Guiding Question: "As your work time goes up, what happens to your total cash? Since both go up, the change is positive!" Misconception Alert: Remind students that counting grid lines can mislead them because the y-axis labels increment by 20, not 1!
Context: Battery levels steadily drain over flight minutes.
Guiding Question: "Is the drone gaining charge or losing charge as minutes pass? How do we write a drop mathematically? (Negative Rise!)" Concept Check: Make sure they write -60 for rise, not positive 60, resulting in -20% / min.
Context: Flat-fee entry cost remains exactly flat regardless of food eaten.
Guiding Question: "How much extra money do you pay for the 2nd tasting? (None, $0). If cost doesn't change, the rise is 0. 0 divided by 3 is exactly 0!" Visual Key: Horizontal lines represent a rate of change of 0.
Context: A perfect vertical beam coordinates height change but zero horizontal change.
Guiding Question: "What is the run (horizontal distance) between the two dots? (0). Try dividing 60 by 0 on your calculator. What happens? We cannot divide by zero, so the slope is Undefined!"
Ask: "If Slope measures steepness, how can we explain the difference between Slip 3 (Zero) and Slip 4 (Undefined) using physical walking? You can walk easily on a flat floor (Slope = 0), but you cannot walk up a vertical brick wall (Slope = Undefined)!"
Created by Lenny, expert educational creator • Teacher Master Reference Sheet • Math Standard CCSS 8.EE.B.5
GUIDED DISCOVERY • PART 3
2D PLANE EXPANSION
Now, let's step up to the **2D Coordinate Plane**! Meet line segment EF with endpoints at E(-5, 4) and F(8, 4). Analyze the color-coded counting jumps below to locate the exact midpoint!
x y Count: 6.5 units Count: 6.5 units E (-5, 4) F (8, 4)
11. Focus on the x-coordinates of E (-5) and F (8). What is their average?
Average x =
\frac{-5 + 8}{2} = ______
12. Focus on the y-coordinates of E (4) and F (4). What is their average?
Average y =
\frac{4 + 4}{2} = ______
13. Combine your average values! Write them as an (x, y) coordinate pair:
Midpoint Coordinate P = ( _____, _____ )
14. Plot your calculated Point P on the coordinate plane. Does it align exactly where the green and red jumps meet? Explain how counting half the total distance helps us find the middle:
____________________________________________________________________________________
Finding a midpoint on a coordinate plane is simply finding the **arithmetic average** of the x-coordinates and the y-coordinates separately!
\[\text{Midpoint } P = \left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\]
Midpoint Mission Guided Handout Math Standard CCSS 8.G.B.8 Page 3 of 3
MIDPOINT SHOWDOWN PACKET | EXIT TICKET • PAGE 5
Name: _______________________ Date: _________ Lesson: ___________________
SWBAT determine midpoints on the coordinate plane and using the midpoint formula
Directions:
Without using your notes from today, carefully read the questions below then answer them. Make sure to show all your work.
1- NSA is planning to build a new high school so that it is evenly located between LPHS and WPHS. LPHS is at coordinate L(-5, 8) and WPH is at W(8, 2).
(a) Graph LW on the coordinate plane.
(b) Where should the new high school be so that it's equal distance between LPHS and WPHS?
(c) How far is the new HS from LPHS?
Work Space:
x y
5
MIDPOINT SHOWDOWN PACKET | CORE ANCHOR NOTES • PAGE 6
Name: _______________________ Date: _________ Lesson: ___________________
SWBAT determine midpoints on the coordinate plane and using the midpoint formula
Topic
Definitions & Methods
Midpoint
midpoint is the middle point of a line
Line
a straight, one-dimensional figure that has no thickness or width and extends forever in opposite directions
Segment
a part of a straight line that has two distinct endpoints and includes all the points on the line between them.
Endpoints
a point that marks the end, start, or boundary of a line segment or a ray
Coordinates
a set of numbers that gives the exact position of a point on a graph or a grid
How do we find midpoint graphically?
Step 1: Annotate your triangle using the points given
Step 2: Find the midpoint of the horizontal and vertical lines of your triangle and trace it to the diagonal.
How do we find midpoint algebraically?
Using the Midpoint Formula: \(\left( \frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2} \right)\)
6
d \(m = -2, \; (0, 4)\)
Slope-Intercept Form Packet • U00L02 NSA Mathematics • Core Geometry Page 3 of 5
GEOMETRY EXIT TICKET | EXIT TICKET • PAGE 4
Name: _______________________ Date: _________ Period: _____ Seat: _____
Directions: Complete the following questions by showing all work and annotations. Keep work organized and box any final answers.
1 Write and Graph Equation:
Line \(m\) passes through the points (-7, 8) and (-2, 3). Write the equation of line \(m\) in slope-intercept form and graph and label it on the plane.
Work Space:
Equation: ____________________
2 Increasing or Decreasing?
Is the line in #1 increasing or decreasing? How do you know?
x y
Slope-Intercept Form Packet • U00L02 NSA Mathematics • Core Geometry Page 4 of 5
GEOMETRY HOMEWORK | HOMEWORK • PAGE 5
Name: _______________________ Date: _________ Period: _____ Seat: _____
Directions: Show all work for full credit. For your graphs, you must clearly label the y-intercept and 2 additional points on the line.
