# Common Fractions, Number Lines, and Geogebra

I don't know how many elementary school teachers use Geogebra in their classrooms, but I suspect it's probably not many. Yesterday I led a follow-up workshop session for a group of mostly K-5 math teachers. I showed how to use Geogebra to make a number line with tick marks noting common fractions.

A thought experiment: suppose you are a K-5 math teacher and want to show students how to compare the fractions 4/9 an 3/7 on a number line. The decimal expansions show these two values are pretty close to one another. 3/7 is approximately 0.428571 and 4/9 is approximately 0.444444. Drawing these two fractions on a number bounded between 0 and 1 might be tough. They may even appear to be the same if the scale isn't discerning enough. What would a person do to find number lines to make teaching materials? Like many people, he or she might go to Google and do something like this:

We may able to find number lines with sevenths and ninths indicated, but they will probably be separate. It may take some time to dig through these images. Should a teacher be at the mercy of materials made by somebody else? What if the teacher could design his or her own materials instead? Enter Geogebra.

Here's a PDF I made with some basic instructions on how to start building number lines: Creating Number Lines with Fractions in Geogebra

# 7 Games for the High School Math Classroom

## Euclid: The Game

Euclid: The Game, designed by Kasper Peulen, takes the basic compass and straightedge constructions featured in geometry class and makes a game of them. There are 20 levels to complete. A teacher can directly access a particular level by modifying the URL. http://euclidthegame.org/Level1.html (for example, change the "1" to a "7") I would have loved the opportunity to learn geometric constructions in this format when I was in high school.

## 2048

The premise of 2048 is very simple. Use the arrow keys to slide tiles about the game board. When two tiles with the same number touch, they merge into a new tile. For example, when two tiles with the number 64 on them touch, they merge into a 128 tile. The objective is to make a "2048" tile, which would require two 1024 tiles, and each of those require two 512 tiles, and each of those requires two 256 tiles, and each of those requires two 128 tiles, and each of those requires two 64 tiles, and each of those requires two 32 tiles, and each of those requires two 16 tiles, and each of those requires two 8 tiles, and each of those requires two 4 tiles, and each of those requires two 2 tiles. Each move in the game yields a new number tile on a random space. A user may continue with the game even beyond the 2048 tile. I have personally witnessed one of my students with a 65,536 tile. This game has some uses in the math classroom. The obvious is knowing the positive integer powers of 2 and its parallels to data storage capacity in computer hard drives. We can also use this game to teach an introduction to game theory and optimal strategies. After playing this game for a while, a user tends to see patterns and trends, cycles of values and positions that appear frequently. Students can intuit optimal strategies through trial and error, but this could also allow students to use formal mathematics to move towards establishing optimal strategies systematically.

## SET (card game)

While the card game isn't free, the daily set puzzle is free. My students avoid trying to explain SET to a novice player. It is much easier to play and learn through trial and error than to learn all the rules before doing anything... (sounds a bit like mathematics, in my opinion). Each card has four attributes - shape, color, shading, and number of shapes. A set is a collection of three cards for which each individual attribute is all the same or all different. Here are two examples of possible sets.

There are some great combinatorial features associated with this game. A great strategy lends itself incredibly well to mathematical statement: For any two selected cards, there exists a unique third card in the deck which completes the set.

## ALICE

Alice isn't a game, but rather a software package which teaches computer programming as if it were a game. Students can use previously constructed environments and characters to create animated videos. Students can use the interface to make their own games. I have used this software in my class for five years. I have had students make a soccer game that plays a victory sequence for the first player to three goals, a tank rolling through a world that enables collision detection (the default objects can pass through walls; writing code to detect collisions requires some great geometric reasoning), and a first person zombie shooter game *complete* with a zoom-in rifle scope. All my work with students has been with Alice 2.0. I plan to use Alice 3.0 this upcoming school year!

## SCRATCH

Scratch is like Alice in that it's a programming interface. I have not used it personally with students, but it's another resource I do plan to explore with some of my students this upcoming school year.

## KenKen

Many are familiar with Sudoku. KenKen is a similar reasoning number puzzle game involving operations in addition to populating the digits.

## Alcumus

I've mentioned Alcumus previously on my blog, but I can't say enough about the role Alcumus plays in addressing learners' needs. Teachers can now register for a teacher account to monitor students' progress through the self-paced, differentiated curriculum. Students can sign up for free, only an email account is needed. Students can take a pre-assessment to determine areas of strength and deficiency. Quests and experience points provide instant feedback, instant gratification, and a mechanism to keep students engaged and to avoid tedium.

# Does the Size of a Fraction Bar Matter?

When we teach children to add, subtract, and multiply, we commonly see a vertical format used in school.

9
+ 6
15

Then, the student learns division. Students usually see the division algorithm represented horizontally.

A student may not, beyond a worksheet with dozens of problems, be comfortable with a vertical format for division.

It almost feels strange to me to the type the problem vertically. Then comes fractions. We did the following exercise in a workshop I led today. I wanted participants to consider the question, “Does the size of a fraction bar matter?”

Below is an image of the problems we worked on in groups. Calculators were allowed. Spirited discussion and abundant disagreement followed.

As the back of the textbook often says, “Answers will vary.” The participants’ responses, split up by problem, appear below.

How can we resolve this conflict? Let’s type it into a reliable computation source, like Wolfram Alpha. It depends on how we type the input.

Above is one way to type the input. Below is another way to type the input.

Why the difference? Because multiplication is commutative, but division is not. 3 x 4 is the same as 4 x 3. But 3 divided by 4 does not yield the same result as 4 divided by 3.

In the original input, where I typed 3/4/1/2, the computer interpreted this as the product of 3 and the multiplicative inverses of 4, 1, and 2, like this:

In the second example, we see the computer associates the fractions ¾ and ½, treating the middle fraction bar as a grouping symbol.

The confusion for me stems from converting between vertical format and horizontal format. We teach students to follow order of operations, to divide in a horizontal statement as they encounter it from left to right. In school mathematics, we commonly write the fractions vertically when we mean to take the ratio between two ratios, specifically ¾ and ½ in this case.

Here is a state assessment practice item, taken from the Nebraska Department of Education website, that demonstrates this understanding of the vertical format when dividing fractions.

Let’s look at this Word document, specifically this problem. The author of the item intends for students to invert 7/3 and multiply. I worry a student may see this problem in a horizontal format as the work shown to the left, which gives a result which is obviously not an available choice, but if we are examining item reliability, I wonder if we should have the conversation about whether this expression should be typed in the following way:

Again, I’m just posing the question about the test item. I want to be careful in how I represent problems with my students. I want my own students to understand the context matters. I want my students to be procedurally fluent with fractions when they solve problems in the world. The conversation in this morning's workshop was great. Participants reflected on how what they say to their students while they teach the students can have unintended consequences on how students view procedures with fractions.

Here’s a link for a great blog entry on this topic:
http://pballew.blogspot.com/2009/01/why-we-flip-and-multiply.html