Reflection: Complex Tasks Make Piecewise Functions Continuous - Section 1: Warm-Up


At first I only chose these problems because I wanted to push students to do some more challenging thinking, and I thought it was a good extension for the piecewise functions unit. I also know that these types of problems show up on the AP Calculus Exam, so I liked that.

As with many challenging problems, only a few students fully understood the problem after their first attempt, so I chose to extend these problems into this next unit, because I thought that they might challenge students in an interesting way. 

It turned out that these problems were really interesting--I never provided students with an algorithm for solving them, and I also did not tell students whether or not their answers were correct. I choose not to give this feedback, because I wanted them to develop methods to check their own answers: some students graphed the functions using the values of k that they had chosen, to determine whether or not the functions were continuous.

Some students struggled to learn the Level A problems, but because we worked on these problems throughout the week, even these students were able to extend their understanding to develop methods to solve the Level B problems. We spent way more time than I anticipated on these problems, but I feel that it was worth it.

I am starting to believe that there are many different problems that can be worthwhile--I don't necessarily think that these problems are especially important in the bigger picture of mathematics, but they bring in a lot of ideas and different ways of thinking, so students are able to engage with the mathematical practice standards in the context of these problems.

  A Fun Challenge with Multiple Representations
  Complex Tasks: A Fun Challenge with Multiple Representations
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Make Piecewise Functions Continuous

Unit 2: Piecewise Functions
Lesson 8 of 12

Objective: SWBAT determine whether or not a piecewise function is continuous and set values for parameters to make functions continuous.

Big Idea: Bring in a lot of different skills and ideas (and preview a calculus concept!) while asking students to solve problems both graphically and algebraically.

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