Thoughts On Didactics Fractions

Fraction misconceptions - MathsPad

Yuvraj (@YuviLite) asked me to write a post service virtually teaching fraction addition. I similar a challenge! Fractions are notoriously hard to teach, riddled alongside misconceptions as well as crucial to acquire right. If I were to endure thorough on this topic, I’d demand to create a hell of a lot of research as well as write a dissertation length weblog post.

I for sure haven't been that rigorous, but I accept works life a few ideas. First read what Yuvraj has written virtually difficulties inwards teaching fractions, as well as thus accept a hold off at my ideas at the end.


When I hear students utter the phrase, ‘uh no non fractions’ a pocket-sized phonation buried deep inwards my subconscious feels the same thing. Fractions is 1 of those topics which seems to endure on the system of function for EVERY twelvemonth as well as yet in that place are nevertheless thus many students who are befuddled yesteryear them as well as this is 1 of the reasons I accept been thinking virtually how I learn the topic.

I am a instructor who believes inwards developing agreement as well as non exactly only reducing mathematics to a serial of procedures. I accept e'er thought this as well as it was drummed into me on my PGCE as well as Masters. Yet fractions has brought me to a crossroads. Before I tell you lot why, allow us hold off at dissimilar strategies of teaching the improver as well as subtraction of fractions. 

The Cross Multiplying (Butterfly Method). This is the 1 I was taught at school. I had no agreement of why it worked but it got me the correct answer as well as at that historic menstruum that is all I cared about. To add together 2 thirds as well as 3 quarters nosotros would multiply the denominators together as well as thus cross multiply the numerators and denominators as shown below.

The Grid Method. I of late discovered this option procedural method. Draw a grid equally shown below. The starting fourth dimension fraction is written on the top as well as the instant along the side. A plus sign is set inwards the starting fourth dimension box equally this is the functioning nosotros desire to comport out. We set the products of the other numbers into the grid. We as well as thus depict around the highlighted boxes as well as this would give us our numerator, inwards this instance 8 + nine = 17. The white box would endure our denominator.

Fraction Strips. In NixTheTricks, the role of fractions strips is advocated to learn the improver as well as subtraction of fractions. For representative to add together a one-half as well as a quarter we would start alongside visual representations equally shown below.

When nosotros set the one-half as well as quarter together nosotros can't limited this equally a unmarried fraction without cutting the pieces into equal parts (quarters inwards this case). The thought hither is that the agreement of a mutual denominator is built upward as well as linked to the method for adding fractions.

Paper Folding. This method is from Mike Ollerton’s fantabulous ‘Getting The Buggers To Add Up’ - read the explanation here (or purchase the book!). 

Fraction Wall. I accept also seen teachers role fraction walls to aid students equally a method of identifying equivalent fractions. Some people role this or shapes to acquire the thought of equivalent fractions across.

My Dilemma
I accept e'er gone downwards the conceptual route. I accept championed the conceptual route, until I got to algebraic fractions. The conceptual route e'er seems to Pb dorsum to the to a greater extent than procedural route, especially for fractions where the denominators are 2 dissimilar prime numbers, as well as for algebraic fractions. If students are to a greater extent than used to the procedural route, they should inwards theory endure able to able this physical care for for algebraic fractions.

Which has led me to question: is it worth the hurting of rewiring students who e'er seemed to accept heard of the cross multiplication method fifty-fifty earlier they accept reached me (even the Year 7s)? I accept spent many hours struggling alongside folding bits of newspaper as well as sharing cakes to embed the thought of equivalent fractions. 

Is it plenty exactly to explicate why the cross multiplication or grid method works as well as and thus ensure fluency inwards the skill?


Thanks Yuvraj. I wonder the same thing. In many topics nosotros fighting to respect the correct residuum of conceptual agreement vs procedural methods. Us pathetic maths teachers are forever fretting virtually it.

My preferred procedural method, for both numerical as well as algebraic fractions, is the no-frills version (find the lowest mutual multiple of the 2 denominators, as well as thus convert both fractions to equivalent fractions alongside that denominator, as well as thus add together the numerators). It works for me, but I learn high powerfulness pupils thus I accept a relatively slow life.

Now for a few teaching ideas. I’m non trying to solve the world’s fraction problems, but I promise you lot may respect some interesting activities hither for your fractions lessons.

Don Steward is a Genius
subtracting fractions activity inwards which students hold off at the diagram below as well as figure out what's going on. As Mr Steward says, this gives the instructor fourth dimension to drinkable their tea! It's a truly clear agency to explicate fraction subtraction using mutual denominators.

Misconceptions as well as Assessment
This blog has some vivid ideas for identifying fraction misconceptions which powerfulness endure peculiarly helpful when you lot start tackling fractions alongside a Year vii class. For example, the author asked his students to explicate what's incorrect inwards each of the 3 pictures below as well as got some interesting responses.

You could as well as thus inquire students to role the same shapes as well as respect an accurate agency to correspond the fractions indicated. Other ideas inwards this weblog include bespeak the students to consummate the sentences 'Fractions are...', 'When you lot add together 2 fractions you lot accept to...' as well as 'You role fractions when you...'. Or fifty-fifty inquire pupils to write downwards 'Everything I Know virtually Fractions' as well as write a listing of what they'd similar to learn. These activities volition aid you lot create upward one's hear your starting dot inwards Year 7. The blog is good worth a read.

Manipulatives as well as Visual Aids
Yuvraj already touched on visual aids. I retrieve beingness told during my PGCE that I only must role Cuisenaire Rods, but non truly having a clue what to create alongside them. There's whole books written virtually this! I works life explanations here as well as here - the method seems rather overcomplicated. But if they appeal to you, as well as thus Nrich provides this online tool for demonstrating Cuisenaire Rods on the Interactive Whiteboard.

This website has a lot of skilful materials on fractions as well as visual aids, including downloadable fraction circles.

‘Number 1 inwards disguise’
Look at the 'NOID' poster below. Yes, it is quite ridiculous. But it’s really slow to follow the thinking hither as well as I wonder if this explanation (or similar) would aid railroad train students' understanding. If you’re looking something a niggling less NOIDy for teaching improper fractions as well as thus this weblog post has some fantabulous ideas using manipulatives.

From cindywhitebcms.blogspot.com

Division
'Ours is non to argue why, ours is but to inverse as well as multiply'. This made me express joy because I've never attempted to explicate fraction partitioning to my pupils beyond the 'KFC' dominion (keep, flip, change). Don Steward has written an fantabulous post virtually ways to explicate why it works - this one from moveitmaththesource.com peculiarly appealed to me. I'll top away on this inwards my dorsum bag inwards instance ever asked the question.

Final thoughts
I'm afraid I don't accept the answer to how best to railroad train conceptual understanding, but in that place may endure some ideas hither that you lot haven't seen before. In price of resources, there's tonnes of worksheets, games as well as activities online - run across my resources library for some recommendations. For to a greater extent than teaching ideas, here's some farther reading:

• The Guardian's 'How to learn fractions' 
• Bruno Reddy's 'With Fractions, Common-Core Training Goes Beyond 'Invert as well as Multiply''
• James Tanton's 'Troublesome Fractions'.

Thank you lot thus much to Yuvraj for his helpful input to this post. Comments as well as tweets would endure much appreciated - we'd dearest to hear your ideas.

Fraction hopscotch inwards the maths corridor - from cisdmathmusings.blogspot.com

5 Maths Gems #15

Welcome to my 15th weekly gems post - this is where I characteristic to a greater extent than or less of the best teaching ideas I've seen on Twitter each week.

