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).

Scalloped Potatoes4 med. potatoes ¼ cup flour 4 tbsp. butter 2 cups milk 1 cup grated cheese Dash of garlic powder and white pepper Salt and pepper to taste Instructions: Preheat oven to 350°. Peel and boil potatoes, then set aside to cool. Make 2 cups of cream sauce by melting the butter and blending in the flour. Stir constantly, slowly adding the milk. Stir until the sauce thickens. Add grated cheese and spices. Slice potatoes and arrange in casserole dish. Pour sauce over potatoes. Sprinkle with paprika and bake for 10 minutes at 350°. Lemon Chicken: ½ lb. boneless chicken breasts 1/8 cup flour ¼ tsp. salt 1 tbsp. butter ½ tsp. lemon pepper seasoning ½ cup of asparagus 1 lemon Instructions: Cover the chicken breasts with plastic wrap and pound until each pieces is about a ¾ of an inch thick. Place the flour and salt in a shallow dish and gently toss each chicken breast in the dish to coat. Melt the butter in a large skillet over medium high heat; add the chicken and sauté for 3-5 minutes on each side, until golden brown, sprinkling each side with the lemon pepper directly in the pan. When the chicken is cooked through, transfer to a plate. Add the lemon slices and chopped asparagus to the pan. Make sure the lemon slices are on the bottom so that they caramelize and pick up the browned bits left in the pan from the chicken and butter. When the asparagus is done and the lemons are golden brown, add the chicken back to the pan and rearrange everything (lemons on top) so it looks nice for serving.

 

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.

 

 

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 Kelley Nguyen. Her topic, from Pre-Algebra: scientific notation.

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

 

First, I would introduce the topic with a comparison towards abbreviating. For example, when text messaging, one could type “Idk” instead of “I don’t know.” For scientific notation, we’re getting a number and abbreviating it using powers of 10.

 

My activity would be a matching game, where there will be a set of pictures and a set of numbers (in basic units). I would ask the students to match each picture with one of the given lengths, e.g. a tree would be 5 meters in height. The students will then guess on more difficult pictures, such as the earth’s width or the length of the Atlantic Ocean from one continent to another. As they start working with these bigger numbers, I will introduce scientific notation, where one can shorten very small or very large numbers with the powers of 10. When it comes to these large numbers, students seem to be scared or uninterested in writing such lengthy numbers.

 

Another fun activity is to give half of my students a number and the rest of my students the numbers in scientific notation. Then, I can then ask them to find their match as they roam around the room.

 

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

 

Scientific notation can definitely help in mathematics when working with very small or very large numbers. When writing numbers such as 3,200,000,000, you can shorten the solution with powers of 10. In this case, we can rewrite the solution to be 3.2 × 10⁹. This also goes for the length and width of very small or very large objects. For example, finding the length of a microchip in meters – that number would be entirely small, using a negative exponent of 10.

 

In science, scientific notation is especially important when dealing with mass, weight, etc. For example, when computing the mass of the sun in kilograms, one wouldn’t answer 1,989,100,000,000,000,000,000,000,000,000 kilograms. Instead, one will write 1.9891 × 10³⁰. With this shorthand notation, students can move on to problems more quickly rather than spending the time to write and count out every zero. As scientists, they will learn that abbreviation is very useful when collecting data or computing expressions.

 

 

What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

 

It has been debated on who discovered scientific notation, but there are many Greek mathematicians and scientists who contributed to the development of this notation. It was first brought about by Archimedes, who studied the Egyptian city of Alexandria. In one project, Archimedes used Greek letter numerals to calculate the number of grains of sand there were in the Archimedean universe. Of course, now, that’s quite impossible to do, but Archimedes did manage to compute that amount and resulted in a very large number. With that being said, that was the start to developing scientific notation and being able to notate very small or very large numbers as short expressions.

 

Other mathematicians and scientists that contributed to scientific notation include Galileo and Copernicus, who both played a big role in the world of science. Galileo used scientific notation when experimenting with the solar system. Copernicus had an idea of scientific notation when he was attempting to make a scaled model of the solar system.

 

References

 

• Pierce, Rod. (12 Feb 2014). “Scientific Notation”. Math Is Fun. Retrieved 2 Sep 2014 from http://www.mathsisfun.com/numbers/scientific-notation.html
• Mattson, Barbara. (1997-2014). “Imagine the Universe!” Imagine the Universe! Retrieved 2 Sep 2014 from http://imagine.gsfc.nasa.gov/docs/dictionary.html#S
• Smith, H. E. (25 Sep 2003). “Physics 11 – The Units of Science”. University of California, San Diego. Retrieved 2 Sep 2014 from http://casswww.ucsd.edu/archive/physics/ph11/lectures/units.html

 

 

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 Donna House. Her topic, from Pre-Algebra: adding a mixture of positive and negative numbers.

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

Version 1

It would be difficult to do well in any future course in mathematics or science without understanding the concept of adding and subtracting positive and negative numbers. This concept is used for temperature, altitude, growth and decay, magnitude, distance, size, profit and loss, and many other topics.

An example from physics would be the students solving a problem involving force. They need to discover how much drag force is needed to stop a drag racing car at the end of the track. The forward force (speed) of the car is positive and needs to be “cancelled out” or reduced to zero in order to stop the car. The students will need to determine if the regular brakes (negative) on the car are sufficient to stop the racer in time, or if additional drag forces (negative) need to be added.

