Saturday, September 29, 2012

Math Using Speedometers

Speed has vast appeal to our students. It embodies risk, excitement, prowess and fun. Speedometers, a basic speed meter that displays instantaneous speed, also offers many applications for educators to teach math. Here are just a few…

Basic Addition & Subtraction while reading meters:
The speed limit for highway 95 is                         Your vehicle is traveling at

What is the difference between the two speeds?
Integer Addition:

Using the same question above, state the difference between the two speeds as a positive integer if speeding and a negative integer if below the speed limit (or vice versa if you would like)
Percentage Above or Below:

--Using the difference in the posted speed and the actual speed, calculate the percentage of difference.
--Using the percentage difference and the posted speed, calculate the actual speed
Elementary Hypothesis Testing:
Based on the error rates of speedometers, what is the likelihood that the car traveling 1 mph over the speed limit is actually speeding?

Sunday, September 23, 2012

Parallel Zombie Tag

The game is simple and fun. It could be used for an activity day and a good way to engage the students’ minds during recess. I will explain the basic way and then add some optional twists

Parallel Zombie Tag.

Zombies have a signature walk in which their arms are extended in a parallel fashion, like so

The game involves dedicating one individual the parallel zombie while the rest are pedestrians. When the pedestrian gets tagged, they two become zombies. Now, the only defense a pedestrian has from the ‘Parallel Zombie’ is their ‘Perpendicular Cross’, in which they hold their fingers like so,

The catch is that while holding up their ‘Perpendicular Cross’, pedestrians cannot move.

*************Added Twists*****************

Given that a pedestrian could hold up their 'perpendicular cross' forever and never get tagged you could set a time limit of 10 seconds. This would ensure that everyone will eventually become a parallel zombie. If you are dealing with more advanced students you could make them count to 30 by 3's or 20 by 2's, etc.

************Additional Notes About The Game*************

I think this game would serve as a great transition into exponential growth or exponents. For example, you could time how long it takes everyone to become a parallel zombie when starting with 1 zombie, 2 zombies, 5 zombies, etc and then review the times with your students. The time involved should exponentially decrease.  

Tuesday, September 18, 2012

Proofs/Formulas With A Memo Pad And Some Detective Work (all grades) Post 1 of 3

Proofs/Formulas With A Memo Pad And Some Detective Work (all grades) Post 1 of 3

How much do we value information that is already packaged and ready for us? Obviously it depends but when it comes to formulas and proofs I believe your students will value their work much more if they value the formula/proof. One way to implement this is to take the formula away from them and make them derive it themselves. Lets start with something basic and then latter we can move on to something more challenging.

Since we have readers from all over the spectrum of mathematics, I would like to provide three examples of deductive reasoning for elementary, middle and high schools stretched out over three posts


The Right Triangle and the Rectangle:

The goal of this exercise is for your students to informally prove [to you] the area formula of a right triangle is true

What Each Student Will Need

  1. Two pieces (two colors) of construction paper
  2. Scissors
  3. Ruler
  4. Tape or glue
  5. Marker
  6. Memo Pad (can be made from stapling scrap paper or using a post-it pad)

The Right Triangle and the Rectangle:

  1. Write At Top Of The Memo Pad “What Mr. [fill in the blank] Accepts To be True”
  2. Write the formula for area of a triangle on the board
  1. Ask the students to read the formula out loud
  2. Tell them you don't believe them. Ask them to prove it!
  3. For the first 5 minutes allow them to play, allow them to experiment ways to prove the formula is true. Ask them to share their ideas with each other and you. Applaud each idea but relay to them that it's still not enough proof. Tell the class that just like a detective doesn't always believe the witness, we will need to build up a case by starting with what we do believe.
  4. Using the materials above, ask each student to draw a 4 by 6 inch rectangle on both pieces of construction paper
  5. Ask them to calculate the area of the rectangle by multiplying its length by its width (4x6 = 24inches squared).

