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 Tracy Leeper. Her topic, from Pre-Algebra: finding points on the coordinate plane.

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

After introducing the topic to the students, I will inform the students that we will be playing a game on the computer. After pulling up the game on the screen and demonstrating how it works, I will then issue a challenge using the maze game. The challenge will be to see how many mines they can avoid while using the least number of moves. Before class, I will play to get my best score, to show the students what I am looking for, and then I will see who can beat my score. To encourage the students to try their best, I will offer extra credit to anyone who can get through the same number of mines, with fewer moves. Multiple attempts are possible, and I will allow students to turn in their best game by the end of the week. By offering extra credit, it will encourage the students to play the game at home as well as in the classroom. This game will be fun for the students, as well as support the topic of finding points on the coordinate plane. A common struggle is confusing the x and y axis, so by playing the game it will reinforce the proper name for the corresponding axis, and which coordinate goes first in the ordered pair.

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

Finding points on the coordinate plane is used in a variety of disciplines. Any type of graph used to represent data, with the exception of a pie chart, uses at least one quadrant of the coordinate plane. Typically, it is quadrant 1, since both numbers are positive. The graph is just labeled to reflect the data shown, instead of using x and y. Scientist use graphs to represent data that has been collected from either observation or experimentation, usually labeled as time and the correlating measurement. In math the coordinate plane is used to represent any function, with x as the input and y as the output, as well as helping to graph things that are not functions, such as circles, and other polygons. As well as adding a third dimension, and including a z axis for graphing 3D objects, such as spheres and cubes. The coordinate plane is also used in other disciplines, such as geography, for determining map coordinates.

How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

Video games have changed tremendously since the days of Pong. The graphics, storylines, characters, and amount of programming required has become much more intricate. One aspect of the games that appeals to players is the moving background that changes and shifts according to where the character is in the game, and how the camera angle is changed by the player. This enables different scenery and perspectives throughout the game. This is done by using points on a 3D graph, and as the character moves, the reference changes according to their position. The fundamental skill for being able to build the game this way, is to first learn how to plot points on a 2D graph. Since most teenagers like video games, and the graphics involved, this would be a good point to make, so the students could see the connection between the math they are learning, and something they really enjoy doing. This same skill is used for calculating GPS coordinates on our phones and computers.

References:

http://www.shodor.org/interactivate/activities/MazeGame/

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 Nada Al Ghussain. Her topic, from Pre-Algebra: solving for unknown parts of rectangles and triangles.

How has this topic appeared in high culture (art, classical music, theatre, etc.)?

In the mathematical classroom it is always easier to engage the left-brainers who excel in numbers, reasoning and logic. My right brainers on the other hand can also shine when engaging them through the underlying foundation of the arts. The Golden ratio is seen in paintings and architecture. It shows how rectangles and triangles can organize the placement of other shapes and figures in an eye pleasing way. Artists and architectures constantly mapped out their masterpieces on blueprints, which required basic calculations that set up the Golden ratio. Artists using the Golden Rectangle would need to find the missing sides to be able to get the correct proportions for the Golden ratio. This is seen in Leonardo Da Vinci’s “The Last Supper” and in the Parthenon building. Rembrandt solved the third side of an acute triangle before he continued work on his self-portrait. He then drew the line from the apex of the triangle to the base, which cuts into the golden section. Finding the part of a triangle and rectangle contributes to creating masterpieces! Students, left and right brained will see beyond paint, color, and stones. As Luca Pacioli, a contemporary of Da Vinci had said, “Without mathematics there is no art.”

How does this topic extend what your students should have learned in previous courses?

Beginning Geometry students, Can with little and quick computational work solve for the unknown parts of any given rectangle and triangle. A great starter for a Pythagorean lesson is to get them to find missing parts using their shoes! Middle school students can take off their shoes as they work in groups and form the two legs of a right triangle. Once they compute the hypotenuse students can check it by adding the right amount of shoes. This lets students interact with each other and with the right triangle. They can see which triangle theorems can be formed, and discuss the type of angles found with the right triangle. Going beyond that, students can shoe in the missing sides of the squares. This sets up The Pythagorean theorem. This engagement can be quick or take a whole lesson. Students find different calculations, theorems, and set them up for figuring out the Pythagorean theorem.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic? Note: It’s not enough to say “such-and-such is a great website”; you need to explain in some detail why it’s a great website.

