Tag Archives: algebra

The Binomial Theorem Jigsaw

Is there a rule in math which encapsulates more great stuff than the binomial theorem?  Increasing powers of x, decreasing powers of y, a bunch of terms to look at…and hey, what’s this….combinations, you say?  I’m in.  But please don’t remove all the fun discovery moments there are to be had here.  Here’s a jigsaw activity which was handed down by some of my “senior” and now-retired colleagues, and is now infused with my need to get kids moving around and discovering things.  I hope you enjoy it.

The students will need some background on a few things here.  In particular, students should already be fluent with multiplying binomials, and have some familiarity with combinations.  Copy the first 4 pages of the binomial theorem Jigsaw Activity and have them ready to go.  Now it’s time for some movement.

Arrange classroom desks into groups of 4.  If your class roster number is not a multiple of 4, then you can have some 5’s, and we will deal with them soon.

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In each group, have the students number themselves from 1 to 4.  It is important that each group have one of each number, and that students know their number. If you have any groups of 5, then allow for two “1’s” in a group.  This will be helpful, as #1’s actually have the trickiest job in this task (but don’t tell them that yet!).

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Next, hand out the packet of tasks.  The packet looks intimidating, but each student is only responsible for completing one page.  All students who are “1’s” are responsible for completing Task 1.  2’s will complete Task 2.  3’s = Task 3.  And 4’s have Task 4.  On your command, students will break from the group, and move to a new location and meet with all oftheir  similarly-numbered classmates to complete the task together.  After all students have completed their task, they will re-join their group and share their findings.  It is important that students understand the need to complete their task thoroughly and thoughtfully, as their group success depends upon it!

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Now, send all the numbered students to meet with their group and complete their task:

SUMMARY OF THE TASKS:

  • In Task 1, students are asked to expand (x+y)^4, starting with (x+y)^2 and working their way up.  This task usually takes the longest for groups to complete, and requires attention to detail.  In the end, groups are asked to list the co-efficients of the terms they get.
  • In Task 2, students are asked to list the sample space for 4 tossed coins.  The task is not difficult, but requires attention to detail in making sure all possibilities are provided.  Groups are then asked to list number of ways to get 0 heads, 1 head, 2 heads, etc., in the 4 coins.  Note, groups often give probabilities here….we want the COUNTS!
  • In Task 3, students complete a number of rows in Pascal’s Triangle.  The task in not tricky, but requires attention to detail in filling out the small boxes.  The group is then asked to list the numbers in row 4 of the triangle.
  • In Task 4, students are asked to compute a number of combinations by formula.  I usually try to have 4-function calculators at the ready for this group, and want them to really focus on the formulas.  In the end, the group is asked to list the combinations of 4 items taken 0, 1, 2 3 and 4 at a time.

The tasks often take about 20-30 minutes for groups to complete and check.  Bringing groups back together and having them share their findings with their teammmates often takes another 20-30 minutes, so this may need to be done over 2 days.

binomial3The big reveal occurs as groups begin to realize that all 4 tasks have the same “answers”:

1-4-6-4-1

We’re now ready to start exploring the amazing connections between 4 ideas (binomials, coins, Pascal’s Triangle, combinations) which seemed quite different math-wise, but have some strong connections.  Now the ideas come fast and furious as we explore the connections.  Page 5 of the packet provides some guiding questions, but don’t feel tied to the linearity of my questions.

Let’s think about (x+y)^5…..

  • How many terms will it have? 6
  • How will the powers of x and y behave?
  • How can we find the coefficients?  Pascal’s Traingle!
  • How else can we find the coefficients?  Combinations!

How about coin flipping:

  • If we flipped 6 coins, how many items would be in the sample space?  2^6
  • How many of these possibilities will have all heads?  1
  • How can we find out the number of ways to get exactly 2 heads?  Pascal’s Triangle

Eventually, this will become the basis for “assembling” the Binomial Theorem.  You’ll be surprised at how much of it your students will be able to piece together after this activity.  Your job is to just help out with some symbols and some common language.

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“Wow” Moments with Wolfram|Alpha

The Siemens STEM Academy offers great resources for teachers, from lesson plans, to blog posts from teachers, to fantastic free webinars.  Full disclosure: I have written for the STEM Academy blog, and been a part of the Academy summer program…but I am but a small fish in a cool ocean of resources!

This week, the Academy hosted a free webinar featuring a demonstration of the dynamic knowledge provided by Wolfram|Alpha.  Having used Wolfram Demonstrations before in my classroom, I was looking forward to learning more about this search tool.  Crystal Fantry provided an hour-long overview of this exciting resource, and ideas for classroom uses.  It’s amazing how many “wow” moments I have these days with the new tech tools our students can have in their hands, but this one goes beyond that.  Knowing that students have access to resources like this should cause us all to think about our roles as math teachers / facilitators….this is a game-changer!