1. Slice The Pie

Paul Rehmet (@prehmet) has built an interactive tool to help students construct numerical fluency. It's a uncomplicated fraction game - Slice The Pie - which helps students railroad train an intuitive sense of the relative sizes of numbers. It's non compatible alongside mobiles yet but would piece of work on your interactive whiteboard. I honey this. Have a play alongside it, I think you'll honey it too. And follow Paul on Twitter because there's to a greater extent than to come upwardly from him.

2. Remembering Inequalities

In my post virtually Ideas from Shanghai I shared this visualisation which helps students cry back which inequality sign is which.

All students demand to exercise is visualise numbers equally blocks - the wider halt of the inequality symbol fits over the bigger stack of blocks. So if x > 2, nosotros know x must last a divulge bigger than 2. 

Chris Smith (@aap03102) pointed out that this doesn't piece of work for negative numbers. 

I'm sure as shooting our friends inwards Shanghai possess got a solution for this but that data hasn't filtered through to us, therefore let's come upwardly up alongside our ain alternative that industrial plant for both positive together with negative numbers. We only demand a agency for students to cry back which symbol is which.

I'm actually grateful to all the lovely creative tweeters who offered ideas - my favourite was this calligram from Chris. 

So if x < - 8, nosotros encounter the < symbol together with think of the alphabetic character L, therefore nosotros know x is less than -8.

Eventually students volition automatically know the important of the symbols on sight, without having to visualise blocks or words.

These methods are far meliorate than thinking of crocodiles - this tin confuse students, equally described inwards Nix the Tricks (extract below).

3. Squares together with Cubes
I'm enjoying La Salle Education's (@LaSalleEd) novel 'Problem of the Day' series.  You tin search the collection past times historic menses or topic. The instance below (Square Boxes) is a prissy logic occupation consolidating noesis of foursquare numbers.

If you're teaching foursquare numbers, you lot powerfulness also similar this 'Square together with Cube Numbers Puzzle' from MathsPad...

... together with this lovely square together with cube numbers display from Clarissa Grandi (@c0mplexnumber).

4. Mathsticks
I lately discovered the brilliant Mathsticks website which is created past times John Duffty (@johnduffty). I registered together with explored - it's to a greater extent than oftentimes than non primary materials but to a greater extent than or less of the ideas powerfulness last helpful at Key Stage 3. Here's a pair of examples:

This Parallel & Perpendicular Slope. Once you've taught students virtually the gradients of parallel together with perpendicular lines, larn them inwards a calculator room (or larn the iPads out, if you lot possess got them) together with give them this worksheet, which I think volition last really slow for them to follow independently. They are given sure as shooting lines which they plot (in Desmos, though I suppose this could last done past times hand) together with therefore they add together the right parallel or perpendicular lines to brand the required shape. It's a actually prissy activity which volition assistance you lot cheque their agreement of this topic.

Extract from Desmos Graph Challenge

Here's my respond to the minute interrogation inwards the extract above.

To extend this task, you lot could possess got students trammel the domains of the functions, equally I've done below. If you lot exercise this, it's of import that they figure out how to calculate the coordinates of the vertices.

If you're teaching parallel together with perpendicular lines therefore also cheque out my post linear graphs for to a greater extent than ideas together with resources.

A asking for misconceptions
When I project design lessons, I abide by it helpful to endeavour to predict the misconceptions that powerfulness come upwardly up. Sometimes I'm surprised past times the mistakes that students make, which are oftentimes the effect of my explanations non beingness clear enough. Knowledge of mutual misconceptions comes alongside sense but I think it would exercise goodness novel maths teachers if they could consult a website to encounter examples of the form of things students larn wrong. 

A not-so-common misconception from 1 of Chris Smith's students

American instructor Michael Pershan (@mpershan) has a bright website Math Mistakes where he compiles, analyses together with discusses the mathematical errors that students make. I desire to prepare something similar for the Great Britain (because our curricula together with vocabulary are therefore different), but I won't possess got fourth dimension to exercise the first-class analysis that Michael does. Instead it would only last a collection of misconceptions, tagged past times topic. But I even therefore think it powerfulness last useful.

