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For Geometry Classes – Blocks 3A, 3B, and 5 (Scroll down for other block classes.)

Lesson 1.1 – Points, Lines, and Planes

Focus Objectives
Identify and model points, lines, and planes.

Identify intersecting lines and planes.
 
Vertical Alignment
Before Lesson 1-1    Use geometric concepts and properties to solve problems.
Lesson 1-1    Identify and model points, lines, and planes. Identify intersecting lines and planes.
After Lesson 1-1    Use numeric and geometric patterns to make generalizations about geometric properties, including properties of polygons.
 
Common Core State Standards
G.CO.1    Know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc.
 
Mathematical Practices
4 Model with mathematics.

6 Attend to precision.


Teach
Scaffolding Questions
Have students read the Why? section of the lesson.
Ask:
What are some other objects that points, lines, and planes could be used to represent? Sample response: Stars can be represented by points, lines can be used to connect the stars to form constellations, and a plane can be used to represent the sky.
What are some other ways that combinations of points, lines, and planes are used? networks and maps Describe a point, a line, and a plane. Can you clearly define these geometric terms? What is the difference between a description and a definition? A point is like a dot, a line is like a long, straight road, and a plane is like a desktop. There are no clear definitions of these terms; a description simply describes, whereas a definition is a detail of specific criteria required for a figure.
 
1. Points, Lines, and Planes 
Examples 1 and 2 show how to name and model points, lines, and planes by using the key concepts provided in this lesson.
 
Additional Examples
1.    Use the figure to name each of the following.
 
a.    a line containing point K
line a,  
b.    a plane containing point L plane B, plane JKM, plane KLM, plane JLM. Reorder the letters in these names to create 15 other acceptable names.
2.    Name the geometric shape modeled by each object.
a.    a 10 × 12 patio plane
b.    a button on a table point
 
2. Intersection of Lines and Planes 
Examples 3 and 4 show how to draw, label, and identify points, lines, and planes in space.
 
Additional Examples
3.    Draw and label a figure for each relationship.
         a.
    ALGEBRA Plane R contains lines     , which intersect at point P. Add point C on plane R so that
it is not collinear with 
         b.
 on a coordinate plane contains Q(–2, 4) and R(4, –4). Add point T so that T is collinear with these points. Sample answer:
 
4.    Use the figure for parts a-d.
 
a.    How many planes appear in this figure? two
b.    Name three points that are collinear. A, B, and D
c.    Are points A, B, C, and D coplanar? Explain. Points A, B, C, and D all lie in plane ABC, so they are coplanar.
         d.
At what point do  intersect? A
 
Focus On Mathematical Content
In geometry, a point is a location without shape or size. A line contains points and has no thickness or width. Points on the same line are collinear, and there is exactly one line through any two points. The intersection of two lines is a point.
A plane is a flat surface made of points. A plane has no depth and extends infinitely in all directions. Points on the same plane are coplanar, and the intersection of two planes is a line.
 
Teach with Tech
Interactive Whiteboard Draw a plane on the board. Select students and give them specific points and lines to draw that are either in the plane or not in the plane.
 
Teaching the Mathematical Practices
  Precision Mathematically proficient students communicate precisely to others. Encourage students to learn and use clear definitions for the mathematical terms used throughout the program.
  Modeling Mathematically proficient students can apply mathematics to solve problems arising in real life. For Exercises 40–42, encourage students to use a cardboard box to model the figure shown.
  Arguments Mathematically proficient students make conjectures and build a logical progression of statements to explore the truth of their conjectures. In Exercise 52, point out to students that these skills in writing arguments will be useful later in this course.
  Arguments Mathematically proficient students can analyze situations by breaking them into cases. In Exercise 59, encourage students to visualize the situation described in the problem.
Resources
Points, Lines, and Planes
Model Points, Lines, and Planes
Draw Geometric Figures
Interpret Drawings
Points, Lines, and Planes
Model Points, Lines, and Planes
Draw Geometric Figures
Interpret Drawings
Points, Lines, and Planes Points, Lines, and Planes
eToolkit
Interactive Classroom: Points, Lines, and Planes
Math Forum
Example 3
Interactive Student Guide: Undefined Terms
Interactive Student Guide Teacher Edition: Undefined Terms
Practice
Differentiated Homework Options
Level    Assignment        Two Day Option
          Day 1        Day 2
  Basic    13–48, 57, 58, 60–85    13–47 odd, 62–65        14–48 even, 57, 58, 60, 61,
66–85
  Core    13–47 odd, 49–58, 60–85    13–48, 62–65        49–58, 60, 61, 66–79
  Advanced    49–79,
(optional: 80–85)         
 
Multiple Representations
In Exercises 54 and 56, students use a table and a graph to investigate a locus of points.
 
