## Shape

Units, Measurements and Drawing Area and Volume Pythagoras and Trigonometry Congruence and Similarity Transformations Shape and Angle Properties (2D) Shape and Angle Properties (3D) Constructions Vectors ## Calculation and MeasuresUnits, Measurements and Drawing
Standard Units of Measurement
Use standard units of mass, length, time, money and other measures, including with decimal quantitiesChange freely between related standard units, such as time, length, area, volume/capacity, mass (KS3) Coming soon. Compound Measures
Use compound units such as speed, unit pricing and density to solve problems (KS3)Convert between related compound units (speed, rates of pay, prices, density, pressure) in numerical and algebraic contexts (KS4) Density and Buoyancy (PHET)A superb interactive for introducing ideas of density; try experimenting with the "mystery" block option where pupils use density to figure out the material used. Archimedes and the Crown (Tyler DeWitt via YouTube)Lovely explanation of density, mass and volume using the golden crown story. Worksheet collection (greatschools.org)**Sheet one**: Simple speed distance time problems in miles per hour.**Sheet two**: More complicated problems involving some conversions.
Drinks Can - Mass, Density, Volume (C Norledge)Context-based problem using DMV calculation. Speed, Distance, Time Calculations (C Norledge)Problems involving use of speed = distance / time, with some conversion of units. Largest Gold Ingot (C Norledge)Use mass, density and volume to calculate the price of the Toi gold ingot. How Fast is the Earth Moving (TwigWorld via YouTube)Using approximation, speed-distance-time and circumference of a circle to estimate the speed of the Earth. Deriving the SUVAT equations (C Norledge)Get pupils to derive the five SUVAT equations by changing the subject. Includes set of problems to work on. Block Density (C Norledge)Make a block with an average density of 9g per cube. Can you find a block with an average density of 8g per cube? What other whole number averages can you make? Scale Drawings and Bearings
Use scale factors, scale diagrams and maps (KS3)Draw and measure line segments and angles in geometric figures, including interpreting scale drawings (KS3) Interpret and use bearings (KS4) Coming soon. Area and Volume
Area of Rectangles, Triangles and Parallelograms
Derive and apply formulae to calculate and solve problems involving perimeter and area of triangles and parallelograms (KS3)Area builder (PHeT)Drag and drop squares to create compound shapes; could be used to model area basics or investigate perimeter and area. Exploring area package (Illuminations)**Interactive area tool**: Interactive resource designed for pupils to discover the formulae for area of a triangle and parallelogram.**Squares and rectangles sheet**: Develop area of a triangle formula and a**couple of practice problems**.**Rectangles and parallelograms sheet**: Transform the rectangles into parallelograms.**Shape cutter interactive**: Useful for demonstrating transformation of a rectangle into a parallelogram and vice versa.
Worksheet from greatschools.org**Sheet one**: Area by counting squares**Sheet two**: Finding area of squares and rectangles by multiplying.**Sheet three**: Finding area using long multiplication strategies.**Sheet four**:**Straightforward simple practice worksheet finding the area of right-angled triangles.**
Area of triangles collection (mathinenglish.com)**Area of triangles**: Simple practice finding the area given base and height.**Area within triangles**: Finding shaded areas using the area of a triangle.**Perimeter and area**: Finding both measurements for triangles.**Height of triangles**: Working backwards to find the height given the area.
Where's the point? (KangarooMaths via SuffolkMaths)Excellent worksheet requiring pupils to convert between metres, km, cm, mm and calculate resulting areas; also good for practising decimal multiplication in context and dealing with unit conversions. Investigations from Median (Don Steward)**Fraction that is a triangle**: Good activity linking area of triangles and fractions.**Triangles in a 5 by 5 square**: Similar to above using 5 by 5 dotty grid.**Dotty triangles**: Which triangle covers the largest fraction of the rectangle?**Area of parallelograms**: Find the link between area, dots on the boundary and dots inside parallelograms with a fixed perimeter.
