#### Triangle area activity

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A challenging activity on the theme of triangle area, suitable for year 11 revision. The initial questions require a knowledge of basic triangle area, Pythagoras’ theorem, SOHCAHTOA, the sine rule and 1/2absinC so a good, challenging revision task. The questions have been designed with a ‘minimally diferent’ element, to draw pupils attention to how subtle changes can have significant implications for selecting methods. There are some follow-up questions, that could be used to shift the focus of the activity. I’d love to hear anyone’s suggestions of further questions.

#### The Maths of Cycling

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A brief look at some of the Maths that underpins cycling performance - how gear ratios, gradient, air resistance and power output effect speed. Requires, ideally, some familiarity with ratios, linear & cubic formulae and functions. Includes some worksheets (at end of presentation) for printing. Nicely timed for the Tour de France. Suggestions for improvement welcome as I wonder if the content is a bit dry!

#### Associative and commutative properties of multiplication

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A complete lesson looking at the associative and commutative properties of multiplication. Activities included: Starter: A simple grid of times table questions, includes ‘reversals’ (eg 7 times 9 and 9 times 7) to get pupils thinking about the commutative property. Main: Visual examples to get pupils thinking about commutativity of multiplication and non-commutativity of division. Pupils could explore further using arrays or Cuisenaire rods. Visual examples to get pupils thinking about associativity of multiplication and non-associativity of division. Pupils could explore further using pictorial representations. Three short activities where pupils make use of the commutative and associative properties of multiplication to make calculations. The last provides opportunities for pupils to create their own puzzles. Plenary: A maths ‘trick’ that uses the same properties. Please review if you buy as any feedback is appreciated!

#### Number pyramids investigation 1

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A complete lesson on number pyramids, with an emphasis on pupils forming and solving linear equations. An excellent way of getting pupils to consolidate methods for solving in an unfamiliar setting, and for them to think mathematically about what they are doing. Activities included: Starter: Slides to introduce how number pyramids work, followed by a simple worksheet to check pupils understand (see cover slide) Main: A prompt to a harder question for pupils to try. They will probably use trial and improvement and this will lead nicely to showing the merits of a direct algebraic method of obtaining an answer. A second, very similar question for pupils to try. The numbers have simply swapped positions, so there is some value in getting pupils to predict how this will impact the answer. A prompt for pupils to investigate further for themselves, along with a few suggested further lines of inquiry. There are lots of ways the task could be extended, but my intention is that this particular lesson would probably focus more on pupils looking at combinations by rearranging a set of chosen numbers and thinking about what will happen as they do this. I have made two other number pyramid lessons with slightly different emphases. Plenary: A prompt to a similar looking question that creates an entirely different solution, to get pupils thinking about different types of equation. Please review if you buy as any feedback is appreciated!

#### Solving two step equations using inverse operations

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A complete lesson on solving two step equations using inverse operations. Does include some decimals, as I wanted to give a more complete example set and make it hard for pupils to just use trial and error to find solutions. As such, I would let pupils use calculators. Activities included: Starter: A set of questions to check that pupils can evaluate two step expressions like 2x+3, given a value of x Main: A prompt to discuss the differences between two equations (a one step and a two step with the same solution), to get pupils thinking about how they could approach the latter. Examples and a set of questions on using inverse operations to reverse a two step flowchart and solve its corresponding equation. These have been deigned to further reinforce the importance of BIDMAS when interpreting an algebraic expression, so the emphasis is on quality not quantity of questions. A more challenging task of pupils trying to make an equation with a certain solution. Designed to be extendable to pupils looking for generalistions. Plenary: A prompt to discuss a few less obvious one-step equations (eg x+8+3=20) Printable worksheets and answers included. Please review if you buy as any feedback is appreciated!

#### Solving basic trigonometric equations

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A complete lesson on solving equations of the form sinx = a, asinx = b and asinx+b=0 (or with cos or tan) in the range 0 to 360 degrees. Designed to come after pupils have spent time looking at the functions of sine, cosine and tangent, so that they are familiar with the symmetry properties of these functions. See my other resources for lessons on these precursors. I made this to use with my further maths gcse group, but could be used with A-level classes too. Activities included: Starter: A set of four questions, effectively equations but presented as visual graph problems, to remind pupils of the symmetry properties of sine and cosine and the fact that tangent repeats every 180 degrees. Main: An example to transition from a visual problem to a formal, worded problem, and a reminder of the symmetry properties of sine and cosine. Five examples of solving trigonometric equations of increasing difficulty, including graphical representations to help pupils understand. A set of similar questions for pupils to do independently. I’ve made this into a worksheet where the graphs are included, to scaffold the work. Includes an extension task where pupils create equations with a required number of solutions. Plenary: A “spot the mistake” that addresses a few common misconceptions. Printable worksheets and answers provided. Please review f you buy as any feedback is appreciated!

#### Equations of tangents of circles

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A complete lesson or two on finding equations of tangents to circles with centre the origin. Aimed at the new GCSE specification, although it could also be used with an A-level group. Activities included: Starter: Two recap questions on necessary prerequisites, the first on equations of circles, the second on equations of perpendicular lines. If pupils really struggled with this I would stop and address these issues, rather than persist with the rest of the lesson. Main: A set of questions on finding the gradient of OP, given some point P on a circle, followed by a related worksheet for pupils to practice. A follow-up ‘reverse’ task where pupils find points P such that the gradient of OP takes certain values. The intention is that pupils can do this task by logic and geometric reasoning, rather than by forming and solving formal equations, although the task could be further extended to look at this. The focus then shifts to gradients of tangents, and finally equations of tangents, with examples and a related set of questions for pupils to practice. An extension task where pupils find the equation of the circle given the tangent. Plenary: A spot the mistake question. Printable worksheets and answers included. Please review if you buy as any feedback is appreciated!

