unit 5: trigonometry introduction to trigonometry · 2019. 12. 5. · 2d 5 old day 1 intro to...
TRANSCRIPT
2D - 5 - OLD - day 1 - Intro to Trigonometry - ANS.notebook1
December 05, 2019
Unit 5: Trigonometry
Introduction to Trigonometry
Learning Goal Identify and utilize the properties of congruent and similar triangles
2D 5 OLD day 1 Intro to Trigonometry ANS.notebook
2
3
a2 + b2 = c2
4
Recall from grade 9
Ratio comparison of 2 or more quantities with the same unit
ex.
Rate comparison of 2 or more quantities with different units
ex.
2
5
6
7
Attachments
MidptCoordPlane.tns
MidptCoordPlane_Student.pdf
MidptCoordPlane_Teacher.doc
MidptCoordPlane_Student.doc
Midpoint.tns
Name ________________________
Class ________________________
©2010 Texas Instruments Incorporated Page 1 Midpoints in the Coordinate Plane
Problem 1 – Midpoints of Horizontal or Vertical Segments
On page 1.3, construct a horizontal segment and a vertical segment. Find the coordinates of the endpoints and then predict the coordinates of the midpoints of the segments.
Endpoints Predicted Midpoint
(_____ , _____) and (_____ , _____) (_____ , _____)
(_____ , _____) and (_____ , _____) (_____ , _____)
Describe how you can predict the coordinates of the midpoint of a horizontal or vertical segment.
Problem 2 – Midpoints of Diagonal Segments
On page 2.2, construct two diagonal segments. Find the coordinates of the endpoints and then make a predication about the coordinates of the midpoints.
Endpoints Predicted Midpoint
(_____ , _____) and (_____ , _____) (_____ , _____)
(_____ , _____) and (_____ , _____) (_____ , _____)
Describe how you can predict the coordinates of the midpoint of a diagonal segment.
Apply The Math
What formula gives the midpoint of a segment with endpoints (x1, y1) and (x2, y2)?
Midpoints in the Coordinate Plane
©2010 Texas Instruments Incorporated Page 2 Midpoints in the Coordinate Plane
Determine the midpoint of a segment with the following endpoints:
1. (3, 10) and (5, 10)
2. (1, 8) and (8, 9)
3. (7, 2) and (4, 4)
4. (–2, 3) and (5, –7)
5. (1.8, 4.9) and (7.2, 2.7)
6. (–3.3, 5.5) and (–5.5, 3.3)
Given an endpoint and midpoint of a segment, find the other endpoint:
7. Endpoint: (3, 1); Midpoint: (3, 4)
8. Endpoint: (2, 5); Midpoint: (5, 6)
9. Endpoint: (–4, 3); Midpoint: (1, 0)
Extension – Trisection Points
On page 3.2, segment PQ has two trisection points, which divide PQ into 3 equal parts. Drag P or Q to change the segments location. Find the coordinates of the endpoints and then make a prediction about the coordinates of the trisection points.
Endpoints Predicted Trisection points
(_____ , _____) and (_____ , _____) (_____ , _____) and (_____ , _____)
(_____ , _____) and (_____ , _____) (_____ , _____) and (_____ , _____)
Describe how you can predict the coordinates of the trisection points of a segment.
SMART Notebook
TImath.com Geometry
TImath.com Geometry
ID: 8612
Time required
40 minutes
Activity Overview
In this activity, students will explore midpoints in the coordinate plane. Beginning with horizontal or vertical segments, students will show the coordinates of the endpoints and make a conjecture about the coordinates of the midpoint. This conclusion is extended to other segments in the coordinate plane.
Topic: Points, Lines & Planes
· Given the coordinates of the ends of a line segment, write the coordinates of its midpoint.
Teacher Preparation and Notes
· This activity is intended to be used in a middle school or high school geometry classroom.
· This activity is designed to be student-centered with the teacher acting as a facilitator while students work cooperatively. Use the following pages as a framework as to how the activity will progress.
· Depending on student skill level, you may wish to use points with integer coordinates, or only positive values.
· The Coordinate Midpoint formula for the midpoint of (x1, y1) and (x2, y2) is
1212
xxyy
æö
ç÷
èø
. This can also be expressed as “The coordinates of the midpoint of a line segment are the averages of the coordinates of the endpoints.”
