Thoughts On Instruction Fractions

Fraction misconceptions - MathsPad

Yuvraj (@YuviLite) asked me to write a post nearly teaching fraction addition. I similar a challenge! Fractions are notoriously hard to teach, riddled amongst misconceptions together with 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 together with write a dissertation length weblog post.

I for certain haven't been that rigorous, but I convey establish a few ideas. First read what Yuvraj has written nearly difficulties inwards teaching fractions, thence convey a expect at my ideas at the end.


When I listen students utter the phrase, ‘uh no non fractions’ a minor vocalization 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 together with yet in that place are nevertheless thence many students who are befuddled past times them together with this is 1 of the reasons I convey been thinking nearly how I learn the topic.

I am a instructor who believes inwards developing agreement together with non exactly only reducing mathematics to a serial of procedures. I convey ever thought this together with it was drummed into me on my PGCE together with Masters. Yet fractions has brought me to a crossroads. Before I tell you lot why, allow us expect at dissimilar strategies of teaching the add-on together with 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 respond together with at that historic current that is all I cared about. To add together 2 thirds together with 3 quarters nosotros would multiply the denominators together thence cross multiply the numerators and denominators as shown below.

The Grid Method. I of late discovered this choice procedural method. Draw a grid equally shown below. The kickoff fraction is written on the overstep together with the minute along the side. A plus sign is set inwards the kickoff box equally this is the functioning nosotros desire to behave out. We set the products of the other numbers into the grid. We thence depict around the highlighted boxes together with this would give us our numerator, inwards this instance 8 + ix = 17. The white box would endure our denominator.

Fraction Strips. In NixTheTricks, the purpose of fractions strips is advocated to learn the add-on together with subtraction of fractions. For illustration to add together a one-half together with a quarter we would start amongst visual representations equally shown below.

When nosotros set the one-half together with 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 together with linked to the method for adding fractions.

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

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

My Dilemma
I convey ever gone downwards the conceptual route. I convey championed the conceptual route, until I got to algebraic fractions. The conceptual route ever seems to Pb dorsum to the to a greater extent than procedural route, especially for fractions where the denominators are 2 dissimilar prime numbers, together with 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 ever seemed to convey heard of the cross multiplication method fifty-fifty earlier they convey reached me (even the Year 7s)? I convey spent many hours struggling amongst folding bits of newspaper together with sharing cakes to embed the thought of equivalent fractions. 

Is it plenty exactly to explicate why the cross multiplication or grid method plant together with thence ensure fluency inwards the skill?


Thanks Yuvraj. I wonder the same thing. In many topics nosotros combat to notice the correct residuum of conceptual agreement vs procedural methods. Us pitiful maths teachers are forever fretting nearly it.

My preferred procedural method, for both numerical together with algebraic fractions, is the no-frills version (find the lowest mutual multiple of the 2 denominators, thence convert both fractions to equivalent fractions amongst that denominator, thence add together the numerators). It plant for me, but I learn high powerfulness pupils thence I convey 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 notice some interesting activities hither for your fractions lessons.

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

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

You could thence inquire students to purpose the same shapes together with notice an accurate means 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 convey to...' together with 'You purpose fractions when you...'. Or fifty-fifty inquire pupils to write downwards 'Everything I Know nearly Fractions' together with write a listing of what they'd similar to learn. These activities volition aid you lot create upward one's heed your starting dot inwards Year 7. The blog is good worth a read.

Manipulatives together with Visual Aids
Yuvraj already touched on visual aids. I recollect beingness told during my PGCE that I only must purpose Cuisenaire Rods, but non genuinely having a clue what to create amongst them. There's whole books written nearly this! I establish explanations here together with here - the method seems rather overcomplicated. But if they appeal to you, thence Nrich provides this online tool for demonstrating Cuisenaire Rods on the Interactive Whiteboard.

This website has a lot of proficient materials on fractions together with 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 together with I wonder if this explanation (or similar) would aid prepare students' understanding. If you’re looking something a trivial less NOIDy for teaching improper fractions thence this weblog post has some first-class ideas using manipulatives.

From cindywhitebcms.blogspot.com

Division
'Ours is non to argue why, ours is but to inverse together with multiply'. This made me express joy because I've never attempted to explicate fraction sectionalization to my pupils beyond the 'KFC' dominion (keep, flip, change). Don Steward has written an first-class post nearly ways to explicate why it plant - this one from moveitmaththesource.com especially appealed to me. I'll leave of absence along this inwards my dorsum steal inwards instance ever asked the question.

