good at math: Search results for trig

Showing posts sorted by relevance for query trig. Sort by date Show all posts

Thursday, September 5, 2019

5 Maths Gems #9

Because I didn't write a gems postal service final weekend, imagine how much I receive got to tell y'all most today! Quick - let's acquire started...

1. Hula Hoops
Who needs technology scientific discipline when y'all tin stick a hula hoop on the board? This uncomplicated idea, shared by @emziecampbell, is for instruction children to tell the time. It would function good for other circle-based topics too. Stick a hula hoop on the board together with purpose it to instruct circle theorems. Stick 2 or 3 on the board to brand a Venn diagram. Hula hoops receive got novelty value, practical application together with toll a mere £2.

2. Stuck Post-Its
Another uncomplicated all the same effective idea, this fourth dimension from @Misswrenmaths: 'stuck post-its'. Sometimes when students are completing exercise questions inward class, y'all terminate upwards repeating yourself numerous times every bit y'all displace from educatee to student. Instead, topographic point the questions on the board together with enquire students to stick a named post-it Federal Reserve notation on whatever questions they ask assist with - together with hence they tin comport on with the remainder of the exercise without having to human face with their manus up. For instance the pic below shows that you'd ask to assist 1 educatee individually with query 9, simply it may move worth going through query 10 with the whole class, or gathering a grouping together to beak over it.

This technique stops duplication of endeavour together with helps the instructor assess understanding, addition it has the produce goodness of getting students out of their seats, albeit briefly. Read the fascinating postal service 'A veteran instructor turned charabanc shadows 2 students for 2 days – a sobering lesson learned' from @grantwiggins and you'll come across why this is important. Influenza A virus subtype H5N1 picayune flake of movement keeps students alert.

While I'm on the bailiwick of post-it notes, I should cite a cool novel app 'Post-it Plus' that was suggested by @dazmck. With this gratuitous app y'all tin receive got a photograph of a wall of post-it notes together with and hence organise them digitally on your telephone or tablet. Amazing.

3. Trig for Victory
@mr_g_walton demonstrated his brilliant inventiveness with his 'Maths Propaganda' posters.

Every poster is first-class hence it was difficult to pick out 3 examples for this post!

In add-on to the posters, Mr Walton suggests revision cards inward the shape of ration booklets together with a schoolhouse broadcast past times the Ministry of Maths! Fantastic ideas. All xvi editable posters together with associated resources tin move downloaded here. Credit also to @El_Timbre for inspiring the 'Trig for Victory' poster.

4. Parabolic Multiplication

qedcat.com

If y'all were at the recent Maths Conference inward Kettering together with hence y'all may receive got been interested inward Johnny Ball's parabolic multiplication. This involves drawing a trouble from 1 betoken on the graph of y = x² to another. Two examples are shown inward the pic - the reddish trouble joins the numbers 3 together with 4, crossing the y-axis at 12, an the greenish trouble joins the numbers 2 together with 2, crossing the y-axis at 4. It's a bizarre multiplication method.

Influenza A virus subtype H5N1 sum explanation is inward this 'Cool Math Essay' from @jamestanton. The algebra is accessible together with satisfying. Figuring it out would brand a proficient chore for vivid GCSE students or 6th formers studying coordinate geometry.

If y'all reach off to move visiting Deutschland together with hence produce acquire to the Mathematikum. There y'all volition discovery a parabola drawn on a wall with a length of rope for visitors to necktie from 1 betoken on the parabola to unopen to other - the betoken at which the rope crosses the y-axis is the production (source: qedcat.com). Does anyone desire to receive got upwards the challenge of putting 1 of these interactive displays on their classroom wall? If so, delight shipping me a photo!

5. Counting Circles
There's been a lot of buzz alongside American teachers on Twitter most counting circles. This is a uncomplicated together with versatile activeness to railroad train divulge feel together with mental maths strategies. The master copy postal service A veteran instructor turned charabanc shadows 2 students for 2 days – a sobering lesson learned' from @grantwiggins and you'll come across why this is important. Influenza A virus subtype H5N1 picayune flake of movement keeps students alert.

While I'm on the bailiwick of post-it notes, I should cite a cool novel app 'Post-it Plus' that was suggested by @dazmck. With this gratuitous app y'all tin receive got a photograph of a wall of post-it notes together with and hence organise them digitally on your telephone or tablet. Amazing.

3. Trig for Victory
@mr_g_walton demonstrated his brilliant inventiveness with his 'Maths Propaganda' posters.

qedcat.com

Last calendar week I also ran my kickoff A veteran instructor turned charabanc shadows 2 students for 2 days – a sobering lesson learned' from @grantwiggins and you'll come across why this is important. Influenza A virus subtype H5N1 picayune flake of movement keeps students alert.

