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Geometric Constructions: Slope of Parallel and Perpendicular Lines

Agenda1. Performance Task Review2. Geometric Constructions: The Basics3. Construction Art Project4. Debrief

DO NOW 10/13:

How can we prove the laser is exitingparallel to the laser when it enters?

+Geometric Constructions

Ancient mathematicians did not have fractions. So they used constructions to determine partial lengths and degrees.

Euclid was known to be the father or modern geometry, and developed many of the rules that govern how we canprove different theorems using constructions.

+Geometric Art is Found All Around the World!

Africa

Middle EastEurope (Ireland)

+Geometric Constructions: Using a Compass and a StraightedgeTwo Tools:

A straightedge ONLY used to draw lines and segments!

A compass is ONLY used to draw circles!

Definitions: Circle – the set of points equi-distant from a given center

point Radius – the distance from the center to the edge of a circle

*MUST BE IN NOTES!*Ex. “Circle A with radius AB”

+Geometric Construction Art Project

The Rules:

1. Start near the center of your paper with a point A and draw a circle with center point A and radius of 1 unit.

2. Draw a point B anywhere on the circle.

3. Using the SAME radius as circle A, draw another circle using B as the center.

4. At the point where those circles intersect (point C), draw another circle (KEEP radius the same!

5. Repeat steps 3&4

6. Choose one circle. Connect all points on the circle to the center of the circle and to each other with line segments.

Your Finished Project Should Have:

At least 7 congruent circles

At least 12 line segments

At least 7 points

Questions to consider (as you color)

Are the segments congruent? Why? How do you know?

+Debrief What shapes did we notice were formed (besides

circles and lines)?

Are there any patterns that we noticed appearing?

Are the segments congruent? Why? How do you know?

Did anyone get lines that appear parallel or perpendicular?

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Construction 1: Parallel LinesAgenda1. Review of Slope2. Construction 1: Steps3. Verifying Slope Algebraically4. Debrief

DO NOW 10/14:

What value of x will prove that the lines are parallel?

+Review: Slope Formula

Given two points as coordinates (x1, y1) and (x2, y2) the slope formula is the “change” (difference) in y divided by the “change” (difference) in x.

m =y2 – y1

x2 – x1

Practice: Find the slope of line a and line b.

*MUST BE IN NOTES!*

+Construction 1:

1. Draw line PQ and a non-collinear point R (NOT on line PQ).

2. Draw a line through point R that intersects PQ. Label the intersection point J.

3. Construct a circle with center point J. Label the intersection points with JR point A and the intersection point with PQ point B.

4. KEEPING the SAME radius, construct another circle at point R. Label the intersection with JR point C. (make sure C is not between JR!)

5. Measure out another circle with a radius that is the length of AB. KEEPING the radius the SAME, move up to point C and draw the new circle. Label the point of intersection of the two circles point S.

6. Draw line RS parallel to PQ!

http://www.mathopenref.com/constparallel.html

“Construct a parallel line through a point”

+Gallery Walk

Trace over parallel lines PQ and RS and flip over your sheet to see the coordinate grid.

Choose two points on each line (P,Q and R,S) and label the coordinates. Tape the construction on the wall.

Using another person’s coordinates, you must use the slope formula to find the slope of both lines PQ and RS.

+Debrief Are the two lines parallel? What do we notice about

their slopes?

How do we know lines are parallel using geometry? How do we know using algebra?

+

Geogebra Review and Slope of Parallel Lines

Agenda:1. HW Review/GeoGebra Recap2. Perpendicular Lines Constructions3. Gallery Walk with Algebraic Slope4. Debrief/Exit Ticket

DO NOW 10/15:

Find the slope of line AB and line CD.

(PSAT Testing Day) P1-P4

+HW Review Peer Correct

LESSON 3.4

Practice A Find the slope of the line that passes through the points.

2.

Find the slope of each line. Are the lines parallel?

4.

6.

10. Line 1: (–1, 3), (–1, 5)Line 2: (0, 0), (0, 6)

 

+Parallel Lines: Graph and Algebra

+Construction 2:

1. Draw line PQ.

2. Draw a circle with center point P that has a radius more than half the length of PQ.

3. Using the same radius, move the compass to Q and draw another circle with center point Q.

4. At the two points where the two circles intersect, label one point A and the other point B.

5. Connect line AB perpendicular to PQ!

6. (You can also label the point where AB crosses PQ point M, since it will be the MIDPOINT!)

http://www.mathopenref.com/constbisectline.html

“Construct the perpendicular bisector of a line”

+Perpendicular Lines: Graph and Algebra

+Gallery Walk

Trace over perpendicular lines PQ and RS and flip over your sheet to see the coordinate grid.