· Trisection Points is an optional extension, which can be used depending on time and student ability. The trisection points of a segment with endpoints (x1, y1) and (x2, y2) are
1212
xxyy
.
· To download the student TI-Nspire document (.tns file) and student worksheet, go to http://education.ti.com/exchange and enter “8612” in the search box.
Associated Materials
Suggested Related Activities
To download any activity listed, go to education.ti.com/exchange and enter the number in the quick search box.
· Division of Integers (TI-84 family) — 1433
· Multiplication of Integers (TI-84 family) — 1434
· Integers (TI-84 family, TI-Navigator Technology) — 4412
Problem 1 – Midpoints of Horizontal or Vertical Segments
Have students open the file and read the directions on page 1.2.
On page 1.3, students are to construct a horizontal segment in the first quadrant. Using the Point On tool (MENU > Points & Lines > Point On) they need to plot two points directly on grid points and then connect the points using the Segment tool (MENU > Points & Lines > Segment).
Direct students to show the coordinates for the endpoints of the segment using the Coordinates and Equations tool (MENU > Tools > Coordinates and Equations).
Note: Press x (or ·) once to select the point whose coordinates you wish to show and then press x (or ·) again to anchor the measurement.
Students should make a prediction about the coordinates for the midpoint of the segment.
To check their predictions, have them select MENU > Construction > Midpoint, construct the midpoint of the segment, and show the coordinates of the midpoint using the Coordinates and Equations tool.
Tell students to hide the coordinates of the midpoint with the Hide/Show tool (MENU > Tools > Hide/Show) before moving the segment.
Students can now drag the endpoints to create a vertical segment and then make a prediction for the new coordinates of the midpoint. They can verify their prediction by using the Hide/Show tool again.
If desired, have students explore the midpoint of a segment whose endpoints do not have integer coordinates by selecting MENU > Window > Window Settings and dividing each value by 10. They can also explore what happens when one or both endpoints are not in Quadrant 1.
Problem 2 – Midpoints of Diagonal Segments
Direct students to advance to page 2.1 and read the directions.
On page 2.2, students are to use the Point On tool and the Segment tool to construct a diagonal segment in the first quadrant.
Students once again need to use the Coordinates and Equations tool to show the coordinates for the endpoints of the segment.
They should use the endpoint coordinates to make a predication about the coordinates of the midpoint.
To check their prediction, students can construct the midpoint and then find the coordinates.
Tell students hide the coordinates of the midpoint with the Hide/Show tool (MENU > Tools > Hide/Show).
Students can drag the endpoints to create a different diagonal segment and then make a prediction for the new coordinates of the midpoint. Students can confirm their prediction by using the Hide/Show tool again.
Discuss how to find the coordinates of the midpoint of a segment if the coordinates of the endpoints are known. Then, challenge students to write a formula or a rule for calculating midpoints.
To observe how the midpoint is related to the endpoints, students should press MENU > Tools > Text and enter the expression (a+b)/2 in a text box.
Then select MENU > Tools > Calculate, click on the expression, and click on the x‑values of the coordinates for the endpoints for a and b. Tell students to repeat the calculation with the y‑values of the coordinates for the endpoints.
Now they can drag the segment endpoints and observe the calculation results as they updates.
Discuss how these calculation results relate to the coordinates of the midpoint.
Extension – Trisection Points
Students will read the directions on page 3.1 and then advance to page 3.2. There is a segment displayed with its two trisection points, which divide the segment into three equal sections.
If needed, the can select MENU > Measurement > Length and measure the lengths of each section to confirm that the segment is trisected.
The Coordinates and Equations tool should be used by students to show the coordinates of the segment endpoints.
Students should make a prediction about the coordinates of the trisection points. To confirm their prediction, students can show the coordinates of the trisection points.
Have students explore the relationship between the coordinates of the segment endpoints and the trisection points by dragging points P or Q to adjust the location of the segment. Challenge them to write a formula for the trisection points.
©2010 Texas Instruments Incorporated Teacher Page Midpoints in the Coordinate Plane
©2010 Texas Instruments Incorporated Page 2 Midpoints in the Coordinate Plane
_1250919735.unknown
_1250919749.unknown
_1250919717.unknown
MidptCoordPlane.tns
Name ________________________
Class ________________________
Problem 1 – Midpoints of Horizontal or Vertical Segments
On page 1.3, construct a horizontal segment and a vertical segment. Find the coordinates of the endpoints and then predict the coordinates of the midpoints of the segments.
