This activity can brighten up the introduction of the scalar product formula: u.v = |u| |v| cos (theta) …using pattern spotting and allowing every student to make a contribution. I have used this activity over several years and spent time refining it to get it right. You can save yourself all this time for less than the price of a pint of beer or a cup of coffee. I promise I won’t spend the money on coffee. The full details of the lesson can be found here. I have improved my resources since the blog post was written: There are more vectors on the accurate graph and more vector-pair cards. I have used proper vector notation on the diagram and labelled the axes. If you need busy-work for fast finishers, they can work out the scalar products of more vector pairs involving the (sqrt(5), 2sqrt(5)) with all of the other vector pairs*. Ask them to find the answers to 3 sig fig.** (* The only vector I have paired it with so far is (5,12). ) (** If they finish that they can work out the angles if the (sqrt(5), 2sqrt(5)) is reflected in the x&y-axes, the line y=x etc… If they finish that ask them how they know that they have finished. There must be a permutations question in that somewhere…)

This is an interactive Geogebra page. I use it for teaching CAIE Mechanics (M1). You can run this page in a browser by clicking here or you can download the file if you want to. Geogebra pages can easily be embedded into other web-sites, including Firefly. You can click-and-drag the spots marked A, B and C to change the triangle of forces and the force diagram on the left will adjust itself to match. I hope the controls are fairly self-explanatory, but get in touch with me if you have a suggestion. You can use it to represent any equilibrium situation with only three forces. The examples I tend to use are: three students fighting over a Take That CD (or whatever) pulling in different directions a box at rest on a rough slope Perhaps you can think of a suitable non-example to include - a cyclist going up a hill, say? Too many forces to consider using a triangle of forces. Another non-example which is not equilibrium maybe? Anyway, once the class is comfortable that a triangle of forces means equilibrium, then Lami’s theorem and the Sine Rule can be shown to be equivalent. Maybe you could challenge your students to convince you that they are equivalent, but they are not allowed to use any trig identities, or you insist they have to refer to the graph of sine in their explanation.

This is an interactive Geogebra page. I use it for teaching CAIE Mechanics (M1). You can run this page in a browser by clicking here or you can download the file if you want to. Geogebra pages can easily be embedded into other web-sites, including Firefly. The slider allows you to select how many forces you want, up to a maximum of 5. You can click-and-drag the circle on the end of each coloured arrow, on the left side, to change the size and direction of each force. On the right, the black arrow shows the resultant of all of the forces. I hope the controls are fairly self-explanatory, but get in touch with me if you have a suggestion. I have used this tool to draw force diagrams for equilibrium and acceleration example problems. Considering different situations, I found it useful to elicit from the class which forces we would need to consider. Then I/the students had to jiggle them about to make the resultant either zero or point the right way. Then the students can copy a beautifully neat force diagram into their notes. It’s also handy for talking about the limitations of Lami’s Theorem or the triangle of forces. I have another Geogebra tool for Lami’s Theorem - take a look in my shop.

This is an interactive Geogebra page. I use it for teaching CAIE Further Mechanics. You can run this page in a browser by clicking here or you can download the file if you want to. You can move the balls by click-dragging the black and red spots at their centres. You can change the direction of motion of the cue ball by moving the black spot at the head of the purple arrow. You can adjust the masses of the balls and the coefficient of restitution using sliders. You can show or hide some other things using the check-boxes. If you show the grid, it makes it easier to line up the balls for a direct collision, when you are introducing the coefficient of restitution. I’m planning to use this to introduce the coefficient of restitution for direct collisions, then later to draw attention to the fact that the red ball moves along the line of action if it doesn’t have a velocity before the collision. At some point I will go into detail about it on: my blog. (I’ll try and remember to include a direct link to that when I write it up, but I won’t mind being reminded if you find I haven’t done it.) Perhaps you are interested in seeing this other resource I made for two particle collisions, which has more detail about vectors and components parallel and perpendicular to the line of action. I also made this one about particle-barrier collisions, and I a have outlined a lesson plan here on my blog.

