An open-ended lesson on number pyramids, with the potential for pupils to practice addition and subtraction with integers, decimals, negatives and fractions, form and solve linear equations in two unknowns and create conjectures and proofs. I used this lesson for an interview and got the job, so it must be a good one! The entire lesson is built around the prompt I’ve uploaded as the cover slide. I have provided detailed answers for some of the responses that pupils could give, so you can get a clear idea of how the investigation might progress. I would spend the lesson responding to pupils’ work and questions, and probably get pupils to make posters of their findings or discuss their work with other pupils. Suitable for a range of abilities. Please review if you buy as any feedback is appreciated!

A complete lesson on solving two step equations of the form ax+b=c using the balancing method. Designed to come after pupils have solved using a flowchart/inverse operations. Activities included: Starter: A set of questions to check that pupils can solve one step equations using the balancing method. Main: A prompt for pupils to consider a two step equation. An animated solution to this equation, showing physical scales to help reinforce the balancing idea. An example-problem pair, to model the method and allow pupils to try. A set of questions with a variation element built in. Pupils could be extended by asking them to predict answers, or to explain the connections between answers after finishing them. A related, more challenging task of solving equations by comparing them to a given equation, plus a suggested extension task for pupils to think more mathematically and creatively. Plenary: A closer look at a question, looking at the two different balancing approaches that could be taken (see cover slide). Depending on time, pupils could go back and attempt the previous questions using the second approach. Printable worksheets and answers included. Please review if you buy as any feedback is appreciated!

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!

A complete lesson on solving two step equations of the form ax+b=c, ax-b=c, a(x+b)=c and a(x-b)=c using the balancing method. Designed to come after pupils have solved using a flowchart/inverse operations. Activities included: Starter: A few substitution questions to check pupils can correctly evaluate two-step expressions, followed by a prompt to consider some related equations. Main: A slide to remind pupils of the order of operations for the four variations listed above. Four example-problem pairs of solving equations, to model the methods and allow pupils to try. A set of questions for pupils to consolidate, and a suggestion for an extension task. The questions repeatedly use the same numbers and operations, to reinforce the fact that order matters and that pupils must pay close attention. A more interesting, challenging extension task in the style of the Open Middle website. Plenary: A set of four ‘spot the misconception’ questions, to prompt a final discussion/check for understanding. Printable worksheets and answers included. Please review if you buy as any feedback is appreciated!

A complete lesson on number pyramids, with an emphasis on pupils forming and solving linear equations. An excellent way of getting pupils to think about equations in an unfamiliar setting, and to create their own questions and conjectures. Activities included: Starter: A mini-investigation on three-tier number pyramids, to set the scene. One combination is best dealt with using a linear equation, and sets pupils up to access the more challenging task to come. Main: A prompt for pupils to consider four-tier number pyramids. Although this task has the potential to be extended in different ways, I have provided an initial focus and provided some responses that pupils could give, so you can get a clear idea of how the investigation might progress. I would spend the rest of the lesson responding to pupils’ work and questions, and probably get pupils to make posters of their findings or discuss their work with other pupils. Please review if you buy as any feedback is appreciated!

At least a lesson’s worth of activities on the theme of quadratic sequences. Designed to come after pupils have learnt the basics (how to use and find an nth term rule of a quadratic sequence). Gives pupils a chance to create their own examples and think mathematically. There are four activities included: Activity 1 - given sets of four numbers, pupils have to order them so that they form quadratic sequences. Designed to deepen pupils understanding that the terms in a quadratic sequences don’t necessarily always go up or down. Activities 2 and 3 - on the same theme of looking at the sequences you get when you pick and order three numbers of choice. Can you always create a quadratic sequence in this way? What if you had four numbers? Could be used to link to quadratic functions. Activity 4 - inverting the last activity, can pupils find possible values for the first three terms and a rule, given the fourth term? A chance for pupils to generate their own examples and possibly do some solving of equations in more than one variable. Where applicable, worked answers provided.

A complete lesson of more challenging problems involving the sine rule. Designed to come after pupils have spent time on basic questions. Mistake on previous version now corrected - please contact me for an updated copy if you have already purchased this. Activities included: Starter: A set of six questions, each giving different combinations of angles and sides. Pupils have to decide which questions can be done with the sine rule. In fact they all can, the point being that questions aren’t always presented in the basic ‘opposite pairs’ format. Pupils can then answer these questions, to check they can correctly apply the sine rule. Main: A set of eight more challenging questions that pupils could work on in pairs. Each one is unique, with no examples offered, and therefore I’d class this as a problem solving lesson - pupils may need to adopt a general approach of working out what they can at first, and seeing where this takes them. Questions also require knowledge from other topics including angle rules, shape properties, bearings, and the sine graph. I’ve provided full worked answers FYI, but I would get pupils discussing answers and presenting to the class. Plenary: A prompt for pupils to reflect on possible rounding errors. Most of the questions have several steps, so it is worth getting pupils to think about how to avoid rounding errors. I’ve left each question as a full slide, but I’d print them 4-on-1 and 2-sided, so that you’d only need to print one worksheet per pair. Please review if you buy as any feedback is appreciated!

