# Category Archives: Algebra 1

## Connecting Factors and Zeros

NCTM’s Principles to Actions suggests Mathematics Teaching Practices for teachers. Two of those are the following.

MTP 1 Establish mathematics goals to focus learning

MTP 6 Build procedural fluency from conceptual understanding

If the goal for students is to use the factors of a quadratic function to determine its zeros, what concepts must students understand to meet that learning goal?

Our team wrote this leveled learning progression for our lesson.

Level 4: I can factor a quadratic function.

Level 3: I can use the factors of a quadratic function to determine its zeros.

Level 2: I can expand the product of two binomials.

Level 1: I can solve an equation in one variable.

Level 1: I can determine the zero(s) of a function from the graph of a function.

We decided to first ensure that students know what a zero is, and we checked this is more than one way on the opener for the day. (See this source for similar Level 1 problems.)

Students had to place a point at the zero of the function.

Almost all students were able to note that the point of interest is where the graph intersects the x-axis. Students had to name the coordinates of the zero of the function, which about half could do.

And then students had to answer a question about a zero in context. A few more than half could do this.

We decided that students also need to be able to solve an equation in one variable.

Which they could easily do. And we also decided that if students are going to meet the learning goal, they are also going to have to be able to multiply binomials. Which you can tell from the results that they could not easily do (Q8 and Q9).

In the lesson, we started with the zeros of a linear function.

What do you notice?

If I give you a similar equation, can you tell me the zero? If I give you a similar equation, can you tell me the zero?

We checked in with students using some Quick Polls. (I noticed that not all are x-intercepts.)

Students showed some improvement as we continued.

How can we tell that (-6,0) is the correct choice using the equation?   We spent a long time on linear functions. Some might think we spent too long.

Then we looked at a quadratic function.

And we related the linear factors to the quadratic visually.

This is part of a Math Nspired activity called Zeros of a Quadratic Function, where there is a lot more flexibility in changing the factors. Our leveled learning progression for the second lesson changed a little:

Level 4: I can factor a quadratic function.

Level 3: I can use the factors of a quadratic function to determine its zeros, and I can use the zeros of a quadratic function to determine its factors.

Level 2: I can rewrite a quadratic function given in factored form to standard form.

Level 1: I can determine the zero(s) of a quadratic function from the graph of a function.

When we checked for student understanding during the opener of the second lesson, we saw that students were able to determine the zero(s) of a quadratic function from the graph of a function.

Lots of students were at Level 1, determine the zeros when given the graph and the equation.

Not as many were at the target – but definitely more than had reached it the day before.

We have worked to build procedural knowledge from conceptual knowledge in our unit on Zeros and Factors. Our standards say that we want students to “Factor a quadratic expression to reveal the zeros of the function it defines”. The standards don’t say that we want students to factor a quadratic expression just for the sake of factoring.

What opportunities are you providing your students to concentrate on relationships rather than just results?

Posted by on April 27, 2015 in Algebra 1

## SMP8: Look for and Express Regularity in Repeated Reasoning #LL2LU

We want every learner in our care to be able to say

I can look for and express regularity in repeated reasoning.

CCSS.MATH.PRACTICE.MP8 But what if I can’t look for and express regularity in repeated reasoning yet? What if I need help? How might we make a pathway for success?

Level 4

I can attend to precision as I construct a viable argument to express regularity in repeated reasoning.

Level 3

I can look for and express regularity in repeated reasoning.

Level 2

I can identify and describe patterns and regularities, and I can begin to develop generalizations.

Level 1

I can notice and note what changes and what stays the same when performing calculations or interacting with geometric figures.

What do you notice? What changes? What stays the same?

We use a CAS (computer algebra system) to help our students practice look for and express regularity in repeated reasoning. What do we need to factor for the result to be (x-4)(x+4)?

What do we need to factor for the result to be (x-9)(x+9)?

What will the result be if we factor x²-121? What will the result be if we factor x²-a²?

We can also explore over what set of numbers we are factoring using the syntax we have been using. And what happens if we factor x²+1? (And then connect the result to the graph of y=x²+1.)

What happens if we factor over the set of real numbers? Or over the set of complex numbers? What about expanding the square of a binomial? What changes? What stays the same? What will the result be if we expand (x+5)²? Or (x+a)²? Or (x-a)²?  What about expanding the cube of a binomial?  Or expanding (x+1)^n, or (x+y)^n? What if we are looking at powers of i?  We can look for and express regularity in repeated reasoning when factoring the sum or difference of cubes. Or simplifying radicals. Or solving equations.

Through reflection and conversation, students make connections and begin to generalize results. What opportunities are you giving your students to look for and express regularity in repeated reasoning? What content are you teaching this week that you can #AskDontTell?

[Cross-posted on Experiments in Learning by Doing]

## 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 Equations – Take 3

What do you do with Systems of Equations in a high school Algebra 1 class?