Perhaps you remember this old TV programme? Well now you can set up your own version for your classroom. Its a strategy game for teams. The team who answers the question correctly wins the tile and each team is trying to build a bridge across the board, so they can block each other. This is version 2 so I’ve made some improvements. Details here if you are interested I’ve included an example with terminology from A-level Mechanics M1. Click once to change a hexagon red and click again to make it white. Keep clicking and it will cycle through the different options. Unfortunately it doesn’t do that flashy thing when you win or play music, but of course you can sing the theme tune when theres a winner.

This is an interactive Geogebra page. I use it for teaching CAIE Further Mechanics. You can run this page in a browser by clicking here or you can download the file if you want to. Geogebra pages can easily be embedded into other web-sites, including Firefly. You can click-and-drag the spots in the centres of the red and blue particles to change their initial and final position on the circle. The spot on the end of the red arrow can be click-dragged to change the initial speed. There are check-boxes for various options. I hope the controls are fairly self-explanatory, but get in touch with me if you have a suggestion. In particular, notice the constrained check-box. If you click this off, you will have the option to see what happens when the particle falls off the circular path. Click-drag the spot in the middle of the blue particle and the blue particle will move along the parabolic trajectory. I’ve found this page to be useful for explaining the difference between the constrained and unconstrained cases. I was using examples of a particle on a string vs a particle on a rod, and this tool allows you to summarise or elicit the key differences between these to situations: Constrained: R = 0 is not interesting R can act away from the centre of the circle and hold the particle up The particle will keep moving on the circular path until its KE = 0 --> it stops Unconstrained: R = 0 is interesting because that is where the particle changes from circular motion to projectile motion R can only act towards the centre of the circle and cannot hold the particle up The particle will leave the circle when R = 0 and it will not reach the KE = 0 position

This is an interactive Geogebra page. I use it for teaching CAIE Further Mechanics. You can run this page in a browser by clicking here or you can download the file if you want to. Clicking on the “Start” button will start the animation of the ball’s motion. You can adjust the sliders to change the initial speed or the value of g. Say, if you want to model playing basketball on Mars, then g=3.8 ish. You can click-and-drag the pink spot to change the angle of projection. You can click-and-drag the blue spot to move the basket. Clicking on the “Trajectory” and “Axes” check-boxes will reveal the trajectory formula and the axes. Then you can adjust the initial speed and direction and the trajectory formula will change in real time. You might want to use it as a game to the finish the lesson preceding your introduction of the trajectory formula. You might want to show the class what computer games used to look like in the old days: Scorched Earth at DOS Games Online If you haven’t played Scorched Earth before, you have to chuck bombs at each other, and the bombs fall with a parabolic trajectory.

This is alternative proof-without-words for the A-level double angle formulae. If you ask me, this is a nicer way to derive them than just say alpha = beta = theta in the compound angle formulae. I suggest getting the students to use the substitution method first and show the circle method to the students who find it easy. If you have a better idea, please tweet me. This proof relies on the circle theorem: “angle at the centre is double the at the circumference” and right-angled trig.

This is a random thing I made. It’s pretty straightforward to set up. set up the formula, say for B2: “=A1+A3” and copy and paste it into all the cells underneath. Choose a cell at the top in the middle somewhere. Type “1”. It should fill in with Pascal’s Triangle automatically. Set the conditional formatting to make empty cells white and colour in multiples of the contents of one cell (I used cell I8) something like this: “=AND(MOD(A2,$I$8)=0,A2<>0)” …or you could download my copy.

This is an interactive Geogebra page. I use it for teaching CAIE Further Mechanics. You can run this page in a browser by clicking here or you can download the file if you want to. Geogebra pages can easily be embedded into other web-sites, including Firefly. (Amendment: Version 2 shows the values of the components) You can click-and-drag the red and purple spots to change the direction of the velocity of the particle immediately before the collision. The blue arrow shows the velocity immediately after the collision. The black spot can tilt the barrier at a different angle. A slider controls the coefficient of restitution. With check-boxes you can show the components of the velocities parallel and perpendicular to the barrier (called x and y), the angles and the grid. I hope the controls are fairly self-explanatory, but get in touch with me if you have a suggestion. You might want to use it to introduce oblique collisions to a class who have already seen direct collisions. See my blog post about using this Update 4/12/2022 - A blog-reader asked how to show the values of the speeds - I’ve added an option for this.

