Interactive Maths - The Interactive Way to Teach Mathematics
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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)
        • The Four Operations (QQI Worksheets)
        • The Four Operations (Video)
        • Timestables Square (QQI)
        • Grid Multiplication (QQI)
      • Missing Numbers >
        • Missing Numbers (QQI)
        • Missing Numbers (10QQI)
        • Missing Numbers (QQI Count Down)
        • Missing Numbers (QQI Relay)
        • Missing Numbers (QQI BINGO)
        • Missing Numbers (QQI Worksheets)
      • 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)

Personal Reflection 29/7/2019

29/6/2019

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Perfect Tau Day
It would be remiss to not mention that today is both a perfect day and tau day (in the US).


Preparing for the 4th Industrial Revolution
I posted a response to a post on preparing students for the future. I don't often get involved in this, but this article in particular riled me up.


Lesson Sheet for 5B HL
I have been using Lesson Sheets with my IB Higher Level class, and I have written a blog about how that is going, and what I do with them.


Context is Important
A short reflection on a question I gave my class that tripped them all up because of the context, and why we should be wary of this when assessing students.
​
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Context is Important

26/6/2019

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We did conditional probability in my IB Higher Level class today, and one of the questions I set them really tripped them up. Not because of the Mathematical content, but rather the context that the question used. Below is the question.
Picture
The issues that arose were from a general lack of knowledge about how tennis works. Many of the students did not know that if you miss the first serve, you get a second serve, for example. One of the interesting misconceptions was that if you miss the second serve you get a third serve. Another, that if you got the first serve in, but then lost the point, you got a second serve.
Once I explained the rules of tennis clearly to the whole class, they all completed the question pretty quickly.
This really highlighted to me the importance of knowledge about the context when answering a question. In Why Don't Students Like School, Willingham gives the oft referenced example of children who know a lot about baseball but with low reading ages having better comprehension of a passage about baseball than those children with high reading ages who know little about baseball. 
Although I knew this, it was not something I had considered in the realm of Maths teaching before, as I just thought it applicable to subjects with reading comprehension. But clearly reading is a big part of Maths as well.
So what does this mean? Context is important. Or, more specifically, knowledge about the context is important. Lacking knowledge about the context can be a real hindrance to being able to solve a problem, even if the mathematical knowledge is there.
This is an argument for removing all context from the sequence of teaching when initially introducing new content and skills, so that students can focus all their attention on what we want them to learn. 
It is also an argument for using lots of questions with contexts after they have mastered the content, and are confident in their abilities to use their new knowledge and skills. This ensures they have seen many contexts, and hopefully, through that and other subjects, they will have knowledge of any contexts that come up later in life (or more likely, in exams).
If I had given that question to a weaker group, it would have thrown them completely, even if they could do the Maths. Fortunately, with a high achieving group, they were secure enough in the Maths to recognise that the problem was the context.
But in future, I am going to be much more aware of the context for questions, and, if necessary, teach the knowledge they need about the context before setting questions, as well as the content knowledge.
UPDATE: I have since read this excellent article from Dylan Wiliam in the IMPACT journal from the Chartered College of teaching, and this quote add nicely to my point.
"The results of a maths test with a high reading demand are difficult to interpret. We can be reasonably sure that students with high scores can do the mathematics that is tested (and the reading required), but for students with low scores, we cannot be sure whether these are due to the fact that they could not do the mathematics, or whether they could not understand the questions. Such a test would support inferences about mathematical competence for some students (good readers) and not for others (poor readers). The problem with such a test is that the variation in scores between students is partly due to differences in their mathematics achievement (which we want), and partly due to differences in their reading ability (which we do not want)."
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Lesson Sheets for IB Higher Level

