Category: Algebra

Bob’s Favorite Things

This year, as I have moved from a role as classroom teacher to a math coach, I find myself sharing some of favorite lessons and approaches with my colleagues. Taking a cue from Oprah, this is the first of a series of my “Favorite Things”, hopefully without the rampant screaming or random things hidden under your seat.

The Monty Hall Problem on NLVM

The famous “stick or switch” problem is one which will generate tremendous discuss in a classroom. The premise is simple: you are offered the choice of one of three doors, behind one of which there is a great prize. After the host reveals a non-winning door, you are offered the chance to switch your guess. Should you stick or switch? The problem rose to a next level of fame with fierce debate in the “Ask Marilyn” article by Marilyn Vos Savant in Parade Magazine. After having included the problem in my prob/stat course for any years, the discussions rose to a crest as the problem was featured in the movie “21”, with even students I did not know stopping me in the hallways to ask me about its logic (note, this video has a number of ads on it):

As an emerging stats teacher, I would play the game using cardboard doors and hand-drawn pictures of cars and goats. In later years, I used an applet at the National Library of Virtual Manipulatives to play the game with my class. Look under Data Analysis and Probability for “Stick or Switch”. What I really like about this applet is that it will play the game 100 times quickly and display the results.

There are many ways to explain this game to your students. One straight-forward approach is to have students attempt to write out the sample space (all possible outcomes) for this game. For example:

Car behind 1. Pick 1. Shown 2. Stay. WIN

Car behind 1. Pick 1. Shown 2. Switch. LOSE

Car behind 1. Pick 1. Shown 3. Stay. WIN

Car behind 1. Pick 1. Shown 3. Switch. LOSE

Patterns in Pascal’s Triangle

In one of my favorite lessons each year, the many, many patterns in Pascal’s Triangle were discussed. This would come after an investigation of combinations, and students would often recognize that each entry in the triangle is actually a combination, along with other patterns like the counting numbers or perhaps the triangular numbers. But how about the Fibonacci sequence and the “square” numbers? What’s the “hockey stick” theorem?

Over the years, I have used a number of sites to display the triangle and invite students to share their ideas. One nice resource which explains patterns in the triangle can be found here.

The cherry on the Pascal sundae would come when I would invite students to approach the board and circle all of the even numbers. Often, only 15 rows or so would be visible, so the chance to make a generalization is rich and there to discuss. How would this look if we colored 100 rows, 200 rows, 10,000 rows? And do we only get patterns for even numbers? How about multiples of 3, or 5, or 24? Check out this site from Jill Britton, which includes an applet that will color multiples in Pascal’s Triangle. Watch the expressions from your students when you reveal multiples of 5 in the first 128 rows of the triangle:

Try different multiples and observe the great symmetries. Prime numbers like 17 and 23 provide the most surprising results. Digging deeper, start a discussion of the modulus function with your class (it’s just the remainder, no big deal….). What happens when we not only color multiples, but also color the remainders similarly. From the Centre for Experimental and Constructive Mathematics, we get this great applet, which will color remainders based on specifications you provide:

Electric Teaching

While I have been using the Monty Hall problem and Pascal’s Triangle in my classrooms for many years, this last resource is fairly new to me. Electric Teaching by David Johns is an excellent site with an effective interface which is ideal for SMART Boards. The site challenges students to match-up equations with data tables and their graphs.

The site has problems for linear, quadratic and trigonometric functions, along with conics and even problems which include derivatives. What a great way to not only have students participate, but verbalize their thoughts as they identify the matches. The YouTube channel has additional resources from Dave, including a tutorial of the Electric Teaching site.

Algebra Technology

TI Publish View – Bringing Interactive Lessons Home

Post author By Bob Lochel
Post date March 19, 2012
No Comments on TI Publish View – Bringing Interactive Lessons Home

In the past few years, Texas Instruments has been aggressive in developing and marketing its Nspire product line. I recall the first time I shared the (now) old blue click pad product, and the oohs and aahs from my students when I showed them how you could trick out the keyboard with an 84 keypad. This was soon followed by the touchpad, and now we have the CX, with its thin design and color screen.

Along with the improved hardware, TI has also improved its software options, providing an opportunity for teachers to create their own lessons and demonstrations on the software. Files can be easily traded and shared with students, or used on a whiteboard as a classroom manipulative. Last month, I had the opportunity to attend a free morning of professional development on the TI Publish View feature. This feature of the Nspire software allows teachers to embed some of the interactive features of Nspire CX files into documents. The TI-Nspire document player then allows students to open these files and navigate the lesson.

In the short example I created below, the coefficients of a polynomial can be adjust using “elevator buttons”, which are sliders used to change the values. Students can then observe the value of the discriminant and look for patterns in the values. Click the link to join in the discriminant insanity!

http://education.ti.com/go/nspireplayer?nspirefile=http://dl.dropbox.com/u/68005919/Discriminant.tnsp

Additional files to try can be found at the TI Activity Exchange. What an interesting way to have students explore on their own. Thanks to Mike Darden from TI for the great session, and Doyt Jones for his continued hard work in bringing these sessions to the Philly area.

Tags algebra, technology

Algebra

Slope and the ADA

The middle school in the district where I work is quite old. Dedicated in 1959, and once serving as the district high school, the building is a Frankenstein of aging classrooms, newer additions, and inconsistent heat. One feature of the building is the network of ramps used to shuttle students from wing to wing, and supplies in and out.

Recently, I worked with a team of 8th grade algebra I teachers to develop an activity which would utilize the many ramps, get kids moving and measuring, and reinforce slope as a measure of steepness. The teachers had great ideas for leading students through measurement activitites. My initial idea of having students choose points along the ramp, then measuring the rise and run between points, was discussed and improved. The teachers used blue painter’s tape to create guiding triangles along the bricks on the walls along two of the ramps. Another teacher noted that railings could be used to connect parallel lines to slopes, and triangles utilizing the railing were also provided.

Students measured the slope of two ramps using the provided triangles, then were led outside, where both a pedestrian ramp and a custodian’s ramp were measured. The outside ramps were additional challenges, as no guiding tape marks were provided. Wacthing student reactions and approaches to these ramps was intriguing. Some students attempted to use the bricks on the building to trace their own triangles, while another group discovered that the level ground along the freight ramp could be used as the “run”.

After the activity, the class discussed and compared their results. In one class, the unusual steepness of one ramp in our building was questioned, and related to the legal limits of handicapped ramps. The class agreed that the ramp seems to be an original part of the building, and that an elevator had been installed alongside the ramp for our disabled friends. Further discussion could include the requirements of the Americans with Disabilities Act, which contains the following requirements for ramps:

The least possible slope shall be used for any ramp. The maximum slope of a ramp in new construction shall be 1:12. The maximum rise for any run shall be 30 in (760 mm) . Curb ramps and ramps to be constructed on existing sites or in existing buildings or facilities may have slopes and rises as allowed in 4.1.6(3)(a) if space limitations prohibit the use of a 1:12 slope or less.

As a follow-up, students found pictures of objects or places which they felt represented interesting slopes. Geometer’s Sketchpad was then used to measure and compare the slopes in their pictures: