**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:**

**a -There are the same number of tiles as the arrangement number in the middle of each side...****b -each side of the square contains the arrangement number of tiles plus two...****c -inside of each arrangement is a square empty space whose dimension is the same as the figure number...**

**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.**

**a - T = (4 x n) + 4****b - T = [4(n + 2)] - 4****c - T = (n + 2)**^{2}- n^{2}

**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.**