21/6/2019

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This year I am teaching IB Higher Level for the second time. My approach to teaching has changed a fair amount in the last three years since I last taught the course, and in particular I am now much more focused on breaking down ideas and giving examples across the range of types of questions. However, with this being a Higher Level class, I am also acutely aware of the Expertise Reversal Effect, and the fact that my students are further along the expertise spectrum than all of the other students I have taught in the last 3 years. 
There are some elements of my teaching I have kept, such as the weekly quizzes. I run these in our single lesson that is after lunch each week. I use past paper questions mostly, with the odd drill style activity (recently we have been doing trigonometry, so have been drilling the exact values), and try to keep to 30-35 marks in the 40 minutes period. 
I started the year with the Last Lesson, Last Unit, Further Back starters as well, but have found that there is too much of a time pressure to include these and the quizzes. Given that students are significantly more focused in the quizzes (there is a little bit of stake there as they do count minimally towards their grade), I have moved away from the longer starters, often just using a single exam question to start the double period, and a prior knowledge priming question in the singles. I am considering going down the route of quick retrieval of key facts and terminology as a starter. ​
But the biggest change is that I have started teaching through lesson sheets. Well, more appropriately, skill sheets. I have focused on breaking each of the units down into the individual skills that students need to master. On each of these I give a starter (which is really just a link to prior knowledge), and then a space for notes. This is followed by a series of examples and your turns on the sheets, and then an exercise (usually just the page numbers from the textbook and 2 ebooks).
With the examples and your turns, I am much less specific about the your turn being very similar to the example, as these kids are good mathematicians, and that would be patronising for them and would not invoke them to think. I even have them one after the other, rather than side by side as I have done with other classes. Below is an example of a set of example and your turns for the Trigonometric Double Angle Identities. As you can see, the jump from example to your turn is significant. Indeed, I have found that often the students need a little help with the your turns, and I will address this with individuals and pairs as I walk around the room.
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One of the great things about having lesson sheets is that I can go through and "do" the sheet as part of my planning. This includes thinking carefully about the notes, both what I want them to write down, and annotations of what I want to say. I also do the examples before class to ensure there is nothing that is going to trip me up, but also to give me the answers to the your turns so I can easily check student work. This is a good example of what Doug Lemov calls Standardize the Format, making it easier to quickly check work as it is in the same place for all students.
In the lesson I model live using my visualiser. In the notes section I will write the key points that they should definitely copy down, but I also expect them to keep notes of the things I am saying as well. I then model the example on the sheet using the visualiser. 
I have also given students folders to store the lesson sheets, quizzes, formula booklets and challenge sheets (UKMT Mentoring sheets). I have a folder with my notes versions too. This means students have an easy set of notes and examples to return to in revision. When we do questions from the textbooks or other exercises, students do these in their books. In reality we do very little of this in class due to time constraints, and they are having to do a lot of that as homework. They are getting practice through the your turns, but not enough to really cement the ideas, and this is a problem I am struggling to overcome at the moment. They do get some practice as part of the weekly quiz.
At the end of each week, I take all the lesson sheets that I have written on live under the visualiser to the library where I scan then in, along with the solutions for the quiz. I then upload them to our class website, where I have a section for each unit. In each unit there are links to the blank lesson sheets, the completed version, the notes from the last time I taught the course (which are similar but a pdf of a smart board file), and any links to other worksheets. There is also a page with links to all the quizzes and solutions.
I have found the process of breaking units down into individual skills to be useful for me to really think about the content. It has also made students more aware of the individual skills they need to work on. When the idea of atomisation has come up on the Mr Barton Maths Podcast a couple of times, Craig has asked how they then pull it all back together. For me this comes in the retrieval practice they get in the weekly quizzes and starters.
The resources are really popular with my students, who have a folder full of organised revision material with links to pages of questions in the textbooks.
Last week I also asked for student feedback, and one of the things they said was they wanted more feedback on their progress, and I have produced Skill Tracker sheets where they can record each time they answer a question correctly on a skill to show their own progress.
The big change for me has been not using a presentation software. I have used both Powerpoint and SMART Notebook successfully for many years now, and this is quite different. Whereas I used to place things I wanted to show them in the presentation, I now have to switch to them from the visualiser. On the other hand, the visualiser gives me the ability to quickly and easily Show Call student work (the Your Turns, for example) and to comment on their answers.
Do you use lesson sheets? What do you include in them? How do you put them together? How do you find using them?
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Getting Student Feedback