#1.) Identify the slope and y-intercept:
\(-4y + 3x = 16\)
a.) \(m = -\frac{3}{4}, \; (0, 16)\)
b.) \(m = \frac{3}{4}, \; (0, 4)\)
c.) \(m = -\frac{3}{4}, \; (0, -16)\)
d.) \(m = \frac{3}{4}, \; (0, -4)\)
#2.) Slope & y-intercept for the equation below:
\(y - 4 = -3(x - 3)\)
Slope: ____________ Y-intercept: ____________
#3.) Intersecting lines:
At what coordinate point do line \(q\) (passes through (2, 7) & (0, 7)) and line \(r\) (passes through (1, 2) & (-4, 7)) intersect?
Intersection Point = ( ______ , ______ )
#4.) Graph the line on the coordinate plane:
\(10x - 4y = 32\)
Slope-Intercept Form Packet • U00L02 NSA Mathematics • Core Geometry Page 5 of 5
1. Find Slope:
\(m = \frac{3 - (-6)}{2 - 5} = \frac{9}{-3} = -3\)
2. Find Equation (use (2,3)):
\(y - 3 = -3(x - 2) \Rightarrow y = -3x + 9\)
Equation: \(y = -3x + 9\)
(2, 3)
YOU DO: ROUND 2 KEY PART B
#3.) Determine the equation of the line below:
y-int: 2
Equation: \(y = -\frac{1}{2}x + 2\) (or \(-0.5x + 2\))
#4.) What is the slope and y-intercept of the line below?
\(3y - 12 = -6x\)
a \(m = -6, \; (0, -12)\)
b \(m = -2, \; (0, \frac{1}{2})\)
c \(m = \frac{1}{2}, \; (0, 6)\)
d \(m = -2, \; (0, 4)\) [CORRECT]
Slope-Intercept Form Answer Key • U00L02 NSA Mathematics • Core Geometry Page 3 of 5
GEOMETRY EXIT TICKET KEY | EXIT TICKET KEY • PAGE 4
MASTER ANSWER KEY
Directions: Complete the following questions by showing all work and annotations. Keep work organized and box any final answers.
1 Write and Graph Equation:
Line \(m\) passes through points (-7, 8) and (-2, 3). Write equation of line \(m\).
Slope \(m = \frac{3 - 8}{-2 - (-7)} = \frac{-5}{5} = -1\)
Using (-2, 3):
\(y - 3 = -1(x - (-2))\)
\(y - 3 = -x - 2 \Rightarrow y = -x + 1\)
Equation: \(y = -x + 1\)
2 Increasing or Decreasing?
Is the line in #1 increasing or decreasing? How do you know?
Decreasing.
We know because the calculated slope is negative (\(m = -1\)). This means that as we move from left to right on the x-axis, the y-values steadily drop downwards.
(-2, 3)
Slope-Intercept Form Answer Key • U00L02 NSA Mathematics • Core Geometry Page 4 of 5
GEOMETRY HOMEWORK KEY | HOMEWORK KEY • PAGE 5
MASTER ANSWER KEY
Directions: Show all work for full credit. For your graphs, you must clearly label the y-intercept and 2 additional points on the line.
#1.) Identify the slope and y-intercept:
\(-4y + 3x = 16 \Rightarrow y = \frac{3}{4}x - 4\)
a.) \(m = -\frac{3}{4}, \; (0, 16)\)
b.) \(m = \frac{3}{4}, \; (0, 4)\)
c.) \(m = -\frac{3}{4}, \; (0, -16)\)
d.) \(m = \frac{3}{4}, \; (0, -4)\) • [CORRECT]
#2.) Slope & y-intercept for the equation below:
\(y - 4 = -3(x - 3) \Rightarrow y = -3x + 13\)
Expand right: \(y - 4 = -3x + 9\)
Add 4: \(y = -3x + 13\)
Slope: -3 Y-intercept: 13 (0, 13)
#3.) Intersecting lines:
At what coordinate point do line \(q\) and line \(r\) intersect?
Line q: horizontal at \(y = 7\)
Line r: slope \(= -1 \Rightarrow y = -x + 3\)
Substitute: \(7 = -x + 3 \Rightarrow x = -4\)
Intersection Point = (-4, 7)
#4.) Graph the line on the coordinate plane:
\(10x - 4y = 32 \Rightarrow y = \frac{5}{2}x - 8\)
(4, 2)
Slope-Intercept Form Answer Key • U00L02 NSA Mathematics • Core Geometry Page 5 of 5
Check for Understanding (CFU): "If your equation has \(y = -11x + 38\), why is it helpful to plot the given coordinates \((4, -6)\) and \((3, 5)\) first to draw your line rather than counting all the way up to 38 on your grid?"
0:42 – 0:50 (8 min) | Exit Ticket Assessment & Final Stamp Independent Exit
Assessment Task: Line \(m\) through \((-5, 7)\) and \((-1, 4)\). Parallel Line \(n\) through \((-1, 6)\).
Students write both equations and graph both lines on the coordinate plane.
Final Check for Understanding (CFU Stamp): "When you look at your graph of Line \(m\) and Line \(n\), how can you visually verify they are parallel? What is identical about them? What is different?" → Answer: They have the exact same tilt/steepness (\(m = -\frac{3}{4}\)), but cross the y-axis at different points (\(b_m = \frac{13}{4}\) vs \(b_n = \frac{21}{4}\)).
Anticipated Errors & Rapid Interventions
Sign Flip in Point-Slope: Writing \(y - 7 = -\frac{3}{4}(x - 5)\) instead of \((x + 5)\).
Prompt: "Subtracting a negative 5 becomes plus 5!"
Parallel Slope Confusion: Attempting to flip or invert the slope for parallel lines.
Prompt: "Parallel lines have the SAME slope! Keep it identical!"
Teacher 50-Minute Pacing Guide • Linear Equations Page 2 of 2