    (Tell them that you accept that the area of a rectangle is the length multiplied by its width but you don't accept the area of a triangle is ½ the base times its height. We are going to add what it is you accept to be true in their memo pad)

  1. Add The Following To Your Memo Pad

                1. The area of a rectangle is its length multiplied by its width(L x W)

  1. Next, ask them to label the width and the length of each rectangle
  2. Now propose the question of what the area would be if you cut the rectangle in half (The answer should be 12 inches squared)
  3. Ask them to prove it by cutting one of the rectangles in half either horizontally or vertically through the middle of the rectangle as shown below and calculating the new area (will be either 6x2 or 4x3 = 12)

  1. Tell them you now believe them, that a rectangle cut in half will yield half the area and add this to your set of Axioms

        1. A rectangle cut exactly in half will yield two pieces, both with half the original area
  1. Now pose the question, is it possible to get a triangle from half a rectangle? Is it possible to get two triangles of equal size from exactly half a rectangle?
  2. Ask them to prove it.
  3. Ask them to cut the other rectangle (on the other piece of construction paper) in half as well but using the following criteria

      1. The rectangle must be cut in half
      2. Each piece must be the same shape and size
      3. The resulting shapes must be two triangles

  1. The students should quickly figure out that if they cut a rectangle in half diagonally they meet each of the criteria above.
  2. Add this to your set of axioms

          1. It is possible to get two triangles of equal size from half a rectangle

Now comes the hard part! Putting it all together and making the students see the solution. Let me show you how I might play-out this scenario if it were my classroom.

Me: “OK!, I get it! guys were right. You've proved the area for a right triangle is 1/2(b)(h). Let's move on to something else”

Class: “Wait, what?...What are you talking about? Show us”

Me: “Well, we agree that

      1. Area of a rectangle is Length x Width right?
(Draw a 4 x 6 rectangle on the board and label the area 24 inches squared)
Class: “yes”

Me: “We've also agreed that
        II.A rectangle cut in half will yield two pieces, both with half the original area.
(Cut the rectangle in half diagonally and label each side 12 inches squared)

Class: “yes”

Me: “And finally, we agree that

          1. It is possible to get two triangles of equal size from half a rectangle.

Me: “Well, a Right triangle is nothing more than rectangle cut in half diagonally--There's the ½ part of the equation. And the base and height are simply new names for the length and width. Thus, it makes sense that ½(b)(h) is the area of a right triangle!

Monday, September 10, 2012

Answering via Informal Proof

There is a lot of adversity toward calculating instruments these days. Although there may be some merit to the accusation, I tend to side with the position that calculators have a role in the classroom and when used properly can allow educators to concentrate on the bigger parts of a math problem. Although this is typically not the place for such a discussion, for those interested in my opinion see my writings here

This posting is about limiting the usage of a calculator to a few buttons. The purpose of the exercise is to show the relationships of addition/subtraction or multiplication/division or even exponents/radicals. We do this by asking our students to solve a problem in one format but show its proof via another format.


Q: What is the product of 5 and 6?

A: 30
Our answer is of course 30. However, instead of accepting this as out answer let's as for an informal proof. Something like this.


Q: What is the product of 5 and 6?

A: 30; because the quotient of 30 and 6 is 5.

How to use this in the classroom:

Create 5 problems you want your students to be able to solve.

1) 8 - 3
2) 6 x 2
3) 8 / 2
4) etc
5) etc

Create a column for their answer and a column for their informal proof. Base their grade on the informal proof.

Problem Answer Informal Proof
1) 8 – 3      ____5___            ___Because 5 + 3 is 8___
2) 6 x 2      _______              ____________________
3) 8 / 2      _______               ____________________
4) etc         _______                 ____________________
5) etc         _______                 ____________________

Wednesday, September 5, 2012

Incorporating Math With Fantasy Football

Fantasy Football is an extremely popular and fun past-time for football enthusiasts. Especially for those, like me, who were never that great at the sport. This post gives you some ideas of ways to connect it with math, however, this is not something you can throw together in the last moment. It will take proper planning and time, but, in the end, your student may be having the most fun they've ever had in a math class.

The game could be built on any theme you would like, combining like terms, simplifying fractions, solving equations, however, the theme of this particular post is factoring.

