CMP Digging Deeper Day 3

June 22, 2016


Algebra, a word that strikes fear in many and is the topic of many jokes about how irrelevant to real life high school mathematics is.  I was good at Algebra, or what I though Algebra was.  I could follow a set of "like" problems so well, that I spent the majority of class writing and folding elaborate origami notes after finishing early (which were occasionally intercepted and read aloud to the class for entertainment.)

Algebra: Function based verses Equation based?

In answering this question we began our work today with Thinking with Mathematical Models Problem 1.3.
Students are given a situation of building trusses with rods (see above picture.) This is a bridge with a length of 7 trusses using 27 rods.  Student think about different length trusses in terms of a pictorial, table, graph, and final an equation.  There are many ways to push algebraic thinking by just beginning with different ways of seeing the picture.


 

Truss	Rods
7	27
1	3
2	7
3	11

Looking at the picture, some students see that for each truss added, four rods are needed. Three to create the triangle and one more connecting rod (except the first one which is missing one (-1). Students may use this picture to find the equation R = 4T-1 without even using a table or graph (although they could certainly use them as well.) This change for each truss can be represented in the  picture, table, graph and equation.

The equation R= 4T -1 shows that for every truss added, there are 4 more rods (except the first)

That is not the only way to see the problem. Students can also count the triangles and multiply by 3 to get the rods and then add the connectors (which will always be one less than the trusses because the top is one less).  R = 3T + T -1.  This is equivalent and gives a powerful visual example as to why.  

They could also count the total number of triangles and subtract out the double triangle pieces (since they have 2 in common for each triangle. R= 3(2n-1) -2(n-1)

We looked at the connections between different representations.  We then examined an entire unit of 7th Grade Moving Straight Ahead without having them sequenced for us. We explore linear functions, but the one piece we had the most trouble with was the equations section using pouches and coins. Where does this fit in?  Then we read a article from Texas Mathematics Teacher explaining the 2 different methods often used to teach algebra, equations and functions.

Texas Mathematics Teacher

CMP is a functions based curriculum that connects solving equations with what students know about functions.  In the case of the truss/rod problem questions are asked if there are 27 rods, how many trusses can be build.  Students can use the table (construct values using rate of change and initial value), graph (find 27 rods on the y axis and find where the line is at 27), or equation (substitute 27 in and solve) to answer the equation that is build from this specific instance. This grounds solving equations in the relationships defined by the function in multiple ways with more meaning to students.

To end our evening, we were given several articles to think about including one on questioning and assessment (with an in-depth example exploring the circle area and circumference standard in 7th grade.)

Questioning our Patterns of Questioning

Assessing for Learning