8/6/2019

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I have been thinking a lot about my approach to teaching. It has changed dramatically over the last few years, but with having an IB Higher Level class for the first time in a few years, I have been forced to reconsider as their level of expertise is much higher. That being said, there is still some in the class who need the explicit instruction of this difficult content.
So yesterday I decided to have a chat with my class about how the course is going so far. I started by asking them two questions:
1. Would they prefer to get rid of the weekly quizzes, or change them to every other week?
2. Do they find the lesson sheets I produce (future blog post on these coming) a useful resource that is worth the time they take to produce?
The answers to both questions were unanimous: keep doing what I am doing.
The students were very happy with the weekly quizzes (a couple did mention they were a little too difficult, so I am going to ensure there are more easier questions in future), and they definitely seemed to be won over by my constant hammering on about the importance of retrieval practice. They could see the benefits of having regular chances to retrieve their knowledge and practice applying it to exam questions.
The lesson sheets were really popular. The comments were that they helped them organise themselves and were great for looking back at for revision. I am glad to hear this since they take a long time to produce, but it is worth it if they are appreciated and helpful.
These were both affirming results. I believed that doing them was beneficial for the students, but it was good to hear they also see the benefits.
After that I turned to some comments they had raised on a recent survey I sent them as part of our Maths Department review. The two main issues that were raised in the survey were that there was too little time for practicing, and that the feedback I give was not detailed enough.
First I addressed the issue of practice. This is something I have been concerned about for a little while myself, and was something I reflected on as a target for this year. We went down from 8 periods (40 minutes each) to 6 periods last year, and this had a huge impact on the amount of practice students get in class. They were given more study periods with this extra time, so I have been assigning more homework than I used to. I explained this to the class, and made the suggestion that we could fit more practice in class time, but I would have to stop doing my tangents on the non-curriculum side of Maths. Thankfully they all (bar one) said they would rather have those and practice at home. It would have been difficult to cut those out, so I am glad they went that way!
I did also explain my main principles for teaching, which are the four quotes I have printed and stuck on my walls:
1. Memory is the Residue of Thought
2. Practice makes Permanent
3. Working Memory is limited
4. Learning occurs over time
I said that I try to give them lots of opportunities for 1 in class through the Your Turns, address 3 through the examples and lesson sheets, and the weekly quizzes and starters are aimed at 4. We left it at 2 was their responsibility if they wanted to learn the material properly, though obviously they do get some practice in class.
For feedback I was a little more contemplative before the lesson. This was not a comment I was expecting, so I took it to heart. I only take in and mark one piece of work a week (the weekly quiz) which is our departmental policy. For homework I expect them to check answers in the back of the textbook, and I will do a walk around checking for any issues they had. But I do not review them. I have decided that I need to be more focused on giving individual feedback whilst they are working in class on the Your Turn questions, and also that I need to make sure I Show Call their answers for these too.
But we talked about how they could become more aware of the areas they need to work on, and I suggested I create a grid which has a row with 4 boxes for each of the skills we learn. They could then tick one of the boxes when they answer a question successfully on that skill, in either the weekly quiz, the starters, exams, or indeed practice at home. That way they could generate a visual of the topics they are doing well at, and see it grow over time. And if they comment when they get it wrong, they can also see the ones they need more practice on. I will review these with them every couple of weeks to get a picture of which skills and topics I need to drop into the quizzes and starters.
I have also been thinking about making use of learning maps as I have read about them in High Impact Instruction by Jim Knight (one of the books we got on the instructional coaching conference I attended in April). They idea of these is that they show the entire unit in simple terms at the start, but that students add to them over the course of the unit. I had a think about one for the upcoming statistics unit, and I am going to try that out. I hope this will also give students a chance to see how they are progressing.
Overall, the conversation with the students was useful. I got some confirmation, but also some ideas to try going forward. I am going to have a similar conversation with my GCSE class next week.
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Personal Reflections 7/6/19

6/6/2019

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INSET Sessions
We started a new term a few weeks ago, and I led a couple of INSET sessions on Monday to kick it off. We started by exploring the second standard of our Principles of Great Teaching. The wording of this is:

All students are expected to participate in questioning sessions, with the use of a "no hands up" policy.
 
We watched a few clips to spark discussion around why asking for hands up is not a great strategy, and then groups shared a few strategies they use to question students. In the last 20 minutes I talked about Doug Lemov's Cold Call technique.
 
