Teaching Indices

Source: dotmaths.com

I've mentioned before that I love teaching indices, and that's because a) there's loads of brilliant resources available b) the concepts are easy to explain c) there's some lovely extension problems and d) this is important mathematics! You can't do calculus without understanding indices.

I know I can plan a sequence of lessons on indices for any age group (Year 8 right through to Year 12) and I'll be able to find good quality engaging resources. Later in this post I'll feature some of my favourites.

Concepts

The concepts relating to indices are relatively easy to explain from first principles. The reasoning behind the index laws is straightforward. We can see that b² x b⁵ = b⁷ simply by writing out the expressions in full (ie b x b x b x b x b x b x b). Students will quickly spot the patterns. The 'anything to the power of zero equals one' rule is also fairly simple to explain using the logic shown below.

Negative powers can be explained by extending the division law (ie what happens when we divide b² by b⁷?), or by considering patterns (see Gems 8 for more on this).

Fractional powers can be explained using the multiplication law.

And then it's just a case of students remembering the laws and practising them until they become fluent. Mim Gosling (@mimgosling) created the picture below to help students remember how fractional powers work -"Fractional indices are like a flower - the bottom's the root, the top's the power!". So if you have a student who gets confused by fractional indices, just remind them that the root is at the bottom (like in a tree).

Source: @StIvoMaths

A lesson on index laws

I taught an introductory lesson on index laws recently - I wrote these slides to structure the class discussion and activities. After we established each index law, the class did three quick practice questions before moving on to the next law. Once we'd covered all three laws, the students completed this 'Powers of y eliminator' from Teachit Maths, which was fantastic. It had just the right combination of independent practice, stretch and engagement. These were the questions that most students got stuck on:

I was quite proud of the slides I made for this lesson - there's an editable version here so feel free to borrow bits, though some formatting may be lost when you download it (the PDF version is much prettier!).

Card Sorts
It's a great idea to have a well-organised supply of sorting activities (eg Tarsia puzzles) in your maths department. Find a clever way to store them (eg in stackable plastic takeaway boxes) and label the boxes clearly. It takes a bit of time to prepare the card sorts. If you've got the budget, it's always best to get them laminated. Print each set on a different colour (eg if there are 10 groups in the class, print the card sets on 10 different colours) - this stops the sets getting mixed up. Use good quality elastic bands - cheap ones degrade quickly - or keep each set in an envelope.

My department has a collection of 56 card sorts ready for use in lessons (sadly, I spent many many hours sitting at the guillotine during my summer holidays!). We call them our 'Black Box Resources' because we keep them in black CD boxes from Ikea. Two of my favourite card sorts are for indices. The first is Indices Dominoes by Teachit Maths - a really nice activity, with the added bonus that the answer is written backwards so students can't cheat by guessing the words. I also like Using Indices from the Standards Unit. Card set A (pairs activity) is really useful when you've taught fractional and negative indices.

You might also like these treasure hunts: simple index laws and negative & fractional indices, both from Teachit Maths.

Worksheets
There's loads of great worksheets and activities for independent practice. I've already mentioned the Powers of y eliminator from Teachit Maths. Here's two more excellent resources that are suitable for practising the three index laws:

• Simplifying Indices Code Breaker - Teachit Maths 
• Collect a Joke - Number Loving

Once you've moved onto fractional and negative indices, there's loads more excellent resources (these might be useful in Year 12 too):

• Mental Powers Code Breaker -Teachit Maths 
• Ever wondered why? - JustMaths 
• Indices activity - SRWhitehouse on TES

STEM Centre provides some fantastic indices activities from Susan Wall. For example, I like 'Why does?' (extract below) in which students are allocated a statement to think about and explain to the class.

Extract from 'Why does?' by Susan Wall

There's also a true or false activity which is designed to address common misconceptions.

Extract from 'True or False' by Susan Wall

Blanks

I featured a 'blanks' task idea in Gems 17. Make up some questions like those shown below and put them up around the room. Provide post-it notes so students can suggest values that fit.

