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 Jason Trejo. His topic, from Algebra: solving one-step algebra problems.

A2) How could you as a teacher create an activity or project that involves this topic?

How can I engage my students with solving for a variable? Off the top of my head, I came up with 3 tried and true surefire ways that would not only further my students understanding but also be a ton of fun for them: Algeblocks with accompanying interactive whiteboard, using a balance and counters, and possibly using snacks (e.g. cookies, chips, candies, etc.)

First things first, the Algeblocks:

Essentially, Algeblocks are made of a variety of cubes and rectangles that represent ones, tens hundreds, thousands, and even the variables x and x². Although obscured in the picture, the Algeblocks mat in the back represents a balance where the fulcrum is “=” and each end of the balance represent both sides of the equation. There is even a place that represents negative numbers! Using the problem “x+4=8”, students would have 8 green blocks to the left of the fulcrum and 4 green blocks with an x block. Students would then add or take away tiles to solve the equation. As for problems such as “4x=16”, the students would display the problem using the blocks and then group the green blocks with the x’s to find there answer. Now that I think of it, I would essentially do the same thing but use either a real balance with any type of manipulative.

 

B1) How can this topic be used in you students’ future courses in mathematics or science?

Being able to solve single step algebraic problems is a foundation to algebra in general, correct? This means that this will continue to pop up regardless of what math class (and even science classes like chemistry). There will always be problems given to students where they will need to solve for a variable and the final step of even the most excruciatingly, horrific looking algebra problems is usually adding, subtracting, multiplying, dividing, etc. to get the “x” all alone. In reality, solving an initial value problem (like I currently do in my Differential Equations class) boils down to one step algebraic solutions.

 

E1) How can technology (YouTube, Khan Academy [khanacademy.org], Vi Hart, Geometers Sketchpad, graphing calculators, etc.) be used to effectively engage students with this topic?

Interestingly enough, I have the perfect example that ties both Khan Academy and the “use of a balance” activity I mentioned earlier. A quick Google search for “one-step equations” gives a link to Khan Academy that allows for a digital balance and you are to solve the equation given with the balance. This would be an amazing tool for teachers to use when they don’t have actual balances for their class or even have their students create a profile on Khan Academy and use it to be able to track extra problems the students can do. Besides Khan Academy, there are even some cheesy yet fun games (like “Equations Pong” off the XP Math website) that would give the students more practice with these equations while feeling like a reward since they are playing a game. Plus, students can go head-to-head in “Equations Pong” and a vast majority of students like to best their friends in anything and everything.

 

References:

Information on Algeblocks: http://www.hand2mind.com/brands/algeblocks

Image of Algeblock Mats: https://cdn.hand2mind.com/productimages/76986_Algeblocks_Mats_BQS-web.jpg

Khan Academy use for subject: https://www.khanacademy.org/math/algebra/solving-linear-equations-and-inequalities/why-of-algebra/e/one_step_equation_intuition

Equations Pong Game: http://www.xpmath.com/forums/arcade.php?do=play&gameid=105

 

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 Emma Sivado. Her topic, from Algebra: rational and irrational numbers.

D.1: What interesting things can you say about the people who contributed to the discovery and/or the development of this topic?

The famous story on the first discovery of irrational numbers is one of violence. We all know the Pythagorean theorem, a²+b²=c² , but what happens if we have a right triangle with height 1 and base 1? The hypotenuse becomes √2. So, √2, what’s the big deal? Well this is where we turn to history for the answer. Hippassus was an ancient greek philosopher who belonged to the Pythagorean school of thought. Now the Pythagorean’s had a saying, “All is number.” What do we think this means? What Pythagoras meant was that everything in the universe had a numerical attribute. For example, one is the number of reason, five is the number of marriage. So one day when Hippassus was playing with the length of the diagonal of the unit square, or the hypotenuse of a right triangle with base 1 and height 1, he discovered the number √2. Hippassus tried to write √2 as a fraction, or rational number, and found it to be impossible. Therefore, √2 is what we call an irrational number. Well this is where the history turns violent. There are numerous stories to explain the death of Hippassus, but all of them point to his ultimate cause of death being the discovery of these irrational numbers. Irrational numbers were so against Pythagoras and the Pythagorean school of thought that they had this man killed!

https://brilliant.org/discussions/thread/discovery-of-irrational-numbers/

http://www.math.tamu.edu/~dallen/history/pythag/pythag.html

 

B.1: How can this topic be used in your students’ future courses in mathematics and science?

