Fractions >
        • Simplifying Algebraic Fractions (Video)
        • Adding and Subtracting Algebraic Fractions (Video)
        • Multiplying and Dividing Algebraic Fractions (Video)
    • Coordinates >
      • Coordinates (GGB)
      • Coordinate Battleship First Quadrant (GGB)
      • Coordinate Battleship All Four Quadrants (GGB)
      • 3D Coordinates (AGG)
    • Equations >
      • Linear Equations >
        • Solving Linear Equations >
          • Solving Linear Equations (QQI)
          • Solving Linear Equations (10QQI)
          • Solving Linear Equations (QQI Count Down)
          • Solving Linear Equations (QQI Relay)
          • Solving Linear Equations (QQI BINGO)
          • Solving Linear Equations (QQI Worksheets)
        • Solving Equations with Algebraic Fractions (Video)
      • Quadratic Equations >
        • Solving Quadratic Equations >
          • Solving Quadratic Equations (QQI)
          • Solving Quadratic Equations (10QQI)
          • Solving Quadratic Equations (QQI Count Down)
          • Solving Quadratic Equations (QQI Relay)
          • Solving Quadratic Equations (QQI BINGO)
          • Solving Quadratic Equations (QQI Worksheets)
        • Solving Quadratic Equations by Factorising (Video)
        • The Quadratic Formula (Video)
        • Problems Involving Quadratic Equations (Video)
      • Simultaneous Equations >
        • Solving Simultaneous Equations >
          • Solving Simultaneous Equations (QQI)
          • Solving Simultaneous Equations (10QQI)
          • Solving Simultaneous Equations (QQI Count Down)
          • Solving Simultaneous Equations (QQI Relay)
          • Solving Simultaneous Equations (QQI Relay Fixed)
          • Solving Simultaneous Equations (QQI BINGO)
          • Solving Simultaneous Equations (QQI Worksheets)
        • Solving Simultaneous Equations Graphically (Video)
        • Simultaneous Equations by Substitution (Video)
        • Simultaneous Equations by Elimination (Video)
        • Simultaneous Equations - One Non-Linear (Video)
    • Sequences >
      • Sequences Activity (QQI)
      • Sequences Activities >
        • Sequences (QQI)
        • Sequences (10QQI)
        • Sequences (QQI Count Down)
        • Sequences (QQI Relay)
        • Sequences (QQI BINGO)
        • Sequences (QQI Worksheets)
      • Generating Sequences (Video)
      • General Term for Linear Sequences (Video)
      • Simple Quadratic Sequences (Video)
      • General Term for Quadratic Sequences (Video)
      • General Term for Cubic Sequences (Video)
      • Geometric Sequences (Video)
      • Common Differences (QQI)
    • Graphs >
      • Straight Line Graphs >
        • Drawing Straight Line Graphs (GGB)
        • Gradient of a Line (GGB)
        • Gradient of a Line 2 (GGB)
        • Parallel Lines (GGB)
        • Perpendicular Lines (GGB)
        • y = mx + c Activity (GGB)
        • Battleships 1 (AGG)
        • Battleships 2 (AGG)
        • Battleships 3 (AGG)
        • Find the Lines 1 (AGG)
        • Regions in Graphs (Video)
      • Non-Linear Graphs >
        • Drawing Curves (GGB)
        • Quadratic Graphs Activity (GGB)
        • Finding Quadratic Functions (Video)
      • Graphs with a Casio GDC (Video)
    • Graph Transformations >
      • Graph Transformations 1 (GGB)
      • Graph Transformations 2 (GGB)
      • Graph Transformations 3 (GGB)
      • Graph Transformations 4 (GGB)
      • Graph Transformations 5 (GGB)
      • Graph Transformations 6 (GGB)
    • Functions >
      • Functions Introductions (Video)
      • Function Graphs and Important Points (Video)
      • Solving Unfamiliar Equations Using Functions (Video)
      • Function Notation Revision (Video)
      • Composite Functions (Video)
      • Inverse Functions (Video)
  • Shape
    • Symmetry >
      • Reflection Symmetry >
        • Reflection Symmetry in Quadrilaterals (GGB)
        • Reflection Symmetry in Triangles (GGB)
        • Reflection Symmetry in Other Shapes (GGB)
      • Rotational Symmetry >
