Rate of Change Lesson Plan

Rate of Change & Slope Lesson Plan Outline
Common Core Standards for Mathematics:
HSF.IF.B.6 Calculate and interpret the average rate of change of a function over a specific interval.
8.F.A.2 Compare properties of two functions each represented in a different way.
MP1 Make sense of problems and persevere in solving them.
MP2 Reason abstractly and quantitatively.
MP3 Construct viable arguments and critique the reasoning of others.
Objective: Students will be able to interpret information from a graph, with or without labels.
Students will be able to find the rate of change from a graph, table, or given points.
Overview
Day1 – 90 min
Students will begin class with a card sort. They will be asked to match scenarios, graphs, and tables. This activity also
asks students to make sense of the situations and complete 2 of the tables with values of their choice and write a story
for a graph. A class discussion will be facilitated, where students will argue their reasoning and collectively determine
the correct answers. Students should be able to recognize that steepness relate to speed and a flat line indicates a
pause in motion, as well as being able to recognize quicker or slower speeds and a pause in motion in a table.
Class then continues with an applet that provides a graphical and visual representation of simple race scenarios.
Students will work independently to correctly complete 10 scenarios (5 story and 5 draw). This activity should solidify
students’ knowledge of steepness relating to speed as well as how to determine who wins from the graph, who is in the
lead, and that the intersection point of the graphs indicates a driver passing another driver.
Day 2 – 90 min
The second day begins with the mountain climbers activity. This activity reinforces the key points from day 1, but now
asks students to create the tables and graphs on their own. The phrase rate of change is introduced in this activity. The
lesson then moves into direct instruction with exploring and calculating rate of change given situations, points, and
tables.
Prior Knowledge
 Plotting points on a graph, general knowledge of where points are located in relation to the origin and other
points
Assessment
 Monitor and check that students are filling in the properties correctly.
o Engage with students during work/explore time to explain how they came to their conclusion(s).
 Class discussion
 Homework
o Demonstrate knowledge and ability to interpret graphs into contextual meaning
o Ability to calculate rate of change and slope
 Quiz
Materials
 Card Sort, 1 per each group of 4
 Card Sort handout, one for each group
 Access to internet and applet for each student
 Homework page for each student
 Notes page for each student
 Projector and way to display examples and
information from computer
 Google Form for card sort story submission
 Google doc for brothers race submission of
correct answers (if choose option 1)

Lesson Outline
Getting Started
Day 1 – Interpreting & Understanding Graphs
Distance/Time Graphs Card Sort (40-45 min)
 Put students into groups of 3
 Pass out card sort cards to each group and/or pass out worksheet
 Explain that students should work together to match the tables, graphs, and situations. (25 min)
o Allow students to work on task independently for at least 7 min before providing any help
o Questions to ask (and things to look for), but do not give answers:
 What would running look like compared to walking on a graph?
 What does the speed look like on the graph? Table?
 What other way could you make the match (perhaps not going straight for the graph but looking
for tables first)
 What does the flat line mean?
 What would stopping look like?
 How would turning around be represented on the graph? Table?
o As students find which table and scenarios need to be filled in/completed on their own, have them fill in
on the indicated places on the handout.
 How did you decide on this situation?
 How does your story/table match the graph?
 Why did you pick that scenario for your story? (this is intended more as a way to connect with
and learn a little more about the students)
 Why these values?
o C: As you notice groups finishing the task have groups submit their stories digitally for the one graph,
bring class together for discussion. (15 min)
 Do NOT give answers. Have students give answers and reasons.
 Once a group responds, they cannot give another answer until every other group has
given one.
 Restate each response reasoning – emphasizing any key points (steeper, flat,
intercepts), their reasoning may be incorrect, but still emphasize to draw attention to
that key feature of the graph, story, or table.
 If class generates incorrect answers at the end they will get stuck and need to problem
solve as a class to determine correct answers.
 Show group stories/responses without names for students to read through and decide which
story they like best AND correctly matches the graph.
o Have class summarize key points and make a list or mind map on the board of these
2 Brothers Race Applet (option 1) (25 min)
 Have students go to media.mivu.org/mvu_pd/a4a/resources/applets/2race.html (It is probably easiest to make
this link available in Google Classroom)
 Show students brief overview of the applet (the 3 tabs, Ryan is red, how to move the dots) to troubleshoot
(Preview this on tablets first!)
 Give students 5 minutes to play with applet
 Have students independently complete 5 of Story and Draw correctly (approx. 15 min)
o Are they capable of screenshotting and submitting these through Google Classroom?
 Only for accountability, not for grading…
 Bring class together. Show a situation (story) and ask students to explain and help you create the graph
correctly.
o One student can only give one point. This should take several students to complete getting through.

