CCSS IP Math I Scaffolded SWB Unit 6.indd

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CCSS IP Math I Scaffolded SWB Unit 6.indd
Student Workbook
with Scaffolded Practice
Unit 6
1
This book is licensed for a single student’s use only.
The reproduction of any part, for any purpose, is strictly prohibited.
© Common Core State Standards. Copyright 2010.
National Governor’s Association Center for Best Practices and
Council of Chief State School Officers. All rights reserved.
1 2 3 4 5 6 7 8 9 10
ISBN 978-0-8251-7778-1
Copyright © 2014
J. Weston Walch, Publisher
Portland, ME 04103
www.walch.com
Printed in the United States of America
WALCH
EDUCATION
2
Table of Contents
Program
pages
Introduction
Workbook
pages
5
Unit 6: Connecting Algebra and Geometry Through Coordinates
Lesson 1: Slope and Distance
Lesson 6.1.1: Using Coordinates to Prove Geometric Theorems with
Slope and Distance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U6-4–U6-31
Lesson 6.1.2: Working with Parallel and Perpendicular Lines. . . . . . . . . . . . . . . U6-32–U6-52
Lesson 2: Lines and Line Segments
Lesson 6.2.1: Calculating Perimeter and Area. . . . . . . . . . . . . . . . . . . . . . . . . . . . . U6-60–U6-84
Station Activities
Set 1: Parallel Lines, Slopes, and Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . U6-105–U6-110
Set 2: Perpendicular Lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U6-117–U6-122
Set 3: Coordinate Proof with Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . . U6-127–U6-132
Coordinate Planes
Formulas
Bilingual Glossary
7–16
17–26
27–38
39–44
45–50
51–56
57–62
63–66
67–90
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© Walch Education
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CCSS IP Math I Teacher Resource
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Introduction
The CCSS Mathematics I Student Workbook with Scaffolded Practice includes all of the student pages
from the Teacher Resource necessary for your day-to-day classroom use. This includes:
•
Warm-Ups
•
Problem-Based Tasks
•
Practice Problems
•
Station Activity Worksheets
In addition, it provides Scaffolded Guided Practice examples that parallel the examples in the TRB
and SRB. This supports:
•
Taking notes during class
•
Working problems for preview or additional practice
The workbook includes the first Guided Practice example with step-by-step prompts for solving,
and the remaining Guided Practice examples without prompts. Sections for you to take notes are
provided at the end of each sub-lesson. Additionally, blank coordinate planes are included at the end
of the full unit, should you need to graph.
The workbook is printed on perforated paper so you can submit your assignments and three-hole
punched to let you store it in a binder.
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CCSS IP Math I Teacher Resource
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Name:
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UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES
Lesson 1: Slope and Distance
Lesson 6.1.1: Using Coordinates to Prove Geometric Theorems with Slope
and Distance
Warm-Up 6.1.1
The graph below shows the average number of miles an airplane travels over the course of its flight.
4200
3900
3600
3300
Distance (miles)
3000
2700
2400
2100
1800
1500
1200
900
600
300
-60 0
60 120 180 240 300 360 420
Time (minutes)
1. What is the average rate of change per hour?
2. How many miles would the airplane travel in 9 hours?
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UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES
Lesson 1: Slope and Distance
Scaffolded Practice 6.1.1
Example 1
Calculate the distance between the points (4, 9) and (–2, 6) using both the Pythagorean Theorem and
the distance formula.
1. To use the Pythagorean Theorem, first plot the points on a coordinate system.
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9
8
7
6
5
4
3
2
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-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
1
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9 10
-1
-2
-3
-4
-5
-6
-7
-8
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-10
2. Draw lines to form a right triangle, using each point as the end of the hypotenuse.
3. Calculate the length of the vertical side, a, of the right triangle.
4. Calculate the length of the horizontal side, b, of the right triangle.
5. Use the Pythagorean Theorem to calculate the length of the hypotenuse, c.
6. Now use the distance formula to calculate the distance between the same points, (4, 9) and (–2, 6).
continued
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UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES
Lesson 1: Slope and Distance
Example 2
Determine if the line through the points (–8, 5) and (–5, 3) is parallel to the line through the points
(1, 3) and (4, 1).
Example 3
Determine if the line through the points (0, 8) and (4, 9) is perpendicular to the line through the
points (–9, 10) and (–8, 6).
Example 4
A right triangle is defined as a triangle with 2 sides that are perpendicular. Triangle ABC has vertices
A (–4, 8), B (–1, 2), and C (7, 6). Determine if this triangle is a right triangle. When disproving a
figure, you only need to show one condition is not met.
Example 5
A square is a quadrilateral with two pairs of parallel opposite sides, consecutive sides that are
perpendicular, and all sides congruent, meaning they have the same length. Quadrilateral ABCD has
vertices A (–1, 2), B (1, 5), C (4, 3), and D (2, 0). Determine if this quadrilateral is a square.
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UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES
Lesson 1: Slope and Distance
Problem-Based Task 6.1.1: Field of Dreams
A square is a quadrilateral with two pairs of parallel opposite sides, consecutive sides that are
perpendicular, and all sides congruent. The bases and home plate of a baseball diamond form a
square. The local recreation department has created a map of its newest baseball field. First base is
represented by the point (7, –3), second base is represented by (4, 6), third base is represented by
(–5, 3), and home plate is represented by (–2, –6). Is this field actually a square? What is the distance
from second base to home plate? What is the distance from third base to first base?
Is this field
actually a square?
What is the
distance from
second base to
home plate? What
is the distance
from third base to
first base?
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UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES
Lesson 1: Slope and Distance
Practice 6.1.1: Using Coordinates to Prove Geometric Theorems with Slope
and Distance
Use the distance formula to calculate the distance between the points indicated.
1. (–1, 1) and (4, 11)
2. (–3, –2) and (6, –1)
A right triangle has a 90˚ angle made up of two perpendicular sides. Graph each triangle and then
determine if each one is a right triangle. Use the slope formula and/or distance formula to justify
your answer.
3. A (–6, 2), B (–2, –2), and C (2, 1)
4. A (–3, 2), B (1, 4), and C (3, 0)
A parallelogram is a quadrilateral with opposite sides parallel. Graph each quadrilateral and then
determine if each one is a parallelogram. Use the slope formula and/or distance formula to justify
your answer.
5. A (–2, –1), B (–1, 3), C (4, 3), and D (3, –1)
6. A (–2, 2), B (1, 8), C (4, 5), and D (3, –2)
A rectangle is a parallelogram with opposite sides that are congruent and consecutive sides that are
perpendicular. Graph each quadrilateral and then determine if each one is a rectangle. Use the slope
formula and/or distance formula to justify your answer.
7. A (–2, –1), B (–3, 1), C (1, 3), and D (2, 1)
8. A (1, –3), B (0, 0), C (4, 4), and D (5, 1)
A square is a parallelogram with four congruent sides and four right angles. Graph each quadrilateral
and then determine if each one is a square. Use the slope formula and/or distance formula to justify
your answer.
9. A (–2, 1), B (1, 3), C (4, 1), and D (1, –1)
10. A (–1, 4), B (0, 6), C (2, 5), and D (1, 3)
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UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES
Lesson 1: Slope and Distance
Lesson 6.1.2: Working with Parallel and Perpendicular Lines
Warm-Up 6.1.2
Civil engineers are planning a new shopping center downtown. They would like the entrance to the
1
center to run perpendicular to the street represented by the equation y = x + 2 .
5
et
Stre
Water
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9
8
7
6
5
4
3
2
1
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
1
2
3
4
5
6
7
8
9 10
1. W
rite an equation that would represent the entrance to the shopping center. Explain
your reasoning.
2. A
second entrance will run parallel to the first entrance. Write an equation that would represent
the second entrance to the shopping center.
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UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES
Lesson 1: Slope and Distance
Scaffolded Practice 6.1.2
Example 1
Write the slope-intercept form of an equation for the line that passes through the point (5, –2) and is
parallel to the graph of 8x – 2y = 6.
1. Rewrite the given equation in slope-intercept form.
2. Identify the slope of the given line.
3. Substitute the slope of the given line for m in the point-slope form of a linear equation.
4. Substitute x and y from the given point into the equation for x1 and y1.
5. Simplify the equation.
continued
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6.1.2
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UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES
Lesson 1: Slope and Distance
Example 2
Write the slope-intercept form of an equation for the line that passes through the point (–1, 6) and is
perpendicular to the graph of –10x + 5y = 20.
Example 3
Find the point on the line y = 4x + 1 that is closest to the point (–2, 8).
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UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES
Lesson 1: Slope and Distance
Problem-Based Task 6.1.2: Pave the Way
The civil engineers are busy again. This time they are expanding one of the well-traveled roads in your
3
town. The center of the road is represented by the equation y = x + 5 . The law says a two-lane road
5
must be at least 24 feet wide. Engineers have mapped out the edges of the road and have one edge
running through the point (16, 1) and the other edge running through the point (4, 21). What are
the equations of the lines that represent the edges of the road? Will this satisfy the minimum width
required by law?
What are the
equations of the
lines that represent
the edges of
the road? Will
this satisfy the
minimum width
required by law?
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UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES
Lesson 1: Slope and Distance
Practice 6.1.2: Working with Parallel and Perpendicular Lines
Write the slope-intercept form of an equation for the line that passes through the given point and is
parallel to the graph of the given equation.