Final thoughts
I'm afraid I don't convey the respond to how best to prepare 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 together with activities online - run into 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 together with Multiply''
• James Tanton's 'Troublesome Fractions'.

Thank you lot thence much to Yuvraj for his helpful input to this post. Comments together with tweets would endure much appreciated - we'd beloved to listen 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 roughly 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 ready 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 upward from him.

2. Remembering Inequalities

In my post almost Ideas from Shanghai I shared this visualisation which helps students squall upward which inequality sign is which.

All students demand to exercise is visualise numbers equally blocks - the wider terminate of the inequality symbol fits over the bigger stack of blocks. So if x > 2, nosotros know x must hold out a number 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 inward Shanghai own got a solution for this but that data hasn't filtered through to us, thence let's come upward up alongside our ain alternative that industrial plant for both positive in addition to negative numbers. We simply demand a agency for students to squall upward 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 run into the < symbol in addition to think of the missive of the alphabet L, thence 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 ameliorate than thinking of crocodiles - this tin give notice confuse students, equally described inward Nix the Tricks (extract below).

3. Squares in addition to Cubes
I'm enjoying La Salle Education's (@LaSalleEd) novel 'Problem of the Day' series.  You tin give notice search the collection yesteryear historic current or topic. The instance below (Square Boxes) is a dainty logic occupation consolidating noesis of foursquare numbers.

If you're teaching foursquare numbers, yous mightiness also similar this 'Square in addition to Cube Numbers Puzzle' from MathsPad...

... in addition to this lovely square in addition to cube numbers display from Clarissa Grandi (@c0mplexnumber).

4. Mathsticks
I late discovered the brilliant Mathsticks website which is created yesteryear John Duffty (@johnduffty). I registered in addition to explored - it's by in addition to large primary materials but roughly of the ideas mightiness hold out helpful at Key Stage 3. Here's a twosome of examples:

This Parallel & Perpendicular Slope. Once you've taught students almost the gradients of parallel in addition to perpendicular lines, larn them inward a reckoner room (or larn the iPads out, if yous own got them) in addition to give them this worksheet, which I think volition hold out real slowly for them to follow independently. They are given sure as shooting lines which they plot (in Desmos, though I suppose this could hold out done yesteryear hand) in addition to and thence they add together the right parallel or perpendicular lines to brand the required shape. It's a actually dainty activity which volition assistance yous banking enterprise check their agreement of this topic.

Extract from Desmos Graph Challenge

Here's my response to the 2nd query inward the extract above.

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

If you're teaching parallel in addition to perpendicular lines in addition to thence also banking enterprise check out my post linear graphs for to a greater extent than ideas in addition to resources.

A asking for misconceptions
When I project design lessons, I uncovering it helpful to travail to predict the misconceptions that mightiness come upward up. Sometimes I'm surprised yesteryear the mistakes that students make, which are ofttimes the number 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 run into examples of the kind of things students larn wrong. 

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

American instructor Michael Pershan (@mpershan) has a bright website Math Mistakes where he compiles, analyses in addition to discusses the mathematical errors that students make. I desire to prepare something similar for the United Kingdom of Great Britain in addition to Northern Ireland (because our curricula in addition to vocabulary are thence different), but I won't own got fourth dimension to exercise the fantabulous analysis that Michael does. Instead it would simply hold out a collection of misconceptions, tagged yesteryear topic. But I notwithstanding think it mightiness hold out useful.

So I own got a asking - could yous kickoff taking photos of misconceptions inward your students' piece of work in addition to emailing or tweeting them to me? Once I've got a small-scale choice I'll kickoff a website in addition to I hope, similar my Pret homework website, it volition grow chop-chop in addition to seek to hold out a helpful resource. Thank yous inward advance!

That's it for today. Please don't forget to vote for me! :), I'll move out yous alongside this cute 'Math nosotros know drama', shared yesteryear @FascinatingVids.

5 Maths Gems #17

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

1. Balancing

Do you lot retrieve inward Gems 13 when I featured the vivid 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 similar problems for purpose on an interactive whiteboard, tablet or computer. The website, from the Education Development Center, allows users to both create together with solve mobiles. The website says that 'the logic of these puzzles is the same every bit the logic behind the solving of equations together with systems of equations, exactly the visual format makes these puzzles, together with the logic they develop, accessible together with appealing...'. 

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

2. Blanks

Sarah Aldous (@mathsfeedback) shared a prissy sentiment for didactics logs that could last used for a publish of topics. The sentiment is that you lot position questions upwards around the room together with students reply on a post-it, exactly their reply must differ from the answers already given.

