Advice for Getting Started with SBG

One of our teacher ed grads emailed one of my colleagues in search of SBG resources for a pre-algebra course he's developing for next year. He wrote:

When I had you for ____, you graded our assessments using a form of standards based grading. I remember receiving a paper back that listed what standards I had mastered and which ones I still needed to work on. I have wanted to try using a standards based grading system ever since I saw it in your classroom. Would you have any resources that you would be willing to share with me?

My colleague replied with several tips and resources, including Matt Townsley's growing list of scholarly articles SBG. He also Cc'd me, and while I was drafting my own response I realized it might be better to write it as a blog post. So here it is, for what it's worth: my list of suggestions and resources for getting started with SBG in the math classroom.

Transformations vs. Order of Operations

The following question was raised by one of the work groups in class today: Why is the order of performing transformations different than the order of operations?

We are studying function transformations like these:

From College Algebra by Coburn & Herdlick

Our explorations in class have supported our book's claim that function transformations must be applied in the following order:

1. horizontal shifts
2. reflections
3. stretches/compressions
4. vertical shifts

Does that sequence conflict with the order of operations? What a great question! Let’s explore it using the absolute value function f(x) = |x| as our parent function and using

g(x) = -2|x-1| + 5.

to define the intended transformation. Consider what happens to a point on the graph of y = |x| under this transformation. Let's use the point (5,5). Where does it end up after the transformation?

To find out, we evaluate f(5)= -2|5-1| + 8. This requires the following sequence of calculations:

1.     5-1 = 4.         that’s the x-1 piece; there’s the horizontal shift*
2.     |4| = 4.           that’s |x-1|; we have just applied the parent function, |x|.
3.     -2*4 = 8.       that’s -2|x-1|; there’s the reflection (-) and vertical stretch (by 2).
4.     8+5 = 13.      that’s -2|x-1|+5; there’s the vertical shift.

*But why is it a rightward shift? That's for another post.

Has the order of operations been maintained?

It is probably easiest to see if we use GEMA rather than PEMDAS to track the order of operations. They reflect the same underlying order of operations, but GEMA seems to produce fewer order of operations misconceptions (sounds like a PhD thesis topic to me!)

GEMA = Grouping symbols first, then Exponents, then Multiplication (and Division, from left to right), and finally Addition (and Subtraction, from left to right).

Now let’s step through GEMA:

G: Grouping symbols. The absolute value bars a type of grouping symbol (so are parentheses and brackets, square root symbols, and even the horizontal line that separates the numerator and denominator in a fraction). First, we work on the expression inside the grouping symbols (absolute value bars). There is only one operation to do in there: subtract 1 (step 1). Now we apply the absolute value bars (step 2), at which point the Grouping symbols are gone and we move on to….

E: Exponents. No exponents to deal with this time. Move on to….

M: Multiply (or Divide): With the || bars gone, the function now reads: f(4) = -2*4 + 5. We multiply by -2 next. This creates the reflection (step 3a) and stretch (3b).

A: Add (or Subtract): Only one thing left to do! (step 4).

Conclusion: It appears the transformations sequence is consistent with the order of operations.

I'm convinced. Are you?

Finding the "Hidden Wows"

I was working with some middle school teachers-to-be who are noticing some of the aspects of algebra we tend to take for granted after years of practice and application, and they are noticing how difficult it can be to anticipate the strategies and struggles of students who are just learning algebra.
 
One way I get around that is to try to hold myself still with a problem for a bit and look for the hidden connections. With kids, I might tell them we're looking for the Hidden Wows.

Suppose I am preparing a lesson for sixth graders to introduce problems of this form:

Solve: ax = b

 My planning might start out sounding a bit like this:

Transformations

Or: Is this the kind of problem where you can have more than one answer?

There are a select few topics in algebra that can get me tied up in knots. One of those is function transformations--you know, horizontal & vertical shifts, vertical stretches & compressions, and horizontal & vertical reflections--specifically, those that involve a multiple transformations.

Here's the task I got hung up on the other day:

The graph of a function f is shown. Sketch the graph of y = 2f(x+1) - 3.

On these types of tasks, it's not the transformations themselves that get me. There are three transformations at work here, and I can describe them easily enough:

1. Horizontal shift one unit left (because of the x+1).
2. Vertical shift three units down (because of the -3).
3. Vertical stretch by a factor of 2 (basically, double the y-values). 

The part that gets me hung up, at least when I have not done these kinds of problems in a while, is... in which order should I apply the transformations?

Because it makes a big difference! To illustrate, let's trace where the point (-2,1) ends up if we shift-then-double vs. double-then-shift:

• Shift-then-double: (-2,1) --left1--> (-3,1) --down3--> (-3,-2) --double y--> (-3,-4).
• Double-then-shift: (-2,1) --double y--> (-2,2) --left1--> (-3,2) --down3--> (-3,-1).

See? (-3,4) and (-3,-1).. we end up in two different spots.

