## Operations on the Number Line with GeoGebra

When learning about fractions and decimals for the first time, students often discover that rules they thought were true for all numbers apply only to whole numbers. For example, "numbers with more digits are always larger" is true for whole numbers but not always true for decimals. NCTM's *Teaching Children Mathematics* had a great article on this idea called *"13 Rules That Expire." *If you have not seen it, you should check it out.

Students need an opportunity to explore these ideas in order to recalibrate their understanding to include fractions and decimals. In graduate school, we used GeoGebra to make an interactive number line.^{[4]}I used "we" but I am honestly not sure who originally created a GeoGebra applet for this purpose. It may have come from my professor, a classmate may have made a version of it, or I may have made it as part of a group. Unfortunately, I just don't remember who to give the proper credit to. I did, however, create this version on my own from scratch, so at least this document is my own work. There are two points, *x* and *y*, that you can drag up and down the number line.

As you change the position of *x* and *y*, GeoGebra does operations on their values and places a point on the number line for the result. You can control which results are shown using the check boxes, and you can add markers for 1 and for -1. Play around for a second to see how it works.

I find that this simple digital manipulative is really useful for investigating some of the "Rules That Expire" related to operations (view on GeoGebraTube).^{[5]}It is easier to work within the GeoGebra document than in the embedded window. But, the window gives you a chance to test it out commitment free. Many students, for example, think that multiplying always results in a product that is larger than both factors. You can have students investigate that idea by asking if it is possible to get the product between the factors.

Have students turn on the multiplication marker and then drag the variables to try to move the product in between the factors. Once they figure out how to make that happen, have them discuss what is happening when the product stops being larger than both factors. Focus the discussion on what is happening when the product switches from being larger than both factors to being between the factors. What must be true at that point?

Here are few other examples of what you could have students investigate with this applet:

- Can you make the sum equal one of the addends?
*(Additive identity)* - Can you make the product equal one of the factors?
*(Multiplicative identity)* - Can you find a place for
*x*so that changing*y*does not change the product?*(Zero property of multiplication)* - Can the sum ever be less than one factor? Less than both?
*(Addition of integers)* - When you divide, can you get a quotient that is larger than one of the factors? Larger than both?
*(Division)*

This tool can also support Always/Sometimes/Never discussions.^{[6]}I've got a post brewing on Always/Sometimes/Never prompts and tasks, but until then you should check out Fawn Nguyen's post. I plan to use Always/Sometimes/Never a lot with my pre-algebra class this year. You may have seen this S/A/N from Fawn Nguyen: "The square root of a number is less than the number." This applet gives you a visual way to investigate that idea.

It is also a great tool for comparing operations. Is the product of two numbers always larger than the sum? Is \(xy < x^y\) sometimes, always, or never true. All of these prompts obviously need to be supported with discussion and reflection, but this GeoGebra tool is great for prompting students to think about arithmetic properties and operations.

**End Thoughts**

- For all you
**Desmos fanboys**out there, I tried making this with Desmos but could not get it to be as functional as it is here. When I added multiple operations at once, it became too cluttered to reasonably use. The issues could be my Desmos skills rather than the platform as I am much more experienced with GeoGebra. If you can make a comparable or (preferably) better version in Desmos, do it! - If you can think of any way to improve this please let me know. I would be excited to improve it or make it more functional