Area of Trapezia and Compound Shapes
Derive and apply formulae to calculate the area of trapezia (KS3)Calculate and solve problems involving areas of composite shapes (KS3)Everything about Perimeter and Area (BGfL)Interactive aimed at KS2, but has lovely animations of splitting up shapes made from rectangles on Levels 2 and 3 of the area section. Circumference and Area of Circles
Calculate and solve problems involving perimeters of circles, areas of circles and composite shapes (KS3)Lesson: Discovering Pi (The Storeroom)My favourite way to introduce any work on circumference or area of circles - pupils use circular objects to investigate links between diameter and circumference, and radius and area. There's a similar step-by-step investigation here on the TES.Curvy Areas (NRich)A great investigation to try, with plenty of practice on calculating the area of semicircles and using compasses. You might also want to look at Yin Yang, which is a simpler version of this activity.Arc Length and Area of Sector
Calculate arc lengths, angles and areas of sectors of circles (KS4) Worksheet generator (Worksheet Math)Excellent comprehensive selection of worksheet generators, including: Circles (Cleave Books via National Stem Centre)Final page of this resource has some nice wordy problems on sector area. Arc and sectors worksheet (nadia_8787 via TES)Lots of problems on arc length and sector area, including a good selection of context and problem-solving questions. Arc length and area of sector worksheets (Maths Worksheets 4 Kids)Large selection of basic practise worksheets with answers; good for homework or revision. Median:**Swinging**: Find the arc length of a swing given the chain length and angle.**Skyhawk**: Similar problem based on the Skyhawk theme park ride in Ohio.**Grate question**: Context-based problem involving dimensions of a fire grate.**Arc length and sector area**: Simple but rich problems to practise finding arc length and sector area.
Arc lengths, sectors and segments loop (Maths Loops via National Stem Centre)Loop card activity including problems on arc length, sector area and a couple of segments problems. Brilliant for revision! Volume and Surface Area of Cuboids and Simple Prisms
Volume and Surface Area of Cylinders
Derive and apply formulae to calculate and solve problems involving volume of cylinders (KS3)Use the properties of faces and surfaces of cylinders to solve problems in 3D (KS3) Coming soon. Volume and Surface Area of Cones, Spheres and Pyramids
Calculate surface areas and volumes of spheres, pyramids, cones and composite solids (KS4)Comparing volumes of cylinders, spheres and cones (PBS Learning Media)Quick interactive showing a sphere and cone being "squashed down" inside a cylinder. Ice cream puddle (NCTM)Looking at melting ice cream in cones and spheres. Gumball machine (Robert Kaplinsky)Work out how many gumballs fit inside a giant gumball machine using volume of spheres. A picture of a cone (ABCTeach)Linked because I genuinely found it nearly impossible to find a decent picture of an unshaded cone without measurements all over it. Using a tennis ball to explain SA of a sphere (Resourceaholic)A brilliant idea for explaining the formula for surface area of a sphere without the need for calculus. Pythagoras and Trigonometry
Pythagoras' Theorem
Use Pythagoras’ Theorem to solve problems involving right-angled triangles (KS3)Apply Pythagoras’ Theorem to find lengths in right-angled triangles and in two-dimensional figures (KS4) Coming soon. Trigonometry in Right-Angled Triangles
Use trigonometric ratios in similar triangles to solve problems involving right-angled triangles (KS3)Apply trigonometric ratios to find angles and lengths in right-angled triangles and in two-dimensional figures (KS4) Teaching Trigonometry I (The Storeroom)Tried and tested set of lessons for introducing trigonometry using similar triangles and trig. tables. Basic practice worksheets (Worksheet Math)Excellent comprehensive selection of worksheet generators, including basic calculator practice, problems using just an angle or side, problems using just one trigonometric ratio, and some good mixtures: Trigonometry Introduction (Don Steward) Label sides on triangles and write ratios. KESH Maths Takeaway (Link)GCSE practice problems booklet (N.B. 1MAo specification) Calculate Angle Size (Don Steward)A few problems using missing angles, includes four right-angled triangles but also isosceles triangle, rectangle, trapezium and sector. Creating measures for steepness (FLA Guide)Real-life application