        • Rotational Symmetry in Quadrilaterals (GGB)
        • Rotational Symmetry in Triangles (GGB)
        • Rotational Symmetry in Other Shapes (GGB)
    • Area and Perimeter >
      • Polygons >
        • Perimeters (GGB)
        • Area of a Triangle (GGB)
        • Area of a Parallelogram (GGB)
        • Area of a Trapezium (GGB)
        • Area of Compound Shapes (GGB)
        • Perimeter and Area (GGB)
      • Circles >
        • Discovering Pi (GGB)
        • Circumference of a Circle (GGB)
        • Area of a Circle (GGB)
        • Running Tracks (GGB)
        • Circle Area Problem (GGB)
        • Circles and Squares (GGB)
      • Area (QQI)
      • Area (10QQI)
      • Tilted Squares (GGB)
      • Difference Between Two Squares (GGB)
    • Volume and Surface Area >
      • Volumes and Surface Areas (QQI)
      • Volumes and Surface Areas (10QQI)
    • Angles >
      • Guess the Angle (GGB)
      • Angles on a Straight Line (GGB)
      • Angles around a Point (GGB)
      • Angles in a Triangle (GGB)
      • Angles in a Quadrilateral (GGB)
      • Angles in a Regular Polygon (GGB)
      • Angles on Parallel Lines (GGB)
      • Striping Angles (GGB)
    • Transformations >
      • Reflection >
        • Reflections (GGB)
        • Reflection Challenge (GGB)
      • Rotation >
        • Rotations (GGB)
        • Rotation Challenge (GGB)
      • Translation >
        • Translations (GGB)
        • Translation Challenge (GGB)
      • Enlargement >
        • Enlargements (GGB)
        • Enlargement Challenge (GGB)
        • Other Scale Factors (GGB)
      • Challenges >
        • Which Transformation (GGB)
        • How Many Transformations (GGB)
        • Find Them All (AGG)
        • Ultimate Challenge (GGB)
      • Matrix Transformations (AGG)
    • Pythagoras Theorem >
      • Pythagoras Theorem Activities >
        • Pythagoras Theorem (QQI)
        • Pythagoras Theorem (10QQI)
        • Pythagoras Theorem (QQI Count Down)
        • Pythagoras Theorem (QQI Relay)
        • Pythagoras Theorem (QQI BINGO)
        • Pythagoras Theorem (QQI Worksheets)
      • Pythagoras Theorem (GGB)
      • Pythagorean Triples (GGB)
      • Pythagoras Proof (GGB)
      • Ladders up Walls (GGB)
      • Pythagoras in 3D (GGB)
      • Finding the Hypotenuse Example (Video)
      • Finding a Shorter Side Example (Video)
    • Trigonometry >
      • Right Angled Trigonometry >
        • Right Angled Trigonometry (QQI)
        • Right Angled Trigonometry (10QQI)
        • Right Angled Trigonometry (QQI Count Down)
        • Right Angled Trigonometry (QQI Relay)
        • Right Angled Trigonometry (QQI BINGO)
        • Right Angled Trigonometry (QQI Worksheets)
        • Discovering Trig Ratios (GGB)
        • Finding Lengths (GGB)
        • Finding Missing Lengths (Video)
        • Finding Missing Angles (Video)
      • Sine Rule (Video)
      • Cosine Rule (Video)
      • Sine and Cosine Rules (Video)
    • Circle Theorems >
      • Angle in the Centre vs Angle at the Circumference (GGB)
      • Angle at the Centre vs Angle at the Circumference (Video)
      • Angles in a Semicircle (GGB)
      • Angle in a Semicircle (Video)
      • Angles in Cyclic Quadrilaterals (GGB)
      • Angles in a Cyclic Quadrilateral (Video)
      • Angles in the Same Segment (GGB)
      • Angles in the Same Segment (Video)
      • Tangents (GGB)
      • Tangents (Video)
      • Alternate Segment Theorem (GGB)
      • Intersecting Tangents (GGB)
      • Intersecting Tangents (Video)
      • Intersecting Chords (GGB)
    • Vectors >
      • Vectors and Scalars (Video)
      • Vector Notation (Video)
      • Resultant Vectors (Video)
      • Resultants of Column Vectors (Video)
      • Scalar Multiplication (Video)
      • Magnitude of a Vector (Video)
    • Miscellaneous >
      • Squares (GGB)
      • Tangrams (GGB)
      • Euler Line (GGB)
      • Geoboards
  • Statistics
    • Probability >
      • Probability (QQI)
      • Probability (10QQI)
