Fractions are a vitally important concept for students. So much so that the 4th grade CCSSM dedicates a whole section to content around them. Despite this, we all see students who have poor conceptual understandings of what fractions represent. Check out these partitioning strategies David Wees's students tried. We have all seen our students struggle with fraction concepts as David's did.
To help students understand what fractions represent, they must have a sense of their relative magnitude. When young students are first learning mathematics, they have to memorize the order of the whole numbers in order to count. As with counting numbers, developing a strong understanding of fractions also means being able to order them based on their value. It is as if our students need to relearn how to count. Despite being countable, though, students cannot memorize fractions' order on the number line as young students memorize the order of the whole numbers.At some point they begin to use place value to help, but they have to memorize at least up to 10 first.
Many years ago, a colleague of mine shared an activity by Marilyn Burns called Put in Order (I have also seen a similar activity from Andrew Stadel for older students). This activity can be found in Marilyn's book About Teaching Mathematics which has MANY great activities.Found on page 277 of the 3rd edition. For this activity, you give students a collection of fractions and they need to do as the name suggests: put them in order.
I have used this activity consistently over the last 5 years and have found it to be a wonderfully engaging and rich task. I have done it in two different ways that I would like to share.
Way #1: Placing Fraction Cards 1-at-a-time at the Board
To set up, I made a collection of fractions (see above) and wrote them on colored pieces of paper chopped in half. After placing 0 and 1 on either ends of the marker rack at the base of my whiteboard, I modeled what I would like each student to do:
- Draw a random card
- Show it to the class
- Place it relative to the other cards on the board (only order matters, relative scale does not)
- Justify your decision to your classmates
One at a time, each student draws a card, places it on the board, and then explains why they think it goes there. Expecting students to justify the position of their card brings out lots of amazing strategies students develop for comparing the fractions.
Students will make errors, but someone in the class will inevitably notice the error and be able to ask a question or provide an explanation to correct the misunderstanding. Sometimes the whole class will have to work together to try to figure out where a card goes and to explain why. Here is a sample from mid-activity:
I love this activity for a few reasons:
- Students develop strategies together. They get to hear how their peers explain ideas and can restate those explanations when they apply to a different comparison.For example, \(\) because elevenths mean dividing a whole into smaller pieces. The same basic argument can be used to show \(\) This hits SMP3 and SMP7 bigtime
- This activity can be differentiated easily! You can either choose cards for students instead of having them draw randomly or you can have the students who are ready for more challenging comparisons go later when there are more fractions to consider
- Students get practice explaining their thinking and justifying their decisions to their classmates
- The cards are chosen specifically to get students to make comparisons using strategies other than making common denominators. I have written previously about different strategies one can use to compare fractions that promote better conceptual understanding
I usually do this activity once early on in my fractions unit, and students beg me to do it over and over again. I keep them waiting though until the end of the unit when I give them much harder sets of cards.
Way #2: Ordering Rational Number Cards in Small Groups
Once my students have investigated negatives, decimals, mixed numbers, and improper fractions, we return to Put in Order but with more challenging sets of cards. I divide them into groups of 2 to 4 and give them a set of cards to order. Here are my 4 sets which each have a loose theme:
Set 1: A little bit of everything (including \(\))
Set 2: Lots of Mixed Numbers and Improper Fractions
Set 3: Difficult Fraction Comparisons
Set 4: Negatives (including some repeating decimals)
When a group thinks they have them all ordered properly, they can ask me to check. If they have even 2 cards out of order, I just tell them "I disagree with your order." This prevents them from just guessing and checking with me over-and-over and incentivizes them to work carefully the first time through.
When they have corrected their set, I ask them to explain to me what had confused them before. This is great for encouraging them to reflect on what strategies work (and don't work) for comparing rational numbers.
A few lingering thoughts:
- I don't use these activities until students have first developed some conceptual models for fractions. For that, I like using fractions strips
- In her text, Marilyn recommends NOT including equivalent numbers, but I prefer to. I have found that the idea of equivalency doesn't seem to click for some of my students until they see two equivalent fractions together in this activity
- These comparisons are hard! Make sure you honor that as your students struggle to understand and justify their orderings. Be sure to try ordering them on your own first
- While I use this activity in the middle of a unit on rational numbers for 5th graders, it would also make a great first-day-of-school activity for older students such as those in Algebra 1
- For Way #2, you could also have all groups work on the same set and then compare and discuss any differences in order. This is the strategy I would use if I was working with older students
- Students LOVE to design their own sets. Just be sure to let them know that they will need to know the correct order of any set they make
- Share any sets you design!
- Print different sets on different colored papers to make sorting easier
- When I finally get to run the math night/coffee/session for parents that I have always been interested in running, this will be on my short list of activities to use. It just encapsulates everything good about how we want math education to look and feel for our students