Homeworks
I've made two Pret homeworks for this topic, both are available here.

Enrichment
There's loads of great extension questions for this topic. I love these mega quadratic equations from Don Steward.

and these mega hard powers questions, also from Don Steward

And two more fantastic rich tasks from Don Steward: Power Sums and Find the Power. 

At A level, and maybe as extension in Year 11, I like to do some work on comparing indices with different bases. For example, I like this ordering starter.

And my worksheet exponential equations has been useful. It ends with this lovely question. 

Finally, I want to share two questions that I've been using for years with GCSE students, either when I teach indices or when I teach algebraic fractions. My students are always horrified by how complex these questions look, but if they're confident with indices then both equations are actually surprisingly easy to solve. I'm not sure where these questions originated so apologies to the creator for not giving credit.

That's it from me - if you know of any other fantastic indices resources, please comment below or tweet me.

How five squared looks in nightmares.

5 Maths Gems #20

Hello. I'm pleased to present my 20th gems post today - a milestone! Twitter has been quiet this week because Christmas took our minds off work for a couple of days, but I've still got some great ideas for you. The Christmas holidays are short and busy so it won't be long before you're back to planning lessons - I hope this post gives you some inspiration for the Spring term.

1. Factorising Quadratics
You know Don Steward's blog is the best thing since sliced bread, right? His catalogue of fantastic rich tasks is updated all the time. I really liked his recent posts on product puzzles. He started with a set of questions like this:

In Question 1 above, you can see that the top left cell has to contain a 3 because it's a factor of both 3 and 6. The rest of the cells can be completed quickly once one common factor has been established. This is a simple example but Don develops the questions to become increasingly difficult, some having multiple solutions. 

The next set of activities extend the same idea to algebra, starting with this:

There's lots of these to complete - excellent practice of factorising linear expressions.

The next stage of this exercise is factorising quadratics.

Conveniently I'm currently planning a Year 10 lesson on factorising quadratics. I want my students to do a lot of practice so will definitely be using this activity. The questions build up to a suitably challenging level of difficulty, ending with this one:

I want to encourage my students to factorise 'harder' quadratics (ie a > 1) by inspection. This is my preferred method (ie 'guess and test') but my students always demand that I teach them a more structured approach (eg 'the Grouping Method') which frustrates me. Their insistence on following an algorithm suggests a lack of confidence. I think the question above turns factorising quadratics into a kind of logic problem. Tackling this question without an algorithm might help my students develop the confidence to factorise harder quadratics by inspection.

One last idea for a lesson on factorising quadratics - I like the problem below from openmiddle.com. There are a number of possible solutions so you could challenge students to find a different solution to the person next to them. 

2. Angle Sense with the Interactive Whiteboard
I've been planning a Year 7 lesson on angles in which I'd like my students to estimate angle sizes. If you were asked to to draw an angle of 210^o freehand, how would you do it? I'd think of it as a straight line plus a third of a right-angle. If you have proportional reasoning skills then it's pretty easy to make an educated guess. An angle estimation activity would work perfectly well without technology (read out a series of angle sizes and ask your students to draw their freehand estimates on paper. They then check their estimates using a protractor - another useful skill). But if you want a similar activity for the interactive whiteboard then you might like this fun Estimating Angles Game from Nrich.

Another interactive whiteboard tool is 'How Far Does it Turn?' from MathsPad. This time your students have to estimate the size of the angle drawn - they could do this on mini-whiteboards so everyone is included in the activity.

While looking at these games, I stumbled across a big range of angle tools for the interactive whiteboard here. Some of these angle games are quite funny - Banana Hunt in particular made me chuckle.

If you like these interactive whiteboard games then you'll find loads at Sheppard Software. It's amusing that there's an Absolute Value Number Balls game - this concept isn't covered until Year 13 in the UK but I bet my students would love to play this game - five minutes light relief in a C3 lesson!

