Monthly Archives: January 2013

It’s the End of Math as We Know It! (and I Feel Fine)

I’m a relatively new iPad user…just scratching the surface of the neat stuff out there, sometimes thinking how cool it would be to be back in the classroom with these tools, sometimes doing the slow burn when I see great tools go un-used. Every now and then I run into an app which has me running to my colleagues like a giddy schoolboy…such as today when a friend tweeted about the MyScript Calculator.

There’s really not much to this app:  you write a math calculation on the screen, and the app recognizes your handwriting, and performs the calculation.  It doesn’t solve equations, it doesn’t factor…it just calculates.   Check out my hairy knuckles as I take it for a test-drive.  Also, note that I understand that there are a number of goofy ways to record an iPad screen…I’m a newbie….baby steps….

It works for iPhone as well.  Take this app around your school, show it off to teachers, and enjoy the reactions and conversations.  Is this the end of the world?  Will we have a generation of kids who can’t make change for a 5?  Hey, it’s just a calculator…a pretty cool one.

So, how do we adapt math instruction in a world where computations are at our fingertips?  Ask better questions!

Building a Better Snowman

In a recent tour of my midde school, I observed a 6th grade teacher working with a class to use compasses.  Their goal: to make a snowman with segments of different, but given radii, 3cm, 2cm and 1cm.  Eventually, this will lead to students having their first experience with circumference, and that sneaky number pi.  But why tie students to fixed radii values?  And just what are the “perfect” measurements for a snowman?  Here are some lesson ideas for letting students explore their own snowmen, using technology, then taking it a step further by considering how much snow our snowmen will need.

Snowmen may come in all different sizes, with different accessories.  But it is pretty well established that the traditional snowman is made of 3 body segments.   Snowmen with anything other than 3 segments are blasphemous.

WHAT ARE THE PERFECT SNOWMAN DIMENSIONS?

Clicking the image to the right will take you to an interactive graph I made using the Desmos calculator.  You can manipulate the diameter of each snowman segment by pulling the sliders provided, but the height of the snowman is fixed at 6 feet.  Like your snowman with equal segments?  Knock yourself out.  Are you more of a bottom-heavy snowman connoisseur?  There’s room on the bus for you , too.

Once you are convinved that you have built the world’s best snowman, please share your slider settings here.  I’d like to feature them in a later blog post.

For a classroom discussion, have students print out their snowman images (Desmos has a snazzy print feature), and compare their snowmen in groups.  Whose snowman would need the most snow to build?  Whose would need the least?  Or, if all of the snowmen are 6 feet tall, then will they use the same amount of snow?

This is a great time to talk about volume, and introduce the formula for the volume of a sphere.  And, since each student has their own product, have them find the volume of each snowman segment, then add them to get their required snow total.

HOW MUCH SNOW DO WE NEED?

If it snows 5″ overnight, will we have enough snow on the ground to build our snowman?  For this next stage, we can have students compare the volume of snow needed for their snowman with the volume of snow on the ground.  For example, if your yard is 10 feet long and 10 feet wide, will 5 inches of ground snow be enough to build our snowman?

BUT, we pack snow while we build snowmen.  How much less is the snow volume in a snowman vs its volume when it is on the ground?  2 times less volume?  5 times less?  10 times less?  I really have no idea.  Next time I have a few inches of snow on the ground, it will be experiment time.  With a class, perhaps debate the correct number and use it for calculations.

To help with calculations, and checking student work, I have created this handy Snowman Calculator on Excel.  You can input your a and b values from the Desmos document, along with the dimensions of a yard or rectangular area.  The volume of your snowman, along with needed snowfalls, are then given.

Think warm!  And then we will start working on sandcastles.

Home on the Range (and the domain!)

A recent benchmark assessment in Algebra 1 I administered to our high school’s 1200 students in grades 9-11 provided some interesting data, as we prepared for the new “Keystone” Exams, which were given for the first time this past December.

The question below is taken from the Algebra 1 Eligible Content and Sample Items document, from PA Department of Education, Standards Aligned System website:

This question was given to over 1,100 students in a 20-question assessment, and only 14% gave the correct answer of B.  Meanwhile, 66% gave the incorrect answer of A.  So, what am I worried about here?  And how can we use this result to improve our approach to domain and range our Algebra 1 courses?