Choose two points on each line (P,Q and R,S) and label the coordinates. Tape the construction on the wall.

EXIT TICKET: Using another person’s coordinates, you must find the slope of both lines PQ and RS using the slope formula and write the slope of each on their sheet.

+Debrief

What pattern do we notice about the slopes of perpendicular lines?

PQ AB

+

Slope of Parallel and Perpendicular LinesAgenda1. HW/Geogebra Review2. Graphic and Algebraic Slope3. Slope Jigsaw4. Debrief

DO NOW 10/15: For each question, use the slope

formula to verify if the two lines are parallel or perpendicular.

Line 1: (–1, 2), (2, 3)Line 2: (0, 0), (3, 1)

1.

2.

Line 1: (1, 1), (3, 3)Line 2: (2, 2), (0, 4)

+GeoGebra Recap

+Parallel Lines: Graph and Algebra

+Perpendicular Lines: Graph and Algebra

+Parallel/Perpendicular Slope Jigsaw

Round 1: Identify and record the 4 coordinates from the graph OR plot the 4 coordinates on the graph.

Round 2: Label the 4 coordinates from Round 1 (x1,y1) or (x2,y2) and set up the 2 slope equations.

Round 3: Solve the slope equations for the 2 lines and identify if they are parallel ( ), perpendicular ( ) or neither (N/A)

+Debrief/Exit Ticket How do we identify parallel and perpendicular lines

geometrically? How to we identify them algebraically?

(Exit Ticket) Find the slope of each line AB. Then use your knowledge of slopes of parallel and perpendicular lines to draw parallel line CD and perpendicular line EF

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Constructing ParallelogramsAgenda1. Quiz Pt. 12. Quiz Pt. 2: Construction Activity3. 1 on 1 Unit 14. Debrief

DO NOW 10/17:

Find the slope of line AB and line CD to prove if they

are perpendicular.

+Quiz Part 2: Constructions

Using the directions provided, construct either parallel or perpendicular lines.

You CANNOT use the ruler or a protractor to measure, only draw circles of straight line. (Remember, we’re working ancient Greek tools here…)

EXTRA CREDIT!

Using either of the two constructions we have done with parallel and perpendicular lines, construct a rectangle.

Must have the following properties1. 2 sets of opposite parallel sides

2. 4 angles measuring 90 degrees

3. 2 sets of congurent opposite sides

+Memory Card Game

If you are finished with your quiz early and have completed your debrief, you may play the memory game

Rules All cards start face down. Each player gets a chance to turn over any two cards. If

they are a match (slopes are perpendicular or parallel) you keep the cards and keep playing.

If they are not a match, you replace them face down and the next player goes.

+Debrief

When constructing the rectangle, why did the opposite sides end up congruent?

+Construction 1b:

1. Draw line PQ and a non-collinear point R (NOT on line PQ).

2. Draw a circle with center point R that intersects PQ at two points. (Make sure your radius is big enough!) Label the intersection closest to P point J.

3. KEEPING the SAME radius, construct another circle with center point J. Label the intersection closest to Q point E.

4. KEEPING the SAME radius, construct another circle with center point E. Label the intersection of circle R and circle E as point S.

5. Draw line RS parallel to PQ!

http://www.mathopenref.com/constparallelrhombus.html

“Construct a parallel line through a point”

+Estimating Slope:

In each example, first estimate if the slope of positive, negative, horizontal or vertical. Then verify using the slope fomula.

m = m = m =

Positive, Negative, Horizontal or Vertical

+

Part C – Equations of parallel and perpendicular lines: Go to http://tube.geogebra.org/student/m59880

1. Write and record the equations of two parallel lines. Have another student verify they are correct and initial.

Equation 1: ____________________ Verified by: _____Equation 2: ____________________ Verified by: _____

2. Write and record the equations of two perpendicular lines. Have another student verify they are correct and initial.

Equation 1: ____________________ Verified by: _____Equation 2: ____________________ Verified by: _____