Scatterplot graphs enable us to build shapes using spreadsheets and to practice transformational geometry. They are surprisingly flexible tools. And since they depend upon a table of value and that table can have both fixed numbers and rules, we can not only build shapes but change them and watch the graph immediately reflect those changes. In this lab we use that capability to get students to explore scaling, reflection, and transposition of a triangle. This is only the beginning and we hope students will take this further exploring symmetries for example.
Author: Ryan McQuade
Multiplying Integers
We have made a big deal of the times table and of other tables.Now we extend the times table to negative numbers and thus to all 4 quadrants of the real number space. We hope to build student intuition about this space and to gain a spatial sense of graphing as well as of multiplication. So we as usual focus on patternmaking and take students through extending the table first left by rows and then down by columns before we have them build the table as a whole. We moved the axes to the outside so that we do not interfere with the table. There are many things you can do with such a table and we urge you to explore it.
Systems of Equations
Solving systems of equations sometimes called simultaneous equations with graphs is simply a matter of finding out where they intersect. One of the most valuable things students can learn is to be able to visualize linear equations and systems of equations so that they can tell the quadrant where the intersection and therefore the solution is. This develops the valuable ability to estimate solutions. We suggest students practice picturing and then graphing systems with a variety of slopes and intercepts. Significant relationships like perpendicular lines should be another focus.
Composition of Functions
One of the most powerful aspects of the mathematics of functions is our ability to treat them as abstract quantities (essentially numbers) and then combine them with standard operations. But with functions we can go further and develop a new operation we call composition or taking a function of a function. We give students the opportunity to explore composition by using different functions and by seeing their result on graphs. This is very powerful and great fun to push imaginations.
Inverse of a Function
Spreadsheets make it very easy to switch axes and add graphs. They enable students to play with what may have been difficult and abstract concepts like the inverse of a function. You may want to approach the inverse of a function by challenging students to fill in a table of values with a rule that creates a mirror of that function. So you can approach the inverse of a function either as the interchange of axes or as a symmetry issue. Either one works well on spreadsheets.