Quadratic Functions – Sample Lessons and Resources

I am starting a monthly feature where I will be focusing on some specific math content areas and providing some resources, in the form of how-to videos (both calculator and Classpad.net) and some ready-to-use math lessons (either PDF or links, depending on the tool used). I know math teachers are always searching for resources that will help them provide more open-ended math activities, where students are collecting and using data, using multiple representations to analyze and solve problems, and where students have to make decisions and support their decisions with mathematics. And integrate technology as well! So, at least once a month I am going to be picking a math content to focus on and provide some technology options as well, sometimes both calculator and online, and sometimes one or the other, depending on content.

This week I would like to focus on quadratic functions and helping students use a real-world context to work with quadratics. I am going to utilize Classpad.net, which is FREE web-based dynamic math software where I can do statistics, graphing, and calculations in one place (geometry as well, but for this activity, our focus does not include geometry). I am using this technology for a few reasons:

1. It’s free, so all of you should be able to access the created activity, including your students, as long as you have a mobile device with internet access.
2. I am able to create a complete activity (i.e. directions, tables, graphs, and place for students to show work) in one place and then share it easily via URL.
3. Everyone who opens the activity can create their own copy of it (as long as you have a FREE account on Classpad.net) by duplicating into their account. Then you can modify, answer the questions, etc. and create it’s new URL to share with others (or for students to share with you). To learn more about duplicating activities, click here.

The Problem

You are fencing in a rectangular area of your yard to create a garden. You have 36 ft. of fencing, of which you plan to use all. You can cut the fencing into whatever lengths are needed, as long as you use all 36 feet.

What dimensions should you use for your garden?

The Lesson

I have created a shared paper on Classpad.net called Quadratic Functions – Area of a Garden which you can access by clicking on the title. The idea behind this problem is that there are actually multiple solutions since the question is rather vague. I did NOT ask what is the largest garden, so students can work on collecting and analyzing the data and come to different conclusions depending on what they think is important. Some might choose largest area for the garden, some might choose largest perimeter, some might only want a rectangle some only a square, etc. By leaving the question a little more open, you are giving students a chance to explain their reasoning and come to multiple solutions based on this reasoning.

In looking at the activity (click the link above), you will note as part of the lesson, students use multiple representations. They first use their prior knowledge about dimensions of a rectangle, perimeter, area, and an understanding of feet and inches to record different dimensions for the garden. In the directions, students are asked to create at least 10 different rectangular gardens that use all 36 feet of fencing, where some of the width and length dimensions are fractional/decimal numbers and where width is sometimes larger than length. They record their dimensions in a table to start with, and then use those table values to calculate area (and perimeter if they choose to do the Extra Challenge), and use those table values to create statistical plots (scatter plots), and from the scatter plots and tables, create functions and graph those functions to fit their data. At different points along the way (after the table and scatter plots, and then after plotting their functions), students are asked to answer the question about what the dimensions they would choose for their garden and back up their reasoning using the information at that time. The idea here is to help them see that each representation provides insight into the dimensions, and some representations help you be a bit more precise or see the relationships between the quantities a little better. And also, depending on your goal for the garden, your reason for choosing certain dimensions may differ from others. There is also an extra challenge at the end (this is a way to support students who finish early, don’t need as much teacher guidance, and/or want to explore more), where students explore how the problem might differ if there was a fixed perimeter.

This is a video that shows using the activity and parts of doing the activity to get a feel for how this looks with students. I would recommend students working in pairs or small groups (3-4). All students can be recording on their mobile devices, or if you have one per group, choose a recorder.

Modifying Graphs with the Prizm – Check it out @NCTM Minneapolis

In preparation for NCTM Regionals in Minneapolis this week, I wanted to do a little show-and-tell with the Prizm. Hoping this sparks some interest and inspires some of you heading to the conference to stop by our booth (#511) at the conference and play with the Prizm.

One of the features of the Prizm that I just love is the ability to dynamically modify graphs, allowing students to visually see the effect of a coefficient on the graph of the function. This ability to modify one coefficient at a time and immediately see the impact on the graph allows students to make conjectures and get a better understanding of the graph and what each coefficient represents.

Here is a little demonstration of how the modify function works on the Prizm using both the standard form and vertex form of a quadratic:

I certainly hope you will stop by the booth Thursday or Friday and come play with us and learn more. I’d certainly love to meet you! Or, drop in on some sessions that utilize some of our products, like the Prizm, Keyboard and fx-55 Plus.

Thursday, November 12

• Session 41 9:45 – 11:00, (M100 DE) Hand-held Technology + Hands-On Activities=CCSS Success -Tom Beatini
• Session 84, 12:30-1:30, (M100 AB) CCSS for Statistics: Paired Quantitative Variables – John Diehl
• Session 101, 1:30-2:45, (200 AB) Thinking Like A Synthesizer – Mike Reiners
• Session 102, 2 – 3:00, (200 C) Connecting the Math through Meaningful Experiences – Jennifer North Morris