Grouping for Writing Equations
In this lesson, we explore patterns with a focus on generating equivalent expressions, graphing discrete functions, and problem solving. Let's begin with a 'square' tile pattern. What observations can you make about the first three arrangements in the sequence?
One strategy for trying to make sense of the pattern is to look for groups of tiles that correspond to the figure numbers, and that are consistent from one arrangement to the next. A tile grouping that works in the second arrangement, for example, should also work in the third, and every other arangement as well. Here are some observations that students have made about this pattern:
There are other observations that can be made, as well. Once we begin to see how the pattern 'works', we can make some predictions about larger figures, and write some equations that represent the total number of tiles in any arrangement in the pattern.
For example, can you use each of the observations above to determine how many tiles are needed to construct the 20th arrangement in the pattern? Maybe the diagrams below will help...
These grouping methods can be used to answer the question by imagining what the 20th arrangement would look like in each instance; in other words, where would you see the groups of 20? Then it's a simple matter to calculate the number of tiles in the 20th arrangement. Click here to see the solutions.
We can use the same groupings to help write equations that represent the total number of tiles in any (nth) arrangement; n is the arrangement number, like 2nd, 3rd, 20th,etc... Here are equations based on the above three grouping methods.
These are called equivalent expressions because they represent the same pattern, and give the same answer when evaluated for the same 'n'. In our example above, using the number '20' in each equation would give the answer 84.
We can also work 'the other way around' - Which arrangement contains 200 tiles?
Using observation 'a', we could reason that if we took away the four corner tiles, then the remaining tiles would be divided into four equal groups containing the arrangement number of tiles:
200 - 4 = 196
196 ÷ 4 = 49
The 49th arrangement contains 200 tiles.
In algebraic terms, the solution would be
T = 200 = (4 x n) + 4
200 - 4 = (4 x n) +4 - 4
196 = 4 x n
196 ÷ 4 = (4 x n) ÷ 4
49 = n
This is just one example of the many patterns we explore in class, and in homework, but the procedure is the same. We try to 'see' groups of tiles that represent the figure number in the arrangements, and then use those groupings to make sense out of the pattern. Writing equations is the generalization of the observations and grouping methods. Join us in the next lesson to see patterns that involve both positive and negative numbers, and an example of graphing a function.