In this exercise, we do some basic work with fractions, and then I present a real-world scenario in which a two-by-four needs to be cut into two pieces with a 4:3 length ratio.
We start out with some calculations like:
1. Find 1/5 of 35
2. Find 2/5 of 40
3. Find 2/3 of 36
Then I show the Two by Four photo on the projector and present the scenario:
I have a project at home, and I need to cut the two by four in the photo into two pieces whose lengths have a 4:3 ratio. Where should I make the cut? How long will the pieces be?
I give students 5 minutes to work on the problem. As I walk around, I scan for exemplars that are correct and may show novel ways of representing or solving the problem. I have two or three of these students show their solutions on the document camera. For the purposes of the lesson, I will also demonstrate the solution (if necessary) in order to emphasize that a 4:3 ratio implies that there are 7 parts total, one piece will be 4/7 of 42 inches, and the other piece will be 3/7 of 42 inches.
The concept of the weighted average is fundamental to this lesson on partitioning line segments. In this section of the lesson, I spend some time developing this concept.
I have students seated in groups of four for this section of the lesson. Each group will get four copies of The Weighted Average (pages 1 and 2 only) and they will be allowed to use calculators. I give my students 15 minutes to work in groups on problems 1 through 3. I explain to them that they may or may not finish. I write the following goals on the board: (a) Each group works together to complete as many problems correctly as they can in 15 minutes and (b) Every group member will understand every problem the group completes.
After the 15 minutes have elapsed, I have my students transition back to row seating and I hand out page 3 of The Weighted Average. As we read through the text together, I provide additional commentary to make sure that my students understand the concept of the weighted average and the arithmetic that is involved.
When we have finished calculating the weighted average in the last example, I return to problems 1 through 3 from the the first two pages of The Weighted Average and solve them using the paradigm of weighted averages. This gives me an opportunity to (a) reinforce the concepts and skills of weighted averages, (b) show and explain how these problems relate to the weighted average and (c) to reveal the solutions to the problems so that students can see how they did.
In this section of the lesson students will be seated in rows as I hand out Partitioning Line Segments. This is a teaching resource, written somewhat in the style of a textbook since this topic is not covered in the textbook I use.
In the first part of the handout, I am trying to get students to see that the midpoint formula (with which they are familiar) is a special case of the partitioning formula. I do this by introducing the paradigm of "coordinate genetics". I explain to students that when we have two endpoints, A and B for example, there are an infinite number of points in between them. If we think of A and B as the "parent" points, then these infinitely many points in between are the possible "offspring" points. The genotype (and phenotype) of the offspring points are directly determined by the relative contributions from the coordinates of the two parent points to the offspring point.
So we start with the case of the midpoint, in which both parent points contribute equally to the offspring.
Then we move to an example in which one parent point contributes relatively more. This gives students their first glimpse at how we express the different relative contributions of the parent points mathematically.
Next, in the part of the handout with the heading, Reasoning, I introduce students to the key decision-making they will have to do as they navigate these types of problems.
When student have understood the reasoning behind the formula, I model two different examples: one that gives a ratio of part to part and another that gives a ratio of part to whole. I model these examples fully so that all students are trained properly. I do give students time to respond to the a. and b. portions of the examples before I start to model.
There are always students who are ready to work on their own at this point. I urge them to begin working on their own. I only ask that they work quietly and independently so as not to cause any distraction while I am addressing the class.
Finally, we get to writing the general formula for partitioning a line segment. The formula has two cases, depending on which endpoint the partitioning point is closer to. I give students a few minutes to work ahead on this. That way, those who can figure it out on their own have a chance to do so. When I notice that some students are stuck and going nowhere, I know it's time to reveal part of the process on the board and then give a few more minutes to see if that sets them sailing. I continue in this fashion until I have revealed the entire process for the first case. Then I have students work independently or with a peer to finish the second case.
In this section of the lesson, I have students seated in groups of 2-4. This is good old-fashioned practice with the help of peers when needed.
Students will be working on Cooperative Practice_Partitioning Line Segments, unabated. When the time has elapsed, I will reveal the answers so that students can go back and re-check anything that they got incorrect.
I give students Independent Practice_Partitioning Segments to take home. This will be their homework assignment after the lesson.
If I have extra time, or if I have advanced students who need enrichment, I have the Extension Task_Centroid as Partitioner ready.
In the previous lesson, we proved that the medians of a triangle must be concurrent. After this current lesson, it is easier for us to look back at our work from the previous lesson and see that the centroid actually partitions each median in the ratio 2:1. This is a useful fact that I will teach students, whether or not we do this task.
This task, though, underscores a neat connection that reinforces the learning from both lessons.