When talking about range, there are two separate issues to consider:

• Do students understand how to express the possibilities of a function’s “output”?
• Have they been exposed sufficiently to the vocabulary which allows them to attach the word “domain” to the inputs and “range” to these outputs?

Where should domain and range be “taught”.  In Pennsylvania, understanding domain and range are part of the Algebra  1 standards for functions:

Identify the domain or range of a relation (may be presented as ordered pairs, a graph, or a table).

So, what’s the problem?  One of the issues I see is that we deal with linear functions so heavily in algebra 1, it is easy for students to begin to believe that every function has “all real numbers” as the domain.  Problems involving non-linear functions often provide natural “ins” for discussing domain and range, but we just don’t get to them until after domain and range have been defined, tested on, and forgotten.  A second issue is that of coverage.  Having students copy definitions into a notebook is simply not sufficient in order to “cover” domain and range.  Students need to see experience the need to communicate domain and range, have a part in developing notation, and see the vocabulary reinforced appropriately in all math courses.  Here’s an station activity you can use with your classes to develop input/output sense.

INPUT/OUTPUT STATIONS

The file with the problems for this activity are here: input/output activity

The file contains 6 stations.  2 of the stations are problem scenarios, 3 are graphs, and 1 gives a function rule.  Here’s one of the scenarios:

A tomato plant is purchased from a local nursery.  When purchased, the plant had a height of 5 inches.  After it is planted, the plant grows an average of 5 inches each week.  After 10 weeks, the plant reaches its maximum height, and we all begin to enjoy the yummy tomatoes!  Let x represent the number of weeks after the plant is placed in the ground, and let y represent the plant’s height.

Place the problems around the room, along with signs for “possible inputs” and “possible outputs”

Next, provide each student (or pairs) with a few potential input and outputs, writing them on a 3×5 card.  3 or 4 of each will suffice, and try to give a variety of positives and negatives, along with a fraction and/or decimal.  Some samples are here.

Have students visit each station, and list items from their card which are appropriate to the scenario.  Soon, both lists begin to populate with inputs and outputs, for all of the stations around the room.

When all students are satisfied that they have placed their values correctly, let’s add a twist.  Assign each partnership a station, having them provide a value NOT appropriate to the problem along with a justification for their choice.

After all students have visited stations and shared their input/output values, we’d like students to summarize the input/output lists.  One method for this is to assign partnerships a different station, and have them write a summary underneath the shared values.  For example, in the tomato problem, we could see:

• Input values: x must be between 0 and 10, inclusive.  No decimals.
• Output values:  y must be between 5 and 55, inclusive.

Now is the time to introduce our friends: domain and range.  And, given the variety of problems we have seen on the board, we will have different means for communicating domain and range.  Sometimes, all real numbers is appropriate, while other times the list is best given as an inequality.  In other problems, a simple list may do.  Do we need to restrict to integers?

There’s no hurry to develop formal symbols for all of the stations right away.  Perhaps complete one a day, and keep the ball rolling by providing problems which cause students to need to talk about restrictions.  And finally, don’t limit discussion of domain and range to just the introduction to functions unit.

You Asked For Piecewise Functions, I Give You Piecewise Functions!

NEW: After popular demand from this post, I have created a tutorial on domain restrictions and piecewise functions.  Enjoy!

UPDATE: Many of my Desmos files are avilable on this page: Desmos File Cabinet Enjoy!

Let is never be said that mathcoachblog doesn’t listen to the needs of its followers!  One of the neat things about having a blog is checking out the routes people take to get to the blog. What search caused them to arrive here?  What countries are my visitors from?  What search phrases cause them to reach the blog?

Every day, without fail, there is a theme which appears in the search terms of blog visitors.  Here is a sampling of terms from just the last week:

• Online piecewise graphing calculator
• Graph a piecewise function online calculator
• Piecewise function calculator online
• Graphing piecewise functions calculator online
• Piecewise functions online grapher
• Online graphing calculator piecewise functions
• How to do a piecewise function on Desmos

OK, folks I get it.  We want to graph piecewise functions.  So, let’s light this candle.

