Fractions with a Calculator – Looking for Patterns

calculatorI have been working with teachers and using manipulatives, both physical and virtual, to help students think about fractions and develop conceptual understanding about fractional operations, versus just memorizing rules or tricks, as we so often do with students. There are fraction circles or fraction strips that work well as physical manipulatives, and there are several virtual manipulatives as well (i.e. DynamicNumber.org for any Sketchpad users out there, and the National Library of Virtual Manipulatives to give just a couple resources).

Manipulatives are a valuable resource in math class as they allow students to visually represent numbers, manipulate them, get hands-on with the math, and make some connections before moving into just the numerical representation alone. When working with fraction manipulatives, from my own experiences and those I have had with students, the manipulatives can constrain the number of possible examples we can provide students (either because a teacher might not physically have enough for all students or the manipulatives themselves only go up to certain values). As an example, most physical fraction circle manipulatives allow you to work with a limited range of fractional values – halves, thirds, fourths, fifths, sixths, eighths, tenths and twelfths. Virtual manipulatives offer more options, which is nice because students should see more than just common fractional pieces or ‘nice’ fractions – sevenths, or elevenths or twenty-fifths as an example. Obviously, the idea of manipulatives is to provide that hands-on experience, visually see what’s happening, and then create conjectures.

Another tool that is often overlooked, particularly at the elementary level, is the calculator. Obviously, when dealing with fractions, you want a calculator that uses natural display, showing fractions in their numerator over denominator form so students recognize the fractional number. I realize many of you might be thinking that the calculator is a bad choice because it provides the answers….but that in fact is an advantage here when trying to help students recognize patterns and develop their own understanding of fractional operations.  We want students to recognize what seems to be happening – test it out on many examples before they come to a conclusion.  A calculator (like the fx-55Plus shown above) is a great way to do this.  If you don’t have manipulatives, you can actually use a calculator like the fx-55Plus to help students understand fractional operations.

Let’s take fraction addition. Obviously, we are going to start with adding fractions with like denominators.  You can put several different problems into the calculator and students can observe both the added fractions and the answers. Students can talk and share what they notice about the multitude of fractions they are adding (all with like denominators). They can make up their own addition problems and see if the pattern or things they notice hold true. Fraction and answers showing up quickly help them discern patterns because they can quickly see many examples, and use ‘funky’ fractions, not just the typical ones we tend to always rely on (i.e. halves, thirds, etc.). It’s even okay that the numerator might occasionally end up larger than the denominator – the pattern still holds true (i.e. the denominator remains the same, the numerators are added together).

With a calculator, you can use messy fractions with not your typical denominators and even numerators larger than the denominator. For addition, our focus is on what patterns do the students see with the numerator and denominator and do those patterns hold true no matter what fractions we are adding? We can get into simplifying the answers at some point, but at first, the focus is on the addition.

Once students have the idea that with a like denominator, you add the numerators, you can then switch it up. Let’s add fractions with unlike denominators.  You can encourage smaller numbers in the denominator and numerator to start, and then once students think they have the pattern, they can ‘test it out’ with some larger digits in the numerator and denominator. The thing here is the denominators are different and so how does the end result differ (if does) from when the denominators are the same? What might be happening? Test it out.

The beauty of the calculator (again, one like the fx-55plus that quickly and easily shows fractions in their natural display), is that students can create many examples to look for patterns and then quickly test their conjectures on different problems to see if it works. You are encouraging critical thinking, problem solving, and communication using a simple tool that provides much more diverse fraction examples than you can provide with manipulatives alone.

My point – when helping students develop number sense, especially with fractions, don’t rule the calculator out as a tool. You should use multiple tools with students to provide them with different ways to develop their own conceptual understanding. Calculators can be a tool, even at the elementary level.

 

 

 

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Questioning In Math – #NCTMRegionals Minneapolis Observations

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John Diehl #NCTMRegionals

Had a nice time in Minneapolis these past two days at the #NCTMRegional. It must get ridiculously cold here in the winter since they built an entire interconnected-Skywalk throughout the city. I think I only went outside twice the entire time I was here – getting in and out of the taxi! (Which was pretty terrific as the first two days were rainy).

I went to a few sessions this time that really got me thinking about the importance of questioning in mathematics. Even when utilizing technology or hands-on manipulatives/resources, the questions we ask the students are vital in order to deepen their understanding and encourage discourse and exploration. Questioning to me is the most important skill a teacher can develop to help their students – more important than any resources or technology that might be available, because it is only through questioning that we  foster rigor and develop deeper thinking to help students understand and make connections in mathematics.

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Teachers talking statistics!

What I loved about two sessions in particular that I attended, a 6-12 Statistics session with John Diehl, and a 3-5 Making Math Meaningful session with Jennifer N. Morris, was the focus on questions and how the same mathematical concept could be appropriate for students in any grade depending on the questions asked. In John’s session, where we were looking at bivariate data, we used data athletes, made a scatterplot on the Prizm, and then had great discussions about lines of fit, variables, causation, association and a multitude of other ideas around helping students understand what the data represents, in context, and how, depending on the question asked, you could address algebra content or calculus content. Students in sixth grade can be do linear regression simply by asking the right questions and allowing them to explore their conjectures with technology, such as the Prizm. The questions lead to discussion and exploration and more importantly, to more questions that the students themselves begin to ask. Something as simple as “should your graph go through (0,0)? What does that actual mean in relation to the data and does that make sense?” helps students apply math content to the real-world context and make sense of the data and the graphical representation (sounds very Common Core to me!)

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Jennifer N. Morris & Origami

In Jennifer’s session, the first part was creating an Origami pinwheel, which seemed relatively simple but the questioning throughout about area of folds, and fractions of the whole, and how do you know, what shape do you have now and what’s the ratio of this shape to the shape before – really demonstrated how a seemingly simple hands-on activity can be full of rigorous mathematics and mathematical connections. And – the same activity would be appropriate for multiple grades – up through geometry even, simply by changing the questions you ask. When we began working with the Casio fx-55Plus, Jennifer did several quick activities using the Random# generator on the calculator, but the questions asked really had the teachers (i.e. students) thinking about numbers, fractions, comparing numbers, estimation, reasonableness, probability. For example, if these random numbers (all fractions, but with varying numerators and ugly numbers like 651/1000) represented the part of a cookie your mom was going to share with you, when would you consider it big enough and why? This led to really interesting discussions on how to determine

Teachers Ordering by Random #

Teachers Ordering by Random #

what fraction you had of a whole and is just being more than 1/2 enough. She had participants stand up front with their calculators & their randomly generated fraction and rearrange themselves in numerical order – not so easy when the fractions are 516/896 and 37/52 for example. Seemingly simple activity, using technology to quickly generate numbers and have really rich discussions that help make mathematical connections. And it came down to the questions asked – engaging, in-context, and appropriate for many grade levels.

Both sessions confirmed for me something I have always believed – the questioning is the easiest way to get your students thinking, talking, applying and connecting math. Learn to ask questions and you can create an engaging learning environment that differentiates the learning and provides students with multiple pathways to make connections.