WatchOut!
Error Analysis In Exercise 58, students should see that a line drawn between two points counts only one time. Camille counted each line twice, once from each point.
Resources
Study Guide and Intervention: Points, Lines, and Planes
Skills Practice: Points, Lines, and Planes
Practice Worksheet: Points, Lines, and Planes
Word Problem Practice: Points, Lines, and Planes
Enrichment: Fano Plane
Graphing Calculator Activity: Points, Lines, and Planes
Points, Lines, and Planes
Points, Lines, and Planes
Points, Lines, and Planes
Points, Lines, and Planes
Fano Plane
Graphing Calculator Activity: Points, Lines, and Planes
Mini-Project: Intersecting Planes
Points, Lines, and Planes Points, Lines, and Planes eSolutions, Geometry Points, Lines, and Planes
Points, Lines, and Planes
Points, Lines, and Planes
Points, Lines, and Planes
Fano Plane
Graphing Calculator Activity: Points, Lines, and Planes
Assess
Name the Math
Discuss how points, lines, and planes are modeled by the objects students see and use every day.
Resources
Self-Check Quiz: Points, Lines, and Planes eSolutions, Geometry
 related to points, lines, and planes. Some examples include:
Naturalist Learners Explain how points, lines, and planes exist in nature. For example, planes can model leaves, lily pads, and the surface of a pond; lines can model spider webs, sunbeams, tree trunks, the edge of a riverbed, and the veins of a leaf.

 

For Geometry B Classes – Blocks 1A and 1B

3.4 – Equations of Lines

Focus

Objectives

Write an equation of a line given information about the graph. Solve problems by writing equations.

 

Vertical Alignment

Before Lesson 3-4    Graph and write equations of lines given characteristics such as two points, a point and a slope, or a slope and y-intercept.

Lesson 3-4    Write an equation of a line given information about the graph. Solve problems by writing equations.

After Lesson 3-4    Identify and sketch graphs of parent functions, including linear.

 

  Common Core State Standards

G.GPE.5    Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

 

Mathematical Practices

4 Model with mathematics.

8 Look for and express regularity in repeated reasoning.

Resources

Review of Slopes of Lines

Teach

Scaffolding Questions

Have students read the Why? section of the lesson.

Ask:

How much is a ticket for traveling 80 miles per hour? $52.50

What are two points on the graph of the equation? Sample answer: (10, 42.5), (15, 52.5) What is the slope of the line? 2

 

1. Write Equation of Lines

Examples 1–4 show how to write a linear equation using the slope-intercept form or point-slope form of a line. Students should be able to use given values to write a linear equation either in slope-intercept form or point-slope form. Example 5 shows how to find a line that is perpendicular to a given line through a given point.

 

Additional Examples

1. Write an equation in slope-intercept form of the line with slope of 6 and y-intercept of 3. Then graph the line. y = 6x  3

 

2.

Write an equation in point-slope form of the line with slope  that contains (10, 8). Then graph the line. 

 

3.    Write an equation of the line through each pair of points in slope-intercept form.      a.

(4, 9) and (2, 0) 

         b. (3, 7) and (1, 3) y = 5x + 8

4.    Write an equation of the line through (5, 2) and (0, 2) in slope-intercept form. y = 2

5.

Write an equation in slope-intercept form for a line perpendicular to the line  through (2, 0). 

y = 5x + 10

 

2. Write Equations to Solve Problems

Example 6 shows how to solve a real-world problem using a linear equation.

 

Additional Example

6. RENTAL COSTS An apartment complex charges $525 per month plus a $750 annual maintenance fee.

a.    Write an equation to represent the total first year’s cost A for r months of rent. A = 525r + 750

b.    Compare this rental cost to a complex which charges a $200 annual maintenance fee but $600 per month for rent. If a person expects to stay in an apartment for one year, which complex offers the better rate? The first complex offers the better rate: the first year costs $7050 instead of $7400.