GRAPHING PIECEWISE FUNCTIONS ON DESMOS

The Desmos knowledge base provides instructions for graphing a piecewise function, and a neat video tutorial.  But I’ll provide a few examples here, and some teaching tips.  Let’s say we want to graph this piecewise function:



In the Desmos calculator, colons are used to separate domain restrictions from their functions.  And commas are used to have multiple function rules in one command.  So, the piecewise function above would be entered as:

The function then appears quite nicely:

Sliders can be used to have students explore the continuity of a piecewise function.  Consider this problem:

For what value(s) of x is the piecewise function below continuous?



In Desmos, start by defining a slider for the parameter “a”.  For mine, I chose to limit the domain to between -10 and 10, and have step counts of .5.  Then, a can be used in the piecewise function.  Click the icon below to play with the document online.  The sharing features are another aspect of Desmos which have improved greatly in the past year.

SO, WHY ARE YOU AVOIDING “EQUALS” IN YOUR FUNCTIONS?

OK, smart guy, yes…yes, I have kinda avoided the equals parts of the domain restrictions.  Something neat (odd, goofy) happens when an equals is used in the domain restrictions.  Let’s graph this function:



Click this link to find out what happened when I tried to enter this function on Desmos.  The Desmos folks tend to monitor these things, so let’s see if they have a suggestion here.

Down the road, I want to take a deeper look at the new table feature, and will report out.  But my early impression is that it is a addition which works seamlessly with the existing, awesome, calculator.

Also, while I’m in a sharing mood, here is a quick file I created to use in an absolute value inequality unit.  Click below to check it out.  Would enjoy your input!

And finally, I started this post by sharing some of the search terms which cause people to find my blog.  Most of the time, I can explain those terms, and why people would end up here.  But this….well….this, I got nothing…..

A Bowl of PASTA with Stats Friends

Today I attended the winter meeting for one of my favorite organizations: PASTA, the Philadelphia Area Statistics Teachers Association.  This group meets a few times a year to discuss best practices in statistics education, and includes a number of AP teachers, many of whom are AP exam readers.  As always, lots of interesting ideas today:

Joel Evans, from my home school, spoke on his first attempts to “flip” his AP Statistics class.  Based on feedback from his students, Joel realized that Powerpoints often dominate his classroom culture.  By flipping, Joel hoped to have students review material before class, then use class time to practice and discuss.   Follow Joel’s flipping story in the slides below.

It is always a pleasure to have Daren Starnes at our meetings.  Daren, one of the co-authors of the ubiquitous The Practice of Statistics textbooks, joins our group often to discuss his ideas for teaching statistics.  Today, Daren shared a presentation, “50 Shades of Independence”.

Daren asked us to think about all of the places where we encounter “independence” in AP Statistics:

• probability of independent events
• independent trials
• independent random variables
• independent observations
• independent samples
• independent categorical variables (chi-squared)

Man, that’s a lot of independence!

Which items from the list above deal with summarizing data?  Which are needed for inference?  How are they related?  How do we help our students understand the varied, and often misunderstood, meanings of independence.

Daren has a knack for leading conversations which invite participants to express and discuss their math beliefs.    Many of the arguments concerning independence, according to Daren, are “overblown”, in that teaching them in a cursory manner often causes us to lose focus on the big picture. That’s not to say that we should discard them, but that, when teaching inference, we should have students focus on items which would cause a hypothesis test to be “dead wrong” if we didn’t mention them, i.e. randomness, justifying normality conditions.

Ruth Carver continued the presentations with some new tech twists on a lesson used by many stats teachers: analyzing sampling distributions by looking at the age of pennies.  A population graph of the ages of 1000 pennies hangs proudly in Ruth’s classroom.

After agreeing that the population is clearly skewed right, we move to the main event – drawing random samples from the population and analyzing the data we get from repeated samples of the same size.  Ruth has developed a lesson for the TI Nspire which generates the samples, and challenges students to think about the behavior of the sampling distributions, now considering the effects of sample size.  Ruth’s presentation allows students to experience and express the differences between:

• Standard deviation of a population
• Sample standard deviation
• Standard deviation of a sampling distribution

Great job Ruth!  Looking forward to more PASTA with my stats friends!

Worksheets and Differentiation – Not Always Mutually Exclusive!

I hate worksheets.