This is an interactive Geogebra page showing an Argand diagram with the first “n” powers of a complex number “z”. Click here if you want to use the tool without downloading it. n is controlled with a slider. z is controlled by dragging a point around with the mouse. You can hide and show the x+yi forms and the modulus of the z, z-squared, z-cubed, etc using checkboxes. I have some ideas about how to use this tool, which I will write about on my Blog soon.

This is an interactive Geogebra page. I use it for teaching CAIE Further Mechanics. You can run this page in a browser by clicking here or you can download the file if you want to. Geogebra pages can easily be embedded into other web-sites, including Firefly. You can click-and-drag the red and green spots to change the velocities of the particles immediately before the collision. The pink and blue arrows show the velocities immediately after the collision. Sliders control the masses of the particles and the coefficient of restitution. I hope the controls are fairly self-explanatory, but get in touch with me if you have a suggestion. You might want to set a homework task for your class, to try and discover some key ideas, for example: what is the significance of the Line of Action? what do you notice about the components of velocity perpendicular to the line of action? How does the applet calculate the velocities after the collision?

This is an interactive Geogebra page. It can be used it for teaching or revising CAIE Mechanics (M1) Friction and Further Mechanics Equilibrium of Rigid Bodies You can run this page in a browser by clicking here or you can download the file if you want to. Geogebra pages can easily be embedded into other web-sites, including Firefly. You can click-and-drag the grey spot to tilt the slope more or less. You can click-and-drag the green spot to increase the weight of the box. There are check-boxes for various options. I hope the controls are fairly self-explanatory, but get in touch with me if you have a suggestion. A slider controls μ, the coefficient of friction and the vectors will all adjust in real time. A common type of problem from CAIE is to establish whether an object will topple or slide first when the supporting surface is tilted, so I created this page to play with this idea a little bit. My plan is to set a centre-of-mass homework task and add a question on on the end where students have to use the applet. I’ll ask them to find the angle where the box is on the point of sliding for different values of μ, and the angle at which it is on the point of toppling. (Potential engineers might want to go away and research coefficients of friction. Are there applications where you might want the point of toppling to come before point of sliding, or vice-versa? What would be the ideal materials to use for the box and surface in these situations? ) Next I’ll ask them about how these angles would be different if I add a massive particles to different positions on the box - larger or smaller and why. Ideally, I can make most of the homework self-marking. I’ll share this here when I create it if it is any good.

This is an interactive Geogebra page. It can be used it for teaching CAIE Mechanics (M1) Newton’s Laws of Motion. You can run this page in a browser by clicking here or you can download the file if you want to. Geogebra pages can easily be embedded into other web-sites, including Firefly. You can click-and-drag the blue spot to tilt the slope more or less. You can click-and-drag the green spot to change the P force’s size and direction, or hide it over the particle. A slider controls the mass of the particle and there are check-boxes for various options. I hope the controls are fairly self-explanatory, but get in touch with me if you have a suggestion. A common mistake students make is to not consider the perpendicular components of extra force when calculating the normal reaction R force. I plan to use it to set up questions with a particle-on-horizontal-surface and a particle-on-slope situation, with the P force either not included, horizontal, vertical, parallel or perpendicular to the slope or at some random angle. I’ll describe the arrangement of the forces, use the tool to draw most of the forces on the projector, perhaps hiding the P force, but describing it. Then ask students to find the magnitude of R, the acceleration of the particle.

This is an interactive Geogebra page. You can use it without downloading anything here. Click and drag the purple X up and down on the screen. You can show how at some positions the peaks and troughs of the sine waves line up as constructive interference, to make bright patches, and in other places the peaks and troughs are out of phase, so destructive interference means that these points are dark. By making the slit smaller you can demonstrate how you need to move the cross further away from the centre to get a dark patch, showing how the diffraction angle is greater. Please get in touch with me if you have more suggestions for how to use this tool, or write me a review if you like it.