The goal of the project is to hold a live draft with your classroom in which the students who complete the hardest math problems will receive the tops pics in the draft. I will provide some advice for how to implement the program toward the end of the post, although, I would love any feedback.

Note 1: It helps to understand the metrics of fantasy football. If you have never experienced fantasy football but are interested in using this exercise, it may be helpful to review the game. Here is a link How to Play Fantasy Football. The classroom activity may also run much smother if you first practice holding a draft with friends or relatives.

Note 2: It will be much easier if you split the class up in 2 teams or, at most, 4 teams. Any more teams than this would pose difficulty in terms of management and time. As it stands, the typical fantasy football team requires 15 players (including 9 starters and 6 backups). Thus, a classroom of 24 students divided into groups of 4 would yield 6 teams. 6 Teams of 15 players each would require 90 Problems. This would be a hard feat to accomplish within a one hour class. I will give strategies for ways around this below. However, 24 students divided into two teams would only yield 30 problems.

Items You Need:
Multiple Large Posters
Cut-outs or printouts of the top fantasy football players
A draft sheet for each student
Access to the top 100 fantasy football pics for each team.
Postcards with problems on the front and answers on the back

How to set it up:
(Honestly I can think of multiple ways to do this of which I will choose the easiest one and, in time add others)

  1. Print off the top 50 fantasy football players (You can find them here)
  2. Proceed to divide the top 50 players into 4 parts (note that two of the parts will need to have 13 players)
  3. Since there will be no kickers or defenses in the top 50 players (as these picks come later in the draft, proceed to subtract four of the players and add two defenses and two kickers).
  4. You should now have 50 players divided into four uneven groups (2 groups of 12, 2 groups of 13)
  5. Next, create (or borrow) 50 problems that you would like your students to be able to solve.
  6. Place the hardest 12 problems in the group with the top 12 picks, the next 12 hardest problems in the second group and so forth.
  7. Using multiple poster boards, create positions for each of the 50 players the students will be drafting from (this could simply be blank squares on a poster board with the players name and position beside it)
  8. Optional, cut out or print out pictures of each player to paste on the poster board by their name and position.
  9. Using index cards write each of the 50 math problems on the front with their answers on the back.
  10. Either keep the postcards in four stacks or tape each postcard in the correct position under the players name

How To Play:
  1. If using two teams, have students from each team huddle up in a circle to figure our who they wish to draft first.
  2. Flip a coin for who drafts first
  3. Have students pick the player they wish to draft. Once chosen, have them select two teammates from their group to go to the back of the room and solve the problem.
  4. Note: I'm not sure if I would allow the students to see the problems prior to picking the player. What you might run into is your smarter students will be solving problems while the other team is drafting. Even still, your team might solve it correctly, but the other team may pick that particular player and thus they did a lot of work for nothing. What's your thoughts?
  5. Using some type of stop watch or countdown, set the time to 2-4 mins based on the toughness of the problem. While your students are on the clock, have the opposite team who is waiting try to figure out the problem as well.
  6. If the student answer it correctly within the allotted time, they get the pick. If they do not, than proceed to ask the opposite team what the correct answer is.
  7. If they answer correctly: they may keep the player (if they so choose to) as well as pick any other player on the board for the opposite team.
  8. If they answer incorrectly, nothing happens and they go back to the drafting.

Note one part 7: Notice that if the drafting team answers incorrectly, the other team has a choice: They may either answer the question correctly to get the player as well as choose a player for the opposing team OR they may pass on the question and proceed to ask for a new question todraft the player they wish.

  1. Students will continue the draft until all positions on their roster is full.
  2. Here are the 15 players and their positions they must choose from.

Draft Sheet Each For Each Student

QB ________________________
RB ________________________
RB ________________________
WR ________________________
WR ________________________
WR ________________________
TE ________________________
K ________________________
Defense ________________________

Any position ________________________
Any position ________________________
Any position ________________________
Any position ________________________
Any position ________________________
Any position ________________________

**I'm anxious to hear any feedback you guys have on this**