I'm reaching out to my Numeracy friends for a resource or two. I'm coming up to teaching division with fractions in my ABE classroom and once again feeling stuck in teaching a procedural skill instead of conceptual understanding or application of division of fractions. What is your "go to" resource for teaching division of fractions/mixed numbers? What do you like about it?
How do I help my students see this isn't just a trick of math?
As far as online lessons go, School Yourself does a decent job at least at getting at understanding why the "trick" is based on reciprocals and inverse operations. Here is that lesson.
cK12 has three resource sets related to dividing fractions, mixed numbers, and whole numbers (CCRS 7.NS.2b)....I haven't looked thoroughly, but mostly procedural.
- Divide Fractions and Whole Numbers (This does have a "Real World" article that is helpful in applying the concept of division as how many sets/groups of a number are in a larger number.)
- Divide Mixed Numbers
- Divide Whole Numbers by Mixed Numbers
As always, I think EMPower Math (Using Benchmarks, Lesson 10) does a really good job at providing conceptual walkthroughs of concepts. If you don't have that program, one freely available program is Visual Mathematics, which I absolutely love and used quite extensively in my teaching days (See Course 1, Lesson 28.)
For operations with decimals and fractions, I used to walk through patterns with students where the either the divisor or dividend increases or decreases and the other number stays the same. For example, we might walk through this patter to understand the fact that if a dividend decreases and the divisor stays the same, the quotient will be smaller:
1,000 ÷ 50 = 20
100 ÷ 50 = 2
10 ÷ 50 = 0.2
1 ÷ 50 = 0.02
Whereas, if the dividend remains the same and the divisor gets larger, the quotient again gets smaller.
100 ÷ 5 = 20
100 ÷ 50 = 2
100 ÷ 500 = 0.2
And then apply this concept to have them theorize about what will happen with fractions. Easiest to use benchmark fractions to start....note the progression of the divisor is 1/2 of the previous divisor):
32 ÷ 16 = 2
32 ÷ 8 = 4
32 ÷ 4 = 8
32 ÷ 2 = 16
32 ÷ 1 = 32
32 ÷ 1/2 = ?
This requires careful consideration of numbers, but is a process that can help them come up with their own rules based on patterns. I would also walk through similar sequences (again with benchmark fractions) to have them see that dividing by a whole number is the same as multiplying by its inverse, again using patterns.
So, there are some options.....hopefully one or more of them prove helpful!!!
Thank you, Jeff, for providing links to so many great resources (not just for dividing fractions, but for LOTS of math topics), all available for FREE!
Thanks Jeff for the great detail in your response, as well as some free on-line programs worth looking at.
I must admit i am guilty in my past high school teaching days in dealing with mainly just the procedures of "division of fractions" and ignoring the concepts.
Now that I am working with ABE students, I find myself asking my wife for math teaching advice (she teaches in a k-8 school).
I am now quick to draw cookies, pies, ( I like desserts _anything I can pictorially split into fractions (division lesson) , looking at patterns for explaining rules of exponents (especially x^0 as a pattern as well as the standard algebraic "proof"). I will now add patterns to benchmark fractions to my teaching tool-bag. Don't assume just because you are "good" with math that it means you know how to teach the "basic" concepts to beginning math learners...thanks for sharing tips and resources!
I like to show that division progression with visuals -- I know a teacher who uses the plastic eggs you can take apart (for putting candies in if that's what you're doing)... showing a dozen eggs divided by 12, 6, 4, 3, 2, 1 ... and then 1/2. Oh, now we have 24 of those halves.
Sometimes I re-visit the idea that dividing is the same as multiplying by the reciprocal going the other way: What's half of 12? We write that as 12 x 1/2 ... and that is the same as 12 divided 2.... I've found spending extra time and attention to the notation is really helpful. I'm writing in that language all the time -- for how many years?
Have you checked out the posters at the math equals love blog?
Here's a "powers display" I put on the wall with her posters, adding a few things of my own for first and second powers.
Sigh, it's been chopped off here ... but she also has horizontal and vertical number lines which I find really handy. Spending time talking about them really helps.
This book from New Readers Press is full of activities for going beyond just procedural math instruction. The section on operations with fractions has a method for picturing division with fractions that might be the kind of thing you're looking for. It's not a super intuitive method, but I've actually never seen division of fractions "explained" anywhere else in adult ed materials. It's always just, here's how you divide fractions.
The tendency for "remedial" math to be overly procedural is real. I think too many times there's this unspoken, perhaps unrecognized framework of "let's cram so you can pass this test because if you didn't learn it by now you won't understand it anyway" which is grossly unfair to the learners. When I taught at The NEw Community School (middle/high school for students w/ specific language learning disorders) it was an amazing shift from that because so many students had memory issues that made the procedural instruction much less effective; building from concrete understanding can make such a difference.
Great topic and question. I'm also always looking to build my ideas on this.
I like using strings of problems combined with creating visual representations. We not only look at a numeric solution for each, we also look at representing them, usually on a number line. Even when students are kind of familiar with the algorithm, I focus on representing them physically and visually for a bit first. I also find the language they use to explain their thinking in the early problems really useful as we get to more complex ones (and this doesn't come out when they are using the algorithm).
For example: If students talk about 10 divided by 5 as "Two moves of 5 to get to 10" on the number line. I'll include "Two moves of 1/4 to get to 1/2." when we get to 1/2 divided by 1/4.
If students say 10 divided by 5 is "How many 5s in 10?" I'll ask "How many 1/4ths in 1/2?" among my questions about 1/2 divided by 1/4.
At first, i usually focus on what is happening in division (whether its whole numbers or fractions) and build on that to make sense of what is happening with dividing fractions.
I can't figure out how to get fractions to show up here properly when I type or paste them, so here is a link to some strings I've been playing with. You'll see multiplication first, scroll down to the purple for the division.
These are all taken and modified from others' ideas; mostly from Rebecca Strom (ABE teacher in MN) who worked with me on the doc I linked, Pamela Harris (Building Powerful Numeracy For Middle and HS Students and #MathStratChat) and Numberstrings.com.
You also asked what we like about the strategy we are using. Before I understood division with fractions conceptually, I could do the procedure correctly, but I could not estimate a sensible answer. The algorithm was a machine I trusted, but I didn't know what it did. Once I could see what the problem was actually asking, the estimating became really accessible. I notice the same a-ha with my students, they are better able to make sense of the problem whether they are using the algorithm or not. Also, making sense of what division (with fractions) is and having a visual for it in mind, seems to make it easier to recognize in context/word problems.