1. (3, 1) and 4x + 2y = 10
2. (–5, 6) and 12x + 9y = 3
Write the slope-intercept form of an equation for the line that passes through the given point and is
perpendicular to the graph of the given equation.
3. (–4, –3) and 8x + 2y = 14
4. (–6, 5) and 2x – 2y = 10
Calculate the shortest distance from the given point to the line indicated.
5. (2, 1) to x – y = –1
For questions 6–10, refer to the following map. Each unit represents 100 yards.
10
9
Bridge
8
Desiree's
House
Union S
7
6
treet
5
4
3
Park
Train
Station
2
1
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
-1
1
2
3
4
5
6
7
8
9 10
-2
-3
-4
-5
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-8
Ma
in
Str
ee
t
-9
-10
continued
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UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES
Lesson 1: Slope and Distance
3
6. Main Street is given by the equation y = − x − 2 . The train station is located at the point (8, 1).
4
What is the equation of the line that represents the train tracks that run parallel to Main Street
through the point (8, 1)?
7. Desiree’s house is located at the point (–8, 6). Her driveway is perpendicular to Union
1
Street, which is represented by the equation y = x + 6 . What is the equation of the line that
5
represents Desiree’s driveway?
8. L ittle River runs parallel to Union Street. There is a bridge located at the point (–5, 9). What is
the equation of the line that represents Little River?
9. T
he center of the park is located at the point (3, 2). A walking trail starting at this point runs
perpendicular to Main Street. What is the equation of the line that represents the walking trail?
10. What is the shortest distance from the train station to Union Street?
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UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES
Lesson 2: Lines and Line Segments
Lesson 6.2.1: Calculating Perimeter and Area
Warm-Up 6.2.1
Mikko wants to make an open-top plywood box to fit on a certain shelf in his closet. He has drawn
the sketch below of the box that he needs.
h = 8 in.
w = 11 in.
l = 18 in.
1. How much plywood would Mikko need to create this box?
2. M
ikko’s dad has brought him a piece of plywood that measures 2 feet by 3 feet. Is this enough
plywood for the project? Explain why or why not.
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UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES
Lesson 2: Lines and Line Segments
Scaffolded Practice 6.2.1
Example 1
Triangle ABC has vertices A (–3, 1), B (1, 3), and C (2, –4). Calculate the perimeter of the triangle.
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9
8
7
6
5
4
B
(1, 3)
3
2
A
(-3, 1)
1
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
-1
1
2
3
4
5
6
7
8
9 10
-2
-3
-4
-5
C
(2, -4)
-6
-7
-8
-9
-10
1. Calculate the length of each side of the triangle. Use the distance formula.
2. Calculate the perimeter of triangle ABC.
continued
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UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES
Lesson 2: Lines and Line Segments
Example 2
Quadrilateral ABCD has vertices A (–3, 0), B (2, 4), C (3, 1), and D (–4, –3). Calculate the perimeter of
the quadrilateral.
10
9
8
7
6
5
B
(2, 4)
4
3
A
(-3, 0)
2
C
(3, 1)
1
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
-1
1
2
3
4
5
6
7
8
9 10
-2
D
(-4, -3)
-3
-4
-5
-6
-7
-8
-9
-10
Example 3
Triangle ABC has vertices A (1, –1), B (4, 3), and C (5, –3). Calculate the area of triangle ABC.
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9
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7
6
5
4
B
(4, 3)
3
2
1
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
-1
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-3
1
2
A
(1, -1)
-4
3
4
5
6
7
8
9 10
C
(5, -3)
-5
-6
-7
-8
-9
continued
-10
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UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES
Lesson 2: Lines and Line Segments
Example 4
Rectangle ABCD has vertices A (–3, –4), B (–1, 2), C (2, 1), and D (0, –5). Calculate the area of
the rectangle.
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9
8
7
6
5
4
B
(-1, 2)
3
2
C
(2, 1)
1
-10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0
-1
1
2
3
4
5
6
7
8
9 10
-2
-3
A
(-3, -4)
-4
-5
-6
D
(0, -5)
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-8
-9
-10
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UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES
Lesson 2: Lines and Line Segments
Problem-Based Task 6.2.1: Building Fences
The local recreation department has created a map of its newest baseball field. The department is
planning to install a rectangular fence around the field. The corners of the field are represented on the
map by the points A (–5, –10), B (13, –4), C (4, 23), and D (–11, 18). How many feet of fencing are
needed for the baseball field? What is the area of the fenced-in field? Each unit on the map represents
10 feet.
How many feet of
fencing are needed
for the baseball
field? What is the
area of the fencedin field?
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UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES
Lesson 2: Lines and Line Segments
Practice 6.2.1: Perimeter and Area
Calculate the perimeter of each of the polygons below.
1. Triangle ABC has vertices A (–2, 1), B (–3, 5), and C (3, 6).
2. Trapezoid ABCD has vertices A (–3, –2), B (3, 4), C (3, –1), and D (1, –3).
3. Quadrilateral ABCD has vertices A (–4, 0), B (–2, 3), C (2, 3), and D (2, 0).
4. Square ABCD has vertices A (–2, 2), B (0, 4), C (2, 2), and D (0, 0).
5. Parallelogram ABCD has vertices A (–5, 4), B (–1, 6), C (5, 2), and D (1, 0).
Calculate the area of each of the polygons below.
6. Rectangle ABCD has vertices A (–5, 2), B (–5, 4), C (4, 4), and D (4, 2).
7. Rectangle ABCD has vertices A (–4, –4), B (0, 2), C (9, –4), and D (5, –10).
8. Right triangle ABC has vertices A (–4, 2), B (–4, 6), and C (9, 6).
9. Triangle ABC has vertices A (–2, 5), B (3, 1), and C (3, 5).
10. Triangle ABC has vertices A (3, 5), B (7, 8), and C (5, –3).
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UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES
Station Activities Set 1: Parallel Lines, Slopes, and Equations
Station 1
At this station, you will find rulers. Use these to help you determine whether or not the following
lines are parallel.
Look at the graph below.
1. Are these lines parallel?
2. Explain two ways you can tell lines are parallel.
3. What is the shortest distance between these two lines?
4. Draw a line that is parallel to the given line below.
5. How do you know your line is parallel?
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UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES
Station Activities Set 1: Parallel Lines, Slopes, and Equations
Station 2
At this station, you will find graph paper, a ruler, a red marker, and a blue marker. As a group,
construct an x- and y-axis on your graph paper.
1. On your graph paper, construct a straight line that passes through (–2, –5) and (7, 1).
What is the slope of this line?
S how how you found this slope by using the red and blue markers to represent rise and run on
your graph.
2. Construct a second line that passes through (0, 4) and (18, 16).
What is the slope of this line?
S how how you found this slope by using the red and blue markers to represent rise and run on
your graph.
3. Construct a third line that passes through (0, 0) and (10, 8).
What is the slope of this line?
S how how you found this slope by using the red and blue markers to represent rise and run on
your graph.
4. O
f the three straight lines you created, which two lines do you think are parallel? Explain your
answer.
continued
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UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES
Station Activities Set 1: Parallel Lines, Slopes, and Equations
For problems 5–8, determine whether the lines are parallel. Answer yes or no. Justify your answers.
y
5.
10
9
8
7
6
5
4
3
2
1
0
–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 –1
–2
–3
–4
–5
–6
–7
–8
–9
–10
1
2
3
4
5
6
7
8
9 10
x
1
2
3
4
5
6
7
8
9 10
x
Are the lines parallel?
y
6.
10
9
8
7
6
5
4
3
2
1
0
–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 –1
–2
–3
–4
–5
–6
–7
–8
–9
–10
Are the lines parallel?
7. One line has a slope of
Are the lines parallel?
16
2
and another line has a slope of
.
3
24
8. One line has a slope of −
Are the lines parallel?
1
1
and another line has a slope of .
2
2
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UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES
Station Activities Set 1: Parallel Lines, Slopes, and Equations
Station 3
At this station, you will find spaghetti noodles, graph paper, and a ruler. Work as a group to
construct the lines and answer the questions.
1. O
n your graph paper, construct a straight line that passes through (–2, –2) and (5, 10). Place a
spaghetti noodle on this line.
2. O
n the same graph, construct a straight line that passes through (–6, 4) and (1, 16). Place a
spaghetti noodle on this line.
By looking at the spaghetti noodles, the two lines may appear to be parallel, but are they?
3. H
ow can you use the points on each line to determine if the two lines are parallel? Show your
work. Are the lines parallel? Why or why not?
4. O
n the same graph, construct a straight line that passes through (5, 10) and (17, 3). Place a
spaghetti noodle on this line.
What is the slope of this line?
Is this line perpendicular to the line you created in problem 1? Why or why not?
Is this line also perpendicular to the line you created in problem 2? Why or why not?
5. Perpendicular lines create four angles that each measure __________________.
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UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES
Station Activities Set 1: Parallel Lines, Slopes, and Equations
Station 4
At this station, you will find graph paper and a ruler. Work as a group to construct the graphs and
answer the questions.
1. On your graph paper, graph the straight line that passes through the points (–4, 5) and (4, –5).
2. What is the slope of this line?
3. U
se the formula for point-slope form to find an equation for this line. Show your work and
answer in the space below.
Point-slope form: y – y1 = m(x – x1)
Equation for the line: __________________
4. I f you were to construct a line parallel to the line in problem 1, what slope would this line have?
Explain your answer.