A complete lesson looking at the effect of multiplying and dividing integers and decimals by 10, 100 and 1000. Activities included: Starter: A prompt for pupils to share any ideas about what the decimal system is. Images to help pupils understand the significance of place value. Questions that could be used with mini whiteboards, to check pupils can interpret place value. Main: A worksheet where, by repeated addition, pupils investigate the effect of multipliying by 10, initially with whole numbers but later with decimals. A slide to summarise these results, followed by some more mini whiteboard questions to consolidate. A prompt for pupils to use a calculator to investigate the effect of multiplying or dividing by positive powers of 10, followed by slides to help pupils reflect on their findings, and provide notes for all pupils. A related game for pupils to play (connect 4). Plenary: A very brief, bulleted summary of the history of the decimal system and the importance of the invention of zero. Printable worksheets included. Please review if you buy as any feedback is appreciated.

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.

A complete lesson on the graphs of sine and cosine from 0 to 360 degrees. I’ve also made complete lessons on tangent from 0 to 360 degrees and all three functions outside 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. Activities included: Starter: Examples to remind pupils how to find unknown angles in a right-angled triangle (see cover slide), followed by two sets of questions; the first using sine the second using cosine. The intention is that pupils estimate using the graphs of sine and cosine rather than with calculators, to refamiliarise them with the graphs from 0 to 90 degrees. Although I’ve called this a starter, this part is key and would take a decent amount of time. I would print off the question sets and accompanying graphs as a 2-on-1 double sided worksheet. Main: Slide to define sine and cosine using the unit circle, with a hyperlink to a nice geogebra to show the graphs dynamically. Or you could get pupils to try to construct the graphs themselves by visualising. A set of related questions that I would do using mini-whiteboards, where pupils consider symmetry properties of the graphs. A mini-investigation where pupils look at angles with the same sine or cosine and look for a pattern. Plenary: An image to prompt discussion about the “usual” definition of sine and cosine (using the terminology opposite, adjacent and hypotenuse) and the fuller definition (using the unit circle) Printable worksheets and answers included. Please review if you buy as any feedback is appreciated!

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!

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!

A complete lesson on the graph of tangent from 0 to 360 degrees. I’ve also made complete lessons on sine and cosine from 0 to 360 degrees and all three functions outside 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 have met the unit circle definitions of sine and cosine. Activities included: Starter: A quick set of questions on finding the gradient of a line. This is a prerequisite to understanding how tan varies for different angles. Main: An example to remind pupils how to find an unknown angle in a right-angled triangle using the tangent ratio, followed by a set of similar questions. The intention is that pupils estimate using the graph of tangent rather than using the inverse tan key on a calculator, to refamiliarise them with the graph from 0 to 90 degrees. Slides to define tan as sin/cos and hence as gradient when using the unit circle definition. A worksheet where pupils construct the graph of tan from 0 to 360 degrees (see cover image). A set of related questions, where pupils use graph and unit circle representations to explain why pairs of angles have the same tan. Pupils can be extended further by making and proving conjectures about pairs of angles whose tans are equal. Plenary: An image to prompt discussion about the “usual” definition of tangent (using the terminology opposite, adjacent and hypotenuse) and the fuller definition (using the unit circle) Printable worksheets and answers included. Please review if you buy as any feedback is appreciated!

A complete lesson on the theorem that opposite angles in a cyclic quadrilateral sum to 180 degrees. Assumes that pupils have already met the theorems that the angle at the centre is twice the angle at the circumference, the angle in a semicircle is 90, and angles in the same segment are equal. See my other resources for lessons on these theorems. Activities included: Starter: Some basics recap questions on the theorems already covered. Main: An animation to define a cyclic quadrilateral, followed by a quick question for pupils, where they decide whether or not diagrams contain cyclic quadrilaterals. An example where the angle at the centre theorem is used to find an opposite angle in a cyclic quadrilateral, followed by a set of three similar questions for pupils to do. They are then guided to observe that the opposite angles sum to 180 degrees. A quick proof using a very similar method to the one pupils have just used. A set of 8 examples that could be used as questions for pupils to try and discuss. These have a progression in difficulty, with the later ones incorporating other angle rules. I’ve also thrown in a few non-examples. A worksheet of similar questions for pupils to consolidate, followed by a second worksheet with a slightly different style of question, where pupils work out if given quadrilaterals are cyclic. A related extension task, where pupils try to decide if certain shapes are always, sometimes or never cyclic. Plenary: A slide showing all four theorems so far, and a chance for pupils to reflect on these and see how the angle at the centre theorem can be used to prove all of the rest. Printable worksheets and answers included. Please review if you buy as any feedback is appreciated!

