Comparing Fractions

OK, so your students have mastered the concept of equivalent fractions and it's time time to start comparing fractions - I have made two sketches to help. 

I find that you really need to keep revisiting the fundamentals when advancing a topic; if students forget the basic ideas then everything suddenly becomes very abstract indeed.   I have designed these sketches so that the ideas of how to find equivalent fractions and equivalent fractions being...well... equivalent, are constantly recapped and reinforced.

First up comparing two fractions:

Chose two fractions to compare by adjusting the sliders.

Click show equivalent fractions.

Move the central sliders to experiment with finding equivalent fractions.

Settle on a common denominator.

Verify the fractions are equivalent by using the top grey slider.

Use the right hand grey slider to make a comparison.

What's the difference?

If you don't like pizza and prefer to eat bars then check the 'model type' box.

I like to mix it up but bars are a fantastic tool for modelling proportions that are easier for students to draw and compare than pizzas! Check out William Emeny's amazing post on bar modelling. I have started to re-design all of my fraction models to offer a bar model but I haven't since tested this one extensively so let me know if you experience any issues.

Next up is comparing multiple fractions and ordering them. Try this:

Show some fractions - how can we go about ordering them?

Maybe some pictures would help (click the 'bars' check-box) - but you wouldn't want to draw them all out by hand every time!

Would finding fractions with a common denominator help? How can we visualise what we are doing here?

Maybe this is too complicated to start with and you just want to compare two or three fractions. Use the left hand check boxes to hide some of the fractions.

You can set up your own fractions - by adjusting the left hand sliders. I always encourage this in my GeoGebra sketches so that you can be dynamic and respond to pupils questions and enable students to visualise any question they like. However in this case, just because it takes a while to dial in all the fractions I have built in five examples that you can access by adjusting the vertical slider on the right-hand side.

I have made the bars drag-able by clicking on the point at the bottom-left corner of each bar. This enables you to put them next to each other to make a closer comparison if you wish.

I have found these sketches to be extremely powerful and would highly recommend you give them a go. If you notices any glitches or have any suggestions for improving them the please just leave a comment. Thanks.

Equivalent Fractions

This week I delivered a training session on GeoGebra to my department, I showed them a variety of different sketches to illustrate how GeoGebra could be used to teach topics from each of the four main areas of the KS3/4 curriculum; algebra, geometry, data and number. We looked at linear graphs, transformations and cumulative frequency distributions but I think it was the fractions sketches that were the most popular.

A colleague who is full of creative teaching ideas and who has been teaching for many years told me that one sketch in particular was possibly the best single resource that he'd ever seen for teaching fractions - that made me smile so I thought I'd better share it here.

Experiment with manipulating the blue sliders to change the left hand fraction. "Will the fraction be bigger or smaller if I increase the numerator"..."Why do you say that?... What will happen to the picture?" demo "What if I increase the denominator ...",  "What if I reduce the denominator so that its less than the numerator?" 

I have used this demonstration several times with different ability groups and students always find these seemingly simple questions very challenging - they really get to the heart of understanding fractions.

Now onto the idea of equivalent fractions...Use the green sliders to perform an operation on the numerator and/or the denominator. "If I double the numerator whats going to happen"  demo, reset "If i double the denominator whats going to happen" demo, reset "What if I double both?"..."Surely not that's crazy we're doubling both the numerator and the denominator; all that effort and the fraction's going to remain the same size. Why is that?" You can use the verify slider to double check the fractions overlay exactly - especially useful when it's a close call.

Rack up the values on the blue fraction a bit and then switch the operation to division and you can explore look at the idea of simplifying fractions.

Great that all works nicely, time for some addition ... "What if I add two to both the numerator and the denominator"..."eh"..."Why doesn't that work?" Ah, if only we lived in Farey Land, our lives as maths teachers would be so much easier!

There are so many questions you can ask with this simple tool that really probe students understanding of equivalent fractions; you can use it at the board or get the students using it independently. My colleague and I were discussing after the session that we should use this for the AS level transition lessons we run in September.We find that even at A-level some students have serious problems when it comes to algebraic fractions as they don't have a basic conceptual understanding of what a fraction is and how they can/cannot be manipulated. I can't think of many other types of resource that you can use from KS2 right the way up to KS5 - I will post up some more of the sketches I use to teach fractions in the near future.