Staircase Problem Case Study

Staircase Problem presented the students with the problem of finding the number of blocks in a staircase, finding the pattern among the steps and finally determining a rule relating the number of blocks and the height of the steps. Mr. Solomon presented the students with paper ‘steps’ to manipulate and build the steps while working, Standards for Mathematical Practice 5-Using appropriate tools strategically, as well as paper for recording questions they had and a table to record answers. While working together, it was obvious that the students had gone through other lessons taught this way as they were interacting, discussing, explaining and questioning each other, Standards for Mathematical Practice 3-Construct viable arguments and critique

As it was seen, students quickly saw a pattern of adding the height of the blocks to the one before, and knowing they were ultimately looking for the rule, focused on finding the general rule instead of focusing on the patterns they could build, Standards for Mathematical Practice 8-Look for and express regularity in repeated reasoning. Ultimately, they were not spending enough time gathering data to develop a formula. In light of this, it may be more beneficial to provide time for the students to build and gather data before you present them with the idea of using the information to find any number thrown at

Students quickly found a pattern between the number of blocks needed given the number of blocks is known for the previous height. Identifying the pattern between two numbers is much easier than finding the computation to find a number. Several students noted that once you got the pattern started, you could find any number by simply adding the numbers continuously into the calculator, Standards for Mathematical Practice 8-Look for and express regularity in repeated reasoning. Clearly, this method will produce an accurate answer, unfortunately, given a large number, it would not be an efficient method. Additionally, they struggled a little when asked to find 215, they simply stated to use 214 to find 215. Finally, after Mr. Solomon got them to see that they didn’t know 214 and would still need to be able to find 215, the concept of a general rule seemed more understandable.
Getting the students away from the recursive rule and on to the general rule takes extra work. Starting with the recursive rule, the students will need to assign variables to define the relationship between the number of blocks and how high the steps are. Start by asking students, what are we looking for and what are using to find the answer? They should quickly see they are looking for the number of blocks at the current step, call it B, the number of blocks in the previous step and the height