Essential Questions |
|---|
What is the difference between a construction, a sketch, and a drawing?
When is it appropriate to use a sketch instead of a drawing?
When is it appropriate to use a drawing instead of a construction?
When is it appropriate to use a construction instead of a drawing?
|
Warm-Up/Opening Activity |
|---|
Define the terms construction, sketch, and drawing. When have you seen these words used? What has been the application of each of these words?
Start a K-W-L (What do you already know?, What do you want to learn?, What did you learn?) chart with the students to find out what they already know (correctly or incorrectly) about the similarities and differences between constructions, sketches, and drawings. Collect the information on the KWL Chart, completing the first two columns. |
Development of Ideas |
|---|
Use concept attainment with examples to build the definitions for each term using the warm-up as a starting point.
Construction – Constructions of figures and lengths with only a compass and a straightedge as tools. Sometimes, the term "ruler" may be used instead of straightedge, but the accepted classical definition is for the use of a straightedge without any markings that could be used to measure.
Classical geometric constructions use only a compass and a straightedge as tools. For the High School Assessment, students may also use patty paper, Mira™, or mirrors.
Drawings – Drawings can use all of the tools listed for constructions and measurement tools can also be used to reflect relationships.
For items that ask the students to draw a geometric figure, students may use a compass, ruler, patty paper, Mira™, mirrors, and/or protractor. Measurement can be part of their strategy.
Sketch – A drawing that can be completed without the use of tools.
Using the definitions, discuss each of the following questions with the class.
- Why are there differences between constructions, sketches, and drawings?
- What is the purpose of each?
- When is it better to use one instead of the other?
- What is the role of each?
- What real world applications does each have?
A key difference between a construction and a drawing is that in constructions all relationships are determined by the use of formal proof, not measured accuracy.
To the Greeks, where the current formalized study of Geometry started, all knowledge was broken down into its most simple parts, which for constructions was the use of basic tools, the compass and the straightedge.
To a mathematician, there is no "exact" measurement or "perfectly reliable" tool, only measurements and tools with less and less error. The key to constructions is that they can be proved to exhibit the relationships shown, not just drawn to show the relationships. That is why constructions and proof are so important in Geometry. Constructions are a way to visualize the formal structure of proof in classical Geometry.
Use the problems on Drawing and Construction Examples as examples of how HSA problems use the terms construction and draw. In the first problem, focus the student's attention on how students need to measure in order to solve the problem, indicating that it truly is a drawing. In the second problem, students will again note that measurements are important in determining the relationship, so that again it is a drawing activity, not a construction. In the third problem, students must justify their answer using properties of mathematics, not measurements, making it a true classical construction problem.
Work with students to understand "Note: Figure not drawn to scale"
Be sure that students do not measure when this phrase is in the problem. Use "Figure Not Drawn to Scale" Examples for examples of problems in which the information in the problem should be used, not any measured relationships from the diagram.
Examine the difference between explanation and justification with constructions. Using the sample problems below, ask some of the students in the class to explain their answer, another group to justify their answer. Explanation/Justification Examples has the following examples for students to work on. Assign half of the class to work on the first bullet (explanation) and the other half to work on the second bullet (justification).
Have the students construct a square, draw a square, and sketch a square. Have the students explain the difference in each.
Follow-Up in the Next Lesson
Discuss with students the answers to the construct, draw, and sketch a square. Emphasize the differences between the three squares, referring to the definitions used from the previous lesson. Encourage the students to edit their work to include this type of information.
Complete the last column of the KWL Chart from the opening activity as a summary of what was learned in the lesson. Be sure that all items that the students desired to know have been instructed. This will connect the opening and closing of the activity.
Supplemental Activities/Resources
High School Assessment Public Release questions
"Construction vs Drawing on the High School Assessment" discussion paper from MSDE
Extension Activities
Have students explore the three classic impossible geometric constructions: trisecting an angle, squaring the circle, and doubling the cube.
Discussion of these problems can be found at: http://mathforum.org/dr.math/faq/faq.impossible.construct.html
Use of computer technology to draw and construct geometric figures.
|
|