If you are interested you can download the powerpoint I used here.
 
After that we had some optional workshops on offer, and I ran one on MARGE (which I have talked about before here). We had 45 minutes, so it was a whistle-stop tour through the 5 principles with some time to reflect on what they meant for classroom practice. I tried really hard to build in the 5 principles to the presentation to model the ideas, and at the end I pointed this out to the group, making the metacognitive explicit. It seemed to go down well, and I think I am slowly getting more people to think about the science of learning. It is a slow process, but my incessant going on about it seems to be making people think.
 
You can find that presentation here.
 
Finally we all came back together to start looking at the Principles of Great Teaching in more depth. My plan has always been to create a document to support the poster front sheet, and I wanted everyone to be involved in creating that document. The purpose is to make the Principles more explicit through explanations and examples. The first stage towards this was to create a rubric for each principle. I started by creating an example one for the first Principle (Challenges All Students) which you can find here. I shared this with all staff, and split them into 15 groups to start putting together rubrics for the other Principles.
 
The purpose of the rubric will be for self-assessment of our teaching. I envisage teachers going through the rubric for a Principle and highlighting the descriptors they feel they are meeting, and  using this to inform their target setting. It will also build into the coaching programme we are starting.
 
By the end of that session we had a starting rubric for most of the Principles. In a future session we shall come back to those, review them in departments, adjust and amend them, so that for the start of 2020 hopefully we have a complete (but not finalised) document.
My High Five
An idea started by Ben Gordon, here are My High Five.
T&L Newsletter Issue 11
Last week I published the 11th issue of our T&L Newsletter. You can find the issue here, and the full back catalogue here.

IB HL Just For Fun
I have decided to continue to open up the world of Maths to my IB Higher Level class. They all did presentations last term on a topic of their choice, and I am planning on doing that again later in the year. But for now, I have also decided to do a Just For Fun lesson at the end of each unit.

As we are finishing of a unit on Trigonometry, I am going to talk to them about the etymology of the trig words, why it is called the CO-sine, and the tangent. And why secant is the reciprocal of cosine not sine (as much as students wish the first letters would match).

The next unit is on exponentials and logarithms, and I am planning on introducing them to fractals and Hausdorff dimensions. I have some other ideas for later units, but I am eager to hear suggestions too
Memorable Teaching
I read Memorable Teaching by Peps Mccrea, which was excellent, and have written a short reflection here, with a sketch note.
Venn Diagrams
I have used a couple of the tasks from mathsvenns.com this week when teaching quadratic functions. I have used them before, but kind of forgot about them, and they are just such an amazingly rich task. In trying to find the functions for each of the regions, students have to think deeply about the ideas. I want to build more of these into my lessons over the next few weeks and months to try to embed the practice of using them. Since I am now doing weekly quizzes with my classes, the need for the Last Lesson, Last Unit, Further Back starters has kind of diminished as they are getting regular retrieval in the quiz. So I am thinking that Venn diagram tasks could become a feature of my starters (on a previous topic so there is some retrieval going on, and time for the maturation of ideas to help in the process).
PD Chats
I have started having 20 minute coaching style conversations with my colleagues about what they are aiming to achieve this year. They have been positively received so far, and it has been an interesting experience for me. I am also trying to set a time when I will catch up with them in a few weeks to see how things are going.

This comes from my delving into the world of coaching, but also thoughts about leadership. I want to have one-to-one conversations with all staff, and really listen to what they have to say. Hopefully this will be informative for the T&L Programme.
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Preparing for the 4th Industrial Revolution

4/6/2019

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This is a response to this post about what schools need to do to prepare students for the 4th Industrial Revolution.
The underlying argument is that students will need to be educated differently to survive in the new world. However, as far as I am aware, the brain structure of children is no different now than it was 4,000 years ago, which would suggest that effective learning happens in the same ways as it always has. 
Many of these ideas are comprehensively argued against by Daisy Christodoulou in her excellent book 7 Myths about Education.
But lets take a look at each of the 8 ideas presented in the post.
1. Redefine the purpose of education
​
The premise of the argument is that we should stop educating children to work in factories and have a single job for life, but rather educate them to be adaptable when they go to work. But that is NOT changing the purpose of education. It is still saying the purpose of education is to prepare students for a life of work. I have come across very few teachers who have such low aspirations for their students. 