I believe that the irrational number would be a great place to introduce a simple proof. Students will have to do proofs in multiple math classes in the future and to give them an example with an interesting story might be a good place to start. For example, after telling the story of the discovery of irrational numbers ask the students how Hippassus might have proven that this was true; possibly his dying words. Then give them an outline or fill in the black of the proof that √2 is irrational. This example I found on homeschoolmath.net is given in good language and gives good explanations of why everything is done in the order it is:

Let’s suppose √2 is a rational number. Then we can write it √2  = a/b where a, b are whole numbers, b not zero.

We additionally assume that this a/b is simplified to lowest terms, since that can obviously be done with any fraction.

From the equality √2  = a/b it follows that 2 = a²/b²,  or  a² = 2 · b².  So the square of a is an even number since it is two times something.

From this we know that a itself is also an even number. Why? Because it can’t be odd; if a itself was odd, then a · a would be odd too. Odd number times odd number is always odd.

Okay, if a itself is an even number, then a is 2 times some other whole number. In symbols, a = 2k where k is this other number. We don’t need to know what k is; it won’t matter. Soon comes the contradiction.

If we substitute a = 2k into the original equation 2 = a²/b², this is what we get:

2	=	(2k)2/b2
2	=	4k2/b2
2*b2	=	4k2
b2	=	2k2

This means that b² is even, from which follows again that b itself is even. And that is a contradiction!!!

WHY is that a contradiction? Because we started the whole process assuming that a/b was simplified to lowest terms, and now it turns out that a and b both would be even. We ended at a contradiction; thus our original assumption (that √2 is rational) is not correct. Therefore √2 is rational.

 

Obviously this would have to be presented slowly, but I believe that the students could do this and understand it.

http://www.homeschoolmath.net/teaching/proof_square_root_2_irrational.php

 

 

I would begin by showing the movie clip from Life of Pi when Pi is reciting all the digits of Pi that he knows, or another video of someone reciting a ridiculous number of digits of pi. Then I would ask the students how many digits of Pi there are? When no one could tell me an exact answer I would introduce the irrational number and explain how the decimals will go on forever because this number cannot be written as a fraction like a rational number. At the end of class you could show the kids the Princeton University Pi Day celebration complete with Einstein look alike contests, and pi reciting competitions to win $314.15!

http://www.pidayprinceton.com/

 

References:

 

 

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 Diana A’Lyssa Rodriguez. Her topic, from Algebra: adding and subtracting a mixture of positive and negative integers.

A2. How could you as a teacher create an activity or project that involves your topic?

Algebra tiles are a fun, hands-on way to help students understand how to add or subtract positive and negative integers. Using a mat with a positive and negative side, students can manipulate the 1-tiles. Using the yellow side of the tile for the positive numbers and the red side for the negative numbers, students pair together opposing colors and take those away. The tiles leftover is the answer to the problem. Here is an example:

Step 1:

Step 2:

Step 3:

B1. How can this topic be used in your students’ future courses in mathematics or science?

Adding and subtracting positive and negative integers is a one of the most crucial foundation skills that students must learn. This concept is demonstrated and needed in almost every math scenario. In its simplest form, students begin to learn this concept around the first grade, 1+1=2. This process is carried over into third grade with multiplication. Then negative numbers are introduced while in sixth grade. Adding and subtracting opposing integers is a continuous concept that consistently builds upon itself, even through algebra, geometry, calculus, or most especially the real world. There is not just one future math course students will use this in; they will use it for the rest of their lives, even if they do not realize it.

 

E1. 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 is always a great resource when trying to engage students. The video below explains how positive and negative numbers work when adding and subtracting them.

A lot of the time students struggle with numbers in general, which makes it harder for them to understand why a concept in math works. This video explains how positive and negative numbers work in relation to each other by using characters from Batman instead of numbers. Using the balance and watching the arrow move in either direction, depending on the type of character that was added into or taken out, allows students to see why positive and negative numbers work the way they do. Once they understand this, it makes working with numbers a whole lot easier. This video also does a wonderful job of maintaining the students’ interest by keeping it related to popular culture by incorporating Batman and the Matrix.