 

Focus on Mathematical Content

This lesson presents two basic forms for the equations of lines. One is called the slope-intercept form. It is written as y = mx + b, where m is the slope and b is the y-intercept. The point-slope form is the second form. It is written as y  y1 = m(x  x1) where (x1, y1) are the coordinates of any point contained in the line.

You can write linear equations to solve real-world problems. Slope often represents a rate of change. This rate can be used to determine cost or other information.

Equations of Lines The equation of a line may be written in many different ways. The point-slope form can be written with any point that the line passes through, or the slope-intercept form can be used. These equations are equivalent by using the algebraic properties of equality.

 

Tips for New Teachers

Using Graphs Some students have difficulty writing a linear equation using the point-slope form from a word problem. The students should make a graph with each axis representing the variables for the given values. This method allows the students to substitute values for the slope and intercept.

 

Teach with Tech

Interactive Whiteboard Drag a coordinate grid on the board. Draw a line on the grid, and have students write the equation of the line in slope-intercept form. Then drag the line to another location on the board and repeat the process. Discuss how the equations are similar and different.

 

Teaching the Mathematical Practices

  Perseverance Mathematically proficient students check their answers to problems using a different method. They can understand the approaches of others to solving complex problems. Encourage students to look for alternate ways to solve problems.

  Critique Mathematically proficient students can distinguish correct logic from flawed reasoning. In Exercise 58, students should recognize that if Josefina simplified her equation, it would be identical to Mark’s answer. Remind students that the slope-intercept form and the point-slope form result in equivalent forms of an equation of a line.

 

WatchOut!

Equations of Lines When converting the point-slope form of the equation of a line into slope-intercept form, remember to distribute across the parentheses.

Resources

Slope and y-intercept

Slope and a Point on the Line

Two Points

Horizontal Line

Write Equations of Parallel or Perpendicular Lines

Write Linear Equations

Slope and y-intercept

Slope and a Point on the Line

Two Points

Write Equations of Parallel or Perpendicular Lines

Horizontal Line

Write Linear Equations

Equations of Lines

Equations of Lines

Interactive Classroom: Equations of Lines eToolkit Math Forum

Example 3

Interactive Student Guide: Equations of Lines

Interactive Student Guide Teacher Edition: Equations of Lines

Practice

Differentiated Homework Options

Level    Assignment        Two Day Option

          Day 1        Day 2

  Basic    13–42, 56–73    13–41 odd, 60–63        14–42 even, 56–59, 64–73

  Core    13–41 odd, 53, 56–73    13–42, 60–63        43–54, 56–59, 64–73

  Advanced    43–69,

(optional: 70–73)         

 

Multiple Representations

In Exercise 52, students use a table, a verbal description, and an algebraic equation to investigate slopes of lines in a system of equations that has one solution, no solution, or infinitely many solutions.

Resources

Study Guide and Intervention: Equations of Lines

Skills Practice: Equations of Lines

Practice Worksheet: Equations of Lines

Word Problem Practice: Equations of Lines

Enrichment: Polygons on a Coordinate Grid

Study Guide and Intervention: Equations of Lines

Skills Practice: Equations of Lines

Practice Worksheet: Equations of Lines

Word Problem Practice: Equations of Lines

Enrichment: Polygons on a Coordinate Grid

Geometry Lab: Graphing Lines in the Coordinate Plane

Equations of Lines Equations of Lines eSolutions, Geometry Equations of Lines

Equations of Lines

Equations of Lines

Equations of Lines

Polygons on a Coordinate Grid

Assess

Yesterday’s News

Have students write how yesterday’s lesson in slopes of lines helped them in learning to write equations of lines. They should give at least two examples that support their reasoning.

Resources

eSolutions, Geometry

Self-Check Quiz: Equations of Lines 

Differentiated Instruction

 

Logical Explain to students that when they find the equation of a graph they should always check their work. Working independently, have students look at the examples in this lesson and substitute points on the line into the final equation. They should see that the substitution results in a true equation.

 

 

Extension Have students define a situation where there is a profit equation and an expense equation, and graph each equation on the same grid. The point of intersection of the two lines is called the “break-even point.” For example, at Ken’s lemonade stand, Ken earns $0.25 per glass of lemonade sold. His expenses are $2.50 for a pitcher and $0.05 per glass of lemonade he makes. Graph the equations y = 0.25x and y = 0.05x + 2.5. The point of intersection (12.5, 3.125) tells Ken he must sell at least 13 glasses of lemonade per pitcher to make a profit. 

 

 

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