#### Circle theorems lesson 3

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A complete lesson on the theorem that angles in the same segment are equal. I always teach the theorem that the angle at the centre is twice the angle at the circumference first (see my other resources for a lesson on that theorem), as it can be used to easily prove the same segment theorem. Activities included: Starter: Some basic questions on the theorems that the angle at the centre is twice the angle at the circumference, and that the angle in a semi-circle is 90 degrees, to check pupils remember them. Main: Slides to show what a chord, major segment and minor segment are, and to show what it means to say that two angles are in the same segment. This is followed up by instructions for pupils to construct the usual diagram for this theorem, to further consolidate their understanding of the terminology and get them to investigate what happens to the angle. A ‘no words’ proof of the theorem, using the theorem that the angle at the centre is twice the angle at the circumference. Missing angle examples of the theorem, that could be used as questions for pupils to try. These include more interesting variations that incorporate other angle rules. A set of similar questions with a progression in difficulty, for pupils to consolidate. Two extension questions. Plenary: A final set of six diagrams, where pupils have to decide if two angles match, either because of the theorem learnt in the lesson or because of another angle rule. Printable worksheets and answers included. Please do review if you buy as any feedback is greatly appreciated!

#### Perimeter problem solving lesson

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A complete lesson of more interesting problems involving perimeter. I guess they’re the kind of problems you might see in the Junior Maths Challenge. Includes opportunities for pupils to be creative and make their own questions. Activities included: Starter: A perimeter puzzle to get pupils thinking, where they make changes to shapes without effecting the perimeter. Main: A set of four perimeter problems (on one page) for pupils to work on in pairs, plus some related extension tasks that will keep the most able busy. A matching activity, where pupils match shapes with different shapes but the same perimeter, using logic. I would extend this task further by getting them to put each matching set in size order according to their areas, to address the misconception of confusing area and perimeter. Pupils are then prompted to design their own shapes where the perimeters are the same. Plenary: You could showcase some pupil designs but much better, use all of their answers to create a new matching puzzle. Printable worksheets and answers included. Please review if you buy as any feedback is appreciated!

#### Pyramids

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A range of resources on the theme of pyramids, but touching on topics including the definition of a pyramid and consideration of their properties, representations (3D sketches, nets, pan views) and volume. Probably a couple of lessons worth of activities. Activities included: Prompts to consider a definition of a pyramid, and a closer look at the definitions of base, apex and height. Introductory slides and a worksheet on different representations of pyramids, plus a link to a geogebra file to look at pyramids dynamically. A venn diagram activity for pupils to consolidate their understanding of the above. A link to a video of someone using three pyramids of water to fill a cuboid (ie to demonstrate the volume rule) A nicely animated related question to calculate volume of a pyramid. A physical ‘puzzle’ where pupils cut out 3 nets of identical pyramids, fold and glue them to create 3D pyramids, then try to arrange them to make a single cuboid, thus demonstrating the volume rule. A mini-investigation into step pyramids, that ultimately shows something similar to the physical puzzle above, but in a much more abstract way, and could even be used with a further maths class to look at summations! Some images of regular and irregular pyramids, to get pupils discussing the measurements they would need in order to calculate volume, followed by a prompt for pupils to create their own examples of pyramids with a given volume. A few examples of pyramids in architecture to finish. Please review if you buy as any feedback is appreciated!

#### Solving basic trigonometric equations beyond the range 0 to 360 degrees

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A complete lesson on solving equations of the form sinx = a, asinx = b and asinx + b = 0 (or using cos or tan) for any range. Designed to come after pupils have spent time solving equations in the range 0 to 360 degrees, and are also familiar with the cyclic nature of the trigonometric functions. See my other resources for lessons on these topics. I made this to use with my further maths gcse group, but could also be used with an A-level class. Activities included: Stater: A set of 4 questions to test if pupils can solve trigonometric equations in the range 0 to 360 degrees. Main: A visual prompt to consider solutions beyond 360 degrees. followed by a second example (see cover image) that will lead to a “dead-end” for pupils. Slides to define principal values for sine, cosine and tangent, followed by a summary of how to solve equations for any range. Three example problem pairs to model methods and then get pupils trying. Includes graphical representations to help pupils understand. A worksheet with a progression in difficulty and a challenging extension to create equations with a required number of solutions. Plenary: A prompt to discuss solutions to the extension task.

#### The graphs of sine, cosine and tangent outside the range 0 to 360 degrees

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A complete lesson on the graphs of sine, cosine and tangent outside the range 0 to 360 degrees. I’ve also made complete lessons on these functions in the range 0 to 360 degrees. Designed to come after pupils have been taught about the ratios sine, cosine and tangent in the context of right-angled triangle trigonometry, and looked at the graphs of sine cosine and tangent in the range 0 to 360 degrees. This could also be used as a precursor to solving trigonometric equations in the further maths gcse or A-level. Activities included: Starter: A worksheet where pupils identify key coordinates on the graphs of sine and cosine from 0 to 360 degrees. Main: A reminder of the definitions of sine, cosine and tangent using the unit circle, with a prompt for pupils to discuss what happens outside the range 0 to 360 and a slide to make this clear. Three examples of using knowledge of the graphs to effectively solve a trigonometric equation. This isn’t formalised, but done more as a visual puzzle that pupils can answer using symmetry and the fact that the functions are periodic (see cover image). A worksheet with a set of similar questions, followed by a related extension task. Plenary: A brief summary about sound waves and how pitch and volume is determined. Printable worksheets and answers included. Please review if you buy as any feedback is appreciated!