      • Probability Tools (Flash)
    • Averages >
      • Averages Activity (QQI)
      • Listed Averages >
        • Listed Averages (QQI)
        • Listed Averages (10QQI)
        • Listed Averages (QQI Count Down)
        • Listed Averages (QQI Relay)
        • Listed Averages (QQI BINGO)
        • Listed Averages (QQI Worksheets)
        • Averages From Lists of Data (Video)
        • Quartiles and Interquartile Range (Video)
      • Averages from Frequency Tables >
        • Averages from Frequency Tables (QQI)
        • Averages from Frequency Tables (10QQI)
        • Averages from Frequency Tables (QQI Count Down)
        • Averages from Frequency Tables (QQI Relay)
        • Averages from Frequency Tables (QQI BINGO)
        • Averages from Frequency Tables (QQI Worksheets)
        • Averages From Frequency Tables (Video)
        • Averages From Grouped Frequency Tables (Video)
      • Averages With A GDC (Video)
    • Statistical Diagrams >
      • Cumulative Frequency (Video)
      • Scatter Graphs and the Mean Point (Video)
      • Scatter Graphs and Linear Regression on a GDC (Video)
      • Correlation and the Correlation Coefficient on a GDC (Video)
  • Post 16 Topics
    • Binomial Expansion >
      • Binomial Expansion (Video)
      • Binomial Theorem (Video)
      • Binomial Coefficients (Video)
      • Binomial Applications (Video)
    • Coordinate Geometry >
      • Coordinate Geometry (QQI)
      • Coordinate Geometry (10QQI)
      • Equation of a Circle (AGG)
    • Differentiation >
      • Differentiating Polynomials >
        • Differentiating Polynomials (QQI)
        • Differentiating Polynomials (10QQI)
        • Differentiating Polynomials (QQI Count Down)
        • Differentiating Polynomials (QQI Relay)
        • Differentiating Polynomials (QQI BINGO)
        • Differentiating Polynomials (QQI Worksheets)
      • Finding Gradients of Curves (QQI)
      • Finding Gradients of Curves (10QQI)
      • Finding Turning Points of Curves (QQI)
      • Finding Turning Points of Curves (10QQI)
    • Trigonometry >
      • Radian and Degree Conversions >
        • Radian and Degree Conversions (QQI)
        • Radian and Degree Conversions (10QQI)
        • Radian and Degree Conversions (QQI Count Down)
        • Radian and Degree Conversions (QQI Relay)
        • Radian and Degree Conversions (QQI BINGO)
        • Radian and Degree Conversions (QQI Worksheets)
      • Trigonometric Exact Values >
        • Trigonometric Exact Values (QQI)
        • Trigonometric Exact Values (10QQI)
        • Trigonometric Exact Values (QQI Count Down)
        • Trigonometric Exact Values (QQI Relay)
        • Trigonometric Exact Values (QQI BINGO)
        • Trigonometric Exact Values (QQI Worksheets)
      • Graphs of Trig Functions (GGB)
  • Starters, Puzzles and Enrichment
    • UKMT Random Question Generator
    • @mathschallenge Random Questions
    • School of Hard Sums Random Questions
    • Random Starter of the Day
    • Mathematically Possible (QQI Starter)
    • Adding Challenge (QQI Starter)
    • Date Starter (QQI Starter)
    • Name That Number (QQI Starter)
    • Matchstick Random Questions
    • Choose 3 Numbers (QQI Starter)
    • What's The Question (QQI Starter)
    • Mathematical Words (QQI Starter)
    • Number of the Day (QQI Starter)
    • Anagrams and Missing Vowels (QQI Starter)
    • Missing Vowels and Word Jumbles (QQI) >
      • Missing Vowels and Word Jumbles Simple Numbers (QQI)
    • Tables (QQI)
    • Target Boards (QQI)
    • Missing Signs (QQI)
    • Random Activities >
      • Exploding Dots
      • Easter Date
      • Easter Tangrams (GGB)
      • Zeller's Algorithm
      • Batman Equation (AGG)
      • Templates
    • Mathematical Videos >
      • Fermat's Last Theorem (Video)
      • Pi Song (Video)
      • Monty Hall Problem (Video)
      • Symmetry, Reality's Riddle (Video)
      • Music of the Primes (Video)
      • Folding Paper (Video)
      • Nature by Numbers (Video)
      • Inspirations (Video)

