The Golden Ratio
Lesson 8 of 8
Objective: SWBAT derive the Golden Ratio and prove why the Golden Rectangle construction works.
Activating Prior Knowledge
This lesson includes a lot of algebra. I don't want to assume that students have retained, or even learned all of it, so I spend some time in this section developing and brushing up on the prerequisite knowledge and skills. To do this, I guide students through the handout Activating Prior Knowledge_Golden Ratio. Some of the items require students give an answer and then I immediately follow up with the correct answer and explanation. Other items have solutions on the handout. For these, I step through the solution line-by-line with the the students to make sure everything is clear.
At various points, I pause to ask higher order thinking questions to be sure that students are understanding and sense-making.
- I pause to hear student answers on item 2c because I want to hear their thinking and rationale.
- On item 3, completing the square, there are several places to pause and check for understanding. What is the goal of completing the square? Why do we add 16? How did we get 2 rad 5?
- In items 4-9, I ask, "Structurally, what do all of these problems have in common?" "How can we generalize here?"
- Items 10 and 11 have detailed explanations, but I want to make sure students don't gloss over them. So I ask questions like "What did that last sentence mean?" and I definitely stop to hear student answers to the questions that are on the handout for these last two items.
In this section of the lesson, my students will learn about a golden rectangle. Then, they will use a specific golden rectangle to find the value of the golden ratio.
In the previous section I reviewed the skills and concepts foundations that students will need to navigate this section of the lesson. I encourage my students to use the completed handout from the previous section as a reference during this section. This move increases their sense of ownership over the knowledge. I don't plan to help them with the algebra, I want to discuss the geometry with them as they work on this task.
For this Activity, I have my students work in groups of 4. My instructions to the groups are as follows
- Make sure everyone in your group understands how to do everything. A person from your group will be randomly selected to present to the class and your group will be scored based on their presentation.
- Make sure that your work on your paper is legible and nicely organized so that it can be used as a visual aid during your presentation.
As my students are working in groups, I walk around making sure the groups are engaged and actually working as groups. If a student seems to be left out of the group, I might have this student and another student compare answers to a question they've both done. Then I'd ask whether their answers agree or if there is something they need to discuss in order to come to an agreement.
When there has been enough time for all groups to finish, I have the students transition back to row seating and I begin to call students randomly for one of the following presentations: 1. Items 1-3; 2. Items 4 and 5; 3. Item 6; 4. Items 7 and 8; 5. Item 9. Usually there will be students who have chosen quadratic formula and those who have chosen completing the square to solve the problem in #6. I make sure to include at least one presenter for each of these methods. I also call multiple presenters for #9.
Deriving the Golden Ratio
In this section we move from a specific case of a golden rectangle to the general case. This idea of starting with a concrete example and then reasoning to the general case has been a recurrent theme throughout the course. We are also progressing from teacher-centered and group collaboration in the previous two sections, to individual student autonomy in this section.
I give a copy of Deriving the Golden Ratio to each student and have them get started. I explain to them that their work will have to speak for itself so be precise, write legibly, and organize all work so that someone else can understand it easily.
While this is an independent task, I do walk around helping students who are stuck. One of the first stuck points is when students have not labeled the diagram in terms of a and b. Some students have trouble expressing the length of segment AB. ab or a+b. I ask them to imagine a and b as numbers, 2 and 5 for example. "What would you do?" They know they should add. "Same rules for variables as for constants", I tell them.
When students have had enough time to finish, I collect the papers. I want to see what they have come up with on their own. Next I will demonstrate the solution process so that students know right away if they did the problem correctly and learn how to do it if they did not know how to do it.
For this section, I give students Golden Rectangle Construction. In the first part of the handout, they follow the directions to use a compass and straightedge to construct a golden rectangle. I walk around the room to make sure that students are performing the construction correctly and I assist with technical difficulties if there are any. Students tend to do fine on this part, though.
On the second side of the handout, I demonstrate for students how we can verify that the construction succeeds in creating a golden rectangle. See the following video to get a feel for the content of that demonstration.
The nice thing about this golden rectangle is that we get two golden rectangles for the price of one. This allows students to engage in a process similar to the one that was demonstrated for them, but using the second rectangle instead of the first. It is not the exact same process so students will have a real opportunity to transfer their new knowledge as opposed to just repeating steps. Check out the video below to get an idea of what students ought to be doing to verify that this second rectangle is also a golden rectangle.
The following are some videos that truly inspired me and could be used as the basis for enrichment around the topic of the golden ratio...enjoy...
This last video is Disney's classic Donald in Mathmagic Land. My students had seen this video before numerous times, but I wanted to add something they hadn't seen. The animation is great and rich in information, but the video moves so fast that students are not likely to catch it. They just have to take the narrator's word for the truth of the statements. I found a way to use Sketchpad (and lots of rewinding and pausing) to verify each statement. The math that went into constructing the figures and verifying the statements could form the basis of a hearty lesson. I don't yet have it in a form that's shareable here, but I'll be working on it. Here's the video. I hope you pause and rewind (repeat) until you really SEE what's being presented...if you haven't already.