So, just what is Wolfram|Alpha?  The site is simple, just enter what you want to search for, and off you go…but this tool is so much more than that.  The “about” from their website provides some insight:

Wolfram|Alpha introduces a fundamentally new way to get knowledge and answers—not by searching the web, but by doing dynamic computations based on a vast collection of built-in data, algorithms, and methods.

So, what the heck does that mean exactly?  Let’s learn by diving in.  And while you can use Wolfram|Alpha for far more than math, this is a math blog so let’s focus in on some math….

Try this: “y=2x +3”.  Let’s start with something simple…what does Wolfram|Alpha give us?

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Fun stuff.  A nice graph, the domain, and alternate form.

How about this: “3x+5=2x-9”

Also nice, a plot of the the functions.  And the equations’s solution..but what’s this…a “step-by-step solution”?  If you are logged in (free accounts) you can step through the solution:

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So, what happens now when you give that worksheet of equations to solve for homework?

There are a lot of other neat computations to explore, try some of these as starters:

  • “y=(x+2)(x-3)”
  • “inverse y=x^2+3x+1”
  • “sin(x)+cos(x)=1”
  • “Integrate x^2 dx from 0 to 5”

WA3But Wolfram|Alpha goes beyond quick lists and computation.  How about “Pascal’s Triangle mod 5”. Or “triangle sides 3, 6, 8”, or try the elusive 17-gon, and see the many facts to check out.

A TOOL FOR RESEARCH AND GENERALIZATION

I have only scratached the surface of the many features, and there are also lots of nooks, crannies and links for you to explore.  I’m eager to use this tool with students as a means to research new ideas, and make some sense of their characteristics.  For example, let’s think about domain and range, as I ranted about in a previous post.  I like that Wolfram|Alpha expresses domains using set notation, and this is a great opportunity to have students research new functions.  Most of what we do in Algebra 1 deals with linear functions, so we get a lot of “all real numbers” domains.  Expose your students to non-linear functions, once they know how to make their x,y tables.  Try these:

  • y = 5 / x
  • y = rt (x-2)
  • y = 1 / (x^2 – 9 )
  • y = 2^x
  • y = x^2 – 4

And what to do with these new functions?  Let’s place them into categories, share our findings, and communicate our ideas.  Give each group 2 or 3 new functions to look at and share their findings on www.padlet.com.  This site, formerly called WallWisher, allows everyone to contribute their ideasd and move them around the canvas.  Here’s a sample of my function domain wall, click the link to contribute your own, play around the wall, and double-click in any empty space on the canvas to contribute.  Or sign up for a free account and create your own wall.

Padlet

Thanks to Kyle Schutt (@ktschutt) and the gang at Discovery Education for providing these great webinars.  Be sure to check out the Siemen’s STEM Academy blog for more great resources, blog posts, and archived webinars.

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.

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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.

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.
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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:
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Who Takes 5 Hours to Mow a Lawn?

Some units and chapters in algebra lend themselves naturally to interesting openers. Interesting scenarios to discuss slopes, systems of equations or quadratic functions are abundant. Finding examples for topics like radicals and complex numbers or rational expressions can be a bit more of a challenge. Addition and subtraction of rational expressions mean that shared work problems can’t be far behind, like this nugget from algebra.com:

One good use for rational equations is the shared work problem. This solution would be of great help in scheduling employees. For example, If Bob can mow a lawn in 3 hours and Joe can do it in 5 hours, how long would it take them together?

A few thoughts come to mind:

  • I’m doubting that the personnel schedulers at WalMart or Jiffy Lube are using rational expressions to schedule their employees.
  • How many of our kids would guess 8 hours, or even 4 hours, as their initial guess?
  • Joe needs to stop lollygagging on the job.

I set out to make a video to encourage discussion of these problems. In a first attempt, my sister and nephew were recruited to each build a Lego tower separately, then together.
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Working together had little effect on the overall time, as the partnership tripped over each other digging into the bucket for Legos, and had trouble coordinating the overall tower construction. This leads to a nice discussion of the assumed independence of the two volunteers in these problems, but made for a pretty bad video.

In the video below, teachers Christine and John were recruited to staple index cards to a stack of 50 “top secret” papers. A shared work ending was also produced. But in a version that was later eliminated, Christine passed papers to John, who then stapled. In order to maintain independence, a new ending was shot where they worked separately, yet simultaneously.

Christine’s final time was 4:50, while John’s final time was 4:29
To find the ideal shared time, we let x = the number of seconds required to complete the job together.