So I possess got a asking - could you lot outset taking photos of misconceptions inwards your students' piece of work together with emailing or tweeting them to me? Once I've got a modest alternative I'll outset a website together with I hope, similar my Pret homework website, it volition grow speedily together with attempt to last a helpful resource. Thank you lot inwards advance!

That's it for today. Please don't forget to vote for me! :), I'll larn out you lot alongside this cute 'Math nosotros know drama', shared past times @FascinatingVids.

5 Maths Gems #17

Ho-ho-hello as well as welcome to my seventeenth weekly gems post. This is where I circular upwards some of the best maths instruction ideas I’ve seen on Twitter. I’ve got some crackers for you lot today. 

1. Balancing

Do you lot recollect inward Gems 13 when I featured the bright balancing problems from Don Steward? My readers loved them! Well this calendar week Damian Watson (@dwatson802) shared the visual algebra puzzle website SolveMe Mobiles which has like problems for purpose on an interactive whiteboard, tablet or computer. The website, from the Education Development Center, allows users to both create as well as solve mobiles. The website says that 'the logic of these puzzles is the same equally the logic behind the solving of equations as well as systems of equations, but the visual format makes these puzzles, as well as the logic they develop, accessible as well as appealing...'. 

And there's already been some proficient feedback on Twitter:

2. Blanks

Sarah Aldous (@mathsfeedback) shared a overnice stance for instruction logs that could live on used for a publish of topics. The stance is that you lot lay questions upwards only about the room as well as students respond on a post-it, but their respond must differ from the answers already given.

Here's my examples of like questions for fractions:

as well as indices:

You larn the idea. Students brand suggestions for the blanks as well as inward doing hence they demonstrate their agreement of the topic. Plus, they receive got to intend creatively if their stance has already been taken.

3. Dictionary
Steven W. Anderson (maths lexicon for kids. This lexicon from Jenny Eather is actually lovely - it's the best maths lexicon I've seen for students. Definitions come upwards amongst examples as well as interactivity. Here's the definition of the news following (check out the website yourself to play amongst the interactive features).

And here's the definition of sector. Again, it's interactive.

And here's 2 to a greater extent than examples - palindromic as well as perpendicular.

4. Simultaneous Equations

I dearest this Linear Simultaneous Equations.

5. Whiteboards (again!)

I've written close vertical whiteboards as well as acrylic sheets earlier (see Gems 7) - I dearest these ideas as well as promise that 1 twenty-four hours I receive got the budget to practise both.  Andrew Stadel (blog post this calendar week close his sense of using large whiteboards - it's good worth a read. One stance is 'Placemat':

If students are working on a business together, they operate individually on split upwards sections of a large whiteboard, as well as hence purpose the centre to tape their terminal answer. This approach helps students compare their operate as well as back upwards each other. It's too a proficient means for the instructor to assess private agreement as well as banking concern gibe that everyone is contributing to grouping discussion.

Another stance is Brain Dump:

Here nosotros receive got an icon inward the middle as well as students write downwards everything they know close it inward their private sections, earlier they compare as well as hash out their answers.

And nosotros receive got Number Talks:

In this trial a publish is written inward the middle as well as students operate inward their private sections to write downwards equally many ways they tin strength out intend of to brand that number. 

Do read Andrew's ship service for the total details as well as links to lots to a greater extent than ideas. If your region doesn't receive got the budget to purchase large whiteboards or acrylic sheets as well as hence you lot could purpose flipchart newspaper for these activities. Whiteboards are much ameliorate though, because of the power of the eraser ('because the markings tin strength out live on easily erased, students are straight off inclined to write without hesitation') - you lot tin strength out read close this inward the fantabulous post Whiteboards vs. Chart Paper by Thomas Ro (@MrT_Ro). 

That's it for this week. I'll locomote out you lot amongst a nifty occupation for your students from UKMT, shared past times Matt Smith (@mathsmatt). Find the angles inward the rhombus, assuming there's 1 equilateral as well as iii isosceles triangles.

