I include Warm ups with a Rubric as part of my daily routine. My goal is to allow students to work on Math Practice 3 each day. Grouping students into homogeneous pairs provides an opportunity for appropriately differentiated math conversations. The Video Narrative explains this lesson’s Warm Up- Transformations of Polynomial Functions which asks students to critique a fictional student's use of transformations.
I also use this time to correct and record the previous day's Homework.
The foundation for this lesson is the polynomial parent functions we used in the previous class. I give students clear expectations for graphing f(x) = x2, f(x) = x3, f(x) = x4, and f(x) = x5. I have found that if you don’t give clear instructions, many students will draw graphs that won't be useful. Minimally I expect the axes and intervals (unless it is an interval of one) to be labeled. For a transformation, I expect them to graph any critical points, such as the vertex (where applicable) as well as points one unit away from that critical point. These points are used to get an accurate picture of any stretch or shrink (Math Practice 6).
As the students graph each of the parent functions, I note the flatness of the middle of the graph as the exponent increases. This often generates good conversation as they talk about why this would happen.
In the previous chapter, we looked at the transformations of f(x)=x2 using the concept of area. Today, we are extending this to cubic functions using volume. The first task asks students to create an algebraic model representing the volume of a cube.
Next, we look at the graph of this model. It is important to note the appropriate domain and range of this function. We do a think-pair-share highlighting the difference between the generic domain and range, and the context-specific version. When they graph the function, I have them use a dotted line to show the parts of the graph that don’t apply to this situation. This allows students to see the total shape of the graph while reminding them of what portion is viable to our model. It is also extremely important to label the x-axis as the side length and the y-axis as the volume as this is fundamental to the conceptual understanding as we build onto this function.
The first transformation asks the students to graph f(x)=2x3 as well as describe it in terms of cubes. In this case, the students draw a picture of the cube(s) and then graph the function. Students may draw two cubes or a prism where one side is twice as long as the other two. I highlight these different solutions as we talk about it as a class (Math Practice 2). Watch for students who double all of the sides. I have them plug in side length to make sure that their model is accurate.
Next, students are given the function f(x) = (x+1)3. In this case, all of the sides of the cube would increase by one. The volume at any given point would be equivalent to the volume of a point on the original cube one unit to the right. This is a great visual representative of the fact that we go the opposite direction in horizontal translations.
The final problem in this section is f(x)=1/2x3 +1. This problem is half of a cube plus a one unit cube added on to the side. This one may be a bit of a challenge for students to model using cubes. The “+1” unit will need to be discussed. Some will want to add it onto the cube as a side length. Some will want to draw a line segment or a square. One way to deal with this is to remind them that 1 is equivalent to 1*1(square) or 1*1*1(cube) and then ask them which would be the most appropriate model here.
Please see the PowerPoint for detailed presentation notes.
I had an interesting conversation with a student today about why I have them model cubic functions with the volume of cubes. We discussed how a model like this really enhances the understanding of some students but not necessarily all students. It is valuable to us teachers to remember this as well. Not every activity or strategy we use will affect every student and that is okay. This is why it is important to have multiple ways to look at topics. I think it is also helpful for students to recognize what methods work for them. This is part of becoming an independent, achieving scholar.
Now we practice graphing a couple of translations using the power functions without the volume model. As a scaffolding measure, I may do a quick review of the different types of transformations. I had my students make a graphic organizer of transformations earlier in the school year that they may use if needed.
I also remind them that the three "anchor" points discussed at the beginning of the lesson which should be graphed for each polynomial.
This final portion of this lesson is an extension. Transformations are limited to fairly simple power functions. Once you get into functions like f(x) = 2x3 - 5x2 + 7x - 3, transformations cease to be useful. It would be extremely difficult, if not impossible, to put this into transformational form. This is why we have other methods to graph polynomials. With that being said, transformations can occasionally still apply to these more advanced functions (Math Practice 7).
Our first example gives the students the function f(x) = x(x - 2)(x + 7). They are then given the
function f(x) = (x + 1)(x - 2)(x + 7) and are asked to describe what transformation occurs between these two functions. I have the students describe these in pairs. One method I use for problems like these is to circulate around the classroom letting them know if they got the correct answer. If they are off, I will give them some quick feedback to help them get on the right track. If used with some enthusiasm, I have found this technique extremely successful with students who can be reluctant or unsure. Once many or most of my students have a good idea about the solution, we discuss it as a class.
The final problem is similar except that the transformation is a stretch of two on the y-axis.
The first three problems of this Homework refer the students back to the volume model. They model the functions for questions one and two by drawing a diagram of the cubes they represent and a graph of their volume. The third question asks them to evaluate whether a negative function could be used to model volume.
The next five problems practice graphing transformations.
The final two ask the students to describe the transformation from one complex polynomial to another (Math Practice 7). The homework can be differentiated by including or not including these problems.
The last two problems on this assignment tested my students’ knowledge of transformations. Problem 9 reflected the graph of the original polynomial over the x-axis and shrunk it by a factor of 2. The complicated nature of this polynomial affected some of the students, but most students successfully identified this transformation. This can be seen in this sample Student Assignment 1. The tenth problem was more complicated. They know that numbers added or subtracted inside of a function shift it horizontally. This problem changed one factor of the polynomial from x+5 to x+7. Many students assumed this would translate the polynomial left 2 additional units. My goal was to have a great conversation the next day discussing horizontal translations. I did receive several insightful answers like the one in Student Assignment 2.
I use an exit ticket each day as a quick formative assessment to judge the success of the lesson.
This Exit Ticket is checking whether students can describe transformations on a cubic function.