A complete lesson for first teaching about corresponding, alternate and supplementary angles. Activities included: Starter: Pupils measure and label angles and hopefully make observations and conjectures about the rules to come. Main: Slides to introduce definitions, followed by a quiz on identifying corresponding, alternate and supplementary angles, that could be used as a multiple choice mini-whiteboard activity or printed as a card sort. Another diagnostic question with a twist, to check pupils have grasped the definitions. Examples followed by a standard set of basic questions, where pupils find the size of angles. Examples/discussion questions on spotting less obvious corresponding, alternate and supplementary angles (eg supplementary angles in a trapezium). A slightly tougher set of questions on this theme, followed by a nice angle chase puzzle and a set of extension questions. Plenary: Prompt for pupils to see how alternate angles can be used to prove that the angles in a triangle sum to 180 degrees. Printable answers and worksheets included. Please review if you buy as any feedback is appreciated!

A complete lesson on the interior angle sum of a quadrilateral. Requires pupils to know the interior angle sum of a triangle, and also know the angle properties of different quadrilaterals. Activities included: Starter: A few simple questions checking pupils can find missing angles in triangles. Main: A nice animation showing a smiley moving around the perimeter of a quadrilateral, turning through the interior angles until it gets back to where it started. It completes a full turn and so demonstrates the rule. This is followed up by instructions for pupils to try the same on a quadrilateral that they draw. Instructions for pupils to use their quadrilateral to do the more common method of marking the corners, cutting them out and arranging them to form a full turn. This is also animated nicely. Three example-problem pairs where pupils find missing angles. Three worksheets, with a progression in difficulty, for pupils to work through. The first has standard ‘find the missing angle’ questions. The second asks pupils to find missing angles, but then identify the quadrilateral according to its angle properties. The third is on a similar theme, but slightly harder (eg having been told a shape is a kite, work out the remaining angles given two of the angles). A nice extension task, where pupils are given two angles each in three quadrilateral and work out what shapes they could possibly be. Plenary: A look at a proof of the rule, by splitting quadrilaterals into two triangles. A prompt to consider what the sum of interior angles of a pentagon might be. Printable worksheets and answers included throughout. Please review if you buy as any feedback is appreciated!

Are you bored of telling students what calculator to get for secondary school maths? Then use this poster!

A complete lesson on using sin, cos and tan to find an unknown side of a right-angled triangle. Designed to come after pupils have been introduced to the trig ratios, and used them to find angles in right-angled triangles. Please see my other resources for complete lessons on these topics. Activities included: Starter: A quick reminder and some questions about using formulae triangles (e.g. the speed, distance, time triangle). This is to help pupils to transfer the same idea to the SOHCAHTOA formulae triangles. Main: A few examples and questions for pupils to try, on finding a side given one side and an angle. Initially, this is done without reference to SOHCAHTOA or formulae triangles, so that pupils need to think about whether to multiply or divide. More examples, but this time using formulae triangles. A worksheet with a progression in difficulty, building up to some challenging questions on finding perimeters of right-angled triangles, given one side and an angle. A tough extension, where pupils try to find lengths for the sides of a triangle with a given angle, so that it is has a perimeter of 20cm. Plenary: A prompt to get pupils thinking about how they are going to remember the rules and methods for this topic. Printable worksheets and answers included. Please review if you buy as any feedback is appreciated! Error on previous version now fixed. If you have bought this already and want the amended version, please message me and I will email the file directly.

A complete lesson on vertically opposite angles. Does incorporate problems involving the interior angle sum of triangles and quadrilaterals too, to make it more challenging and varied (see cover image for an idea of some of the easier problems) Activities included: Starter: A set of basic questions to check if pupils know the rules for angles at a point, on a line, in a triangle and in a quadrilateral. Main: A prompt for pupils to reflect on known facts about angles at the intersection of two lines, naturally leading to a quick proof that vertically opposite angles are equal. Some subtle non-examples/discussion points to ensure pupils can correctly identify vertically opposite angles. Examples and a set of questions for pupils to consolidate. These start with questions like the cover image, then some slightly tougher problems involving isosceles triangles, and finally some tricky and surprising puzzles. A more investigatory task, a sort-of angle chase where pupils need to work out when the starting angle leads to an integer final angle. Plenary: An animation that shows a dynamic proof that the interior angle sum of a triangle is 180 degrees, using the property of vertically opposite angles being equal. Printable worksheets and answers included. Please do review if you buy, as any feedback is helpful!