This is not "the kind of problem you can have two different answers to" (Cathy Humphreys; clipped from one of the videos in this book).

When will I use this? (Laws of exponents?)

The Question: 

I received the following email from my Mom the other day. (She's been a middle school teacher since the early '90s). She wrote:

Jon, I received The Question today from one of my 7th graders today: "When am I ever going to need to know how to do this???" He is working on zero and negative exponents and had to solve problems like this:

Write an equivalent expression for 

The answer was

He understands how to do the problems--just wants to know why he needs to know, why it's not a waste of his time.

My Response:

Good questions! I wrote a blog post about the matter... let me know what you tell him. I'd love to hear how he responds.

http://profjonh.blogspot.com/2014/05/when-will-i-use-this-laws-of-exponents.html

That Blog Post (AKA, This Blog Post):

My colleague answers that question this way:

I don't know when or if you will ever need this particular concept. It depends on what you do with your life and what technological advances are made in the future. But you know what you will need to be able to do, regardless? You will need to problem solve. You will need to think critically (reason and prove). You will need to be able to communicate quantitative thinking to others. You will need to use representations to support your thinking and share your thinking. And you will need to make connections in order to consolidate your understanding. Mathematics is a discipline that provides opportunities to practice and strengthen all of these skills. So, as we solve for x, I want you to monitor your thinking because that's what's really important.

(Read his full blog post at: http://deltascape.blogspot.com/2012/05/when-will-we-ever-use-this.html)

He's talking about the Process Standards from NCTM (2000), which are now reflected in CCSS Standards for Mathematical Practice:

But just in case that argument doesn't satisfy your young math skeptic, you might want to share the following list of websites with him (see below). It turns out rational exponents are at the heart of a great many useful scientific formulas, not to mention a lot of really cool (and really beautiful) mathematics!

Good luck!
- Jon

Seven applications of rational exponents:

• Chemistry:
     http://chemwiki.ucdavis.edu/Physical_Chemistry/Kinetics/Rate_Laws/The_Rate_Law
• Modern Physics (Albert Einstein):
     http://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalence#Mass-velocity_relationship
• Classical Physics (Isaac Newton, born on Christmas Day, 1642!):
     http://www.logrog.net/users/lkovach/physics/Universal%20Gravitation/Days%201-3.pdf
• Rocket Science:
     http://exploration.grc.nasa.gov/education/rocket/isentrop.html
• Astronomy:
     http://scioly.org/wiki/images/c/c6/Formula_Sheet.pdf
• Wall Street investment trading:
     http://fr.accuracy.com/visuels/28t-1271338388.jpg
• Renewable Energy (Wind):
     http://www.raeng.org.uk/education/diploma/maths/pdf/exemplars_advanced/23_wind_turbine.pdf

My Intermediate Algebra Targets (Su13)

I used these targets as part of a my standards-based grading effort in a summer offering of our Mth110 course (Intermediate Algebra) at GVSU.

They are not perfect. Some changes I would make for next time include:

• Reducing the overlap between certain pairs of targets (Target 1.2 and Target 1.14 are examples of that).
• Splitting some targets that merge two concepts or skills in one target (Targets 1.5 and 1.11 are two examples of that).

But overall these worked well and were well-received by my students.

Feel free to use or modify for your own non-commercial educational purposes. For all other purposes, please contact me and let me know what you have in mind.

Intermediate Algebra Learning Targets

...why rubrics? (part 1)

Why I use rubrics, #1: Rubrics help me to focus on proficiencies, not deficits, and they support my efforts to give feedback that can be used to improve future performances.

In earlier posts, I discussed the "In the cups" performance task and my use of a proficiency-based assessment system in my intermediate algebra course. In this post, I'll share an example to illustrate how the use of an evidence-based rubric has supported my implementation.

I can show you more than that

In a previous post, I talked about my grading system based on the mantra, "Show me what you can do." Here's a quick example of a student's response to a quiz item that shows why this proficiency-based approach feels so right:

One of my learning targets was:

___1.5. I can solve linear systems and represent the solution symbolically and graphically.

I stated that objective at the top of the quiz, along with several others, then gave the following task:

The solutions to the following system of equations are provided. Show that you can use the elimination and substitution methods (use each one once) to solve these problems.

{y = 3x+6
{2x + 4y = -4              solution is (-2,0)  

{7.5x - y = 10
{15x - 4y = 10            solution is (2,5)

What it Means to Solve (Again): Finding Wormholes

In two previous posts, I have explored what it means to solve linear inequalities and what it means to solve quadratic equations. The latter post describes a classroom moment in which my students and I contemplate what it means to find and visualize the (complex) solutions of a quadratic equation that has no real solutions. I wrote:

"A picture formed in my mind of an invisible, ethereal, wormhole-style thread binding the two curves. How could we represent the functions so that the "intersection" would be visible?"

This post documents my post-class exploration of that issue.