5. H
ow can you use your graph paper and the slope you found in problem 2 to construct a line
parallel to the line in problem 1?
Draw this line on your graph paper.
continued
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UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES
Station Activities Set 1: Parallel Lines, Slopes, and Equations
6. U
se the formula for point-slope form to find an equation for this line. Show your work and
answer in the space below.
Point-slope form: y – y1 = m(x – x1)
Equation for the line: __________________
7. Construct another parallel line on your graph paper.
8. U
se the formula for point-slope form to find an equation for this line. Show your work and
answer in the space below.
Point-slope form: y – y1 = m(x – x1)
Equation for the line: __________________
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UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES
Station Activities Set 2: Perpendicular Lines
Station 1
At this station, you will find graph paper and a ruler. As a group, construct an x- and y-axis on the
graph paper.
1. Construct a line that passes through (–2, 5) and (8, 5).
What type of line have you created?
2. On the same graph, plot and label the point (3, 12).
3. H
ow can you use perpendicular lines to find the distance between the line you created in
problem 1 and the point you plotted in problem 2?
4. W
hat is the distance between the line you created in problem 1 and the point you plotted in
problem 2? Explain your answer.
5. On a new graph, construct a line that passes through (2, 10) and (2, 0).
What type of line have you created?
6. On the same graph, plot and label the point (6, 3).
continued
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UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES
Station Activities Set 2: Perpendicular Lines
7. H
ow can you use perpendicular lines to find the distance between the line you created in
problem 5 and the point you plotted in problem 6?
8. W
hat is the distance between the line you created in problem 5 and the point you plotted in
problem 6? Explain your answer.
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UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES
Station Activities Set 2: Perpendicular Lines
Station 2
At this station, you will find a protractor.
For problems 1 and 2, work as a group to determine whether or not the two lines are perpendicular
using the coordinate graph. Do NOT use your protractor.
y
10
9
8
7
6
5
4
3
2
1
1.
0
–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 –1
–2
–3
–4
–5
–6
–7
–8
–9
–10
1
2
3
4
5
6
7
8
9 10
x
9 10
x
Are the lines perpendicular? Explain.
y
2.
10
9
8
7
6
5
4
3
2
1
0
–10 –9 –8 –7 –6 –5 –4 –3 –2 –1 –1
–2
–3
–4
–5
–6
–7
–8
–9
–10
1
2
3
4
5
6
7
8
Are the lines perpendicular? Explain.
continued
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UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES
Station Activities Set 2: Perpendicular Lines
3. What strategy did you use to determine whether or not the lines are perpendicular?
For problems 4 and 5, use your protractor to determine whether or not the lines are perpendicular.
4.
Are the lines perpendicular? Explain.
5.
Are the lines perpendicular? Explain.
6. How did you use your protractor to determine whether or not the lines were perpendicular?
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UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES
Station Activities Set 2: Perpendicular Lines
Station 3
At this station, you will find graph paper and a ruler. As a group, create an x- and y-axis on the graph
paper. Work together to answer the questions.
1. On your graph paper, construct a line that passes through points (–4, 0) and (0, 8).
What is the slope of this line?
2. W
hat is the slope of a line that is perpendicular to the line you created in problem 1? Explain
your answer.
3. U
se the point-slope form of an equation, y – y1 = m(x – x1), to find the equation of a line that is
perpendicular to the line from problem 1, and that passes through the point (–2, 4). Show your
work and answer in the space below.
Equation: __________________
5
4. W
rite the equation for the line that is perpendicular to y = − x + 10 and passes through
6
(1, 2). Show your work and answer in the space below.
Equation: __________________
5. O
n your graph paper, graph the two lines in problem 4 to justify that the lines are
perpendicular.
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UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES
Station Activities Set 2: Perpendicular Lines
Station 4
At this station, you will find 18 index cards. Each index card has an equation written on it. Your job is
to pair the equations into sets of parallel and perpendicular lines.
1. W
ork with your group members to find the parallel and perpendicular lines. Record your
results below.
Parallel lines
Perpendicular lines
2. What was your strategy for working through this activity?
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UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES
Station Activities Set 3: Coordinate Proof with Quadrilaterals
Station 1
At this station, you will find a geoboard and rubber bands.
1. W
ith your group, use the geoboard to construct a polygon with only one set of parallel sides.
Draw this figure below.
2. Construct a polygon with two sets of parallel sides. Draw this figure below.
3. Construct a polygon with at least one set of perpendicular sides. Draw this figure below.
4. H
ow did you apply what you know about parallel and perpendicular lines to successfully
complete this activity?
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UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES
Station Activities Set 3: Coordinate Proof with Quadrilaterals
Station 2
At this station, you will find graph paper and a ruler. Work as a group to construct the quadrilaterals
and answer the questions.
1. On your graph paper, construct a quadrilateral that has vertices (1, 1), (2, 4), (8, 4), and (9, 1).
What type of quadrilateral did you create?
2. On the same graph, construct a quadrilateral that is congruent to the quadrilateral in problem 1.
What are the vertices for this new quadrilateral?
Explain why the quadrilaterals in problems 1 and 2 are congruent.
3. O
n the same graph, construct a quadrilateral that is NOT congruent to the quadrilateral in
problem 1.
What are the vertices of this new quadrilateral?
Explain why the quadrilaterals in problems 1 and 3 are NOT congruent.
4. Graph a quadrilateral with vertices (0, 0), (2, 5), (–2, 5), and (0, 7).
What type of quadrilateral did you create?
continued
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UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES
Station Activities Set 3: Coordinate Proof with Quadrilaterals
5. J acob claims that a quadrilateral with vertices (–2, 2), (0, 6), (–2, 8), and (–4, 6) is congruent to
the quadrilateral in problem 4. Is Jacob correct? Why or why not? Graph the quadrilaterals on
your graph paper to justify your answer.
I f Jacob is incorrect, what vertices would make this quadrilateral congruent to the quadrilateral in
problem 4?
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UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES
Station Activities Set 3: Coordinate Proof with Quadrilaterals
Station 3
At this station, you will find graph paper and a ruler. Work as a group to construct the quadrilaterals
and answer the questions.
1. On your graph paper, graph a quadrilateral that has vertices (2, 4), (8, 4), (2, 10), and (8, 10).
hat are three names for this quadrilateral? Justify each name using properties of that type of
W
quadrilateral.
2. On your graph paper, graph a quadrilateral that has vertices (–1, 0), (–4, 0), (–1, 7), and (–4, 7).
What is the name of this quadrilateral?
3. Is the quadrilateral in problem 1 a regular quadrilateral? Why or why not?
4. Is the quadrilateral in problem 2 a regular quadrilateral? Why or why not?
5. What strategy did you use for problems 1–4?
continued
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UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES
Station Activities Set 3: Coordinate Proof with Quadrilaterals
6. What is the definition of a regular quadrilateral?
7. Are all rhombi and rectangles regular quadrilaterals? Why or why not?
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UNIT 6 • CONNECTING ALGEBRA AND GEOMETRY THROUGH COORDINATES
Station Activities Set 3: Coordinate Proof with Quadrilaterals
Station 4
At this station, you will find graph paper and a ruler. Work together to construct the quadrilaterals
and answer the questions.
1. On your graph paper, construct a parallelogram with vertices (0, 0), (–6, 0), (–7, 4), and (–1, 4).
2. On the same graph, construct a similar parallelogram.
What are the vertices of the parallelogram?
Why is this new parallelogram similar to the parallelogram in problem 1?
3. O
n a new graph, construct a trapezoid with vertices (1, 1), (4, 1), (2, 4), and (3, 4). Construct a
second trapezoid with vertices (0, 0), (6, 0), (2, –6), and (4, –6).
Are the two trapezoids congruent or similar? Explain your answer.
4. O
n a new graph, construct a kite. Construct a second kite that is similar, but not congruent to
the first kite.
What are the vertices of the first kite?
What are the vertices of the second kite?
Why are the kites similar?
5. When would you use similar quadrilaterals in the real world?
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60
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Formulas
ALGEBRA
General
Exponential Equations
(x, y)
Ordered pair
y = abx
(x, 0)
x-intercept
(0, y)
y-intercept
General form
x
Symbols
y = ab t
Exponential equation
y = a(1 + r)t
Exponential growth
y = a(1 – r)t
Exponential decay
≈
Approximately equal to
≠
Is not equal to
 r
A= P1+ 
 n
a
Absolute value of a
Compounded…
a
nt
Compounded
interest formula
n (number of
times per year)
Yearly/annually 1
Square root of a
Semi-annually
2
Quarterly
4
Slope
Monthly
12
ax + b = c
One variable
Weekly
52
y = mx + b
Slope-intercept form
Daily
365
ax + by = c
General form
Linear Equations
m=
y2 − y1
x2 − x1
Functions
y – y1 = m(x – x1) Point-slope form
f(x)
Notation, “f of x”
Arithmetic Sequences
f(x) = mx + b
Linear function
an = a1 + (n – 1)d Explicit formula
f(x) = bx + k
Exponential function
an = an –1 + d
(f + g)(x) = f(x) + g(x) Addition
Recursive formula
(f – g)(x) = f(x) – g(x) Subtraction
Geometric Sequences
(f • g)(x) = f(x) • g(x) Multiplication
an = a1 • r
Explicit formula
(f ÷ g)(x) = f(x) ÷ g(x) Division
an = an – 1 • r
Recursive formula
n–1
F-1
Formulas
63
Formulas
Properties of Equality
Property
In symbols
Reflexive property of equality
a=a
Symmetric property of equality
If a = b, then b = a.
