How accurate is the distance formula?

Vernier is a company that produces software and equipment to be used in education. For this specific lesson, motion detectors are used to gather data. Students have the opportunity to work in groups developing a model to determine the accuracy of the distance formula. By working in groups, students have the opportunity to practice standards in mathematics such as construct variable arguments and critique the reasoning of others and model with mathematics. The activity is also aligned with the CCSS.MATH.CONTENT.HSG.GPE.B.7
Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula.^*

In this activity, students use the distance formula to find the distance between two points. Using the motion detectors, students can create a “live” cartesian plane and determine the coordinates of 5 points.

Students have to find the distance not only with the distance formula but also with a meter/yard stick. They will be recording their data in a table (Figure 1).By making measurements with different tools, students have the opportunity’s to develop ideas that connect to the concept of distance between two points.  After the data is collected, students need to compare/contrast both measurements,  determine which one is more accurate and explain their reasoning .

Figure 1

Find more about this the activity here: from-here-to-there

Hungry to Learn

Using Geogebra to Bring Out the Inner Mathematician.

Written by Jenell Sellers

INTRODUCTION

Times have changed, can anyone deny that? Gone are the days in which primary and secondary students sit row by row and take notes to a lecture. The image of school and education is changing and morphing into an environment where adolescents are learning how to collaborate with their colleagues and learn as a group instead of in solitude.

Now, instead of rows we have desks put into groups. Instead of the students only holding pencils to paper, they are holding manipulatives (algebra tiles, fractal cards, hand-made unit circles, etc.). This is all to help students become effective citizens. Yes, times are changing indeed.

What we are seeing in our students is that they want to learn; they want to understand what is going on within the math class, it’s just that they don’t think that they can. They’ve seen math taught may a couple of different way, but it just hasn’t really sunk in yet and they start to think that maybe math is really only supposed to be taught a certain way. This is where we start to break the boundaries that they, or maybe other teachers, have set up around them; around their mind and ability to learn.

One change that is also being implemented is trying to introduce technology into the classroom. Our students, being part of the Millennial generation, are growing up with technology at their fingertips. It is our way of connecting with them and bringing their world into our classroom instead of bringing the students into our world of pure mathematics and computations.

One piece of technology that is immensely helpful within the classroom to help students see a better visual of what they are learning within the classroom is Geogebra. Geogebra can be used for Algebra I and II, Geometry, and even Trigonometry. Geogebra is a free application that can be downloaded from the Geogebra website (www.geogebgra.org) to the aspiring mathematician’s home computer or laptop. For the schools that have laptops in the classroom it can be especially effective so that every student can explore the program. Throughout this article we will be examining how Geogebra may be implemented within the Algebra II classroom.

STEP BY STEP, WE PROCEED…

The beauty about Geogebra is that it is amazingly simple to use. When the application is first opened we see the x and y axes, just waiting for an equation to be plotted. Only by going under the Graphics tab, we can make the grid show so that we can see each individual intersection of the x and y values; this makes it significantly easier for the students to follow along with the values that are shown when you plug in a specific graph, especially if it is being projected in front of them.

Let’s say that we are in an Algebra II class setting where we are learning how to translate graphs. This can be an especially hard lesson for students because it can be so hard to visualize the moves of a graph when the teacher is explaining; “If the value is inside the parentheses you move the vertex left or right in the opposite direction of what the value is. And when it’s outside the parentheses you move the vertex up or down the number of units according to the value.” Umm, what? Let us see a visual, shall we?

Take the equation, . When we type this into the input bar at the bottom of the screen within Geogebra we get the following visual:

This function alone is not that hard for students to plot on a graph. However, when it comes to the translations the students feel that it’s a lot trickier that what it really is.

Now, let’s take the equation         . The students are told within their class that the horizontal shift is going to be ‘the opposite sign of what is in the parentheses’. So instead of shifting over to the right 2 units, we go to the left 2 units. Like this:

Plugging this into the input bar, we can see that the graph does indeed shift over to the left when we add 2 within the parantheses. After showing the students the parent equation, the students can see how the graph physically moves.

Now let’s look at the final transition of this graph. Typing  into the input bar we see the following:

During this final transition, the students are able to see that when the value (minus 4) is outside of the parentheses then we actually move the graph down 4 units.

Not only might it help for the students to see each individual shift by itself, but it would also be well worth it to have them see each individual shift put altogether. Like so:

It is of upmost importance that the students are able to see each of these translations, rather than just hear about them and the instructions. The lecture piece is important, but without seeing what is physically going on with the graphs, then it is easier for the students to get lost.

ACHIEVING THOSE STANDARDS

This lesson would most definitely prepare the way for students to achieve a couple of common core state standards: “factor a quadratic expression to reveal the zeros of the function it defines” (CSS.MATH.CONTENT.HSA.SSE.B.3.A) and “understand that a function is a rule that assigns to each input exactly one output. The graph of a function is the set of ordered pairs consisting of an input and the corresponding output” (CCSS.MATH.CONTENT.8.F.A.1). Not only will they be able to be prepared to achieve these standards and more, but they will also be able to apply the learning targets to go along with the lessons. The learning targets that could go along with the lesson mentioned here could be something along the lines of “I can write the equation for a translated parabolic function” or “I can show that a parent function is still one-to-one.”

CONLCUSION

As time moves along, change comes with it. We have made several innovations for cars, computers, phones, and so much more. The future generations are continually learning, but they are learning at a far different level than what students used to. It is about time that we change our teaching methods as well to match the changes that have over the years in the education system. Geogebra, among many other applications, is one way that we can further assist our students in visualizing the mathematics of the world.