To use this tool without downloading anything, click here. I wrote a detailed description of how I use this in a lesson on my blog. In brief: There is a slider which chooses the type of locus you want to focus on. Type 1: | z - a | = r You can control the position of the points representing a and z. Type 2: | z - a | = | z - b | You can control the positions of the points representinga, b and z. Type 3: Arg(z-a) = alpha You can control the positions of a and z. You can use the page to test whether points on the plane satisfy a given equation or inequality. You can join the dots to find the right shapes: Type 1: | z - a | = r A circle of radius r centred at (Re(a),Im(a)) Type 2: | z - a | = | z - b | The perpendicular bisector of (Re(a),Im(a)) and (Re(b),Im(a)) Type 3: Arg(z-a) = alpha The half-line which is alpha radians anticlockwise from the Real-axis direction beginning at (Re(a),Im(a)) I hope this is useful for you. Please feel free to comment on my blog if you have suggestions for improving the tool. Do you have any ideas for memorising these different types of locus? Like an acronym or a mnemonic? If you find it useful, please write a review on TES.com. Thanks!

This is an interactive Geogebra page. I use it for teaching CAIE A-level Mechanics M1 . You can run this page in a browser by clicking here or you can download the file if you want to. Geogebra pages can easily be embedded into other web-sites, including Firefly. You can click-and-drag the green and yellow spots to change the initial and final of the velocities and the time taken. All the values shown on the velocity-time graph will update in real time. With check-boxes you can show or hide the values on the graph, the equations with values substituted in and the grid. I hope the controls are fairly self-explanatory, but get in touch with me if you have a suggestion. By the time students come to me, they have already seen the suvat equations before and many use them instead of using v-t graphs when I teach them, out of reluctance to draw graphs, and when they do draw graphs, they are often rubbish. Suffice to say, the students here are more confident with algebra than with graphs. I plan to use this to encourage students to use area and gradient methods for velocity-time graphs. I’ll say something like “no, look the equations are switched off” if they try to use an algebraic method to solve a problem. When we cover suvat, I may include a homework task which asks students to make v-t graphs of each of their solutions after they have finished their algebraic method, and they can use screen-shots from this tool. Then I’ll ask them to draw one for themselves and they can peer-assess. If you think of another way to use this tool, please leave a comment on my blog - I’d be delighted to hear about it.

This is a GeoGebra tool to help teach the Transformation Matrices topic. You can use it in the browser without downloading any software here. The purple vectors a and b can be dragged around with the mouse, and these control the transformation matrix. (a is the image of the i unit vector (1,0) and b is the image of the j unit vector (0,1)) This transformation is applied to object PQRS to give the image P’Q’R’S’. The object points can also be dragged around with the mouse. The image can be hidden and revealed. You can also show the matrix and the matrix multiplication using a checkbox - sorry it doesn’t align perfectly, but it is hopefully sort of ok. I’ve included an inksplat so I can use it to show the calculation but hide most of the answer, as a lesson starter. (This image belongs to Pixabay.) I will be writing down some ideas about how to incorporate this into a lesson on my blog. Please leave a comment if you have suggestions or ideas to make this better.

This is an interactive GeoGebra page controlled by the mouse. Change the mass of particles and move them around and the Centre Of Mass (COM) will move instantaneously. You can use this page without downloading anything by clicking here. I hope the controls are pretty straightforward. You can drag the points A, B, C, D, E around with the mouse. A slider controls how many particles you have and their masses. Hide and show the COM with the checkbox, so you can set up a question on the board and then reveal the answer. Another idea is to hide one of the points behind an image (you can Ctrl-C, Ctrl-V an image into Geogebra easily) and then the class have to work it out using the COM. Other ideas for using it? How could I make this better? Please get in touch on Twitter or comment on my blog.

This is an interactive GeoGebra page built using only Further Mechanics level formulae. You can use the page without downloading if you click here. You can control the simulation using the mouse and dragging points around: The circle on the end of the arrow can be used to adjust the initial velocity. Points A and B control the position and inclination of the ground. Sliders control the g value (in case you are not on planet Earth) and the coefficient of restitution (how bouncy the bounces are). You can show and hide the axes, the trajectory curves and the velocity arrows where the bounces take place.