Transitive property of equality
If a = b and b = c, then a = c.
Addition property of equality
If a = b, then a + c = b + c.
Subtraction property of equality
If a = b, then a – c = b – c.
Multiplication property of equality If a = b and c ≠ 0, then a • c = b • c.
Division property of equality
If a = b and c ≠ 0, then a ÷ c = b ÷ c.
Substitution property of equality
If a = b, then b may be substituted for
a in any expression containing a.
Properties of Operations
Property
General rule
Commutative property of addition
a+b=b+a
Associative property of addition
(a + b) + c = a + (b + c)
Commutative property of multiplication
a•b=b•a
Associative property of multiplication
(a • b) • c = a • (b • c)
Distributive property of multiplication over addition a • (b + c) = a • b + a • c
Properties of Inequality
Laws of Exponents
Property
Law
General rule
If a > b and b > c, then a > c.
Multiplication of
exponents
Power of exponents
bm • bn = bm + n
If a > b, then b < a.
If a > b, then –a < –b.
If a > b, then a ± c > b ± c.
If a > b and c > 0, then a • c > b • c.
If a > b and c < 0, then a • c < b • c.
If a > b and c > 0, then a ÷ c > b ÷ c.
If a > b and c < 0, then a ÷ c < b ÷ c.
(b )
m n
= b mn
( bc )n = bnc n
Division of exponents
bm
n
Exponents of zero
Negative exponents
F-2
Formulas
64
= b m− n
b
b0 = 1
b− n =
1
b
n
and
1
b
−n
= bn
Formulas
DATA ANALYSIS
IQR = Q 3 – Q 1 Interquartile range
Q 1 – 1.5(IQR) Lower outlier formula
Q 3 + 1.5(IQR) Upper outlier formula
y – y0
Residual formula
GEOMETRY
Symbols
( )
Translations
d 
ABC
Arc length
T(h, k) = (x + h, y + k) Translation
∠
Angle
Reflections

Circle
rx-axis (x, y) = (x, –y) Through the x-axis
≅
PQ
Congruent
ry-axis (x, y) = (–x, y) Through the y-axis
Line
ry = x (x, y) = (y, x)
PQ
PQ
Line Segment
Rotations
Ray

R90 (x, y) = (–y, x)
Parallel
⊥
Perpendicular
•
Point

Triangle
A′
Prime
°
Degrees
Through the line y = x
Counterclockwise 90º
about the origin
R180 (x, y) = (–x, –y) Counterclockwise
180º about the origin
R270 (x, y) = (y, –x) Counterclockwise
270º about the origin
Congruent Triangle Statements
Side-Side-Side (SSS)
Side-Angle-Side (SAS)
A
C
 ABC ≅  XYZ
G
D
B
Z
Angle-Side-Angle (ASA)
F
X
T
Y
W
J
E
V
 DEF ≅ TVW
Q
S
H
R
GHJ ≅ QRS
F-3
Formulas
65
Formulas
Pythagorean Theorem
Distance Formula
a2 + b2 = c2
d = ( x2 − x1 ) 2 + ( y2 − y1 ) 2
Distance formula
Area
A = lw
Rectangle
1
A = bh
2
Triangle
MEASUREMENTS
Length
Volume and Capacity
Metric
Metric
1 kilometer (km) = 1000 meters (m)
1 liter (L) = 1000 milliliters (mL)
Customary
1 meter (m) = 100 centimeters (cm)
1 centimeter (cm) = 10 millimeters
(mm)
Customary
1 mile (mi) = 1760 yards (yd)
1 gallon (gal) = 4 quarts (qt)
1 quart (qt) = 2 pints (pt)
1 pint (pt) = 2 cups (c)
1 cup (c) = 8 fluid ounces (fl oz)
1 mile (mi) = 5280 feet (ft)
1 yard (yd) = 3 feet (ft)
1 foot (ft) = 12 inches (in)
Weight and Mass
Metric
1 kilogram (kg) = 1000 grams (g)
1 gram (g) = 1000 milligrams (mg)
1 metric ton (MT) = 1000 kilograms (kg)
Customary
1 ton (T) = 2000 pounds (lb)
1 pound (lb) = 16 ounces (oz)
F-4
Formulas
66
PROGRAM OVERVIEW
Glossary
English
Español
acute angle an angle measuring less than
90˚ but greater than 0˚
algebraic expression a mathematical
statement that includes numbers,
operations, and variables to represent a
number or quantity
algebraic inequality an inequality that
has one or more variables and contains at
least one of the following symbols: <, >,
≤, ≥, or ≠
altitude the perpendicular line from a
vertex of a figure to its opposite side;
height
angle two rays or line segments sharing a
common endpoint; the symbol used is
∠
angle of rotation the measure of the
angle created by the preimage vertex to
the point of rotation to the image vertex.
All of these angles are congruent when a
figure is rotated.
A
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angle-side-angle (ASA) if two angles
and the included side of one triangle are
congruent to two angles and the included
side of another triangle, then the two
triangles are congruent
arc length the distance between the
endpoints of an arc; written as d 
ABC
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area the amount of space inside the
boundary of a two-dimensional figure
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( )
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ángulo agudo ángulo que mide menos de
90˚ pero más de 0˚
expresión algebraica declaración
matemática que incluye números,
operaciones y variables para representar
un número o una cantidad
desigualdad algebraica desigualdad que
tiene una o más variables y contiene al
menos uno de los siguientes símbolos: <,
>, ≤, ≥, o ≠
altitud línea perpendicular desde un
vértice de una figura hasta su lado
opuesto; altura
ángulo dos semirrectas o segmentos de
línea que comparten un extremo común;
el símbolo utilizado es ∠
ángulo de rotación medida del ángulo
creada por el vértice del preimagen
hasta el punto de rotación del vértice
del imagen. Todos estos ángulos son
congruentes cuando una figura está
rotada.
ángulo-lado-ángulo (ASA) si dos ángulos
y el lado incluido de un triángulo son
congruentes con los dos ángulos y el lado
incluido de otro triángulo, entonces los
dos triángulos son congruentes
longitud de arco distancia entre los
extremos de un arco; se expresa como
d 
ABC
( )
área cantidad de espacio dentro del límite
de una figura bidimensional
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PROGRAM OVERVIEW
Glossary
English
arithmetic sequence a linear function
with a domain of positive consecutive
integers in which the difference between
any two consecutive terms is equal
asymptote a line that a graph gets closer
and closer to, but never crosses or
touches
base the factor being multiplied together
in an exponential expression; in the
expression ab, a is the base
bisect to cut in half
box plot a plot showing the minimum,
maximum, first quartile, median, and
third quartile of a data set; the middle
50% of the data is indicated by a box.
Example:
causation a relationship between two
events where a change in one event is
responsible for a change in the second
event
circle the set of points on a plane at a
certain distance, or radius, from a single
point, the center. The set of points forms
a two-dimensional curve that measures
360˚.
circular arc on a circle, the unshared set
of points between the endpoints of two
radii
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Español
secuencia aritmética función lineal
con dominio de enteros consecutivos
positivos, en la que la diferencia entre dos
términos consecutivos es equivalente
asíntota línea a la que se acerca cada vez
más un gráfico, pero sin cruzarlo ni
tocarlo
base factor que se multiplica en forma
conjunta en una expresión exponencial;
en la expresión ab, a es la base
bisecar cortar por la mitad
diagrama de caja diagrama que muestra
el mínimo, máximo, primer cuartil,
mediana y tercer cuartil de un conjunto
de datos; se indica con una caja el 50%
medio de los datos. Ejemplo:
causalidad relación entre dos eventos en la
que un cambio en un evento es responsable
por un cambio en el segundo evento
círculo conjunto de puntos en un plano
a determinada distancia, o radio, de un
único punto, el centro. El conjunto de
puntos forma una curva bidimensional
que mide 360˚.