Endpoints Predicted Midpoint
(_____ , _____) and (_____ , _____) (_____ , _____)
(_____ , _____) and (_____ , _____) (_____ , _____)
Describe how you can predict the coordinates of the midpoint of a horizontal or vertical segment.
Problem 2 – Midpoints of Diagonal Segments
On page 2.2, construct two diagonal segments. Find the coordinates of the endpoints and then make a predication about the coordinates of the midpoints.
Endpoints Predicted Midpoint
(_____ , _____) and (_____ , _____) (_____ , _____)
(_____ , _____) and (_____ , _____) (_____ , _____)
Describe how you can predict the coordinates of the midpoint of a diagonal segment.
Apply The Math
What formula gives the midpoint of a segment with endpoints (x1, y1) and (x2, y2)?
Determine the midpoint of a segment with the following endpoints:
1. (3, 10) and (5, 10)
2. (1, 8) and (8, 9)
3. (7, 2) and (4, 4)
4. (–2, 3) and (5, –7)
5. (1.8, 4.9) and (7.2, 2.7)
6. (–3.3, 5.5) and (–5.5, 3.3)
Given an endpoint and midpoint of a segment, find the other endpoint:
7. Endpoint: (3, 1); Midpoint: (3, 4)
8. Endpoint: (2, 5); Midpoint: (5, 6)
9. Endpoint: (–4, 3); Midpoint: (1, 0)
Extension – Trisection Points
On page 3.2, segment PQ has two trisection points, which divide
PQ
into 3 equal parts. Drag P or Q to change the segments location. Find the coordinates of the endpoints and then make a prediction about the coordinates of the trisection points.
Endpoints Predicted Trisection points
(_____ , _____) and (_____ , _____) (_____ , _____) and (_____ , _____)
(_____ , _____) and (_____ , _____) (_____ , _____) and (_____ , _____)
Describe how you can predict the coordinates of the trisection points of a segment.
©2010 Texas Instruments Incorporated Page 1 Midpoints in the Coordinate Plane
©2010 Texas Instruments Incorporated Page 2 Midpoints in the Coordinate Plane
_1250933532.unknown
December 05, 2019
Unit 5: Trigonometry
Introduction to Trigonometry
Learning Goal Identify and utilize the properties of congruent and similar triangles
2D 5 OLD day 1 Intro to Trigonometry ANS.notebook
2
3
a2 + b2 = c2
4
Recall from grade 9
Ratio comparison of 2 or more quantities with the same unit
ex.
Rate comparison of 2 or more quantities with different units
ex.
2
5
6
7
Attachments
MidptCoordPlane.tns
MidptCoordPlane_Student.pdf
MidptCoordPlane_Teacher.doc
MidptCoordPlane_Student.doc
Midpoint.tns
Name ________________________
Class ________________________
©2010 Texas Instruments Incorporated Page 1 Midpoints in the Coordinate Plane
Problem 1 – Midpoints of Horizontal or Vertical Segments
On page 1.3, construct a horizontal segment and a vertical segment. Find the coordinates of the endpoints and then predict the coordinates of the midpoints of the segments.
Endpoints Predicted Midpoint
(_____ , _____) and (_____ , _____) (_____ , _____)
(_____ , _____) and (_____ , _____) (_____ , _____)
Describe how you can predict the coordinates of the midpoint of a horizontal or vertical segment.
Problem 2 – Midpoints of Diagonal Segments
On page 2.2, construct two diagonal segments. Find the coordinates of the endpoints and then make a predication about the coordinates of the midpoints.
Endpoints Predicted Midpoint
(_____ , _____) and (_____ , _____) (_____ , _____)
(_____ , _____) and (_____ , _____) (_____ , _____)
Describe how you can predict the coordinates of the midpoint of a diagonal segment.
Apply The Math
What formula gives the midpoint of a segment with endpoints (x1, y1) and (x2, y2)?
