Hi, I'm an HSE teacher and I've been teaching fractions recently, funny how some subjects just pop up over and over. I need help, I'm trying not to forget anything when I'm actually showing this overview of what you can do with fractions. Anyway, I have a quick list of things I teach and a basic description of what each item means. Is there something I'm missing or should I teach it in a different way?
First, define fractions, improper fractions, and mixed numbers.
All Fractions, Ratios, and Rates are division problems. We use fractions instead of decimals for some things because it is exact. An example is 1/3 of a cup of milk vs. .333333 cups of milk.
Proper fractions are less than one and more than zero.
Improper fractions and mixed numbers are more than one.
There are four rules to fractions.
1. All Fractions are Division. - Examples include 1/2= .5 2/4=.5 4/8=.5 1/4=.25 2/8=.25 1/3 = .33333
2. Any number divided by 1 is itself. - Examples include 2/1 37/1 365/1 Question what makes a whole number a fraction?
3. Any number divided by itself is 1. Examples include 2/2 8/8 47/47 478/478
4. Any number multiplied by its reciprocal is 1. Examples include 1/2 and 2/1 47/83 and 83/47 56 and 1/56 1/27 and 27/1
Reciprocal means in Latin means "back and forth" or "to and fro" ( I still don't know what a fro is other than a cool hairstyle)
Then I only teach three operations of fractions in order; Multiplication, Division, Addition and Subtraction.
Multiply straight across, then apply the rules to "Simplify" or "Reduce" the fraction. I introduce how to divide and multiply the top and bottom by the same number and why that is really only changing the look of the fraction and not the value.
Examples include 1/2 x 3/5 5/8x 4/4 3/8x7/8 etc.
Divide by flipping the second fraction and then multiply. Nothing fancy and the same fractions are used as multiplication. I even just erase the answers and the x and replace it with a ÷.
Addition and Subtraction
Simply make sure the bottoms are the same and then add or subtract the tops.
I start with the same denominators as the examples to show how it can be done. 5/8+1/8 2/3+1/3 34/47+13/47
I move on to using the easiest common denominator and then the greatest common denominator. I draw pizzas on the board for the first couple so they can see how they fit together.
1/2 + 1/3 1/8 + 3/4
This is an overview that I give right before the actual section on fractions begins. It takes about 7 minutes from beginning to end and I repeat it during the lesson at spaced intervals.
All that being said, is there anything I'm missing on a rough overview of what a student is going to learn or review?
I appreciate you reaching out through the discussion board asking for help. You've written a lot here, but in terms of what's missing, I would say that equivalent fractions (especially visual representations of equivalent fractions) and finding fractions on the number line are both very rich topics to explore that I didn't see on your list. You might check out the College & Career Readiness Standards for Adult Education: Content Progressions - for other ideas - see pages 2-7 for the fraction standards and how they build from Level B to Level C. Graham Fletcher is an elementary math specialist, but his fraction progression video is just over 7 minutes long and can be very helpful in visualizing how a conceptual understanding of fraction develops and builds. In HSE, we have more to teach than his visualizations cover, but they show a coherence and a useful order for early fraction concepts.
What are fractions? Defining/implying fractions, ratios, and rates (a) as all division problems and (b) as the same, is an incomplete story and might be confusing to students. We use fractions to talk about how many parts of a whole, rates are relationships between two quantities, and ratios show relative size of two (or more) quantities.
I appreciate that you are thinking about why we use fractions, but I think the precision or how exact something is depends on the context, not necessarily the representations we use. Asking students where they see fractions - either from experience or even having them bring in examples is an easy way to get examples of real-world uses of fractions.
In terms of operations with fractions, I think all students, but especially adult education students deserve a conceptual understanding and some practice with visual models for representing the things they are doing. Procedures without a conceptual understanding are hard to remember and difficult to use or adapt.Think of how all the rules sound when you put them next to each other - “When we add or subtract fractions, we have to find a common denominator, but not when we multiply or divide. And once we get a common denominator, we add or subtract the numerators, but not the denominators, despite the fact that when we multiply, we multiply both the numerators and the denominators, and when we divide, we divide neither the numerators nor the denominators.” from Creating, Naming and Justifying Fractions by Daniel Siebert and Nicole Gaskin. A great resource for exploring multiple solution methods, including useful (and artful) visual models, check out Sarah Lonberg-Lew’s writing on the TERC SABES Blog - here’s an entry looking at dividing with fractions. https://www.terc.edu/adultnumeracycenter/will-this-be-on-the-test-june-2021/
Other Resources The TERC EMPower math book series is a strong resource and their books dealing with fractions - Using Benchmarks: Fractions and Operations & Split It Up: More Fractions, Decimals, and Percents - are great. https://store.bwwalch.com/empower-math/
One of my favorite instructional routines at the moment are Fraction Talks. A Fraction Talk is a discussion-rich routine that invites multiple ways of identifying fractions. Nat Banting created the Fraction Talks website, which has a huge collection of images to inspire discussion and exploration, as well as suggestions for using this routine in your class. MATH FOR LOVE also has some great Fraction Talk images using pattern blocks.
Of course, there are a lot of other instructional routines that lend themselves to working with fractions. The Adult Numeracy Network has a collection of instructional routines - in addition to fraction talks, you might also check out Clothesline Math & SPLAT!
OPEN MIDDLE has some fantastic problems in all topics, and fractions are no exceptions. You can find some of them here:
yours in productive struggle,
Fractions it seems are an Achilles heel for so many adults trying to comprehend math.
Thanks for the insights and resources.
Past two weeks no classes at the college, so I've just been coaching students who Need To Do Better on the Placement Test.
Pretty painfully consistent: if they went to school in this country, they don't know from fractions.
I'm off work today but we had some really good practice with buildng that understanding of fractions beign division, 5/5 being 1 (lots of "what's 3/5 + 2/5?" then Wha'ts 1 minus 1/5?" that was *really* helpful. I'll share it (she said, glancing at the binder beside her that *is that course* but I have to go to a 100th birthday party in a few :P )
I also have a way for adding fractions w/ diff denominators that many of mine find really helpful. When I contemplated how many people could change mixed number to improper confidently... BUT NOTHING ELSE!!... I realized there's a whole visual-kinesthetic element of that. So I teach the complicated stuff that way. I'll find those notes Tuesday too ;)
https://www.lightandsaltlearning.org/ also has some awesome ideas.
Back at this soon and I want to share some COOL stuff I learned at the conference this week...
I ***love*** gfletchy's progressions but I find it *very* useful w/ older learners to include the "what's 1/2 of 100" early on, especially since lots of problems have that framework.
You are correct, Fractions are and Achilles heel for so many.
Thanks for the tips and resources.