FlashMaths.co.uk is another great website for interactive whiteboard activities. Flash Maths was created by Jonathan Hall (@studymaths) who brings us a plethora of fantastic tools on StudyMaths.co.uk. If you haven't seen it before, check out MathsBot.com which is his simple (but brilliant) worksheet generator.

3. Big Questions

Billy Adamson (@Billyads_47) shared a fantastic set of mathematical thinking prompts 'The Big Questions'. Here's a few examples: 

Lovely open questions from Billy to generate discussion and develop understanding. There's some more good examples of open questions here:

4. Trigonometric Problem Solving

Our Year 13s' problem solving skills are tested when they're asked to simplify expressions involving trigonometric identities in C3 (like the example below).

I find that my students get frustrated when they can't spot a 'way in' straight away. They give up quickly. There's actually a pretty standard set of starting points, as described on www.intmath.com (@intmath). 

I struggle to help my students feel confident in tackling these problems, so I really like this activity from @mjfenton. Here's an extract:

The idea is that we start with a lot of structure and gradually give fewer hints until students are able to solve the problems themselves. The steps might seem logical to us, but we're experienced problem solvers. 

It's a good idea for maths teachers to try to solve unfamiliar problems every now and then (like the example below from ‏@dannytybrown) to remind ourselves that mathematical problem solving often requires patience, creativity and multiple attempts. We all experience frustration in problem solving, just like our students do, but we know that the satisfaction of eventually finding the solution is well worth it.

5. Dividing with Decimals

I've mentioned before that I love MathsPad's resources - plenty of them are free and the rest come at a cost of only £3 per month. Whether your school subscribes or not, it's worth registering for email updates in order to keep track of all the new resources. This month, the interactive resources on Decimal Calculations caught my eye. It always surprises me how many students will happily say that 40 divided by ½ equals 20. Activities like the one shown below will help tackle this misconception and encourage students to think before they answer.

That's it for this week. I'll leave you with a video from 1977 - 'Congruent Triangles' by Bruce and Katharine Cornwell (another gem found on @MathMunch). Happy New Year!

5 Maths Gems #19

Hello and welcome to my 19th gems post. This is where I share some of the best teaching ideas I've seen on Twitter each week. Most of my gems have come from across the Atlantic today.

1. Polygraph
There's been a lot of buzz about Desmos' new Polygraph activities. They're mathematical versions of Guess Who, designed to 'foster the pleasure and the power of words without the drudgery of the lists'. There are currently four Polygraph activities:

Let's look at Polygraph: Parabolas. Students work in pairs (a picker and a guesser) on separate tablets, phones or computers. The picker selects a graph from a set of parabolas. The guesser sees the same set of parabolas - their task is to identify which graph the picker has chosen by asking a series of yes/no questions. For example they might ask about the number of roots or the location of the vertex or y-intercept. The guesser types their questions and the picker responds with yes, no, or don't know. The guesser keeps asking questions until they are able to identify the chosen graph.

Students then have the opportunity to analyse and refine their questioning, and you have a class discussion to formalise the mathematical vocabulary.

As with Desmos' awesome Water Line activity that I described in Gems 13, it's incredibly quick and easy to get started. I'm going to be teaching quadratics to Year 10 shortly and can't wait to use the Parabolas activity. And I plan to use the Linear Graphs version with my Year 9s too.

Extract from Polygraph: Lines

2. Point, Point, Gradient

Tina Cardone (@crstn85) shared this lovely activity 'Point, Point, Slope' by Michael Fenton (@mjfenton). It's a simple but effective lesson idea in which students have to use any digit from 2 to 9 to create two points - you could give them post-its or number tiles to shuffle around. You provide criteria, such as 'create two points which determine a line with the greatest possible gradient'. Read the full post for more ideas. There's loads you could do with this activity, for example you could ask your students to use six different digits (from 0 - 9) to create three points which lie on the same line. It's a good way to develop understanding of linear graphs and I think it would work well at both GCSE and AS level.