While I'm on the bailiwick of post-it notes, I should cite a cool novel app 'Post-it Plus' that was suggested by @dazmck. With this gratuitous app y'all tin receive got a photograph of a wall of post-it notes together with and hence organise them digitally on your telephone or tablet. Amazing.

3. Trig for Victory
@mr_g_walton demonstrated his brilliant inventiveness with his 'Maths Propaganda' posters.

qedcat.com

I also featured inward the 'Behind the Mathematician' serial from @WorkEdgeChaos - writing that postal service was a slap-up chance for me to reverberate on my instruction together with mean value most what motivates me together with what I discovery challenging.

I desire to terminate today's postal service past times encouraging y'all to majority a ticket for @LaSalleEd's next National Maths Teacher Conference, which is on Sat 14th March 2015 inward Birmingham. Click here to majority your gratuitous ticket - I'll come across y'all there.

Thursday, October 24, 2019

5 Maths Gems #9

qedcat.com

Teaching Trigonometry

In today's ship I focus on ideas as well as resources for instruction trigonometry. H5N1 recent conversation alongside a pupil made me aware that I demand to alter my approach to instruction this topic.

What's upward alongside Hannah?
Hannah is a Year 12 who I privately tutor. She is bright, articulate as well as hard-working. I was her maths instructor inward Years 10 as well as eleven (she has directly moved to about other school). Last calendar week I asked Hannah to factorise a quadratic. This is what she did:

x² + 13x + 36

= x² + 4x + 9x + 36

= x(x + 4) + 9(x + 4)

= (x + 9)(x + 4)

I'm certain you'll care that this is a rather convoluted method. At about indicate (I suspect from a previous mortal tutor), Hannah learnt a method for factorising quadratics alongside a > 1 as well as has chosen to apply this method to all quadratics. She gets the right answer, therefore there's cypher incorrect alongside this method (albeit a petty slow), but when I saw her do this it raised an of import issue. What I realised, to my horror, was that I'd taught Hannah for 2 years as well as had no thought that this was how she factorised quadratics. She'd been getting the answers right therefore I hadn't examined her methods inward detail. In a shape of 24 GCSE students who I solely come across for 3 hours a week, I suppose I focus most of my efforts on helping students when they acquire answers wrong. But my lack of awareness of Hannah's odd method highlights gaps inward my formative assessment.

What's this got to do alongside trigonometry? Well inward a subsequent conversation alongside Hannah she asked me near the upcoming changes to GCSEs, which volition touching on her younger sister. I told her that her sis volition demand to know how to uncovering exact values of trig ratios such every bit sin30 as well as cos45. Hannah didn't sympathize what I meant past times 'trig ratio'. She saw no link betwixt what she saw every bit 2 distinct topics - ratio as well as trigonometry. She had absolutely no agreement of what sin30 is, though she is really competent at solving GCSE trigonometry problems. Again, this develop off warning bells inward my head. I demand to instruct trigonometry differently.

Does it matter?
My primary purpose is to ensure that my students possess a toolkit of mathematical methods alongside which to solve problems accurately as well as efficiently, as well as an agreement of the mathematical concepts that underlie those methods.

Hannah got a skilful A* inward her GCSE - this suggests I succeeded inward equipping her alongside the required skills as well as cognition for that qualification. But from my recent conversations alongside her, I realise that I had mixed success alongside the underlying concepts. I'm non going to crunch myself upward near it, but it gives me a focus - I demand to think to a greater extent than near how I instruct as well as assess for conceptual understanding.

Introducing trigonometry alongside similar triangles
When I innovate trigonometry I usually acquire my students to stair out triangles as well as await for patterns (like this activeness from Teachit Maths). Next fourth dimension I innovate trigonometry, I'm going to evidence something a flake different. I'll demonstrate students these 3 triangles as well as inquire what they select inward common:

I promise they'll location that they are similar triangles as well as that the ratio of the summit to the diagonal is 1:2 (this would endure a skilful fourth dimension to innovate the terminology opposite:hypotenuse).

Then I'll demonstrate them the triangle below as well as inquire for the length of the hypotenuse. I desire them to realise that because it is similar (ie equiangular) to the 3 triangles above, nosotros know that the ratio opposite:hypotenuse is 1:2. So the length of the hypotenuse must endure 70.

We could country 'the ratio of reverse to hypotenuse inward whatever right-angled alongside an angle of thirty degrees is 1:2'. This is a flake of a mouthful therefore instead mathematicians country 'sin30 = ½' (there's an interesting article here near the origins of damage sine, cosine as well as tangent).