Is there anything worse than a math classroom where the pace and expectation are dictated by the almighty worksheet?  OK class, continue working on the blue worksheet, and homework will be on the pink worksheet.  Tomorrow, we will do test review with the aqua worksheet.  And then we will have a whole new chapter packet to work on….blah….

Don’t get me wrong…I’m not anti-practice.  Much of math is like learning to play the piano, you need to expend some sweat in order to master skills.  But, like playing the piano, all students will master the skills differently, with different timelines.  And, like piano players, some students will handle rigor and improvisation quicker than their peers.

So, how do we provide students with appropriate practice, while at the same time allowing students to have some say in their learning, assess their own progress, and provide for differentiation? Here are two strategies for you to try:

POINT-VALUE ASSIGNMENTS

In this strategy, students are not required to complete all assigned problems (unless they choose to). Instead, problems are assigned individual point values, and students complete enough problems to earn the assigned number of points. Easier problems have smaller point values, while more challenging ones are worth more.

Here’s an example, which use the Linear Functions Review given here (pdf): Linear Functions Review

This review has 18 problems, increasing in difficulty.  One way to assign point values would be:

• Problems 1-7, 1 point each: these problems can be done mostly by looking at the linear pattern and providing a quick answer.
• Problems 8-14, 2 points each: these problems mostly ask students to match function rule to situations.
• Problems 15-18, 3 points each:  open-ended, and all require students to develop a function rule.

For this assignment, I would ask students to complete 12 points worth of problems.  This would require students to reflect upon their understanding, and provide differentiation.  How could students complete the assignment?

• Students at a basic level could complete all 1-point problems, but would then also need to complete at least 3 of the 2-point problems (of their choice).
• Students comfortable with the material could complete a mix of 1, 2 and perhaps 3-point problems.
• Students at the advanced level could complete only all 4 of the 3-point problems.

The worksheet provided here was created uses the fantastic site Problem Attic, developed by EducAide software.  The site has a large bank of problems from various state, national and international assessments, and allows users to create their own customized assessments.  Definitely worth checking out!

CHOOSE-YOUR-OWN PATH

Many textbooks (particularly high-school texts) will arrange their problems sets into A, B and C levels.  Do I need to see students complete all problems from a set?  If a student demonstrates mastery of a C-level problem, do I really need to see them complete many A and B level problems?  This strategy allows students to choose the best path for completing an assignment, using this template:

In this assignment, all students start with a B problem, then choose their own path for completing the assignment, by selecting one of 3 colored paths.   This could mean completing a few A problems, with a few B problems.  Other students many choose the series of B problems, with a few A’s sprinkled in.  Ambitious students may choose the challenging C problem to complete.

With both strategies, students are challenged to reflect upon their own learning, make appropriate choices, and take responsibility for their progress.  Classroom expectations don’t change at the drop of a hat, and may take a few conversations and failed attempts before working the way you like.  But they payoff, increasing student responsibility and reflection, are worth the pain.

Ring in the New Year with Fun Classroom Lessons!

Now is a good time to reflect upon the past year, and think about all of the professional growth I have made through the people whose ideas I have shared and experienced through the twitter-sphere and blog-o-sphere (are these actual words?), and to send thanks from all of the new math friends I have made.  I took a look back at all of my posts from the previous year, and here are 5 great activities you can use tomorrow is your classroom.  Share them, adapt them, expand upon them…it’s all good.  Just pay it forward and share your best works, or leave a comment /contact me and let me know if you use them!  Enjoy.

Conic Sections Drawing Project – this was the most popular post of the year.  For algebra 2 or pre-calc, this project just got better with the Desmos online calculator, which is my favorite new tool of the past year.

Tapping Into the Addition of Bubble Wrap – bubble wrap, iPads, and slope meet for a fun exploration.   Look at rate of change through student-produced data.

Tall Tales for Probability – Featuring the poker chip drawing game, and examples from the Amazing Race and craps.  Probability should be fun.  Make it so!

Let’s Play Plinko! – I have used Plinko as an introduction for binomial distributions for years, but in this presentation from last summer’s Siemens STEM Academy, tech tools like PollEverywhere and Google Drive are used to increase interaction.

Composite Functions and ESP – Use this activity with middle-schools and see if they can develop the pattern.  For high school, have students write and justify their own ESP puzzles.  Also features Doceri, another favorite new tool of mine, for iPad.