Technology is information at our fingertips. Calculator Soup has a Triangle Theorems calculator that can calculate AAA, AAS, ASA, ASS, SAS, and SSS. This would be a great and quick way for students to explore triangles. As a teacher I would ask the students to make an acute SSS triangle using the digits 1through 10 for the sides. I then can ask them if a given side was 20 and the other two were between 1through 10, would I still have an acute triangle? Many quick questions can be used from this calculator. It has the students think about the relationship of the sides and angles as they form triangles. There are also Square, Rectangle, Parallelogram, and a Polygon calculator too. For the parallelogram, different angle measurements can be given to change the side length. Good ways to have students differentiate between rhombus and parallelograms. Calculator Soup is quick visual for students to help them understand the relations between different squares and triangles.

References:

http://www.goldennumber.net/art-composition-design/

http://britton.disted.camosun.bc.ca/goldslide/jbgoldslide.htm

http://psychology.about.com/od/cognitivepsychology/a/left-brain-right-brain.htm

http://www.mathsisfun.com/activity/pythagoras-theorem-shoes.html

http://www.regentsprep.org/regents/math/algebra/at1/pythag.htm

http://www.calculatorsoup.com/calculators/geometry-plane/triangle-theorems.php

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 Kristin Ambrose. Her topic, from Pre-Algebra: adding and subtracting fractions with unequal denominators.

What interesting (i.e., uncontrived) word problems using this topic can your students do now?

Cooking is a great example of where you frequently add and subtract fractions with unequal denominators. For example, here is a real-world word problem I came up with for adding and subtracting fractions in cooking:

You are making dinner tonight and you’re having Lemon Chicken with Scalloped Potatoes. The recipes for these can be found below (and yes they are real recipes that you can use in real life).

You only have a half a cup of flour left in your pantry. Looking at the recipes above, do you have enough flour to make dinner? Or do you need to go to the grocery store to buy more flour?

In order to solve this problem students would first have to add the different amounts of flour for each recipe (1/4 + 1/8 = 3/8). Then students would have to subtract this amount from the amount of flour they had to see if they would have enough (1/2 – 3/8 = 1/8). Since 1/8 cup of flour would be left, they have enough flour to make dinner.

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

It would be difficult to do mathematics without knowing how to add and subtract fractions with unequal denominators. In mathematics when adding or subtracting fractions, it doesn’t always work out nicely where the denominators are the same, so it’s important to be able to solve problems even when the denominators are different. One example of this is summations. Take ; what this equation really means is to add 1/2+1/4+1/6+1/8=25/24 or 1 1/24. Therefore adding fractions with unequal denominators could arise in summations. Also, in Algebra students will study quadratic functions and the factors of quadratic functions often take a form similar to something like (x+a)(x-b), with a and b being numbers. Students will have to know how to multiply these factors out and simplify the expressions. For example, a set of factors could be . When multiplied out students will have . Students will have to know how to subtract 2/3 from 1/2 in order to simplify the expression.

How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

YouTube can be a good source for finding videos to engage students in a topic. In particular, I found a short, funny video that reminds students of the significance of fractions. Here is the link to the video: https://www.youtube.com/watch?v=CBy8QbZyzy4. It makes a difference when a superhero only saves half of your stuff and not all of it. Just like you wouldn’t want only half your things saved, you wouldn’t want to add 2/3 of a cup of flour to a recipe that only calls for 1/4 a cup, or you wouldn’t want to fill up 2/3 of your tank of gas if it was already 1/2 of a tank full. Understanding fractions and how to add and subtract them is an important part of daily life.

I also found another video that demonstrates where fractions can come into play in science. Here is the link to the video: https://www.youtube.com/watch?v=hLGDJFGAmic. The YouTube channel ‘Numberphile’ in particular has many interesting videos involving numbers and mathematics, and would be a great resource for finding interesting videos to engage students.