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