arco circular en un círculo, conjunto
de puntos no compartidos entre los
extremos de dos radios
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PROGRAM OVERVIEW
Glossary
English
clockwise rotating a figure in the direction
that the hands on a clock move
coefficient the number multiplied by a
variable in an algebraic expression
common difference the number added to
each consecutive term in an arithmetic
sequence
compass an instrument for creating
circles or transferring measurements that
consists of two pointed branches joined
at the top by a pivot
compression a transformation in which
a figure becomes smaller; compressions
may be horizontal (affecting only
horizontal lengths), vertical (affecting
only vertical lengths), or both
conditional relative frequency the
percentage of a joint frequency as
compared to the total number of
respondents, total number of people
with a given characteristic, or the total
number of times a specific response was
given
congruency transformation a
transformation in which a geometric
figure moves but keeps the same size and
shape
congruent figures are congruent if they
have the same shape, size, lines, and
angles; the symbol for representing
congruency between figures is ≅
congruent angles two angles that have the
same measure
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Español
sentido horario rotación de una figura en
la dirección en que se mueven las agujas
de un reloj
coeficiente número multiplicado por una
variable en una expresión algebraica
diferencia común número sumado a cada
término consecutivo en una secuencia
aritmética
compás instrumento utilizado para crear
círculos o transferir medidas, que consiste
en dos brazos terminados en punta y
unidos en la parte superior por un pivote
compresión transformación en la que
una figura se hace más pequeña; las
compresiones pueden ser horizontales
(cuando afectan sólo la longitud
horizontal), verticales (cuando afectan sólo
la longitud vertical), o en ambos sentidos
frecuencia condicional relativa porcentaje de una frecuencia conjunta
en comparación con la cantidad total
de respondedores, cantidad total
de personas con una determinada
característica, o cantidad total de veces
que se dio una respuesta específica
transformación de congruencia transformación en la que se mueve una
figura geométrica pero se mantiene el
mismo tamaño y la misma forma
congruente las figuras son congruentes si
tienen la misma forma, tamaño, rectas y
ángulos; el símbolo para representar la
congruencia entre figuras es ≅
ángulos congruentes dos ángulos con la
misma medida
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PROGRAM OVERVIEW
Glossary
English
congruent sides two sides that have the
same length
congruent triangles triangles having the
same angle measures and side lengths
consistent a system of equations with at
least one ordered pair that satisfies both
equations
constant a quantity that does not change
constant ratio the number each
consecutive term is multiplied by in a
geometric sequence
constraint a restriction or limitation on
either the input or output values
construct to create a precise geometric
representation using a straightedge along
with either patty paper (tracing paper), a
compass, or a reflecting device
construction a precise representation
of a figure using a straightedge and a
compass, patty paper and a straightedge,
or a reflecting device and a straightedge
continuous having no breaks
coordinate plane a set of two number
lines, called the axes, that intersect at
right angles
correlation a relationship between two
events, where a change in one event is
related to a change in the second event. A
correlation between two events does not
imply that the first event is responsible
for the change in the second event; the
correlation only shows how likely it
is that a change also took place in the
second event.
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Español
lados congruentes dos lados con la
misma longitud
triángulos congruentes triángulos con las
mismas medidas de ángulos y longitudes
de lados
consistente sistema de ecuaciones con
al menos un par ordenado que satisface
ambas ecuaciones
constante cantidad que no cambia
proporción constante el número que
cada término esta multiplicado por en una
secuencia geométrica
limitación restricción o límite en los
valores de entrada o salida
construir crear una representación
geométrica precisa mediante regla de borde
recto y papel encerado (papel para calcar),
compás o un dispositivo de reflexión
construcción representación precisa de
una figura mediante regla de borde recto
y compás, papel encerado y una regla de
borde recto, o un dispositivo de reflexión
y una regla de borde recto
continuo sin interrupciones
plano de coordenadas conjunto de dos
rectas numéricas, denominadas ejes, que
se cortan en ángulos rectos
correlación relación entre dos eventos
en la que el cambio en un evento se
relaciona con un cambio en el segundo
evento. Una correlación entre dos
eventos no implica que el primero sea
responsable del cambio en el segundo;
la correlación sólo demuestra cuán
probable es que también se produzca un
cambio en el segundo evento.
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correlation coefficient a quantity
that assesses the strength of a linear
relationship between two variables,
ranging from –1 to 1; a correlation
coefficient of –1 indicates a strong
negative correlation, a correlation
coefficient of 1 indicates a strong positive
correlation, and a correlation coefficient
of 0 indicates a very weak or no linear
correlation
corresponding angles angles of two
figures that lie in the same position
relative to the figure. In transformations,
the corresponding vertices are the
preimage and image vertices, so ∠A and
∠A′ are corresponding vertices and so
on.
Corresponding Parts of Congruent
Triangles are Congruent (CPCTC) if two or more triangles are proven
congruent, then all of their corresponding
parts are congruent as well
corresponding sides sides of two figures
that lie in the same position relative
to the figure. In transformations, the
corresponding sides are the preimage
and image sides, so AB and A′B′ are
corresponding sides and so on.
counterclockwise rotating a figure in the
opposite direction that the hands on a
clock move
curve the graphical representation of the
solution set for y = f(x); in the special case
of a linear equation, the curve will be a
line
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coeficiente de correlación cantidad que
evalúa la fuerza de una relación lineal
entre dos variables, que varía de –1 a
1; un coeficiente de correlación de –1
indica una fuerte correlación negativa,
un coeficiente de correlación de 1 indica
una fuerte correlación positiva, y un
coeficiente de correlación de 0 indica una
correlación muy débil o no lineal
ángulos correspondientes ángulos de dos
figuras que se ubican en la misma posición
relativa a la figura. En las transformaciones,
los vértices correspondientes son los
vértices de preimagen e imagen, de
manera que ∠A y ∠A′ son los vértices
correspondientes, etc.
Las partes correspondientes de
triángulos congruentes son
congruentes (CPCTC) si se comprueba que dos o más triángulos
son congruentes, entonces todas sus partes
correspondientes son también congruentes
lados correspondientes lados de dos
figuras que están en la misma posición
relativa a la figura. En las transformaciones,
los lados correspondientes son los de
preimagen e imagen, entonces AB y A′B′
son los lados correspondientes, etc.
en sentido antihorario rotación de una
figura en la dirección opuesta a la que se
mueven las agujas de un reloj
curva representación gráfica del conjunto
de soluciones para y = f(x); en el caso
especial de una ecuación lineal, la curva
será una recta
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dependent a system of equations that has
an infinite number of solutions; lines
coincide when graphed
dependent variable labeled on the y-axis;
the quantity that is based on the input
values of the independent variable; the
output variable of a function
diameter a straight line passing through
the center of a circle connecting two
points on the circle; twice the radius
dilation a transformation in which a figure
is either enlarged or reduced by a scale
factor in relation to a center point
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dependiente sistema de ecuaciones con
una cantidad infinita de soluciones; las
rectas coinciden cuando se grafican
variable dependiente designada en el eje
y; cantidad que se basa en los valores de
entrada de la variable independiente;
variable de salida de una función
diámetro línea recta que pasa por el centro
de un círculo y conecta dos puntos en el
círculo; dos veces el radio
dilatación transformación en la que una
figura se amplía o se reduce por un factor
de escala en relación con un punto central
discrete individually separate and distinct
distance along a line the linear distance
between two points on a given line;
written as d ( PQ )
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discreto individualmente aparte y distinto
distancia a lo largo de una recta distancia
lineal entre dos puntos de una determinada
línea; se expresa como d ( PQ )
distance formula formula that states the
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fórmula de distancia fórmula que establece
distance between points (x1, y1) and
(x2, y2) is equal to ( x2 − x1 ) 2 + ( y2 − y1 ) 2
domain the set of all inputs of a function;
the set of x-values that are valid for the
function
dot plot a frequency plot that shows the
number of times a response occurred
in a data set, where each data value is
represented by a dot. Example:
la distancia entre los puntos (x1, y1) y
(x2, y2) equivale a ( x2 − x1 ) 2 + ( y2 − y1 ) 2
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dominio conjunto de todas las entradas de
una función; conjunto de valores x que
son válidos para la función
diagrama de puntos diagrama de frecuencia
que muestra la cantidad de veces que se
produjo una respuesta en un conjunto de
datos, en el que cada valor de dato está
representado por un punto. Ejemplo:
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drawing a precise representation of a
figure, created with measurement tools
such as a protractor and a ruler
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elimination method adding or subtracting
the equations in the system together so
that one of the variables is eliminated;
multiplication might be necessary before
adding the equations together
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end behavior the behavior of the graph
as x approaches positive infinity and as x
approaches negative infinity
endpoint either of two points that mark
the ends of a line segment; a point that
marks the end of a ray
equation a mathematical sentence that
uses an equal sign (=) to show that two
quantities are equal
equidistant the same distance from a
reference point
equilateral triangle a triangle with all
three sides equal in length
explicit equation an equation describing
the nth term of a pattern
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explicit formula a formula used to find
the nth term of a sequence; the explicit
formula for an arithmetic sequence is
an = a1 + (n – 1)d; the explicit formula for
a geometric sequence is an = a1 • r n – 1
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exponent the number of times a factor
is being multiplied together in an
exponential expression; in the expression
ab, b is the exponent
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dibujo representación precisa de una figura,
creada con herramientas de medición tales
como transportador y regla
método de eliminación suma o
sustracción conjunta de ecuaciones
en el sistema de manera de eliminar
una de las variables; podría requerirse
multiplicación antes de la suma conjunta
de las ecuaciones
comportamiento final el comportamiento de la gráfica al aproximarse x a infinito
positivo o a infinito negativo
extremo uno de los dos puntos que marcan
el final de un segmento de recta; punto
que marca el final de una semirrecta
ecuación declaración matemática que
utiliza el signo igual (=) para demostrar
que dos cantidades son equivalentes
equidistante a la misma distancia de un
punto de referencia
triángulo equilátero triángulo con sus
tres lados de la misma longitud
ecuación explícita ecuación que describe
el enésimo término de un patrón
fórmula explícita fórmula utilizada para
encontrar el enésimo término de una
secuencia; la fórmula explícita para una
secuencia aritmética es an = a1 + (n – 1)d;
la fórmula explícita para una secuencia
geométrica es an = a1 • r n – 1
exponente cantidad de veces que se
multiplica un factor en forma conjunta
en una expresión exponencial; en la
expresión ab, b es el exponente
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exponential decay an exponential
equation with a base, b, that is between
0 and 1 (0 < b < 1); can be represented by
the formula y = a(1 – r) t, where a is the
initial value, (1 – r) is the decay rate, t is
time, and y is the final value
exponential equation an equation that
has a variable in the exponent; the
general form is y = a • b x, where a is the
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decaimiento exponencial ecuación
exponencial con una base, b, que está entre
0 y 1 (0 < b < 1); puede representarse con la
fórmula y = a(1 – r) t, en la que a es el valor
inicial, (1 – r) es la tasa de decaimiento, t es
el tiempo, y y es el valor final
ecuación exponencial ecuación con una
variable en el exponente; la forma general
es y = a • b x, en la que a es el valor inicial, b
initial value, b is the base, x is the time,
es la base, x es el tiempo, y y es el valor final
and y is the final output value. Another
de salida. Otra forma es y = ab t , en la que t
form is y = ab t , where t is the time it
es el tiempo que tarda la base en repetirse.