5 Maths Gems #19

Hello in addition to welcome to my 19th gems post. This is where I part approximately of the best didactics ideas I've seen on Twitter each week. Most of my gems convey come upwards from across the Atlantic today.

1. Polygraph
There's been a lot of buzz almost Desmos' novel Polygraph activities. They're mathematical versions of Guess Who, designed to 'foster the pleasance in addition to the ability of words without the drudgery of the lists'. There are currently 4 Polygraph activities:

Let's expect at Polygraph: Parabolas. Students move inwards pairs (a picker in addition to a guesser) on split upwards tablets, phones or computers. The picker selects a graph from a ready of parabolas. The guesser sees the same ready of parabolas - their trace is to position which graph the picker has chosen past times cry for a serial of yes/no questions. For instance they mightiness inquire almost the seat out of roots or the place of the vertex or y-intercept. The guesser types their questions in addition to the picker responds alongside yes, no, or don't know. The guesser keeps cry for questions until they are able to position the chosen graph.

Students in addition to then convey the chance to analyse in addition to refine their questioning, in addition to y'all convey a flat discussion to formalise the mathematical vocabulary.

As alongside Desmos' awesome Gems 13, it's incredibly quick in addition to slow to larn started. I'm going to survive didactics quadratics to Year 10 before long in addition to can't hold back to purpose the Parabolas activity. And I computer program to purpose the Linear Graphs version alongside my Year 9s too.

Extract from Polygraph: Lines

2. Point, Point, Gradient

Tina Cardone (@crstn85) shared this lovely activity 'Point, Point, Slope' past times Michael Fenton (@mjfenton). It's a unproblematic precisely effective lesson stance inwards which students convey to purpose whatsoever digit from 2 to ix to create 2 points - y'all could rank them post-its or seat out tiles to shuffle around. You furnish criteria, such every bit 'create 2 points which create upwards one's heed a trace alongside the greatest possible gradient'. Read the amount post for to a greater extent than ideas. There's loads y'all could create alongside this activity, for instance y'all could inquire your students to use half dozen unlike digits (from 0 - 9) to create iii points which prevarication on the same line. It's a expert agency to railroad train agreement of linear graphs in addition to I cry back it would move good at both GCSE in addition to AS level.

Michael's spider web log is amount of fantabulous ideas. For instance in Two Fractions students house digits in addition to operators inwards the boxes below to brand an facial expression alongside the greatest possible value.

This reminds me of the Blanks activities I featured inwards Gems 17, which y'all tin rank the axe read to a greater extent than almost inwards Sarah Aldous' post Gems 13 I talked almost making your students 'Number Experts'. I shared 2 websites that are useful for finding out properties of numbers (numbergossip.com in addition to numdic.com). Both websites are fantabulous - 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 in addition to Roman Numeral representations of 28. I've immediately discovered a novel website - the Number Property Calculator - which I similar because of the agency it gives data almost the important of the seat out properties. Here's a small extract from what comes upwards when I search for 28:

I flora this website when I was browsing through sometime posts on Math Munch. Math Munch is a 'Weekly Digest of the Mathematical Internet'. It's packed amount of fantastic resources in addition to ideas thus it's definitely worth subscribing to have weekly e-mail updates, in addition to y'all tin rank the axe follow them on Twitter also (@MathMunch).

5. Drawing Euclid

Andy Shaw (@Squidworm74) pointed me inwards the direction of these lovely Euclid videos on YouTube past times Shoo Rayner (@shoorayner). They are fantastic videos in addition to I can't hold back to part them alongside my Year vii students. Check out the amount playlist here. Here's an instance almost unlike types of triangles:



That's it for this week. I promise you've flora approximately inspiration inwards this post. I'll move out y'all alongside this bright video past times the always awesome Vi Hart - The Gauss Christmath Special. Thanks to Chris Smith (@app03102) for sharing this.




Have a wonderful Christmas!