What (else) it means to solve

In class yesterday, I had an opportunity to reinforce what it means to solve equations and inequalities (a connection I've discussed in a previous post). My students and I were working on a task that required that we verify that x = 2 and x = 0 are the two solutions of the quadratic equation -2x^2 + 4x + 1 = 1. How can we do that? A chorus of replies: "plug them in and check." 

"Right," I said, "We need to see if we have found both (all) of the values that make the equation true."

Learner #1 raised his hand and asked, "Doesn't it also mean finding the points where the parabola intersects the line y = 1?" I know from a previous conversation (described in this previous post) that this was how he learned to solve quadratics in high school: enter the parabola into Y1, enter the other expression into Y2, and use the CALC -> INTERSECT to solve.

Created with Geogebra

So I was happy to affirm the connection.

A short time later, in the same lesson, we were exploring the function f(x) = (x-3)^2 + 5. We had determined that because the parabola opened upward and had its vertex at (3,5), so there must not be any solutions to the quadratic equation f(x) = 1. Furthermore, we agreed, the equation f(x) = 5 would have just one solution (at the vertex) and f(x) = k for any k > 5 would have two real solutions.

Created with Geogebra

Then Learner #2 offered: "So if we ever find that there's no solution, we can conclude that the curves do not touch, right?" It was a nice extension of Learner #1's remark.

But it depends a bit on what you mean by "no solution," doesn't it? The situation is exactly analogous to the 1st grader who asks the question, "Teacher, what's 1 minus 3?" to which the teacher response helpfully, "Honey, you can't do 1 minus 3." I like to imagine a precocious 1st grader reply: "Yeah, maybe you can't, but I can! It's negative 2!"

In the Cups

Our department has a shared e-folder where we can share tasks we develop for use in specific courses. I've used it a lot this summer to support my planning for the intermediate algebra course I'm teaching. The activities there range from routine to inspiring--and both types have been useful!--which reminds of a quote I saw recently: Good teachers borrow, great teachers steal.

So I have been drawing heavily on the excellent activities there. Some tasks I use almost verbatim, and others I modify to suit my own purposes. One of the activities I found was a nice math modeling activity called "In the Cups". It is designed to support students with creating and using linear models to make predictions and solve problems.

The original document presents the following data to students:

 

Number of Cups	Height of stack (in cm)
1	7
2	7.6
3	8.2
4	8.7
5	9.2
7	10.1
8	10.6

The task proceeds to ask students to predict the height of 6-cup, 10-cup, and 25-cup stacks, and eventually invites students to find a model expressing the stack height as a function of the number of cups in the stack.

I noticed that the data in the table do not quite conform to a perfectly linear relationship--notice that the unit rate of change varies from 0.6cm to 0.5cm per cup--which got me thinking. I wondered if the author had introduced some random variation into the stack heights presented in the table in attempt to make the task seem more authentic. Or perhaps he or she had really measured some cups and the data were authentic approximations. Either way, I liked it.

I started measuring a stack of dixie cups I had in my office, then I realized that by doing the measuring myself, I was about to rob my students them of an opportunity to do some critical thinking. So I decided to replace the table of values with a photograph. 

That one picture (of 27 cups) contains all of the information needed to model the relationship, and it invites students to consider questions about the accuracy of the approximation. Replacing the table with the photograph felt like a more authentic way to present the task to my students.

What it means to solve

"Do we understand" was on my mind as I was grading intermediate algebra quizzes over lunch today. In particular, I wondered: do my students really understand what it means to solve an equation or inequality? It was clear enough that they knew how to do it, but do they know what they are accomplishing in the process?

One of the learning targets I have identified for my students reads:

I can explain what it means to solve equations, inequalities, and systems, and I can use this knowledge to check my answers for reasonableness and correctness. 

I'll be honest: I really love that learning target. It was inspired based on one of the Common Core State Standards for Mathematics, specifically, one of the Grade 6 Expressions and Equations standards:

CCSS-6.EE.5 [Students will] understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? ... (emphasis added)

That concept is fundamental to the work we do in algebra. Consider the wide array of situations where the sole directive is "solve". Come to think of it, one of my old doctoral advisors once described "solve" as one of the four universal directives of algebra homework sets. (Can you come up with the other three? Answers appear at the end of the post.)

In the case of "solve", the meaning is universal and boils down to "find the set of [thing(s)] that make this [thing] true". Consider: 

• "Solve 3x+4=7" means "Find the values of x that make the equation true."
• "Solve {3x + y = 4, x - 2y = -1}" means "find the ordered pairs that simultaneously make both equations true."
• "Solve y'(x) - y(x) = 0" means "find the family of functions that make the differential equation true."
• "Solve Ax = λx" means "find the real numbers λ and corresponding vectors x, called the eigenvalues and eigenvectors of A, that make the matrix equation true."
• "Solve Ax = 0" means "describe the set of vectors x, called the null space of A, that make the matrix equation true."