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?
Easing the Hurry Syndrome 
howardat58
April 27, 2015 at 8:55 pm
So I had a look at Laughbaum’s stuff. A case of a one track mind, I felt! One observation on the drip example. 1000ml at the start, 4ml per minute. How much after 0 minutes? Student says 1000.
How much after 1 minute? Student says 996. How much after 2 minutes? Student says 992 and can explain why. Teacher says So after 3 minutes there is 988 = 1000 – 4×3 left. Then this suddenly becomes a linear function f(x) = -4x + 1000, destroying the carefully constructed development. What is the matter with f(x) = 1000 – 4x ???????? Ah, but we don’t write it like that. ( !!!!!!!!! ). I tried to laugh, but I wanted to cry.
Regarding your lesson, at the point when the “ball is thrown upward from a height of 8ft”, I guess by a basketball player, the wind blows it sideways, and its trajectory is shown in the plot. This is a plot of height against sideways displacement, but the graph has TIME on the x axis, and the question is about the time at which the ball reaches the ground. How did TIME get in there? (Well, I know, but it would take several classes on the behaviour of projectiles and Newton’s laws to convince a skeptical student).
If the question had been “How far away was the ball when it landed?” then ALL these major difficulties are avoided. SMP ….. attend to precision……..
One more thing. I didn’t see a definition of a zero of a function. I guess they were supposed to know that a zero of a function is a value of the input which yields zero for the output.
jwilson828
April 28, 2015 at 9:30 am
I am grateful for your careful reading and for your suggestions. It makes me realize that while I sometimes feel vulnerable posting what we are doing because of your scrutiny, we are also reaping the benefits of your expertise with improved questions. Thank you!
howardat58
April 28, 2015 at 4:26 pm
Thankyou too. I really admire what you are doing. I was a little worried about the tone of my above comment.
About the parabola, I thought more about it, and saw that as the ball is moving at a constant velocity in the horizontal direction it is ok, with explanation, to measure its position in seconds. Why it is a parabola? They’ll just have to wait a bit !