Midpoints in the Coordinate Plane
©2010 Texas Instruments Incorporated Page 2 Midpoints in the Coordinate Plane
Determine the midpoint of a segment with the following endpoints:
1. (3, 10) and (5, 10)
2. (1, 8) and (8, 9)
3. (7, 2) and (4, 4)
4. (–2, 3) and (5, –7)
5. (1.8, 4.9) and (7.2, 2.7)
6. (–3.3, 5.5) and (–5.5, 3.3)
Given an endpoint and midpoint of a segment, find the other endpoint:
7. Endpoint: (3, 1); Midpoint: (3, 4)
8. Endpoint: (2, 5); Midpoint: (5, 6)
9. Endpoint: (–4, 3); Midpoint: (1, 0)
Extension – Trisection Points
On page 3.2, segment PQ has two trisection points, which divide PQ into 3 equal parts. Drag P or Q to change the segments location. Find the coordinates of the endpoints and then make a prediction about the coordinates of the trisection points.
Endpoints Predicted Trisection points
(_____ , _____) and (_____ , _____) (_____ , _____) and (_____ , _____)
(_____ , _____) and (_____ , _____) (_____ , _____) and (_____ , _____)
Describe how you can predict the coordinates of the trisection points of a segment.
SMART Notebook
TImath.com Geometry
TImath.com Geometry
ID: 8612
Time required
40 minutes
Activity Overview
In this activity, students will explore midpoints in the coordinate plane. Beginning with horizontal or vertical segments, students will show the coordinates of the endpoints and make a conjecture about the coordinates of the midpoint. This conclusion is extended to other segments in the coordinate plane.
Topic: Points, Lines & Planes
· Given the coordinates of the ends of a line segment, write the coordinates of its midpoint.
Teacher Preparation and Notes
· This activity is intended to be used in a middle school or high school geometry classroom.
· This activity is designed to be student-centered with the teacher acting as a facilitator while students work cooperatively. Use the following pages as a framework as to how the activity will progress.
· Depending on student skill level, you may wish to use points with integer coordinates, or only positive values.
· The Coordinate Midpoint formula for the midpoint of (x1, y1) and (x2, y2) is
1212
xxyy
æö
ç÷
èø
. This can also be expressed as “The coordinates of the midpoint of a line segment are the averages of the coordinates of the endpoints.”
· Trisection Points is an optional extension, which can be used depending on time and student ability. The trisection points of a segment with endpoints (x1, y1) and (x2, y2) are
1212
xxyy
.
· To download the student TI-Nspire document (.tns file) and student worksheet, go to http://education.ti.com/exchange and enter “8612” in the search box.
Associated Materials
Suggested Related Activities
To download any activity listed, go to education.ti.com/exchange and enter the number in the quick search box.
· Division of Integers (TI-84 family) — 1433
· Multiplication of Integers (TI-84 family) — 1434
· Integers (TI-84 family, TI-Navigator Technology) — 4412
Problem 1 – Midpoints of Horizontal or Vertical Segments
Have students open the file and read the directions on page 1.2.
On page 1.3, students are to construct a horizontal segment in the first quadrant. Using the Point On tool (MENU > Points & Lines > Point On) they need to plot two points directly on grid points and then connect the points using the Segment tool (MENU > Points & Lines > Segment).
Direct students to show the coordinates for the endpoints of the segment using the Coordinates and Equations tool (MENU > Tools > Coordinates and Equations).
Note: Press x (or ·) once to select the point whose coordinates you wish to show and then press x (or ·) again to anchor the measurement.
Students should make a prediction about the coordinates for the midpoint of the segment.
To check their predictions, have them select MENU > Construction > Midpoint, construct the midpoint of the segment, and show the coordinates of the midpoint using the Coordinates and Equations tool.
Tell students to hide the coordinates of the midpoint with the Hide/Show tool (MENU > Tools > Hide/Show) before moving the segment.
Students can now drag the endpoints to create a vertical segment and then make a prediction for the new coordinates of the midpoint. They can verify their prediction by using the Hide/Show tool again.
If desired, have students explore the midpoint of a segment whose endpoints do not have integer coordinates by selecting MENU > Window > Window Settings and dividing each value by 10. They can also explore what happens when one or both endpoints are not in Quadrant 1.
Problem 2 – Midpoints of Diagonal Segments
Direct students to advance to page 2.1 and read the directions.
On page 2.2, students are to use the Point On tool and the Segment tool to construct a diagonal segment in the first quadrant.
Students once again need to use the Coordinates and Equations tool to show the coordinates for the endpoints of the segment.