of missing angle problem solving; using trigonometry to find the angle of elevation of a staircase and use this to compare the steepness of six different staircases. There are some resources for this on Median.Resource Collection (Stacy Brookes)A huge collection of links to plenty of trigonometry resources. Interactive unit circle (Maths is Fun)Brilliant for demonstrating the link between trigonometry in right angled triangles, the unit circle and graphs of all three trigonometric functions. Interactive triangle for trigonometry (Next Level Maths via TES)Useful for generating problems to solve or just for demonstration to pupils. Trigonometry loop cards (dsc505 via TES)Two sets of loop cards to practise finding missing angles and sides using trigonometry. Set B is slightly harder than Set A. Excellent for finishing off the topic or as quick revision. Exact Trigonometric Values
Know the exact values of sin θ, cos θ and tan θ for θ = 0, 30, 45, 60, 90 degrees (KS4)Exact values triangles worksheet (Stacy Brookes)Simple write-on worksheet for deriving the exact trigonometric ratios from standard triangles Codebreaker (Stacy Brookes)Quick codebreaker worksheet using exact values. Unit circle (Maths is Fun)Unit circle interactive for demonstrating sine, cosine and tangent graphs and relationships - may be useful for some additional enrichment. Hand trick for exact trigonometric values (Math TP via YouTube)Short video demonstrating the way to work out exact trigonometric values for sine and cosine using your hand; I use this with sixth formers as a quick trick for exams to avoid drawing out the standard triangles, so this may be equally applicable to pupils at GCSE. Pythagoras and Trigonometry in 3D (H)
Apply Pythagoras’ Theorem and trigonometric ratios to find angles and lengths in right-angled triangles (and, where possible, general triangles) in three dimensional figures (KS4 Higher only) Inscribed pyramid 2 (Math Arguments 180)A nice problem using Pythagoras in 3D with a pyramid inside a cuboid. Sine and Cosine Rules (H)
Know and apply the sine rule and cosine rule to find unknown lengths and angles (KS4 Higher only)Coming soon. Area of a Triangle using 1/2 ab sin C (H)
Know and apply area = 1/2 ab sin C to calculate the area, sides or angles of any triangle (KS4 Higher only)Coming soon. ## Properties and ConstructionsCongruence and Similarity
Congruence and Similarity
Know and use the criteria for congruence of triangles (KS3)Identify and construct congruent triangles (KS3) Apply angle facts, triangle congruence, similarity and properties of quadrilaterals to derive results about angles and sides and use known results to obtain simple proofs (KS3) Apply the concepts of congruence and similarity, including relationships between lengths, in similar figures (KS4) Coming soon. Similar Areas and Volumes (H)
Apply the concepts of congruence and similarity, including the relationships between areas and volumes in similar figures (KS4 Higher only)The Unbelievable Bulk (alutwyche via TES)A great investigation about "The Unbelievable Bulk" and "Doctor Tanner" - students find how cross the Bulk is based on his size by calculating the scale factor of enlargement. This also has a nice extension about surface area and volume. NRICH**Growing rectangles**: Investigating how enlargement affected area and volume by looking at different sizes of rectangles. Great for extension or whole-class investigation**Who Is The Fairest Of Them All:**Brilliant for a bit of enlargement practise, with scope to develop further.
Transformations
Translations, Rotations and Reflections
Identify properties of, and describe the results of translations, rotations and reflections applied to given figures (KS3)Describe translations as 2D vectors (KS4) Transformation package (TristanJones via TES)Lovely bank of worksheets - pupils complete transformations on a selection of shapes to spell out a word in the centre of the page. Always popular whenever I use these. Transformation puzzle (ECBayliss via TES)Similar puzzle with a combination of rotations, reflections and translations. Transformation interactives (EChalk via Eduqas)**Reflections**: Superb interactive for demonstrating reflections, with moveable shapes and mirror lines.**Rotations**: Similar interactive for demonstrating rotations.
Transformation interactives (TeacherLED) **Translation**: Simple interactive for demonstrating translations in words or as vectors.**Virtual tracing paper**: Great for demonstrating how to use tracing paper to complete reflections and rotations.
