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?

What do you notice on this page?

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

We checked in with students using some Quick Polls.

What do you notice about the answers for this first poll?

(I noticed that not all are x-intercepts.)

Students showed some improvement as we continued.

The answers are all x-intercepts.

We asked questions like …

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?