  • Christine’s rate is 1 / 290 of the job completed per second
  • John’s rate is 1 / 269 of the job completed per second

Since we want one job to be completed, this leads to the equation:

Solving for x yields an ideal solution of 2:19, so the partnership’s time of 2:10 is not too surprising.  The subjects admitted that they were a bit more competitive to do well working together than when they were separated.  Also, my quick appearance during the shared portion on the video is due to the team needing more index cards, and not any funny business!  What would happen if Christine showed up a minute late?  How long would it take them to complete 2000 cards?

Hopefully, we can encourage some discussion and debate, and move away from Joe and his 5-hour lawns.

Factoring – Sending Out the Bat Signal!

One of the joys of my job is having mathematically interesting chats with my colleagues about how they approach  specific problems with their classes.  These conversations often begin as one-on-one discussions, but usually evolve into calling multiple people into the fray to give their two cents.  This semester, a teacher in my department is tackling an Accelerated Algebra II class for the first time.  Having taught academically talented kids for many years, my advice to him was to constantly challenge his students, perhaps using problems like those from the American Mathematics Competitions as openers.  But while offering up academic challenges can keep a teacher’s mind sharp, there is the risk of having that “hmmmm…” moment…..that uncomfortable feeling where you’re not quite sure what the correct response to a student question is.

The discussion today came from a review of factoring, and a problem which seems innocent enough:

Factor x6– 64

Take a moment and think about how you would factor this….show all work for full credit…

Enjoy a few lines of free space as you consider your work….

And…time….pencils down….

The interesting aspect of x6– 64 is that it is both a difference of cubes and a difference of squares.  I used the neat algebraic interface on purplemath.com to do some screen captures and make the algebra look pretty here.  In this case, the calculator factors this expression as a difference of squares, (x3– 8 ) (x3 + 8),  which then become both a sum and difference of cubes and can both factor further:

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But, the initial expression is also a difference of cubes, and can be factored as such.  It is verified below:

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The plot thickens as the discussion then centers about the “remnants” we get when we factor a difference of cubes.  We can verify that the two “remnants” (underlined in red) from the first factorization are factors of the remnant of the second method (underlined in green):

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So, what’s happening here?

The extra, messy, factor we get when we factor a sum or difference of cubes is up for discussion here.
According to Purplemath:

The quadratic part of each cube formula does not factor, so don’t attempt it.

But we don’t have a quadratic here (though we could perform a quick substitution and consider it is one), we have a 4th degree polynomial.  Even the algebra calculator on the site doesn’t care for this quirky 4th power expression:

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So, I am looking to my math peeps for some thoughts:

  1. Is there an order to consider when a polynomial meets 2 special cases?  Should we look at sum of cubes or squares first?
  2. Does anyone have any insight on x4+4x2 + 16?

Good night, and good factoring…

Tapping Into the Addiction of Bubble Wrap

Engaging students in discussion of mathematics in applied situations is a rewarding experience. Seeing students immerse themselves in a task and offering to share their results makes a math class hum with excitement. But finding the right scenario, the right “hook” which will drive discussion can be an effort. While we hope to link math to real-life science and engineering, sometimes the silliest data collection experiments create a buzz in class.

I give you the Bubble Wrap Challenge.

The past week, I worked with a 7th grade teacher on a slope activity. By the end of the unit, students would be expected to compute the slope of a line via a formula. In my experience, students tend to understand indivdual aspects of slope,as they are often taught in pieces, but have difficulty shifting between meanings. What do we want students to understand about slope?

Slope between data points can be computed using a formula
Slope can indicate steepness
Slope can indicate a rate of change

As students entered the class, and the teacher took attendance, I was playing on a SMART Board with a Virtual Bubble Wrap Applet. You’ll need Java for it.   I challenge you to play with it for less than 10 minutes, and without calling 3 friends over to play.  Can’t be done.  Try Manic Mode for the extra-special dose of stress relief.

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After some initial playing, we sought to find the Inter-Galactic Bubble Wrap Champion of period 2.  Each group was given an Ipad loaded with a similar app and 60 seconds to play the game.  While a student played, a partner wrote down the player’s score every 10 seconds.  Results were then plotted and a connected line-graph made.

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Discussion then centered around finding not only the overall bubble-popping rate, but debating the 10-seconds intervals when Aiden was the most, and least successful, at bubble-popping. A second contestant was then added to the mix…

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Which player was the fastest popper? Who was the best in a short period? Students were soon able to compute rates for segments, without prior knowledge of the slope formula. The teacher later introduced the formal formula. The payoff comes when students volunteer that we can identify the “best” popping rates by looking for steep segments, and lower popping rates in shallow segments.

Now back to popping some bubbles…..