# Tag Archives: MathNspired

## Piecewise Functions

We started a unit on piecewise functions in Algebra 1 with the following leveled learning progression:

Level 4: I can sketch a graph of a piecewise-defined function given a verbal description of the relationship between two quantities.

Level 3: I can interpret key features of a piecewise-defined function in terms of its context.

Level 2: I can determine the domain and range of a function given a context.

Level 1: Using any representation of a function, I can evaluate a function at a given value of x, and I can determine the value of x for a given value of f(x).

We started with an opener to ensure that students were successful with Levels 1 and 2 so that we could reach our target (Level 3) during the lesson.

Questions 1-4 gave us evidence that most students could evaluate a function at a given value of x and determine the value of x for a given value of f(x) using any representation of a function.

Questions 5-6 gave us evidence that our students needed more support determining the domain and range of a function given a context.

Without showing the results from the opener that are pictured above, we talked all together about the context, reading the graph, but not explicitly discussing the domain and range. When we sent the question as a Quick Poll, we saw evidence that more students could determine the domain and range of a function given a context.

We probably could have spent more time on Level 2 in class. But we didn’t. Instead, we had to provide additional support for Level 2 outside of class, through homework practice, zero block, and after school help.

To open our discussion of piecewise functions, we showed this picture from the front of the Jackson airport parking garage.

What do you notice? What do you wonder?

Students wrote down a few observations individually, then shared their thoughts with a partner. We selected some for our whole class discussion. In particular, it was helpful that one student specifically said, “pay depends on time”.

How much would you pay for a certain number of hours of parking?

Similar to an idea from the Internet Plans Makeover, we asked students to choose a number between 0 and 24. If you park that many hours, how much will you pay?

We asked students to check work with a partner before submitting. The result wasn’t quite as disastrous as when we tried the Internet Plans Makeover.

Showing the grid helps some.

And having a whole class discussion about the stipulations of the sign helped even more.

Did anyone choose a non-whole number?

What would happen if you parked for 1.5 hours?

Or 2 hours and 20 minutes?

We also used the Math Nspired lesson Dog Days or Dog Years with good success. What we are still trying to decide is which comes first … the structure from the Dog Days or Dog Years lesson about creating piecewise functions? Or the less structured conceptual introduction from the cost of parking at the airport? I’m not sure it’s wrong (or even better) to start with either one. But we still wonder, as the journey continues …

Posted by on March 30, 2015 in Algebra 1

## Systems of Linear Equations

We are working on Systems of Linear Equations in our Algebra 1 class.

The first day, our lesson goals were the following:

Level 4: I can create a system of linear equations to solve a problem.

Level 3: I can solve a system of linear equations, determining whether the system has one solution, no solution, or many solutions.

Level 2: I can graph a linear equation y=mx+b in the x-y coordinate plane.

Level 1: I can determine whether an ordered pair is a solution to a linear equation.

We used the Mathematics Assessment Project formative assessment lesson Solving Linear Equations in Two Variables to introduce systems.

Then we asked students to draw a sketch of a system of equations with one solution, a sketch of a system of equations with no solution, and a sketch of a system of equations with many solutions.

We used the Math Nspired activity How Many Solutions to a System to let students explore necessary and sufficient conditions for the equations in a system with one, no, or many solution(s). We used a question from the activity so that students could solidify their thinking about the relationship between the equations for different solutions.

We sent the questions as Quick Polls so that we could formatively assess their thinking and correct misconceptions.

We like using this type of Quick Poll (y=, Include a Graph Preview) while students are learning. The poll graphs what students type so that they can check their results and modify the equation when they do not get the expected result.

(Note: on the second day, we asked again for students to create a second equation to make a system with one, no, and many solution(s), but the given equation is in standard form.)

On Day 2 of Systems, the learning goals are slightly different.

Level 4: I can solve a system of linear equations in more than one way to verify my solution.

Level 3: I can create a system of linear equations to solve a problem.

Level 2: I can solve a system of linear equations, determining whether the system has one solution, no solution, or many solutions.

Level 1: I can determine whether an ordered pair is a solution to a linear equation.