Mathematical Diversions

1/10/2020

0 Comments

 
Due to COVID-19, we are unable to run our IB and IGCSE exams this November. Local restrictions make it impossible. It has been tough for students and teachers to have 2 years worth of work count for nothing. Our students will not get an IGCSE set this year (as an international school, no systems in place like there were for the UK). The IB students will have grades awarded purely on coursework. 
The announcement that there would be no IB exams this year was made the day before I was due to finish teaching the syllabus, and my IB HL class were keen to finish that last little bit of vectors. But we still had two weeks left of term left, and with no exams, the usual rush of exam papers was pointless.
So between me and the other HL teacher we decided to offer two separate options: she taught the calculus option (we did a different option, but many students were interested in this) and I did a series of lessons on random mathematical diversions. 
Here I will share those diversions, along with the resources, in case anybody ever feels like using them.

Taxicab numbers
We started by looking at this problem.
Picture
I blogged about this problem in the past here when a student brought it to me. I ran the session basically as an open problem, with students in breakout rooms in Zoom, and me popping between them.

Benford's Law
Next we took a look at Benford's Law. 
I asked students to think about a set of data. They could choose anything, but I gave some ideas like
  • Populations
  • Number of goals scored in a season
  • Average playing time per player
  • House numbers on a street
  • Cost of items in a weekly shop
Once they all had an idea of their data set, I asked them to think about the first digits of each data point in the set, and to decide what the probability distribution would be for them. That is, what is the probability if you choose a random point in your data set, the first digit is a 1 (or 2,3,…)
We had a brief discussion about this, with the first answer being the expected 11% each as they are all equally likely. One student suggested that they would be clumped around a number (probably the mean) value.
After a brief discussion, I told them to go away and find the data set they had thought about in the first place.
We entered them all into this Google Sheet.
And then I added the data sets one by one to Autograph.
Obviously, with any activity like this you are open to it failing dramatically, but below you will see all the data sets plotted together, and Benford's Law falls out beautifully. There is even an excellent non-example.
Picture
The non-example was of weights of NBA players, and we discussed why this did not fit the pattern (weights will all fall within the range 70 - 120 kg approximately, so the first digits will be 7, 8, 9, 1, and the mean is in the 80s, so more 8s as this is a normal distribution.
But what about the other data sets? Why do they follow this same pattern?
I finished the session by explaining Benford's Law, the percentages it predicts and the formula, and how it can be used to spot people who have made up a data set.

Exploding Dots
Exploding Dots was a part of the Global Maths Project a couple of years back, set up by James Tanton. It is definitely worth checking out the website.
However, I prefer to teach it a bit more actively, and so created a version that I can present to students, with questions for them to do along the way. You can find a blank PDF of this here. 
It starts from the very beginning of school level Maths, with counting (in different bases), followed by the four operations. It introduces the idea of zero pairs to perform subtractions, and then builds up to unknown bases: that is polynomials. Within an hour you can take a group from counting to performing polynomial divisions.
I have done this with other classes before, and it went down well with this class too. By the end of the first double period we were answering questions like the one below (admittedly, they do this in the course, but we did the more traditional long division).
Picture
Picture
​We took this further to polynomial division that creates infinite polynomials, and the students wondered what would happen in other situations.
Picture
It was so popular, that we did a second double period on Exploding Dots, and this was all new stuff that I hadn't done with classes before.
We looked at decimals and fractions in the exploding dots model, which allowed us to look at fractions in different bases.
Then we looked at fractional bases, and explored what they might look like.
Picture
Picture
I love the exploding dots model. The students all commented on how visual it was, an how easy it was to understand what is going on. I really need to make more of this in 'normal' teaching, and not just as an enrichment activity.

Bayes Theorem
We study this in the course, but a couple of students asked to look at it in more detail. I was fairly lazy with this one, relying on some excellent resources available online.
First I sent students to this page to read the examples. We discussed the importance of the size of the population, and then did a few of the questions at the bottom of the page.
Then we watched this excellent video from 3Blue1Brown which visualises the whole thing beautifully.

Chinese Postman
I taught D1 once before leaving the UK, but it has been a while since I have done any decision maths. I thought this was a wonderful opportunity to take a look at the Chinese postman problem. I based the lesson on the plan from the Standards Unit, and turned it into a Desmos Activity. It was a very discursive session, so I paced them through the activity to start with, and also talked about the ideas whilst demonstrating and collating their ideas on a whiteboard.
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Continued Fractions
Based on a couple of articles from nrich, I put together an activity on continued fractions. We started with evaluating them (like the one below).
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We quickly moved on to look at infinite continued fractions
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And generalised this
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We took a quick look at how we can write any rational number as a continued fraction by using reciprocals.
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And then how we can use continued fractions of surds to get pretty good rational approximations for them
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 Finally we took a look at some of the continued fraction representations of other irrational numbers such as pi and e.

It was fun to get to explore some different aspects of mathematics with the class. I really must try to build it into my teaching more often, and not as enrichment, but as an integrated part of teaching Mathematics. If you have any go to activities like this, I would love to hear about them.
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    Dan Rodriguez-Clark

    I am a maths teacher looking to share good ideas for use in the classroom, with a current interest in integrating educational research into my practice.

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