I'll begin by telling students the objective: write linear expressions (and equations) to represent patterns.
Students may need examples of patterns that are described by algebraic expressions. Examples may include $25 per hour could be written as 25h. If you are x years old and have a sibling that is 3 years older, your sibling's age can always be found by x + 3.
The patterns today are geometric in nature.
The example pattern shows how students are to solve the remaining problems. You may want to have pattern blocks available for students to use. It may suffice just to show the pattern using a document camera or SmartBoard.
We'll examine the first 3 stages and look for a pattern of how many pieces are needed at each stage. The first few are shown in the given table. The table is just an input/output or function table with a middle column. The middle column is key to creating the expressions.
Students will quickly see that each stage has two more pieces than the previous stage. I will ask students to either draw or model stages 4 and 5.
Now I will ask students to describe the pattern. If necessary, I will tell them that the missile is made up of the body and the flames. The body is made up of the triangle, the square, and the trapezoid. The flames are made up of the non-rectangular parallelograms. Under the "What I See" column I will write "BODY + FLAMES".
We will then notice that the stages grow in this manner (starting with stage 1):
Stage 1 --> 3 + 2
Stage 2 --> 3 + 4
Stage 3 --> 3 + 6
Stage 4 --> 3 + 8
I'll ask: What do you notice about the body? Students will notice that the number always remain at 3. Then I'll ask: What do you notice about the flames? Students may notice that the flames increase by 2 each stage. More importantly I'll ask: How does the stage number relate to the flames? Eventually students will notice that the flames are twice the stage number.
When we reach stage n, watch out for students who treat this stage as if it is stage 5. Explain because there is a variable n, this could be any stage. I'll give students a minute to see if they can write an expression that would find the total parts based on stage n.
If they are stuck, I'll ask: "What did we notice about the body?" Answer: it always is 3. "What did we notice about the relationship between the stage number and the number of flames?"
This should lead students to writing an expression: 3 + 2n or 2n + 3. Some may have written 3 + n + n. This is also okay. Especially if a group wrote that they saw "Body + Flame + Flame".
The third question could be solved in a number of ways. The simplest may be to write an equation 103 = 2n + 3. Solve for n. Some may use a guess and check method by trying different stage numbers until they come to a total of 103 pieces.
However students solve it, it is a perfect moment for students to explain their thinking (MP3).
The next four problems students will work on with partners. You may want to supply the orange square tiles for problem 1, toothpicks for problems 2 & 4 - I would avoid the rounded toothpicks as they will roll around on the desks, yellow hexagons for problem 3.
As students are working on each problem, I will walk around to check progress and ask questions as necessary. For each problem, I'll be especially interested in finding different descriptions for "What I See." This will allow for a good conversation about equivalent expressions at the end of the activity.
I will also be on the lookout for groups that try to treat the variable input as the next number in the sequence. For groups struggling to make an algebraic expression, I will ask them to look at the relationship between the input value (left most column) and the values in the "What I See" column.
On problem 3, some students may want to count the sides that meet. Remind them that perimeter is the distance around the outside of a shape, so those inner sides should not be counted. Though they will be counted in problem 4 since the shapes are made of toothpicks. If you don't have toothpicks around, you could always just use the term line segments.
After most groups have completed all 4 patterns, we will discuss solutions. I will make sure to bring up a variety of expressions that are equivalent, to show that different people had different yet valid ways of seeing the patterns.
Students almost always enjoy this activity and its a great way to develop their ability to use equations or expressions to model solutions to problems (MP4).
Students will now take a three-part exit ticket. I will count this as 5 points. 1 point for completing the table correctly in part 1, 2 points for a valid expression in part 2, and 2 points for a correct solution to 3 with supporting work or an explanation.
The pentagon pattern can be difficult to see. I may walk student through the first 3 stages of the pattern, to make sure they are correctly finding the perimeter. Other than that, they are on their own!
A score of at least 4 of 5 points will be considered a successful exit ticket.