We started with a task from Dan Meyer’s Makeover series, Internet Plans. Part of this makeover was a complete overhaul of the context. Students write down a number between 1 and 25. They view this flyer (from Frank Nochese).

And then they decide which gym membership to choose, using the number they wrote down as the number of months they plan to work out.

When I saw the C’s and B’s on the number line on the blog post, I actually thought that the image was beautiful – a clear indication that one plan is better than the other two for a while, and then another plan is better than the other two. This image absolutely leads to the question of which one becomes better when – and thinking about why plan A is never the better deal.

Our students started calculating. Several asked what \$1 down meant before they could get a calculation. Then they shared their work on our class number line.

We were dumbfounded at the results. Really. It was hard to know where to start. We were prepared to help student develop the question with correct calculations. We hadn’t thought about what we would do when students couldn’t figure out which plan was the better deal for a specific, given number of months.

What would you do next if this is how your students answered?

We started with the lone C at 4 months, hoping someone would claim it and share his calculations. He did: I divided \$199 by 12 months and then multiplied by 4 to get the cost for 4 months.

Another student said he couldn’t do that because you had to pay \$199 for 12 months no matter how many months you actually used it.

Another student said that A was the better deal because they were planning to get every other month free by working out more than 24 days each month. We discussed how realistic it is to work out at a gym 24 days a month.

We did eventually look at a graphical representation of each plan. And talked about what assumptions had been made using these graphs.

Can you tell which graph is which plan?

Can you write the equations of the lines?

In the first class we never made it to actually calculating the point(s) of intersection (class was shorter because of a pep rally), but other classes did calculate the point(s) of intersection.

It’s not a waste of time to think about Plan C costing \$199 for 12 months.

But then we really should have created a piecewise function to represent the cost for 13-24 months as well.

The number lines weren’t much better in subsequent classes.

So maybe the students got to practice model with mathematics, just a little, even though we have a long way to go before students will be able to say, “I can create a system of linear equations to solve a problem”.

The teachers were definitely reminded of how much work we have to do this year, as the journey continues …

## Completing the Square to find the Center and Radius of a Circle

From CCSS-M: Expressing Geometric Properties with Equations G-GPE

Translate between the geometric description and the equation for a conic section

1. Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation.

I like the way this standard is worded. I am glad that it is not enough for our students to write an equation of a circle – instead, we must provide an opportunity for our students to derive the equation of the circle (that will be another post), and we have a reason to complete the square (to find the center and radius, which is the topic of this post).

As I was preparing this lesson, I wracked my brain to think of how my students could discover “completing the square” instead of me just giving them an algorithm to follow. I decided to try starting with a circle in center-radius form, expanding that to the general form (or standard form, depending on which textbook you are using) – and then letting them figure out how to go backwards. Of course we had to spend the first few minutes of the lesson going back over how to expand binomials (they were only in algebra last year) – but I didn’t just tell them that, either. I posed a quick poll to start the lesson:

While they were answering, I was monitoring the student responses.

I often go on and show students the correct answer (indicated by a green bar) when I show the results to a Quick Poll. In this case, however, I unchecked “show correct answer” before showing the class results. When the vote is split like this in our classroom, my students have learned this means that they have to get up, find another student in the room, and try to convince that student why they chose what they did. I send the poll again to see if anyone has been convinced to change the response.

And as you can tell, the results were still not that great. So they had to find a different person to convince of their answer. And I sent the poll one more time. And all but a few very stubborn boys got it correct.

After the Quick Poll, we used the TI-Nspire CAS – for students to enter into the practice of look for regularity in repeated reasoning.

I felt like they really needed to have a handle on expanding binomials before they could be proficient completing the square.

Once they had determined how to go backwards (the phrase “divide by 2 and square” was their phrase, not mine),

we looked at a visual representing of completing the square from MathNspired.

My final Quick Poll was evidence that students were beginning to make sense of why and how we were completing the square to find the center and radius of a circle.

Even though they will eventually need to do this without technology, we use the technology while we are learning to make more sense of the mathematics.

And so the journey continues ….

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Posted by on May 11, 2013 in Circles, Coordinate Geometry, Geometry