Version 2 (Written as an Engage)

How much force is needed to stop a drag racing car? If you do not stop the car in time, it will crash into the wall, or the fence, or maybe even the water tower and then where would you declare your undying love to Betty Sue? If you stop the car too quickly, you will lose the race and Betty Sue won’t love you anymore. And all this happened because you did not know about adding and subtracting positive and negative numbers!

To stop that racing car you will need to know how much drag force is needed to cancel out the forward force (speed) of the car. Since the forward force is positive, the drag force is negative. But the regular brakes may not give enough drag force to stop the car in time. You may need to add some negative numbers! Remember, your entire future with Betty Sue depends on adding positive and negative numbers!

 

C1. How has this topic appeared in pop culture?

What if you thought you won the lottery, but found out you were wrong? In November of 2007 a scratch-off lottery card game in England was pulled from the shelves because customers did not understand the concept of negative numbers. Many people tried to claim their prizes only to be told they did not win. Why was there so much confusion? The cards involved negative numbers!

The “Cool Cash” scratch-offs had a cute picture of a penguin on the front. One scratched off windows trying to reveal a temperature that was lower than the temperature revealed on the card. All of the temperatures revealed were below zero and had negative signs. The problem was that many people could not understand whether -6 was larger or smaller than -8. What do you think?

http://www.manchestereveningnews.co.uk/news/greater-manchester-news/cool-cash-card-confusion-1009701

 

 

D3. How did people’s conception of this topic change over time?           

Negative numbers have struggled for recognition since ancient times. Negative numbers were ignored by mathematicians for centuries, and considered to be false, non-existent, or simply absurd. Whenever a solution was found to be negative, it was discarded as nonsense. They got no respect. Eventually, the concept of adding and subtracting negative and positive numbers was used to indicate debt and payment, but not much else. Mathematicians just did not quite understand what negative numbers are, even though the negatives cried out to be seen as real numbers.

As time passed, negative numbers began to be recognized as useful, but were generally considered imaginary. They were not accepted as real numbers until the middle of the 18^th century, but were still commonly ignored as solutions. The negatives protested peacefully. In the 19^th century, negative numbers were finally accepted, but still not widely liked. However, their usefulness caused them to be recognized and they happily indicated the weather, the distance below sea level, and whether or not a golfer’s score was below par.

Today negative numbers have a very good relationship with positive numbers, and are loved by many people. The usefulness of adding and subtracting positive numbers cannot be denied. (Just try to break a world record in racing without this concept!)

http://webspace.ship.edu/kgmcgi/m400/Presentations/Chapter 5 Something Less Than Nothing.ppt

http://en.wikipedia.org/wiki/Negative_number#History

 

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 Delaina Bazaldua. Her topic, from Pre-Algebra: solving two-step algebra problems.

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

I have a love for TED ED videos because of how the videos can explain math, science, etc. with real world examples which is often foreign to students. Bill Nye has always been a hero of mine growing up; his witty ways to communicate math and science to students is admirable. With that being said, when I found, http://ed.ted.com/on/vUO3lcyK#watch, I was really excited that Bill Nye and TED ED made a video that included a subject that was seemingly abstract to students and related it to something very common such as, in this case, cupcakes. Bill Nye takes the viewer on an errand he has to run to pick up cupcakes for his niece and nephew. Of course, since they’re siblings, they have to have an equal amount of cupcakes or World War III may happen. This creates balance between the equal sign. From there, he and we determine the amount of cupcakes in each box (the x) that he is giving to his niece and nephew.

 

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

 

Every child loves playing games and students in Pre Algebra are no exception to this assumption. In order to manipulate math into games, the resource I found used Bingo as a game to play with a high school class: http://makingmathfun.wikispaces.com/file/view/Two-Step+EQ+BINGO.pdf. I find this as an exceptional game for students to receive practice solving two-step algebraic equations because they may not necessarily realize they’re learning math in the process of playing even though they inevitably are. I am a strong believer in making something seemingly difficult much more fun so that it can be enjoyed by more people. If Bingo is fulfilling this dream, then I am doing my job because passion in math through a game for example leads to understanding of the material and to hard-working students. Playing games to teach algebra makes math seem like less of a chore and hassle, which unfortunately, it is often perceived as. If I can, as a teacher, change this perspective, I could have an effect on students’ lives for the rest of their education career and possibly even their life.

 

 

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

 

As I had previously mentioned, algebra is often viewed by students as abstract and unrelated to the real world. I felt like I needed to include word problems that translate to things that happen in life such as the TED ED/Bill Nye video example that portrayed two-step algebraic equations; math isn’t just simply numbers, but instead is applicable to everyday activities. I found a great PDF file, http://cdn.kutasoftware.com/Worksheets/PreAlg/Two-Step%20Word%20Problems.pdf, which includes 14 word problems that students are familiar with. Another great characteristic about word problems is you can receive a deeper understanding about what a student knows and doesn’t know based on what numbers they write from the word problem that forms their equation. Way too often teachers give students the numbers they need to work with instead of allowing the students to figure out the numbers on their own from a problem that they may actually encounter in life. This habit becomes a disadvantage and a hindrance to students which is why they feel that math is foreign to the world around them and become frustrated with “a pointless subject.” These two reasons make word problems extremely important and useful for students and I believe the worksheet I chose is perfect for accomplishing the goal of allowing students to learn with relevant scenarios.

 

References:

http://ed.ted.com/on/vUO3lcyK#watch

http://makingmathfun.wikispaces.com/file/view/Two-Step+EQ+BINGO.pdf

http://cdn.kutasoftware.com/Worksheets/PreAlg/Two-Step%20Word%20Problems.pdf