Title: **Sketching graphs of polynomial functions**

Course: Advanced functions and introductory calculus

Age level: 17-18

Idea: introduce the notion of sketching graphs of polynomial functions to ESL students. Move the emphasis from textual instruction to one based on visuals, mathematical language (which can be more or less understood by members of all cultures, given appropriate amount of schooling)

At the end of the lesson, students are expected to determine, using the first and the second derivative, the shape of the function being sketched, the local minima and maxima, and the points of inflexion. Students are expected to apply that knowledge to graph the required functions.

For part of the lesson, students can be broken up into groups based on their first language, where students with greater English profficiency can interpret for their less articulate classmates. The groups can be especially producive when learning to use graphing technology to sketch polynomial functions. In such groups, studetns can take turns at operating the graphing calculator and teaching each other how to do that.

The main teaching strategy for this lesson would be to go from prior to new knowledge. Students will be led from their knowledge base of first and second derivatives and the concepts of local minima and maxima towards a system that synthesizes all these elements.

The instruction will rely heavily on visual aids, mathematical language, and oral language. Also, the lesson will involve, as much as possible, relevant examples that relate the subject matter to real life.

The level of knowledge attained at the end of this section is best assessed by means of a test or a quiz. Because this section is highly graphical, the test or the quiz can be composed avoiding the complex language of word problems, and therefore will ose no significant challenge to ESL learners.

For ESL purposes, the lesson will have to rely more heavily on visuals (which abound in this section) and on mathematical language, and less on formal written English. Whatever formal written English is to be used, it is to be kept as simple as possible without compromising the content of the section.

An up-to-date Ontario Calculus textbook.

TI-83/84 graphing calculator.

Submitted by: **Gregory Sperlin**