I'll write sin30 = ½ on the board, as well as therefore repeat this procedure for triangles alongside unlike angles, starting alongside 50^o:

I'll inquire my students to uncovering the length of the reverse side inward the 3rd triangle as well as I promise they'll locomote it out based on their realisation that when the angle is 50^o, the ratio opposite:hypotenuse is 0.766:1.

I'll locomote on alongside to a greater extent than sets of examples as well as halt upward alongside a listing on the board that looks a flake similar this:
sin10 = 0.174 sin30 = 0.5 sin50 = 0.776 sin lx = 0.866 sin80 = 0.985 etc

Then I'll convey a 30^o triangle dorsum into play. I'll inquire them to locomote out the summit of this triangle

Source: openlibrary.org

Hopefully someone volition laissez passer on the right reply of 40cm - as well as and therefore comes the of import questioning. How did they know? Did they squall upward that the ratio is 1:2 when the angle is 30^o? Or did they cheque the listing on the board? Would they endure able to do it from retentivity if the angle is 50^o? That ratio is much harder to remember. What if the angle was 52^o? We haven't worked that 1 out yet. Well, what nosotros demand is a reference listing of all the ratios, similar a bigger version of our listing on the board. Then we'll endure able to locomote out lengths inward whatever right-angled triangle.

Here I select a choice of where to acquire side past times side alongside this lesson: either brand a mini-project out of this (where my shape brand their ain mass of trigonometric ratios) or merely demonstrate them a develop of trigonometric tables ("here's 1 I made earlier!").

I think students volition gain a amend agreement of trigonometric ratios if they utilization tables, instead of calculators, to solve trigonometric problems for a few lessons (click here to come across an instance of a student's workings using this method). There's an online version of the tables here. Once students actually sympathize what the ratios are as well as how to utilization them, let on that all the numbers from these tables are saved inward the retentivity of a scientific calculator.

I haven't tried this approach withal but I know other teachers do something similar. For example, the author of this blog has his shape create their ain trigonometric tables. He says, 'I select constitute that past times using a trig tabular array my students concentrate on the concepts beingness studied rather than the calculator'.

The cardinal indicate is that students demand to sympathize that trigonometric ratios stand upward for the ratios of the sides inward right-angled triangles.

I'm interested to withdraw heed how others innovate trigonometry therefore delight comment below or tweet me.

Resources resources resources
Now let's await at about skilful resources for instruction trigonometry. In my resource library I've made about recommendations therefore if you're planning whatever lessons on trigonometry, do select a await there. I've also constitute a few extras for you lot today. Did I advert that I dearest resources?!

Dan Walker has produced a brilliant PowerPoint on right-angled triangle trigonometry which starts off past times introducing the ratios every bit I've described above. The whole PowerPoint is splendid character as well as good worth a look.

Resources guru Don Steward gives us a make of fantastic activities. To practise calculating sides as well as angles inward right-angled triangles, I similar resources similar these:

Bearings
My students select a melt-down when I laissez passer on them a occupation that involves trigonometry as well as bearings. I suspect this is because my schoolhouse doesn't instruct bearings really good at Key Stage 3 (up until lately it wasn't fifty-fifty on our system of work), therefore they never know which angle they're meant to endure calculating. In the inquiry below they demand to utilization alternate angles as well as trigonometry to calculate the bearing of town H5N1 from town B:

And inward this to a greater extent than challenging question, they're asked to calculate the bearing of C from A. It tin flame endure done using right-angled triangles but it's much quicker to utilization the Sine as well as Cosine Rules.

The Sine Rule
MathsPad has a skilful make of trigonometry resources. I don't usually recommend resources that aren't gratuitous but I'm a large fan of MathsPad - a subscription costs £3 a calendar month (please don't pay out of your ain steal - inquire your boss). I especially similar the Sine Rule Codebreaker - it contains lots of practise questions inward a to a greater extent than engaging format than touchstone worksheets.

TROL
If you lot haven't seen this resource before, written by Frank Tapson for TROL (teacher resources online), do select a await through it. It contains lots of helpful practise questions, including about on 3D trigonometry.

Spaghetti Graphs
Finally, I'm looking frontward to the side past times side fourth dimension I instruct trigonometric graphs later watching Chris Smith's brilliant video inward which he makes a spaghetti sine graph. I can't hold off to evidence this.

Well that's it from me - I promise that's given you lot about ideas for instruction trigonometry. Please do allow me know if you lot select whatever non bad ideas to share.

Thursday, September 5, 2019

Teaching Trigonometry

x² + 13x + 36

= x² + 4x + 9x + 36

= x(x + 4) + 9(x + 4)

= (x + 9)(x + 4)

Source: openlibrary.org