Enlargement by Positive Scale Factor
Construct similar shapes by enlargement, with and without coordinate grids (KS3)Interpret and use fractional scale factors for enlargements (KS4) Spell a Word (whidds via TES)These really are great - students carry out the required transformations which creates a picture of a related word (in this case, SIZE). Whenever I've used these, students have found them very engaging. The worksheet on enlargement does include two negative scale factors, but students can "guess" these at the end, leading to a good discussion about how negative scale factors work.Coming soon. topic or as quick revision. Inside Maths (via STEM Centre)Video with many practical applications of enlargement, as well as flagging up and looking at misconceptions with a few common exam questions. Middle of Centres (Transum)A nice little starter looking at finding the centre of enlargement. Note that you need to scroll down to the "advanced" problem. Photographic Enlargements (STEM centre)Useful for a bit of quick-fire questioning as a starter or plenary. Your computer will tell you when you download the application that it might be a virus - don't worry, it's not (it's flagged because it's an infrequently downloaded .exe file). There are also some explanatory notes available here. Requires registration with the National STEM centre, which is free and full of great resources.Enlargement by Negative Scale Factor (H)
Interpret and use negative scale factors for enlargements (KS4 Higher only)Fractional and negative SF - spell a word (alicecreswick via TES)Five question grids with problems on fractional and negative enlargements, which spell out the letters MATHS. 8-bit Mario negative enlargements (solvemymaths)Great practice using negative scale factors to produce an 8-bit picture of Mario. Negative scale factor enlargement worksheets (eChalk)Free printable worksheets - saves a considerable amount of time drawing graphs and grids. Enlargement with Negative Scale Factors (alutwyche via TES)Quick, simple printable worksheet. Enlargement worksheet - negative scale factors (HolyheadSchool via TES)Standard worksheet with lots of negative scale factor practice. Diagrams will need copying into books. Maths takeaway booklet - Enlargements by negative scale factor (KESH)Exam questions for Edexcel 1MAO specification; valid for all 2015 GCSE specs. Revision booklet (CorbettMaths)QR linked to video revision; old spec but still valid. Enlargement scale factor root 2 (Don Steward)While not a negative enlargement problem, I've included this here as it's a really nice, challenging problem with lots of scope for links to other topics. Enlargements teaching tool (EChalk via Eduqas)This interactive has options to select negative scale factors, and show/hide centre of enlargement, scale factor and coordinate axes; the best I've found for demonstrations. Invariance (H)
Describe the changes and invariance achieved by combinations of rotations, reflections and translations (KS4 Higher only)Invariance activity sheet (Peter Mattock via TES)A RAG worksheet around identifying invariant points on different transformations, incorporating a CLOZE activity (fill in the blanks), a matching activity, and a Venn Diagram activity Shape and Angle Properties (2D)
Angles Around a Point, on a Line and in a Triangle
Apply the properties of angles at a point, angles at a point on a straight line and vertically opposite angles (KS3) Derive and use the sum of angles in a triangle (KS3) Describe, sketch and draw using conventional terms and notations: points, lines, parallel lines, perpendicular lines, right angles (KS3) Coming soon. Angles in Parallel Lines
Understand and use the relationship between parallel lines and alternate and corresponding angles (KS3)Coming soon. Properties of Triangles
Use the standard conventions for labelling the sides and angles of triangle ABC (KS3) Derive and illustrate properties of triangles (for example, equal lengths and angles) using appropriate language and technologies (KS3) Identifying types of triangle (greatschools.org)Straightforward simple practice worksheet identifying equilateral, isosceles, scalene and right-angled triangles. Properties of Quadrilaterals
Derive and illustrate properties of quadrilaterals and other plane figures (for example, equal lengths and angles) using appropriate language and technologies (KS3)Properties and Angles in Polygons
Describe, sketch and draw using conventional terms and notations: regular polygons, and other polygons that are reflectively and rotationally symmetric (KS3) Deduce the angle sum in any polygon, and to derive properties of regular polygons (KS3) Circle Properties
Derive and illustrate properties of circles using appropriate language and technologies (KS3) Identify and apply circle definitions and properties, including: centre, radius, chord, diameter, circumference, tangent, arc, sector and segment (KS4) Coming soon. Circle Theorems
