Flickerbill's Growth

Last week, we completed our investigation on why do we need algebraic expressions. 

Most students are able to find the pattern when the sequence is chronological. When the sequence skips values and jumps from 10 to 100 things begin to change. It is not ten times as much as the value for term 10. 

Here you can sort of see how some students were postulating that the value for term ten should have doubled what the value was for term five (they are highlighted yellow) but then quickly the student could see that actually that was not the case. 

So we tried using the comparison of terms 2, 4, and 6. You can see those highlighted in orange. The values are not increasing by doubling or tripling. 

After more discussion, and much encouragement to stop looking just at the column of the height and its numbers (the recursive pattern) but to look across the columns to the co-variant values. The age and height correspond in value to each other.

We truly felt accomplished after quite a few students were able to determine the algebraic expression!!! Lots of high-fives and Woot Woots were going on!

Staircase problem day two

Students were presented with another staircase problem. This time, their task was to determine how many sides are present based on the number of stairs. Students created an input/output table showing the function.

Students used craft sticks to build the staircase, counting the sides as they went.

As the students approached ten, they realized that it was pretty hard.A lot is going on here, but there are a few takeaways.

1. For the most part, students agreed on the outcomes for terms 1-5.

2. Students started to see a pattern emerge.

3. When students "jumped" from 5 to 10, the output for ten was varied. Some students thought that they could use what they knew about the fifth term to help them determine the tenth term.  Another student pointed out that if we look at the second and fourth term's outputs, they are not just "doubling" them.

4. A student noticed the terms were all factors of its corresponding output. This led to another pattern... 1X4=5, 2X5=10, 3X6=18, etc... 

5. Students recognized that if we could figure out an expression or a rule, we could easily compute the 100th term without having to build with the sticks or making a really elaborate table.

We will keep thinking on how we can create a rule, using an expression, to help us compute with greater values.

Introducing variables and expressions

Students were introduced to a series of "staircases". You can find detailed ideas on how to use in the classroom, across multiple grade levels, here .  You can tell that I am an exceptional artist when it comes to representing something in 3-D.

Students went to work. They recorded the height of the staircase and the number of cubes that matched each term. While most students used cubes, many were able to establish some sort of pattern pretty quickly and abandoned the cubes.

It is hard to see here but a student is trying to draft an expression in word form. 

Another student suggests that we try using a variable to express the pattern. We tried using one for n but it was not quite something that worked when n changed. 

I introduced how we can use a variable, or a letter, preferably n to help us communicate an expression. The wheels were turning. We shared our ideas as a class and tested them out. 

A few students were beginning to make connections as to how this could be represented and that it might take more than one operation to do so. The investigation will continue!

Please note, the learning target was purposefully absent. I have moments where I think it is helpful and can support student understanding. However there are times that I think it blows the whole inquiry down the drain.  Want to consider another perspective? Read this blog post here: Joe Bower's Blog