Polar plot

Source: http://www.xkcd.com/1230/

Engaging students: Finding points on the coordinate plane

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Rebekah Bennett. Her topic, from Pre-Algebra: finding points on the coordinate plane.

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Applications: How could you as a teacher create an activity or project that involves your topic?

 

For this topic, the first thing that came to mind was battleship. The game was introduced to me when I was around 8 or 9 years old. The mathematical content that the game expresses never really occurred to me until I became older and made a connection. The game board for battleship is simply one quadrant of the coordinate plane and the players call out coordinates which are found on the game board. This is the same as finding a point on the coordinate plane but in a much more fun way of doing so.

For those of you who do not know what the game is, here is a quick clip from Seinfeld where they are playing the game.

To make things interesting, we will play Human Battleship. For this activity you would need a large area that can be marked off as a grid, such as a gym or field. Each group will have at least 4 students (ships) that they can place strategically on their side. Since there is no barrier between the sides, the captains must face the opposite direction to ensure they have not seen the opponent’s ships locations. Now each captain will take turns calling out coordinate points and having them recorded by their co-captain. The shipmates must go to each point and yell hit or miss, marking a hit with a red flag and miss with a white flag. When a ship is sunk the shipmates will make a bombing sound so that both captains know they are a down a ship. The students will continue to do this until one team has all their ships sunk and the other is declared the winner.

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Curriculum: How can this topic be used in your students’ future courses in mathematics or science?

This topic is used continually throughout mathematics and builds up to something more every day in math. It is a basis for learning how to work with graphs. Students learn how to plot points now and then later they learn how to create graphs according to the points. With graphs, they will learn how to move points along the coordinate plane, learning new vocabulary such as; translation, rotation, reflection, stretch and shrink. Students will then learn how to draw a line using slope to connect one point to another and find the distance between those 2 points. The x and y values work as an input, output function. All these things are based on the simple concept of plotting points which we use in every day math.

This topic is also used throughout the scientific world. The student learns how to make scatter plots and line graphs. Also, science uses functions as well. In science students record data in a table using an x and y value but are typically labeled according to a real life experiment such as growth and amount of water. When conducting research or displaying data the student uses the same techniques for graphs that were learned in math and applies them to science, which builds more and more everyday as well.

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History: How was this topic adopted by the mathematical community?

During the European Renaissance, mathematics was split into two separate subjects of geometry and algebra. They didn’t coincide. Algebraic equations were only used in algebra and people only drew pictures in geometry. Rene Descartes changed the whole outcome and combined both subjects together developing a brighter future for mathematics.

Descartes’ method involved two number lines. The student was already introduced to the basic number line in elementary and then introduced to a number line with negative numbers during 8th or 9th grade completing the number line. Knowing that the students have full knowledge of a number line, Descartes decided to put two number lines together. The traditional number line is horizontal and rotated the other number line 90 degrees (vertical) where both of the number lines intersect at zero. These two lines are called axes; such as x-axis (horizontal line) and y-axis (vertical line). Since a number line stretches in both directions, the axes will have arrows on each end. The whole area, side to side, top to bottom, and stretching infinitely in all directions creates a plane. When constructing two axes within a plane, it is then converted to a Cartesian Plane. The name “Cartesian” was derived from the name “Descartes.” From creating a plane, the student can now find a point on the plane using the coordinate pair they are given.

Sources:

http://simple.wikipedia.org/wiki/Cartesian_coordinate_system

http://www.math.wichita.edu/history/men/descartes.html

http://plato.stanford.edu/entries/descartes-mathematics/

Engaging students: Finding points in the coordinate plane

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission comes from my former student Derek Skipworth. His topic, from Pre-Algebra: finding points in the coordinate plane.

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A. Applications – How could you as a teacher create an activity or project that involves your topic?

When I think of the coordinate plane, one of the first things that come to mind is mapping.  When I think of my teenage years, I think of how I always wanted more money.  By using these two ideas together, an activity could easily be created to get the students involved in the lesson: a treasure map!

The first part of the activity would be providing the students with a larger grid.  Then provide them with a list of landmarks/items at different locations (i.e. skull cave at (3,2)) that would then be mapped onto the grid.  By starting out with one landmark, you could also build off previously identified landmarks, such as “move 3 units East and 4 units North to find the shipwreck.  The shipwreck is located at what coordinates?”   These steps could also be based off generic formulas with solutions for x and y.  After all landmarks were identified, there would be a guide below that would trace out a path to find the treasure, which is only discovered after the full path is completed.

treasuremapCourtesy of paleochick.blogspot.com

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B. Curriculum: How can this topic be used in your students’ future courses in mathematics or science?

One of the good things about the exercise above is that it integrates several different ideas into one. A big one that stands out to me is following procedures.  This is vital once you get into high school sciences.  By building the map step-by-step, which each one building off the previous step, you cannot find the treasure without replicating the map exactly if you miss/misinterpret a step along the way.

As far as the coordinate plane, finding locations on the plane is important when graphing functions.  Being able to find the intercepts and any asymptotes gives you starting points to work with.  From there you generally only need a few more points to create a line of the function based off plotted points.  This also has applications in science/math when creating bar graphs/line graphs and similar graphs.

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D.  How was this topic adopted by the mathematical community?

As discussed in my Geometry class this semester (Krueger), the Cartesian plane opened up a lot of doors in the world of Geometry.  Euclid had already established a great working knowledge of a vast amount of Geometric ideas and figures.  One thing he did not establish was length.  In his teachings, there were relative terms such as “smaller than” or “larger than”.  No values were ever assigned to his figures though.  By introducing the Cartesian plane (and in effect, being able to plot points on said plane), we were able to actually assign values to these figures and advance our mathematical knowledge.  The Cartesian plane acts as a bridge between Algebra and Geometry that did not exist before.  Because of this, we can know solve problems based in Geometry without ever even needing to draw the figure in the first place (example: Pythagorean Theorem).