They should use the endpoint coordinates to make a predication about the coordinates of the midpoint.
To check their prediction, students can construct the midpoint and then find the coordinates.
Tell students hide the coordinates of the midpoint with the Hide/Show tool (MENU > Tools > Hide/Show).
Students can drag the endpoints to create a different diagonal segment and then make a prediction for the new coordinates of the midpoint. Students can confirm their prediction by using the Hide/Show tool again.
Discuss how to find the coordinates of the midpoint of a segment if the coordinates of the endpoints are known. Then, challenge students to write a formula or a rule for calculating midpoints.
To observe how the midpoint is related to the endpoints, students should press MENU > Tools > Text and enter the expression (a+b)/2 in a text box.
Then select MENU > Tools > Calculate, click on the expression, and click on the x‑values of the coordinates for the endpoints for a and b. Tell students to repeat the calculation with the y‑values of the coordinates for the endpoints.
Now they can drag the segment endpoints and observe the calculation results as they updates.
Discuss how these calculation results relate to the coordinates of the midpoint.
Extension – Trisection Points
Students will read the directions on page 3.1 and then advance to page 3.2. There is a segment displayed with its two trisection points, which divide the segment into three equal sections.
If needed, the can select MENU > Measurement > Length and measure the lengths of each section to confirm that the segment is trisected.
The Coordinates and Equations tool should be used by students to show the coordinates of the segment endpoints.
Students should make a prediction about the coordinates of the trisection points. To confirm their prediction, students can show the coordinates of the trisection points.
Have students explore the relationship between the coordinates of the segment endpoints and the trisection points by dragging points P or Q to adjust the location of the segment. Challenge them to write a formula for the trisection points.
©2010 Texas Instruments Incorporated Teacher Page Midpoints in the Coordinate Plane
©2010 Texas Instruments Incorporated Page 2 Midpoints in the Coordinate Plane
_1250919735.unknown
_1250919749.unknown
_1250919717.unknown
MidptCoordPlane.tns
Name ________________________
Class ________________________
Problem 1 – Midpoints of Horizontal or Vertical Segments
On page 1.3, construct a horizontal segment and a vertical segment. Find the coordinates of the endpoints and then predict the coordinates of the midpoints of the segments.
Endpoints Predicted Midpoint
(_____ , _____) and (_____ , _____) (_____ , _____)
(_____ , _____) and (_____ , _____) (_____ , _____)
Describe how you can predict the coordinates of the midpoint of a horizontal or vertical segment.
Problem 2 – Midpoints of Diagonal Segments
On page 2.2, construct two diagonal segments. Find the coordinates of the endpoints and then make a predication about the coordinates of the midpoints.
Endpoints Predicted Midpoint
(_____ , _____) and (_____ , _____) (_____ , _____)
(_____ , _____) and (_____ , _____) (_____ , _____)
Describe how you can predict the coordinates of the midpoint of a diagonal segment.
Apply The Math
What formula gives the midpoint of a segment with endpoints (x1, y1) and (x2, y2)?
Determine the midpoint of a segment with the following endpoints:
1. (3, 10) and (5, 10)
2. (1, 8) and (8, 9)
3. (7, 2) and (4, 4)
4. (–2, 3) and (5, –7)
5. (1.8, 4.9) and (7.2, 2.7)
6. (–3.3, 5.5) and (–5.5, 3.3)
Given an endpoint and midpoint of a segment, find the other endpoint:
7. Endpoint: (3, 1); Midpoint: (3, 4)
8. Endpoint: (2, 5); Midpoint: (5, 6)
9. Endpoint: (–4, 3); Midpoint: (1, 0)
Extension – Trisection Points
On page 3.2, segment PQ has two trisection points, which divide
PQ
into 3 equal parts. Drag P or Q to change the segments location. Find the coordinates of the endpoints and then make a prediction about the coordinates of the trisection points.
Endpoints Predicted Trisection points
(_____ , _____) and (_____ , _____) (_____ , _____) and (_____ , _____)
(_____ , _____) and (_____ , _____) (_____ , _____) and (_____ , _____)
Describe how you can predict the coordinates of the trisection points of a segment.
©2010 Texas Instruments Incorporated Page 1 Midpoints in the Coordinate Plane
©2010 Texas Instruments Incorporated Page 2 Midpoints in the Coordinate Plane
_1250933532.unknown