Michael's blog is full of excellent ideas. For example in Two Fractions students place digits and operators in the boxes below to make an expression with the greatest possible value.

This reminds me of the Blanks activities I featured in Gems 17, which you can read more about in Sarah Aldous' post Logarithm Questions Around the Room.

3. Owning Facts
I liked Davis Wees' (@davidwees) tweet about owning multiplication facts.

There's probably an interesting psychological discussion to be had here - would relying on someone else make students less likely to remember facts, or would the association create a memory trigger as David suggests? I think it's the latter. Like Colin, I once had a sixth form student called Suzy who was always very vocal in class about 'not forgetting the plus C'. I still hear Suzy's voice whenever I do indefinite integration!

4. Number Properties
Back in Gems 13 I talked about making your students 'Number Experts'. I shared two websites that are useful for finding out properties of numbers (numbergossip.com and numdic.com). Both websites are excellent - for example, numbergossip.com tells me that 28 is a perfect, composite, happy number, and numdic.com (which wins the prize for funniest URL) gives me the binary and Roman Numeral representations of 28. I've now discovered a new website - the Number Property Calculator - which I like because of the way it gives information about the meaning of the number properties. Here's a small extract from what comes up when I search for 28:

I found this website when I was browsing through old posts on Math Munch. Math Munch is a 'Weekly Digest of the Mathematical Internet'. It's packed full of fantastic resources and ideas so it's definitely worth subscribing to receive weekly email updates, and you can follow them on Twitter too (@MathMunch).

5. Drawing Euclid

Andy Shaw (@Squidworm74) pointed me in the direction of these lovely Euclid videos on YouTube by Shoo Rayner (@shoorayner). They are fantastic videos and I can't wait to share them with my Year 7 students. Check out the full playlist here. Here's an example about different types of triangles:



That's it for this week. I hope you've found some inspiration in this post. I'll leave you with this brilliant video by the ever awesome Vi Hart - The Gauss Christmath Special. Thanks to Chris Smith (@app03102) for sharing this.




Have a wonderful Christmas!

Back to school

In two weeks I'll return to school from maternity leave. I've done this before, but not mid-way through a school year. I'll only be working three days a week but teaching eight different classes, all shared with another teacher. It's going to be a challenge.

On top of the anxieties associated with teaching lessons for the first time since May, I'll be getting my seven month old baby settled into nursery - I know from experience that this can be difficult and emotional. Although these challenges are a little daunting, I'm excited to try out all the new teaching ideas that I've picked up from all the blogs and tweets I've read over the last six months. I'm going back to school with fresh enthusiasm for teaching maths. In this post I want to tell you about a few of the things I've got lined up for January.

1. Exercise Books
I despair when I see teachers spending their hard-earned meagre salaries on supplies for work, but I'm guilty of it too. I've stocked up on pretty stationery, folders, board pens, post-it notes... and this time I've gone a step further and bought my own exercise books!

As an experiment, I'm going to try out these maths books 2.0 from Design Thinking. One of the features of these books is the checklist, key words, further questions and key facts sections at the bottom of each page. I think my students will love having these special books and will take more pride in their work. I also think I'll look forward to marking these books (at first!), to see what words, facts and questions my students have chosen to write in the boxes.

Including postage these cost me £40 for 60 books so I'll try them with two classes and see how it goes. I'm crossing my fingers that they arrive in time for the start of term! (and that my husband doesn't find out that I spent my own money on supplies for school!).

2. Expectations

I'm planning to set out my presentation expectations with all my students at Key Stage 3 and 4 using a poster: 'Five Expectations for a Beautiful Maths Book', an idea stolen from @danicquinn - see gems 14 for details.

