Relationships & Rules

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Objective

SWBAT use a table to determine an expression or rule that describes the relationship between numbers in a pattern.

Big Idea

Using an expression can help to discover a number pattern.

Warm-Up

15 minutes

We open with a quick check in -- students complete an evaluating expressions problem.  The majority of last week's teaching was focused on simplifying expressions.  To start this week, I ask students to apply what they remembered.

I choose a problem that asks students to critique "an example" of other students' work, and include here an analysis of a student response. Having students tackle this sort of problem gives me so much information because they have to analyze the work to determine which problem is incorrect and explain why. While problem solving, students are also refining the precision of their mathematical communication (MP1, 3, 6).  This problem is rigorous, because it requires the students to evaluate, which is a higher level thinking skill than application.

Launch & Guided Practice

20 minutes

I write a basic pattern on the board, then ask the students to determine the rule used to complete this table.

 Step Total 1 3 2 6 3 9 n

They quickly recognize that the rule is nx3 because they know their basic facts.  Even though they are familiar with the rule from the start, I ask them to build this pattern using tiles.  Together they add 3 tiles each time I say "next step".

After 3 steps, pause to make sense of what is happening.

• How many tiles did we start with? (zero)
• How many tiles did the pattern increase by each time there was a new step (3)

Write zero and 3 on the board.  Explain that these numbers are an important part of finding the rule.  Students know the rule is nx3.  Push their thinking to develop a rule that includes the 3 (the amount the pattern increases by) and the 0 (the number of tiles that the pattern started with).

Students agree that the rule can be expressed as n x 3 + 0.

Repeat this activity with a few more simple patterns, using basic facts so the students can identify the rule quickly and easily.  Revisit the two focus questions each time.

• Where did the pattern start?
• What did it change by?

The new pattern I write on the board requires two operations, but I don't tell the students this.

 Step Total 1 4 2 7 3 10 n

Allow time for students to think about the pattern and identify what they know.  Here, students recognize that the pattern increases  by 3.  Ask them to determine the 100th step.  This is challenging if they don't have the entire rule.  Refocus on the 2 questions

• Where did the pattern start?
• What did it change by?

so students can agree that the pattern, although it doesn't say on this table, started at 1.

The two numbers that they can use to develop a rule are 1 ( the number the pattern started at) and 3 (the number it increased by).  Students recognize the rule is n x 3 +1.

Continue with a few more examples before allowing them to practice on their own.

Now You Try

25 minutes

Now it's time for students to practice finding a rule for patterns that include more than one operation.  They are given a worksheet with seven input output tables, and rules that include combinations of all the operations.

I specifically didn't use any examples with subtraction ex: x 3 - 1 because I want the kids to be challenged to think about the two focus questions.

• Where did the pattern start?
• What did it change by?

The first input output table had an example of -3.  This challenged the students to rely on strategies, not just guessing and checking different combinations.

I allow students to use models to assist their thinking (MP4). Some images are included here of how models were used to help students who are struggling.

Group Share

5 minutes

During the group share, students are encouraged to share any struggles, challenges they over came, or interesting discoveries from today's task.

I was surprised by the number of students who explained their strategy for today's task, rather than discussing each problem separately.

"I got stuck on the first problem, even though it looked like it would be easy because there are only a few rows, it was harder than the others we had done together because the table didn't start at 1.  I noticed the tables in the middle row started a 1 and 0.  I did these first because they were easier.  Then, I was used to thinking like this so I did the first two".

When I made this handout, I had not considered the placement of these tables.  Looking back, I think it worked out well to challenge the students thinking and also their strategic approaches to tasks.  Now, in  the future, I will know this challenge and also provide students who are having trouble accessing these types of problems with a hint to start in the second row.

I learn from my students every day, it is in these group share moments that I often learn the most.