What is the 100th Step?

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SWBAT use patterns to determine an expression (rule) that can be used to extend a pattern.

Big Idea

Students gain understanding that finding the rule is key to exploring a pattern, as they seek the 100th number (and beyond) in a pattern.


5 minutes

Today class opens with a Math Museum.  

I ask the students to draw a pattern on their paper, they can use color if they want to and I suggest that they include a question about their pattern.  I tell the students they will have 3 minutes to do this.

To prepare for our "show",  students clear their desks of everything besides the pattern they created.  This makes for less clutter for the exhibit.  Before we begin our museum walk, we have a quick discussion about expectations at a museum.  This is a familiar routine for the students, so it only requires a brief reminder.  The students remember that museums are usually quiet to allow others to think. Then we agree on a direction of travel and the students walk around the room looking at the patterns.  They are preparing to answer the question, "What did you notice when looking at the pattern?"

After 5 minutes for the Math Museum, students head back to their seats and share what they have noticed.  

They notice that there are patterns with numbers, shapes, colors, some with variables, some with rules.  

I let the students know that the purpose of visiting the math museum today is to notice that pattern means many different things, and our tour reminds us that there are many different types of patterns that we can work with. 

Launch & Guided Practice

15 minutes

The students and I play Function Machine together.  There are many different function machines available online.  I like this one because there are options for finding the input, finding the output, and finding the function.  I start with finding the input, then move to output, and then function.

As the students are finding the function, I ask them each to explain their thinking as they are solving.  I prompt them by saying, "What are you thinking".  

I make connections between the term "rule",  algebraic expression, and function by saying them interchangeably; sometimes saying "rule, function, expression" all at the same time.

After each student has a turn, I write a table on the board with 3 columns.  I use the same function from one of the examples on the website. Then, I erase the middle column (function) and slide the output column over.  I show the students that input/output tables are just the same at the function questions from the game, they just have the function column hidden. Determining the function is the same as finding the rule.

Next, I put another input/output pattern table on the board.  I ask the students to remember the problem solving thinking everyone shared while working with the Function Machine.  I have a record of these on the board.

  • You can try to narrow down the operations that might be used by looking at the input and output to see if the numbers increase (you know its add or multiply) or decrease (you know its subtraction or division).  A student reminded us that if you are dividing by a number less than one, the quotient could be larger and so not to rule out decimals!
  • You can turn the equation around to have the inverse operation help you find the answer.


The students appear ready to go, so I ask, "Are all input output tables set up vertically?" 

Now You Try

25 minutes

For today, I have the students complete tables from two lessons in the book.  The text book breaks the lessons into adding and subtracting expressions and then multiplication and division expressions.  

I use the tables from the book, but create my own sheet so that the operations are mixed.  This requires the students to have to think about the operation that is being used in each table.  Since all of these tables represent expressions with only one operation, I consolidate them to increase the rigor.  

Some students are able to determine the rule for these patterns right away.  Others have to be more methodological.  There are students who might think they have the rule right away, but need to then check and make sure the rule applies to the whole pattern, not just one step.  Completing tables using rules and relationships requires students to attend to precision (MP6).

I have the students work in groups today, not pairs, so I can easily pull individuals from the group to work with me.  As I move around the room, I find the students who are having more difficulty and call them to the small table.  We work on a few tables together. The area that appeared to pose the greatest difficulty was turning the equation around to help determine the rule.  

For these students, I help them write the equation they are trying. For example, 3 x ? = 60.  Then, by reminding students about inverse operations they recognize that 60/3 can help them get the answer.  This is not a quick fix, as it shows a deeper lack of understanding, so it is critical to work on a few together because the problems cover all 4 operations.  I will keep in mind, as we move on to patterns with multiple operations, that these students may need more differentiation and support.

At the end of the lesson, I praise the students for their hard work and great thinking.  They pat themselves on the back for being able to do much more than their book expected they could do.

Ticket Out

5 minutes

Students develop their own input/output pattern, write the rule, write the 100th step, and then write the expression.  This is easy because they make it up!  To make it more challenging and to help me see the strategies they use, I ask students to then explain how another student could use this table to determine the rule and the 100th step.

I use these reflections to prepare for our next lesson when I will share some strong examples with the class to encourage students to explain their thoughts when writing about math.