3. Hashtag Marking

I featured my ideas for hashtag marking in gems 12. I'm planning to have a go at this - each of my students at Key Stage 3 and 4 will be issued with a Hashtag Marking Key (I'll print it on A5 so they can stick it in their maths book or planner). I have a feeling I'm going to have a huge marking workload this year so hopefully this will save me some time.

Extract from Hashtag Marking Key

4. Work Review Log
Based on an idea in this tweet, I plan to introduce a feedback log at Key Stage 5. I normally set a piece of assessed work for my A level students every week. I always put a lot of effort into providing written feedback, so I'd like to ensure my students are reading and reflecting on my comments. I'll ask them to keep this Work Review Log at the front of their folders and I'll give them an opportunity to complete it every time I return a piece of work.

Extract from Work Review Log

The other thing I want to do better this year is ensuring that my A level students are doing homework on their own. I'm happy for my students to help each other in class and on non-assessed homeworks (eg 'finish the textbook exercise'), but if I set a piece of work for marking it should be reflective of each student's individual level of understanding, not that of a group of friends. In the past I've had sixth form students getting As in their homeworks throughout the year (because they have a clever friend, or even a tutor) and then doing terribly in their actual exam.

5. Mistake Log
All my students will receive a Mistake Log to stick in the front of their book or folder. The idea is that whenever they make a mistake, whether because of a careless slip or a misconception, they record that mistake. Again, like the Work Review Log, I hope it helps ensure that students are reading my feedback and reflecting on how they can improve. I think they'll find this log useful when they come to revise.

6. #ByYourSide

I really think every teacher should ensure their subject knowledge is up to scratch as a matter of priority. And what better opportunity to test your own subject knowledge than sitting the qualification that you're responsible for teaching? I hope to sit the Maths GCSE exams with my students this summer. I think they'll appreciate the act of solidarity, and hopefully my teaching will benefit too. Although I'm very confident of my GCSE knowledge, I'll feel huge pressure to get 100% so I better be careful not to make any silly mistakes! I also intend to sit A level maths this summer. This is a much scarier prospect. For C3 (which I've never taught) and C4 I'll certainly have to revise and practise. I'm convinced this will make me a better A level teacher. I already fear being stumped by a tricky C4 integration question! I did a statistics degree so haven't studied pure maths since I did my A levels in 1999. But if I teach it then I should know it like the back of my hand shouldn't I? I hope I get permission to do these exams because I think it will be a really worthwhile exercise. Anyone with me? :)

7. Apps
I'm thinking about taking my iPad into school, if I can prise it away from my toddler. I'm really keen to try Idoceo because teachers on Twitter rave about it. But my school has started using Mint Class this year and I want to avoid duplication. So Idoceo might have to wait until next September, when I'll be at a new school.

I'm also keen to try Plickers and Quick Key, both of which I've written about in my gems posts, and I'm considering using iVisualiser too.

8. Resources
I've been lucky enough to receive a couple of lovely freebies recently - one of my favourite resource websites MathsPad have given me a free subscription that I can't wait to use! I'm also interested in Create a Test. The nice people at Create a Test have given my school a three month free trial of their test-writing tool. I wonder whether other schools are in the same situation as us - we give all our Key Stage 3/4 students three or four tests a year (including an end of year exam) and this is how we determine maths sets for the next year (I should mention at this point that I'm not supportive of a 'teach to the test' culture in maths and would love to hear alternative approaches). Anyway, because these are 'high stakes' tests, we want to avoid any risk of papers being seen in advance, so we create new tests every year. This is really time consuming so we'd benefit from a good quality test-writing tool with a large bank of questions. I'll report back on how we're getting on with Create a Test in a couple of months.

So that's a few of the highlights of my current 'going back to school' resources and plans. Now I have the most important thing to do - I need to start planning my lessons. I'll be doing circle theorems with Year 10, which is convenient because I recently wrote a post about teaching circle theorems. Unfortunately I've also been assigned 'constructing triangles' with my Year 8s - my least favourite topic in the world! Any ideas to make this more bearable would be gratefully received!