Writing Formulas
In this lesson, we build on the ideas developed in the Variables and Formulas lesson in a more formal way. Some of the ideas developed through applying algebraic formulas to cube patterns include odd and even number sequences, square numbers and square roots, and cubic numbers and and cube roots. The basic ideas are still the development of strategies for grouping for counting and using the methods to make predictions and generalize. We also develop some ideas about equivalent algebraic expressions. All of these concepts won't be dealt with on this page, but we'll explore one pattern to demonstrate the basic approach.
Here are the first three arrangements in a cube pattern sequence...
Once we have established the pattern's next few figures, we can begin to look at grouping strategies so we can describe larger arrangements, like the 29th or 53rd. One way of showing our thinking about these strategies is to draw an nth sketch. Some possibilities:
Notice that these are abstract sketches of the major elements of the pattern, and that it is not necessary to draw every cube, or to make the sketches complicated. In fact, simplicity is desireable, and makes the communication more clear!
Just like in the last lesson, we can now use variables to represent the arrangement numbers and the volumes (number of cubes) of the arrangements. Let's use 'V' for the volume, and 'n' for the arrangement number. We could represent each of the grouping strategies above with an algebraic formula:
All of these formulas describe the pattern groupings, and for any given arrangement number would give the same volume. Therefore, we call them equivalent expressions. In class discussion, we look for similarities in the expressions, and discuss simplifying expressions as well.
We explore lots of patterns in this lesson, here are a couple for you to try your hand at. What grouping methods can you see? How would you represent them with formulas? Talk them over with your child and see what ideas they can share with you.