Our students have some experience with systems from grade 8, but not as much as they eventually will with full implementation of our new standards.

NCTM’s Principles to Actions lists Establish mathematics goals to focus learning as one of the Mathematics Teaching Practices. We can tell that having a common language to talk about what we are doing is helping our students communicate to us about what they can and can’t (yet) do.

We started our unit on Creating Equations & Inequalities with the following leveled learning progression and questions:

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

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

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

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

NCTM’s Principles to Actions lists Elicit and use evidence of student thinking as one of the Mathematics Teaching Practices. We need to know what students are thinking so that we can move their thinking forward. We created a leveled learning formative assessment so that we could see where students are.

Level 1: I can determine whether an ordered pair is a solution to a linear equation.  We have this y= Question configured to generate a graph preview. The equations students enter graph as they are typed, so the students are able to check their thinking as they go. (Note that we don’t have to configure the question to graph the equation as it is entered; we are choosing to do so while students are learning.)

The following are the results from one of our Algebra 1 classes. This is my favorite question.

How can we use the equation to know whether it contains the point (2,3)? How can we use the graph to know whether it contains the point (2,3)? What happens when we give students a point & ask them to create two equations that contain the point?

Do they know that they are creating a system?

Does it help them to know that eventually we will give them the equations and ask them for the point?

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

And results from the second question:

Level 3: I can solve a system of linear equations. What’s significant about the green point?

What does it have to do with the given equations? At least half of our students don’t yet understand what the solution is. And only a very few are able to solve the system from the question on 3.3, which is okay. This is the first day of the unit. We have the information we need to know how to proceed with the lesson. I wrote about some of the tasks we used on the first two days of the unit here.

On the third day, we tried Dueling Discounts from Dan Meyer’s 101 Questions, which went better than Dan’s Internet Plans Makeover.   And more: Maybe it’s time to generalize our results?

On the last day we tried Candy & Chips from 101 Questions, which also went well.

NCTM’s Principles to Actions lists Build procedural fluency from conceptual understanding as one of the Mathematics Teaching Practices. We know we aren’t there yet, but we are definitely making progress as the journey continues …

## 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 …

## Graphing Stories

How do you and your coworkers effect change in the classroom? Two years ago, I asked our principal whether we could schedule a geometry class during first block with four teachers and about 25 students. It was our first year to implement our new CCSS Geometry standards, and we needed to try it together. I have learned over the years that it doesn’t hurt to ask – he might say no, but he might also said yes. Well, he said yes, and as you can imagine, sharing a class together has been important for us both as learners and as teachers. As one of the teachers reflected recently, “Participating in a class all year with a team of teachers is the best professional development I have ever had.” Last year, our Algebra 2 team had a shared class to implement their new standards, and this year, our Algebra 1 team shares a class. I visit as often as I can.

We are building our course as we go, using all sorts of resources. We are using the framework from EngageNY, and we are using some of the activities and tasks in their lessons. We have started with a unit on Graphing Stories.

On the first day, we used Growing Patterns from an NCTM Article Coloring Formulas for Growing Patterns.

Lesson Goals (written with Jill Gough this summer):

Level 4: I can represent the number of tiles in a figure in more than one way and show the equivalence between the expressions.

Level 3: I can represent the number of tiles in a figure using an explicit expression or a recursive process.

Level 2: I can apply patterns to predict the number of tiles in a later figure.

Level 1: I can describe the pattern and draw a figure before and after the given figures.

How do you see the pattern growing?

How many tiles are in H(5)? H(100)? H(N)?

Some students completed a table of values, and some students drew a graph.

On the second day, we used a Mathematics Assessment Project formative assessment lesson, Interpreting Time-Distance Graphs, with a focus on rate of change.

Lesson Goals:

Level 4: I can calculate average rate of change from a graph and a table.

Level 3: I can calculate average rate of change from a graph or a table.

Level 2: I can match a distance-time graph with a story and a table.

Level 1: I can annotate a graph using away from home and towards home, how fast and how slow. On the third day, we used Graphing Stories Video 2 to begin the lesson.

Lesson Goals:

Level 4: I can create a believable story for a given graph.

Level 3: I can calculate average rates of change for an elevation graph.

Level 2: I can create an elevation graph from a video, labeling the axes with appropriate units of measure.

Level 1: I can identify time intervals for each piece of an elevation graph.

Level 1: I can calculate the slope of a line.

Whose graph is believable?

We used additional scenarios from Engage NY Algebra 1, Module 1, Lessons 1-2.

There were more videos and scenarios on Day 4 from Engage NY Algebra 1, Module 1, Lessons 3-4.

The lesson from Day 5 comes from David Wees’ webinar at the Global Math Department on Strategic Inquiry.

We have read Timothy Kanold’s blogpost on Leaving the Front of the Classroom Behind. And we are trying. And we are most grateful to our administrators for letting us try it together before we have to try it alone.