We do not teach them so they can get jobs. We teach them to give them an education that will allow them to have options later in life. We gift them the knowledge and skills that have been useful to the development of society over hundreds (or more) years. We give them the chance to explore beauty in nature, in art, in science. We provide opportunities for them to find themselves, and learn to work with others. Are some of these things useful in our working lives? Maybe. But that is not why we teach them.

Mark Enser argues this point well here.
2. Improve STEM education
I am a Maths teacher, so of course I think Maths is important. But is it more important than languages or arts? To some students, maybe. But, as I argue above, the purpose of education is to provide a broad insight into the world.
​
But, the author barely argues this point, but rather goes on to say that we should be teaching humanities as well (which is what we already do) and then further changes tact saying we should teach critical thinking and collaboration, citing this rather famous WEF Report. Blake Harvard does a good job of breaking down this report here. And I refer back to my argument above again. This suggests we should teach things that employers want, not what is best for students. Perhaps if employers want these skills, they should invest in training their employees.

Anyway, what a whirlwind. From stating we should improve STEM teaching we end up at teaching creativity and collaboration. All in one paragraph. A muddled argument at best. 
3. Develop Human Potential
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With machines able to do all the manual and repetitive labour, we need humans to be more creative and do the things machines can't do. So the argument goes. But creativity is hugely domain specific, and requires a large amount of background knowledge. Schools do teach students to be creative, by teaching them enough content and skills that they can be creative. As an Art teacher once said to me, "You need to know the rules to be able to break them". He was describing the work of Pablo Picasso, and how he had spent years learning the basics of drawing and painting before being able to create the masterpieces we now celebrate him for. We need to go through those steps to be able to become the creative and adaptable thinkers able to compete with the machines.

And don't even get me started on what happens when the machines learn enough content to become creative!
4. Adapt to lifelong learning models
​
"The illiterate of the 21st century will not be those who cannot read and write, but those who cannot learn, unlearn and relearn." - Future Shock, Alvin Toffler.

What nonsense. Those who cannot read and write will not be able to "learn, unlearn and relearn" (at least not as effectively as those who can). And we can all learn, unlearn and relearn naturally anyway. Everyone of us is in built with the ability to learn. What makes us more able to learn, is knowing lots of stuff. The more we know, the more we can learn, and the more we learn, the more we will know. This is the Matthew Effect.

And then we hear that we need people to be lifelong learners as the jobs of the future don't exist yet. Another old argument that has been argued against many times (e.g. here). But again, this supposes that schools are not teaching kids to be lifelong learners. That is one of the primary aims of many teachers. I want my kids to be successful in whichever path they choose, and the best way to do that is to give them a broad education of the stuff that has helped generations of previous thinkers to advance society.
5. Alter Educator Training
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Finally one I agree with. We need to incorporate more cognitive science into teacher training so that teachers are aware of how the brain works and learns.

Oh wait. No. We need to become facilitators. That is not altering educator training. That is what most teacher training currently is. Both initial training and training we receive later in our careers.

"Failure needs to be embraced as an essential step to learning." Another dangerous idea. Whilst we can learn from mistakes, they are not what we want to happen, and they are not essential. They are likely to happen, and can be used to in ways to remove misconceptions. But a 'perfect' explanation with examples and models can lead to learning with no mistakes. The danger in this phrase is that mistakes will happen, and we want students to learn from their mistakes. But we should not be aiming to make mistakes.
6. Make schools makerspacesHonestly, I do not know enough about this to make an informed comment. But my gut feeling is that these makerspaces could be useful, unless they are taking away time from other stuff (like teaching of classes), which I suppose is the implication.
7. International Mindfulness
​
I actually do agree with this one. Although I think many teachers and schools do already do this.
8. Change higher education
​
I am not sure how this is something that schools are supposed to do. But the argument once again boils down to colleges being a place to prepare people for work, and schools prepare you for college, so the people in charge of businesses should decide what they want and schools and colleges should fit in with giving them the mindless drones (who are all very creative about things they know nothing about) that they want.
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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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