x
x
takes for the base to repeat.
exponential function a function that has
a variable in the exponent:
• the general form is f (x) = ab x, where a
is the initial value, b is the growth or
decay factor, x is the time, and f (x) is
the final output value
• can also be written in the form
f(x) = b x + k, where b is a positive
integer not equal to 1 and k can equal
0; the parameters are b and k. b is the
growth factor and k is the vertical
shift.
exponential growth an exponential
equation with a base, b, greater than 1
(b > 1); can be represented by the
formula y = a(1 + r) t, where a is the initial
value, (1 + r) is the growth rate, t is time,
and y is the final value
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función exponencial función con una
variable en el exponente:
• la forma general es f (x) = ab x, en la
que a es el valor inicial, b es el factor
de crecimiento o decaimiento, x es el
tiempo, y f (x) es el valor de salida
• también puede expresarse en la forma
f(x) = b x + k, en donde b es un entero
positivo diferente de 1 y k puede ser
igual a 0; los parámetros son b y k.
b es el factor de crecimiento y k es el
desplazamiento vertical.
crecimiento exponencial ecuación
exponencial con una base, b, mayor que
1 (b > 1); puede representarse con la
fórmula y = a(1 + r) t, en la que a es el valor
inicial, (1 + r) es la tasa de crecimiento, t es
el tiempo, y y es el valor final
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expression a combination of variables,
quantities, and mathematical operations;
4, 8x, and b + 102 are all expressions.
extrema the minima and maxima of a
function
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factor one of two or more numbers
or expressions that when multiplied
produce a given product
first quartile the value that identifies the
lower 25% of the data; the median of the
lower half of the data set; written as Q 1
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formula a literal equation that states
a specific rule or relationship among
quantities
function a relation in which every element
of the domain is paired with exactly one
element of the range; that is, for every
value of x, there is exactly one value of y.
function notation a way to name a
function using f(x) instead of y
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geometric sequence an exponential
function that results in a sequence of
numbers separated by a constant ratio
graphing method solving a system
by graphing equations on the same
coordinate plane and finding the point of
intersection
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growth factor the multiple by which a
quantity increases or decreases over time
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expresión combinación de variables,
cantidades y operaciones matemáticas;
4, 8x, y b + 102 son todas expresiones.
extremos los mínimos y máximos de una
función
factor uno de dos o más números o
expresiones que cuando se multiplican
generan un producto determinado
primer cuartil valor que identifica el 25%
inferior de los datos; mediana de la mitad
inferior del conjunto de datos; se expresa Q 1
fórmula ecuación literal que establece
una regla específica o relación entre
cantidades
función relación en la que cada elemento
de un dominio se combina con
exactamente un elemento del rango;
es decir, para cada valor de x, existe
exactamente un valor de y.
notación de función forma de nombrar
una función con el uso de f(x) en lugar de y
secuencia geométrica función
exponencial que produce como resultado
una secuencia de números separados por
una proporción constante
método de representación gráfica resolución de un sistema mediante
graficación de ecuaciones en el mismo
plano de coordenadas y hallazgo del
punto de intersección
factor de crecimiento múltiplo por el que
aumenta o disminuye una cantidad con
el tiempo
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half plane a region containing all points
that has one boundary, a straight line that
continues in both directions infinitely
histogram a frequency plot that shows
the number of times a response or range
of responses occurred in a data set.
Example:
image the new, resulting figure after a
transformation
included angle the angle between two sides
included side the side between two angles
of a triangle
inclusive a graphed line or boundary is
part of an inequality’s solution
inconsistent a system of equations with
no solutions; lines are parallel when
graphed
independent a system of equations with
exactly one solution
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semiplano región que contiene todos los
puntos y que tiene un límite, una línea
recta que continúa en ambas direcciones
de manera infinita
histograma diagrama de frecuencia que
muestra la cantidad de veces que se produce
una respuesta o rango de respuestas en un
conjunto de datos. Ejemplo:
imagen nueva figura resultante después de
una transformación
ángulo incluido ángulo entre dos lados
lado incluido lado entre dos ángulos de
un triángulo
inclusivo línea graficada o límite que forma
parte de una solución de desigualdad
inconsistente sistema de ecuaciones
sin soluciones; las líneas son paralelas
cuando se las grafica
independiente sistema de ecuaciones con
una solución exacta
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independent variable labeled on the
x-axis; the quantity that changes based
on values chosen; the input variable of
a function
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inequality a mathematical sentence
that shows the relationship between
quantities that are not equivalent
inscribe to draw one figure within another
figure so that every vertex of the enclosed
figure touches the outside figure
integer a number that is not a fraction or a
decimal
intercept the point at which a line
intersects the x- or y-axis
interquartile range the difference
between the third and first quartiles; 50%
of the data is contained within this range
interval a continuous series of values
inverse a number that when multiplied by
the original number has a product of 1
irrational numbers numbers that cannot
a
be written as , where a and b are
b
integers and b ≠ 0; any number that
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cannot be written as a decimal that ends
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variable independiente designada en el
eje x; cantidad que cambia según valores
seleccionados; variable de entrada de
una función
desigualdad declaración matemática que
demuestra la relación entre cantidades
que no son equivalentes
inscribir dibujar una figura dentro de otra
de manera que cada vértice de la figura
interior toque la exterior
entero número que no es una fracción ni
un decimal
intersección punto en el que una recta
corta el eje x o y
rango intercuartílico diferencia entre
el tercer y primer cuartil; el 50% de los
datos está contenido dentro de este rango
intervalo serie continua de valores
inverso número que cuando se lo multiplica
por el número original tiene un producto de 1
números irracionales números que no
a
pueden expresarse como , en los que a
b
y b son enteros y b ≠ 0; cualquier número
que no puede expresarse como decimal
or repeats
isometry a transformation in which the
preimage and image are congruent
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joint frequency the number of times a
specific response is given by people with
a given characteristic; the cell values in a
two-way frequency table
finito o periódico
isometría transformación en la que la
preimagen y la imagen son congruentes
frecuencia conjunta cantidad de veces
que personas con una determinada
característica brindan una respuesta
específica; valores de celdas en una tabla
de frecuencia de doble entrada
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laws of exponents rules that must be
followed when working with exponents
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like terms terms that contain the same
variables raised to the same power
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line the set of points between two points P
and Q in a plane and the infinite number
of points that continue beyond those
points; written as PQ
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line of reflection the perpendicular
bisector of the segments that connect the
corresponding vertices of the preimage
and the image
line of symmetry a line separating a
figure into two halves that are mirror
images; written as l
line segment a line with two endpoints;
written as PQ
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line symmetry exists for a figure if for
every point on one side of the line of
symmetry, there is a corresponding point
the same distance from the line
linear equation an equation that can be
written in the form ax + by = c, where a,
b, and c are rational numbers; can also
be written as y = mx + b, in which m is
the slope, b is the y-intercept, and the
graph is a straight line. The solutions to
the linear equation are the infinite set of
points on the line.
linear fit (or linear model) an
approximation of data using a linear
function
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leyes de los exponentes normas que
deben cumplirse cuando se trabaja con
exponentes
términos semejantes términos que
contienen las mismas variables elevadas a
la misma potencia
línea recta conjunto de puntos entre dos
puntos P y Q en un plano y cantidad
infinita de puntos que continúan más allá
de esos puntos; se expresa como PQ
línea de reflexión bisectriz perpendicular
de los segmentos que conectan
los vértices correspondientes de la
preimagen y la imagen
línea de simetría línea que separa una
figura en dos mitades que son imágenes
en espejo; se expresa como l
segmento de recta recta con dos
extremos; se expresa como PQ
simetría lineal la que existe en una figura
si para cada punto a un lado de la línea de
simetría, hay un punto correspondiente a
la misma distancia de la línea
ecuación lineal ecuación que puede
expresarse en la forma ax + by = c, en la que
a, b, y c son números racionales; también
puede escribirse como y = mx + b, en donde
m es la pendiente, b es el intercepto de y, y
la gráfica es una línea recta. Las soluciones
de la ecuación lineal son el conjunto infinito
de puntos en la recta.