Apply and prove the standard circle theorems concerning angles, radii, tangents and chords, and use them to prove related results (KS4 Higher only)Coming soon. Shape and Angle Properties (3D)
Faces, Vertices and Edges
Use the properties of faces, surfaces, edges and vertices of cubes, cuboids, prisms, cylinders, pyramids, cones and spheres to solve problems in 3D (KS3)3 dimensional shapes (greatschools.org)Pupils identify vertices on common 3D shapes. They could also identify faces and edges, and cut out/stick the sheet into their exercise books with the properties listed. Plans and Elevations
Construct and interpret plans and elevations of 3D shapes (KS4)Isometric drawing tool (Illuminations)An absolutely fantastic interactive isometric drawing tool; create shaded isometric drawings or 3D cube models, rotate and automatically generate plans and elevations. Constructions
Constructions
Derive and use the standard ruler and compass constructions (perpendicular bisector of a line segment, constructing a perpendicular to a given line from/at a given point, bisecting a given angle)Recognise and use the perpendicular distance from a point to a line as the shortest distance to the line (KS3) Coming soon. Shape: Vectors
Vectors
Vector calculationsApply addition and subtraction of vectors, multiplication of vectors by a scalar, and diagrammatic and column representations of vectors (KS4) Vector proofs (H)Use vectors to construct geometric arguments and proofs (KS4 Higher only) Probability
Probability Basics
The probability scale Record, describe and analyse the frequency of outcomes of simple probability experiments involving randomness, fairness, equally and unequally likely outcomes, using appropriate language and the 0-1 probability scale (KS3) Understand that the probabilities of all possible outcomes sum to 1 (KS3) Mutually exclusive events Apply the property that the probabilities of an exhaustive set of mutually exclusive events sum to one Simple sets and Venn diagrams Enumerate sets and unions/intersections of sets systematically, using tables, grids and Venn diagrams (KS3) Theoretical and Experimental
Generate theoretical sample spaces for single and combined events with equally likely, mutually exclusive outcomes and use these to calculate theoretical probabilities (KS3) Use a probability model to predict the outcomes of future experiments; understand that empirical unbiased samples tend towards theoretical probability distributions, with increasing sample size (KS4) Independent and Conditional
Calculate the probability of independent and dependent combined events, including using tree diagrams and other representations, and know the underlying assumptions (KS4) Calculate and interpret conditional probabilities through representation using expected frequencies with two-way tables, tree diagrams and Venn diagrams (KS4 Higher only) Statistics
Presenting Data
Simple charts and graphsConstruct and interpret frequency tables, bar charts and pictograms for categorical data, and vertical line (or bar) charts for ungrouped and grouped numerical data (KS3) Pie chartsConstruct and interpret pie charts for categorical data (KS3) Time-series graphsInterpret and construct tables and line graphs for time series data (KS4) Scatter graphsDescribe relationships between two variables and illustrate using scatter graphs (KS3) Recognise correlation and know that it does not indicate causation (KS4) Draw estimated lines of best fit, make predictions Interpolate and extrapolate apparent trends whilst knowing the dangers of so doing Cumulative frequency and box plots (H)Construct and interpret cumulative frequency graphs, and know their appropriate use (KS4 Higher only) Use box plots to interpret, analyse and compare the distributions of data sets (KS4 Higher only) Histograms (H)Construct and interpret histograms with equal and unequal class intervals and know their appropriate use (KS4 Higher only) Processing Data
Ungrouped data - averages and spreadDescribe, interpret and compare single-variable distributions through measures of central tendency (mean, mode, median) and spread (range, consideration of outliers) (KS3) Grouped data - averages and spreadDescribe, interpret and compare single-variable grouped distributions through measures of central tendency (mean, mode, median) and spread (range, consideration of outliers) (KS3) Interpret, analyse and compare distributions through appropriate measures of central tendency (including modal class) (KS4) Quartiles and inter-quartile range (KS4 Higher only)Interpret, analyse and compare distributions using quartiles and inter-quartile range (KS4)Data collection and sampling
Sampling (KS4) Infer properties of populations or distributions from a sample, whilst knowing the limitations of sampling (KS4) Apply statistics to describe a population (KS4) |
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