This lesson has two stages with students working on different kinds of In/Out tables and building general rules for the tables. In the first stage, students write general descriptions for two functions. Though students may find recursive patterns for the tables, you'll want to push them to find a closed formula.
In the second stage of the lesson, students will write equations for six In/Out tables. Here, the focus is on having students write equations, focusing on symbolic notation, and helping students move back and forth between the In and Out columns. Students already have practice finding the Out when they have the In. Now they will also focus on finding the In when they have the Out value.
Today's opening will look at two different patterns and include a discussion about the patterns. You can begin class by having the first pattern up on the board. This activity is called Squares and Scoops and is taken from IMP's Year 1 textbook (page 40). The first pattern shows squares that are stacked to different heights. The pattern involves consecutive sums. For example, a 4 high stack would need 4 + 3 + 2 + 1 number of squares. There is also an accompanying In/Out table with the number of squares in stack dependent on the height of the stack.
Students may see the pattern very quickly but might have trouble describing it. This is my focus for the first part of the lesson: getting students to write a general description for how to find the number of squares for an n-high stack. This may be the first time students have seen this language, but let them know they will see it over and over again this year in Algebra 1. Talk to them about the importance of being able to generalize a pattern. How would they describe how to find the number of squares for a 100-high stack (100 + 99 + 98 + ... + 1 squares)?
Some students may notice the recursive pattern in the Out column of the In/Out table. If so, I applaud them on this noticing and then push them to find a functional pattern. Remind them that again if they were looking for that 100-high stack, they would have to find all the In values leading up to that number if they continued on with their recursive pattern.
The second pattern we will definitely explore is a multiplicative one. The pattern is developed from a permutation problem. Students are asked how many different ways can they arrange different numbers of scoops of ice cream on a cone. I like to introduce this problem using manipulatives. I also like to start with four scoops of ice cream. I use four different color mini-cubes. Ask students to show you how many ways they can arrange the "scoops." This is a fun way to be able to see permutations. You can ask students how they know they have found all the arrangements and ask them to share out how they kept track. I like this problem because it allows you to highlight different ways students think and is accessible to all students.
Next, I move my students back to the In/Out table and ask them again to write a general formula. Again, I applaud when they notice the recursive pattern but try to move them toward describing how they found find the Out when given the In (that is how they would find the number of ways to arrange, when given the number of scoops). If time permits, I show students factorial notation.
Teacher's Note: There are lots of places where you can allow this opening to go over the 25 minutes. If you find you want to spend more time on this piece, you could move the second part of the class to homework.
The next activity returns students to In/Out table work. This activity is called Another In-Outer and is on page 41 of IMP's Year 1 textbook. There are six In/Out tables. Half of the tables do not have simple numerical values. For example, one involves shapes and another involves pictures. All of the tables have missing values both on the In and Out sides. You can have students work in small groups or pairs to find the patterns and fill in missing values. For the tables that involve pictures, students will write descriptions for how to find the Out when given the In. For the tables that involve numbers, students will write algebraic equations. Highlight for students the beauty of being able to write an algebraic equation to describe a table.
A few things that I am looking for are:
Once students have written or algebraic descriptions for each table, I have them share out how they found them. As we discuss the task I want to help students highlight how they found the missing values in the tables. For example, one of the equations for a table is y = -3x. When the In is given as 13, students need to find the corresponding y value. When the Out is given as -30, students need to figure out what the In would be. I emphasize to students that they are able to "do" and "undo" their algebraic thinking.
I allow my students some time to reflect on their work with patterns at the end of class. It is usually interesting to hear from them which patterns they found most challenging and which patterns they liked the best. I have created a Pattern Reflection Sheet that asks them these questions.
Sometimes, after I review this type of reflection activity I like to share some of the results with the class. When the work has been particularly interesting, I like to type up some of the student responses and share them with the whole class at the beginning of your next lesson.
Note: This lesson is derived from the Interactive Mathematics Program. For more information about how I use IMP in my classroom, please visit my IMP Resource Page.
Program, I. (2008, June 3). Squares and Scoops. Retrieved from the Connexions Web site: http://cnx.org
Program, I. (2008, June 3). Another Inner Outer. Retrieved from the Connexions Web site: http://cnx.org