ajuste lineal (o modelo lineal) aproximación de datos con el uso de una
función lineal
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linear function a function that can be
written in the form f(x) = mx + b, in which
m is the slope, b is the y-intercept, and
the graph is a straight line
literal equation an equation that involves
two or more variables
marginal frequency the total number
of times a specific response is given, or
the total number of people with a given
characteristic
mean the average value of a data set, found
by summing all values and dividing by
the number of data points
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mean absolute deviation the average
absolute value of the difference between
each data point and the mean; found
by summing the absolute value of the
difference between each data point and
the mean, then dividing this sum by the
total number of data points
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measures of center values that describe
expected and repeated data values in a
data set; the mean and median are two
measures of center
measures of spread a measure that
describes the variance of data values, and
identifies the diversity of values in a data
set
median 1. the middle-most value of a data
set; 50% of the data is less than this value,
and 50% is greater than it
2. the segment joining the vertex to the
midpoint of the opposite side
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función lineal función que puede
expresarse en la forma f(x) = mx + b, en la
que m es la pendiente, b es el intercepto
de y, y la gráfica es una línea recta
ecuación literal ecuación que incluye dos
o más variables
frecuencia marginal cantidad total de
veces que se da una respuesta específica,
o cantidad total de personas con una
determinada característica
media valor promedio de un conjunto de
datos, que se determina al sumar todos
los valores y dividirlos por la cantidad de
puntos de datos
desviación media absoluta valor
promedio absoluto de la diferencia
entre cada punto de datos y la media; se
determina mediante la suma del valor
absoluto de la diferencia entre cada
punto de datos y la media, y luego se
divide esta suma por la cantidad total de
puntos de datos
medidas de centro valores que describen
los valores de datos esperados y repetidos
de un conjunto de datos; la media y la
mediana son dos medidas de centro
medidas de dispersión medidas que
describen la varianza de los valores
de datos e identifican la diversidad de
valores en un conjunto de datos
mediana 1. valor medio exacto de un
conjunto de datos; el 50% de los datos es
menor que ese valor, y el otro 50% es mayor
2. segmento que une el vértice con el
punto medio del lado opuesto
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midpoint a point on a line segment that
divides the segment into two equal parts
midsegment a line segment joining the
midpoints of two sides of a figure
natural numbers the set of positive
integers {1, 2, 3, …, n}
negative function a portion of a function
where the y-values are less than 0 for all
x-values
non-inclusive a graphed line or boundary
is not part of an inequality’s solution
non-rigid motion a transformation done
to a figure that changes the figure’s shape
and/or size
obtuse angle an angle measuring greater
than 90° but less than 180°
one-to-one a relationship wherein each
point in a set of points is mapped to
exactly one other point
order of operations the order in which
expressions are evaluated from left to
right (grouping symbols, evaluating
exponents, completing multiplication
and division, completing addition and
subtraction)
ordered pair a pair of values (x, y) where
the order is significant
outlier a data value that is much greater
than or much less than the rest of the
data in a data set; mathematically, any
data less than Q 1 – 1.5(IQR) or greater
than Q 3 + 1.5(IQR) is an outlier
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punto medio punto en un segmento de
recta que lo divide en dos partes iguales
segmento medio segmento de recta que une
los puntos medios de dos lados de una figura
números naturales conjunto de enteros
positivos {1, 2, 3, …, n}
función negativa porción de una función
en la que los valores y son menores que 0
para todos los valores x
no inclusivo línea graficada o límite que no
forma parte de una solución de desigualdad
movimiento no rígido transformación
hecha a una figura que cambia su forma
o tamaño
ángulo obtuso ángulo que mide más de
90˚ pero menos de 180˚
unívoca relación en la que cada punto de
un conjunto de puntos se corresponde
con otro con exactitud
orden de las operaciones orden en el que
se evalúan las expresiones de izquierda
a derecha (con agrupación de símbolos,
evaluación de exponentes, realización de
multiplicaciones y divisiones, sumas y
sustracciones)
par ordenado par de valores (x, y), en los
que el orden es significativo
valor atípico valor de datos que es mucho
mayor o mucho menor que el resto de
los datos de un conjunto de datos; en
matemática, cualquier dato menor que
Q 1 – 1,5(IQR) o mayor que Q 3 + 1,5(IQR)
es un valor atípico
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parallel lines that never intersect and have
equal slope
P
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parallel lines lines in a plane that either
do not share any points and never
intersect, or share all points; written as
AB PQ
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parallelogram a quadrilateral with
opposite sides parallel
parameter a term in a function that
determines a specific form of a function
but not the nature of the function
perimeter the distance around a twodimensional figure
perpendicular lines that intersect at
a right angle (90˚); their slopes are
opposite reciprocals
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perpendicular bisector a line constructed
through the midpoint of a segment
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perpendicular lines two lines that
intersect at a right angle (90°); written as
AB ⊥ PQ
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point an exact position or location in a
given plane
point of intersection the point at which
two lines cross or meet
point of rotation the fixed location that
an object is turned around; the point can
lie on, inside, or outside the figure
polygon two-dimensional figure with at
least three sides
positive function a portion of a function
where the y-values are greater than 0 for
all x-values
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paralelas líneas que nunca llegan a
cortarse y tienen la misma pendiente
líneas paralelas líneas en un plano que
no comparten ningún punto y nunca
se cortan, o que compartentodos
los
puntos; se expresan como AB PQ
paralelogramo cuadrilátero con lados
opuestos paralelos
parámetro término en una función que
determina una forma específica de una
función pero no su naturaleza
perímetro distancia alrededor de una
figura bidimensional
perpendiculares líneas que se cortan en
ángulo recto (90˚); sus pendientes son
recíprocas opuestas
bisectriz perpendicular línea que se
construye a través del punto medio de un
segmento
líneas perpendiculares dos líneas que se
cortan en ángulo recto (90˚); se expresan
como AB ⊥ PQ
punto posición o ubicación exacta en un
plano determinado
punto de intersección punto en que se
cruzan o encuentran dos líneas
punto de rotación ubicación fija en torno
a la que gira un objeto; el punto puede
estar encima, dentro o fuera de la figura
polígono figura bidimensional con al
menos tres lados
función positiva porción de una función
en la que los valores y son mayores que 0
para todos los valores x
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postulate a true statement that does not
require a proof
preimage the original figure before
undergoing a transformation
properties of equality rules that allow
you to balance, manipulate, and solve
equations
properties of inequality rules that allow
you to balance, manipulate, and solve
inequalities
quadrant the coordinate plane is
separated into four sections:
• In Quadrant I, x and y are positive.
• In Quadrant II, x is negative and y is
positive.
• In Quadrant III, x and y are negative.
• In Quadrant IV, x is positive and y is
negative.
quadrilateral a polygon with four sides
quantity something that can be compared
by assigning a numerical value
radius a line segment that extends from
the center of a circle to a point on the
circle. Its length is half the diameter.
range the set of all outputs of a function;
the set of y-values that are valid for the
function
rate a ratio that compares different kinds
of units
rate of change a ratio that describes how
much one quantity changes with respect
to the change in another quantity; also
known as the slope of a line
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postulado afirmación verdadera que no
requiere prueba
preimagen figura original antes de sufrir
una transformación
propiedades de igualdad normas que
permiten equilibrar, manipular y resolver
ecuaciones
propiedades de desigualdad normas que
permiten equilibrar, manipular y resolver
desigualdades
cuadrante plano de coordenadas que se
divide en cuatro secciones:
• En el cuadrante I, x y y son positivos.
• En el cuadrante II, x es negativo y y es
positivo.
• En el cuadrante III, x y y son negativos.
• En el cuadrante IV, x es positivo y y es
negativo.
cuadrilátero polígono con cuatro lados
cantidad algo que puede compararse al
asignarle un valor numérico
radio segmento de línea que se extiende
desde el centro de un círculo hasta un
punto de la circunferencia del círculo. Su
longitud es la mitad del diámetro.
rango conjunto de todas las salidas de una
función; conjunto de valores y válidos
para la función
tasa proporción en que se comparan
distintos tipos de unidades
tasa de cambio proporción que describe
cuánto cambia una cantidad con respecto
al cambio de otra cantidad; también se la
conoce como pendiente de una recta
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ratio the relation between two quantities;
can be expressed in words, fractions,
decimals, or as a percent
rational number a number that can be
a
written as , where a and b are integers
b
and b ≠ 0; any number that can be
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proporción relación entre dos cantidades;
puede expresarse en palabras, fracciones,
decimales o como porcentaje
número racional número que puede
a
expresarse como , en los que a y b son
b
enteros y b ≠ 0; cualquier número que
puede escribirse como decimal finito o
written as a decimal that ends or repeats
ray a line with only one endpoint; written
as PQ
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real numbers the set of all rational and
irrational numbers
reciprocal a number that when multiplied
by the original number has a product of 1
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rectangle a parallelogram with opposite
sides that are congruent and consecutive
sides that are perpendicular
recursive formula a formula used to
find the next term of a sequence when
the previous term or terms are known;
the recursive formula for an arithmetic
sequence is an = an – 1 + d; the recursive
formula for a geometric sequence is
an = an – 1 • r
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reflection a transformation where a
mirror image is created; also called a flip;
an isometry in which a figure is moved
along a line perpendicular to a given line
called the line of reflection
regular hexagon a six-sided polygon with
all sides equal and all angles measuring
120˚
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periódico
semirrecta línea con un solo extremo; se
expresa como PQ
números reales conjunto de todos los
números racionales e irracionales
recíproco número que cuando se lo
multiplica por el número original tiene un
producto de 1
rectángulo paralelogramo con lados
opuestos congruentes y lados
consecutivos que son perpendiculares
fórmula recursiva fórmula que se utiliza
para encontrar el término siguiente de
una secuencia cuando se conoce el o los
términos anteriores; la fórmula recursiva de
una secuencia aritmética es an = an – 1 + d;
la fórmula recursiva para una secuencia
geométrica es an = an – 1 • r
reflexión transformación por la cual se
crea una imagen en espejo; isometría en
la que se mueve una figura a lo largo de
una línea perpendicular hacia una recta
determinada llamada línea de reflexión
hexágono regular polígono de seis lados
con todos los lados iguales y en el que
todos los ángulos miden 120˚
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regular polygon a two-dimensional figure
with all sides and all angles congruent
relation a relationship between two sets of
elements
relative maximum the greatest value of a
function for a particular interval of the
function
relative minimum the least value of a
function for a particular interval of the
function
residual the vertical distance between an
observed data value and an estimated
data value on a line of best fit
residual plot provides a visual
representation of the residuals for a set
of data; contains the points (x, residual
for x)
rhombus a parallelogram with four
congruent sides
right angle an angle measuring 90˚
rigid motion a transformation done to a
figure that maintains the figure’s shape
and size or its segment lengths and angle
measures
rotation a transformation that turns
a figure around a point; also called a
turn; an isometry where all points in the
preimage are moved along circular arcs
determined by the center of rotation and
the angle of rotation
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polígono regular figura bidimensional
con todos los lados y todos los ángulos
congruentes
relación conexión entre dos conjuntos de
elementos
máximo relativo el mayor valor de una
función para un intervalo particular de la
función
mínimo relativo el menor valor de una
función para un intervalo particular de la
función
residual distancia vertical entre un valor
de datos observado y un valor de datos
estimado sobre una línea de ajuste óptimo
diagrama residual brinda una
representación visual de los residuales
para un conjunto de datos; contiene los
puntos (x, residual de x)
rombo paralelograma con cuatro lados
congruentes
ángulo recto ángulo que mide 90˚
movimiento rígido transformación que
se realiza a una figura que mantiene su
forma y tamaño o las longitudes de sus
segmentos y las medidas de ángulos
rotación transformación que hace girar
una figura alrededor de un punto;
isometría en la que todos los puntos de la
preimagen se mueven a lo largo de arcos
circulares determinados por el centro de
rotación y el ángulo de rotación
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S
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scale factor a multiple of the lengths
of the sides from one figure to the
transformed figure. If the scale factor is
larger than 1, then the figure is enlarged.
