Category: Algebra

Categories
Algebra

Hooks for Inverse Functions

While browsing through Dan Meyer’s most recent post on great classroom action, I found a link to a picture which put a smile on my face, from the blog Your Poisonous Cousin (cool name):

Staplers

A high school colleague of mine uses the Dr. Seuss story “The Sneetches” to hook students into a conversation about inverse functions.  This AP Calculus Blog has a nice summary of inverses and pictures of our Sneetch friends.

Enjoy the Sneetches with this video, discuss the star-applying machine, and the charlatan who sells its eventual inverse.

Categories
Algebra

Encourage Generalization and Communication with these Math Challenges

A comment from a recent post of mine on differentiation asked what I do with students who complete tasks early.  In every course I have ever taught (usually Algebra 2, Prob/Stat, Algebra) I have used weekly problem-solving challenges, no matter what the level of student.  Often, my intent in these problems was to develop written communication skills in mathematics, and have students begin to reflect upon their own writing style.  Have students complete the challenge, critique their writing and provide a path for improvement, and have students turn in their best works as part of a portfolio at the end of the semester.

In this post, I focus on tasks involving counting, number theory, and algebra.  The problems here are ones I have assigned, graded, revised, and enjoyed over the past 15 years.  I’ll have some more tasks in a later post.  Click the title to download the PDF document.

PATHS:  How many ways are there from start to finish?  I love this problem, because there are multiple ways to approach it.  Combinations give the result, but there is also a Pascal’s Triangle approach, or as a permutation with identical items.  And Polya’s strategy of starting small and working your way up is key to this one.

SOME ZEROES:  I have always enjoyed giving this problem, as you you can have rich conversations about simple number facts and the commutative property.  And the student explanations will range from the ridiculous to the intriguing.  When I started giving this problem 15 years ago, some students would use Excel to try to simply compute the answer, which often “broke” Excel and gave a wrong answer.  I have given up on trying to follow the technology, and have given a similar problem as a follow-up on a quiz.

AVERAGE SPEED:  One of my favorites because of its simple premise, and a result that is counter-intuitive.  Also, can be easily differentiated.  For some students, choosing distances and testing serves as a good starting point, while students with advanced algebraic skills can dive right into the abstract.

LAST DIGIT:  A premise simple enough for grade 6, yet complex enough to challenge older students if you ask for a general formula.  It’s also easy to adapt this problem and use it as an opener for class.

INVERSES:  In this challenge, students must find a matrix which is its own inverse, of which there are many, many possibilities.  How will your students ensure that their matrix is unique?

Feel free to contact me for solutions, tips, or more ideas.

Categories
Algebra

Absolute Value Inequalities and the Human Number Line

In most Algebra 1 courses, the topic of Absolute Value inequalities comes at the end of a longer unit on inequalities.  We shade our number lines, attend to our open or closed circles, and start to hit the wall a bit with the routine.  So, as we begin to think about introducing absolute values, let’s get our students up and moving.  Here’s how:

THE HUMAN NUMBER LINEIntegers

Print out or scribble out cards with the integers -12 to +12, or use my handy integer card set.  This will give 25 cards, and you can adjust the cards based on the size of your class.  Give each student, except 1, an integer card.  The student who does not get a card will act as the observer during the activity, and will verify the class’ actions.

In a hallway, or outside on a nice day, have students sit in order from lowest to highest.  The students are making a human number line.  It is important that their card be clearly visible at all times.   The class observer should verify that all students are seated in order, with somewhat equal space between them.

THE ACTIVITY

With students seated, the teacher holds up an inequality.  Any student holding an integer which is a solution to the inequality will stand, thus making a human solution set.  The job of the observer is to verify the correctness of his class-mates solution.  My inequality cards file starts with two warm-up problems, to make sure the instructions are understood, before we start to head into the absolute value inequalities.

Inequalities

One of my colleagues used this activity with his class recently, having students step forward if they were a solution.  He also added a twist I hadn’t thought of: having students hold their hands over their head to make an “open circle”, if they were a boundary number.

class

As the class builds the solution sets for the absolute value inequalities, have the observer describe the graph.  What do greater-than problems “look” like?  How about less-than problems?  What sorts of problems tend to veer off (to infinity) in both directions?  What sorts of problems are bounded?  Here are some other teaching tips and ideas for this activity:

  • Have students trade cards, or totally re-mix after 2 problems.  If you don’t, the students with “end” cards can simply follow the crowd.
  • This is a great time to find a class leader to be observer, or uncover a hidden talent of a shy student.  Keep those cards visible.  Give them the responsibility to keep things orderly.
  • If you have room, take pictures of the human number lines, and use them later as a review, or to keep around the class to build the team spirit.
  • Using this activity a lot, or with many classes?  I always thought it would be neat to have integer shirts for this, and to use through the year.

Shirt

BACK TO CLASS

After we have developed some ideas about absolute value inequalities and their solution sets, it’s time to start formalizing our thoughts.  If you need more hands-on practice, click on the graph link below to try a Desmos demonstration with sliders.

Compare the absolute value function (in blue), to the constant function (in green).  The comparison (in red) allows us to look at make greater/less-comparisons.

Let me hear about your Human Number Line experiences!