Here's my examples of similar questions for fractions:

together with indices:

You learn the idea. Students brand suggestions for the blanks together with inward doing hence they demonstrate their agreement of the topic. Plus, they receive got to intend creatively if their sentiment 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 alongside examples together with interactivity. Here's the definition of the intelligence following (check out the website yourself to play alongside 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 together with perpendicular.

4. Simultaneous Equations

I honey this Linear Simultaneous Equations.

5. Whiteboards (again!)

I've written well-nigh vertical whiteboards together with acrylic sheets earlier (see Gems 7) - I honey these ideas together with promise that 1 twenty-four hr current I receive got the budget to practice both.  Andrew Stadel (blog post this calendar week well-nigh his sense of using large whiteboards - it's good worth a read. One sentiment is 'Placemat':

If students are working on a chore together, they operate individually on split sections of a large whiteboard, hence purpose the centre to tape their terminal answer. This approach helps students compare their operate together with back upwards each other. It's likewise a skilful means for the instructor to assess private agreement together with depository fiscal establishment lucifer that everyone is contributing to grouping discussion.

Another sentiment is Brain Dump:

Here nosotros receive got an picture inward the middle together with students write downward everything they know well-nigh it inward their private sections, earlier they compare together with hash out their answers.

And nosotros receive got Number Talks:

In this instance a publish is written inward the middle together with students operate inward their private sections to write downward every bit many ways they tin compass the sack intend of to brand that number. 

Do read Andrew's postal service for the total details together with links to lots to a greater extent than ideas. If your division doesn't receive got the budget to purchase large whiteboards or acrylic sheets hence you lot could purpose flipchart newspaper for these activities. Whiteboards are much amend though, because of the power of the eraser ('because the markings tin compass the sack last easily erased, students are at nowadays inclined to write without hesitation') - you lot tin compass the sack read well-nigh this inward the fantabulous post Whiteboards vs. Chart Paper by Thomas Ro (@MrT_Ro). 

That's it for this week. I'll exit you lot alongside a smashing work for your students from UKMT, shared past times Matt Smith (@mathsmatt). Find the angles inward the rhombus, assuming there's 1 equilateral together with 3 isosceles triangles.

5 Maths Gems #19

Hello in addition to welcome to my 19th gems post. This is where I percentage but about of the best education ideas I've seen on Twitter each week. Most of my gems accept come upward 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 operate inwards pairs (a picker in addition to a guesser) on split tablets, phones or computers. The picker selects a graph from a prepare of parabolas. The guesser sees the same prepare of parabolas - their employment is to position which graph the picker has chosen past times quest a serial of yes/no questions. For instance they mightiness inquire almost the release of roots or the place of the vertex or y-intercept. The guesser types their questions in addition to the picker responds amongst yes, no, or don't know. The guesser keeps quest questions until they are able to position the chosen graph.

Students hence accept the chance to analyse in addition to refine their questioning, in addition to yous accept a degree discussion to formalise the mathematical vocabulary.

As amongst Desmos' awesome Gems 13, it's incredibly quick in addition to slow to larn started. I'm going to endure education quadratics to Year 10 presently in addition to can't hold off to purpose the Parabolas activity. And I programme to purpose the Linear Graphs version amongst 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 elementary but effective lesson view inwards which students accept to purpose whatsoever digit from 2 to nine to create 2 points - yous could laissez passer them post-its or release tiles to shuffle around. You furnish criteria, such equally 'create 2 points which create upward one's heed a draw amongst the greatest possible gradient'. Read the amount post for to a greater extent than ideas. There's loads yous could create amongst this activity, for instance yous could inquire your students to use 6 unlike digits (from 0 - 9) to create 3 points which prevarication on the same line. It's a expert means to educate agreement of linear graphs in addition to I recall it would operate good at both GCSE in addition to AS level.

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

This reminds me of the Blanks activities I featured inwards Gems 17, which yous tin laissez passer notice 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 splendid - 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 right away discovered a novel website - the Number Property Calculator - which I similar because of the means it gives data almost the important of the release properties. Here's a small extract from what comes upward when I search for 28:

I constitute this website when I was browsing through onetime 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 hence it's definitely worth subscribing to have weekly e-mail updates, in addition to yous tin laissez passer notice follow them on Twitter likewise (@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 off to percentage them amongst 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 constitute but about inspiration inwards this post. I'll travel out yous amongst 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!