If the scale factor is between 0 and 1,
then the figure is reduced.
scatter plot a graph of data in two
variables on a coordinate plane, where
each data pair is represented by a point
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segment a part of a line that is noted by
two endpoints
sequence an ordered list of numbers
side-angle-side (SAS) if two sides and
the included angle of one triangle are
congruent to two sides and the included
angle of another triangle, then the two
triangles are congruent
side-side-side (SSS) if three sides of one
triangle are congruent to three sides of
another triangle, then the two triangles
are congruent
sketch a quickly done representation of a
figure; a rough approximation of a figure
skewed to the left data concentrated on
the higher values in the data set, which
has a tail to the left. Example:
20
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28
32
36
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factor de escala múltiplo de las longitudes
de los lados de una figura a la figura
transformada. Si el factor de escala
es mayor que 1, entonces la figura se
agranda. Si el factor de escala se encuentra
entre 0 y 1, entonces la figura se reduce.
diagrama de dispersión gráfica de
datos en dos variables en un plano de
coordenadas, en la que cada par de datos
está representado por un punto
segmento parte de una recta comprendida
entre dos extremos
secuencia lista ordenada de números
lado-ángulo-lado (SAS) si dos lados y
el ángulo incluido de un triángulo son
congruentes con dos lados y el ángulo
incluido de otro triángulo, entonces los
dos triángulos son congruentes
lado-lado-lado (SSS) si los tres lados de
un triángulo son congruentes con los tres
lados de otro triángulo, entonces los dos
triángulos son congruentes
bosquejo representación de una figura
realizada con rapidez; aproximación
imprecisa de una figura
desviados hacia la izquierda datos
concentrados en los valores más altos
del conjunto de datos, que tiene una
cola hacia la izquierda. Ejemplo:
20
24
28
32
36
40
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skewed to the right data concentrated on
the lower values in the data set, which
has a tail to the right. Example:
20
24
28
32
36
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slope the measure of the rate of change
of one variable with respect to another
yy22 −− yy11 yy rise
rise
variable; slope =
; the
== ==
xx22 −−xx11 run
xx run
slope in the equation y = mx + b is m.
slope-intercept method the method
used to graph a linear equation; with
this method, draw a line using only two
points on the coordinate plane
solution a value that makes the equation true
solution set the value or values that make
a sentence or statement true; the set of
ordered pairs that represent all of the
solutions to an equation or a system of
equations
solution to a system of linear
inequalities the intersection of the half
planes of the inequalities; the solution
is the set of all points that make all the
inequalities in the system true
square a parallelogram with four
congruent sides and four right angles
straightedge a bar or strip of wood,
plastic, or metal having at least one long
edge of reliable straightness
Español
desviados hacia la derecha datos
concentrados en los valores más
bajos del conjunto de datos, que tiene
una cola hacia la derecha. Ejemplo:
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pendiente medida de la tasa de cambio
de una variable con respecto a otra;
yy22 −− yy11 yy rise
rise
pendiente =
== =;=la pendiente
xx22 −−xx11 xx run
run
en la ecuación y = mx + b es m
método pendiente-intercepto método
utilizado para graficar una ecuación lineal;
con este método, se dibuja una línea con
sólo dos puntos en un plano de coordenadas
solución valor que hace verdadera la ecuación
conjunto de soluciones valor o valores
que hacen verdadera una afirmación o
declaración; conjunto de pares ordenados
que representa todas las soluciones para
una ecuación o sistema de ecuaciones
solución a un sistema de desigualdades
lineales intersección de los medios planos de
las desigualdades; la solución es el conjunto
de todos los puntos que hacen verdaderas
todas las desigualdades de un sistema
cuadrado paralelograma con cuatro lados
congruentes y cuatro ángulos rectos
regla de borde recto barra o franja de
madera, plástico o metal que tiene, al
menos, un borde largo de rectitud confiable
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substitution method solving one of a pair
of equations for one of the variables and
substituting that into the other equation
symmetric situation in which data is
concentrated toward the middle of the
range of data; data values are distributed
in the same way above and below the
middle of the sample. Example:
20
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28
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36
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método de sustitución solución de un par
de ecuaciones para una de las variables y
sustitución de eso en la otra ecuación
simétrico situación en la que los datos se
concentran hacia el medio del rango de
datos; los valores de datos se distribuyen
de la misma manera por encima y por
debajo del medio de la muestra. Ejemplo:
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system a set of more than one equation
system of equations a set of equations
with the same unknowns
system of inequalities two or more
inequalities in the same variables that
work together
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term a number, a variable, or the product
of a number and variable(s)
third quartile value that identifies the
upper 25% of the data; the median of the
upper half of the data set; 75% of all data
is less than this value; written as Q 3
transformation a change in a geometric
figure’s position, shape, or size
translation moving a graph either
vertically, horizontally, or both, without
changing its shape; a slide; an isometry
where all points in the preimage are
moved parallel to a given line
T
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sistema conjunto de más de una ecuación
sistema de ecuaciones conjunto de
ecuaciones con las mismas incógnitas
sistema de desigualdades dos o más
desigualdades en las mismas variables
que operan juntas
término número, variable o producto de
un número y una o más variables
tercer cuartil valor que identifica el 25%
superior de los datos; mediana de la
mitad superior del conjunto de datos; el
75% de los datos es menor que este valor;
se expresa como Q 3
transformación cambio en la posición, la
forma o el tamaño de una figura geométrica
traslación movimiento de un gráfico en
sentido vertical, horizontal, o en ambos,
sin modificar su forma; deslizamiento;
isometría en la que todos los puntos de la
preimagen se mueven en paralelo a una
línea determinada
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trend a pattern of behavior, usually
observed over time or over multiple
iterations
triangle a three-sided polygon with
three angles
two-way frequency table a table that
divides responses into categories,
showing both a characteristic in the table
rows and a characteristic in the table
columns; values in cells are a count of
the number of times each response was
given by a respondent with a certain
characteristic
undefined slope the slope of a vertical
line
unit rate a rate per one given unit
variable a letter used to represent a value
or unknown quantity that can change or
vary
vertical shift number of units the graph
of the function is moved up or down; a
translation
whole numbers the set of natural numbers
that also includes 0: {0, 1, 2, 3, ...}
x-intercept the point at which the line
intersects the x-axis at (x, 0)
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tendencia patrón de comportamiento, que
se observa por lo general en el tiempo o
en múltiples repeticiones
triángulo polígono de tres lados con tres
ángulos
tabla de frecuencia de doble entrada tabla que divide las respuestas en dos
categorías, y muestra una característica
en las filas y una en las columnas; los
valores de las celdas son un conteo de la
cantidad de veces que un respondedor
da una respuesta con una determinada
característica
pendiente indefinida pendiente de una
línea vertical
tasa unitaria tasa de una unidad
determinada
variable letra utilizada para representar
un valor o una cantidad desconocida que
puede cambiar o variar
desplazamiento vertical cantidad de
unidades que el gráfico de la función
se desplaza hacia arriba o hacia abajo;
traslación
números enteros conjunto de números
naturales que incluye el 0: {0, 1, 2, 3, ...}
intercepto de x punto en el que una recta
corta el eje x en (x, 0)
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y-intercept the point at which a line or
curve intersects the y-axis at (0, y); the
y-intercept in the equation y = mx + b is b.
Y
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intercepto de y punto en el que una recta
o curva corta el eje y en (0, y); el intercepto
de y en la ecuación y = mx + b es b.
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