Introduction to Constructions
Lesson 8 of 14
Objective: SWBAT use a straight-edge and compass to perform constructions: a line through two points, a circle of a given radius, a copy of a segment, adjacent segments, a triangle with given sides. Students will understand what construction tools are and how to use them.
The Warm-Up prompt for the lesson asks students to think why it might be impossible to draw a line. This allows me to introduce geometric constructions as pencil-and-paper models of geometric objects, which are ideal concepts.
As I am introducing the topic of constructions, I display the lesson Agenda and Learning Targets.
I distribute copies of the Guided Notes for this activity.
I use direct instructions for introducing constructions. Prior to this lesson, my students have been working with construction tools for a week, and I have been giving some informal instruction. Now, however, I am instructing them in the techniques that will allow them to use a compass and straight-edge with efficiency and precision (MP5, MP6).
The rationale for the guided notes is:
- Each construction is described step-by-step. The instructions use formal mathematical language and geometric notation, which I encourage students to read. They are responsible for learning the language of geometry.
- Each step of a construction is accompanied by an illustration. We create the illustrations in class.
- I lead the class through each construction. I describe the step of the construction using formal language, then demonstrate the step. Students then perform the step.
- Since a construction typically has 3 or more steps, students get to practice the steps of the construction 'pyramid-fashion'. To create the illustration for the 2nd step, they perform that step. To create the illustration of the 3rd step, they perform the 2nd and 3rd steps. And so on.
- The first step of each construction describes what we begin with. Often, students will start by constructing a line and picking a point on the line to make a ray. At other times, an existing object--such as the side of an angle--will provide the starting materials.
I tell students that they are free to work ahead if they feel confident. However, I will not be able to rescue them if they get lost. If they get confused, they should just sit tight, pay attention, and wait for the class to catch up.
During this lesson, we learn the following constructions:
- Construct a line through two given points (I demonstrate proper technique for using a straight-edge)
- Construct a circle (or arc) with a given radius and center (I demonstrate proper technique for using a compass)
- Construct a copy of a line segment
- Construct adjacent line segments
Students find these constructions pretty straightforward. (In fact, they have performed several of them many times in the past week.) So, this section is really about getting organized and teaching students the routine for learning constructions.
Students practice the constructions they have just learned using the Rally Coach format. I display the instructions on the front board. I distribute the Activity, and students perform the exercises on their own paper.
I am on the lookout for: Are students using the scale of their ruler?
Some will not want to get out of their comfort zone, so they will actually measure a line segment (in centimeters, say), perform multiplication when required, and use the ruler scale to measure a copy of the segment. After I see that students have diligently performed that procedure once or twice, it is not hard to persuade them that a compass measures length much more efficiently (and just as precisely) (MP5, MP6).
The goal of this activity is to have students apply what they have learned about the properties of geometric objects to solve a basic construction problem: constructing a triangle using three given sides.
I am on the lookout for:
Are students having trouble starting the triangle?
I tell them to start with any line segment. They can put it anywhere. Just, don't work too close to the edge of the paper.
Are students stumped about how to position the remaining sides of the triangle?
Some may actually be using guess and check. This is an opportunity for a class discussion about the properties of a circle! I bring the class together and ask them to focus on the front board, where I draw one side of the triangle. (If a student has made a good start, however, I use student work which I project on the front board using a document camera.)
How should we place the second side? Well, what do we know? We know that the second side has to connect to the side we have, so we know where to put one endpoint. We also know how long the side should be, but we don't know what direction it should take. Sketching three or four possible sides of the correct length, I show how the other endpoint of the side--being a set distance from the vertex--must lie on a circle. So, we can use a circle to represent all the possible locations for the second endpoint (MP4). From here, students can usually fill in the rest.