o May be student volunteers or you pick and call on a student(s)
o Do this 2-3 times, until you have verified they understand all major points (stopping, passing, speed,
winning, ‘cheating’ in the race)
2 Brothers Race Applet (option 2) (25 min)
 Show students the applet, as a class, and walk though how it works once or twice.
 Put up cards around the room prior to beginning, or as explaining task (5 min)
 Pass out student sheets
 Have students complete the handouts as they walk around the room (approx. 15 min)
 Bring class together. Show a situation (story) and ask students to explain and help you create the graph
correctly.
o One student can only give one point. This should take several students to complete getting through.
o May be student volunteers or you pick and call on a student(s)
o Do this 2-3 times, until you have verified they understand all major points (stopping, passing, speed,
winning, ‘cheating’ in the race)
Summary
 Add any new points or concepts to the mind map/summary from earlier
o Passing
o ‘cheating’
o Speed
Questions
Homework
 Easy – Journey to the Bus Stop
 More difficult – Four Sisters Race

Day 2 – Rate of Change & Slope
Mountain Climbers (30 min)
 Pass out and have students begin working through the Mountain Climbers activity with a partner.
 About 10-12 minutes into the task, display the following on the board
This is data collected about Ms. Catapano’s drive to __________. How fast was Ms. Catapano driving?
Time 10:00am 11:00am Noon 1:00p 2:00p 3:00p
Miles
Traveled
20 65 105 145 185 225
o Monitor and look to provide support for students
 Which way do you think the labels go on the graph? Why?
 How did you determine the intervals on each axis?
 Bring class together. Discuss task.
o How did they determine the RoC (speed)?
o How is the RoC (speed) represented in the table? Graph?
o Does it matter that Carlos was going up and Vijay was going down? How can we show the up/down
direction in the rate of change?
o What does the negative mean? (referring to Vijay’s RoC)
o How are RoC and speed different?
 Additional class questions
Pass out RoC Notes Page
 Bring students attention to first 4 problems on the notes are similar to what we discussed with the activity
 Encourage and give students a few minutes to try the problems.
o Encourage them to make sense of the units
o Go through problems 1-4 with students
 Focus on the units! Miles per gallon or gallons per mile make more sense? Repeat for eat
problem.
 May require an example (e.g. make more sense to say $13 per lunch or 1.4 lunches per
$20?
 Fill in box and discuss
o RoC is the blank line
o Connect the vertical and horizontal to the graphs and math you’ve been working on
 Discuss the dependent vs independent variables (time always independent, x, does the number
of messages send depend on the number of days passed or does the number of days passed
depend on the number of messages sent? Money remaining depend on number lunches bought
or number of lunches purchased depends on money remaining?)
 Do problem 5 together, then have students do 6-8.
o Check as class
o #8 likely won’t make a lot of sense to students if they do not understand acceleration, but it’s there to
promote understanding that time is always independent
 Problems 9-11, refer back to the activity and how did they find rate of change (in the problem, the table, and
graph). Reiterate that just because the problem and information look different, doesn’t mean that we can’t
figure it out.
o Help students work through the process together as a class or have them work in small groups.
o Push them to make sense of it how they feel comfortable (put info into a table, sketch out a graph, etc)
 Problems 12-15, it will be helpful to refer back to the mountain climbers and how the students described RoC
being present in the table. Then rewriting the points as a table, similar to the activity.

Day 3 – Slope
This third day is all about slope. Calculating slope from points, tables, and graphs and using the term. Bring up the
notion of zero slope vs undefined slope by addressing if 0 is in the numerator or denominator. By the end of the
class period students should be able to calculate slope from any of the 3 (table, points, graph), and plot and list 2
points with a given RoC (sneak this terminology back in, don’t always default to slope now), and likely graph a
line given a point and a slope.

Activity – Interpreting Graphs Name:
1Standard: Period:
Activity Interpreting Distance Time Card Sort
Do Not Write On the Cards. If it asks you to write something please write it on this half sheet.
Graph Interpretation Table
A
C
E
G
I
B
D
F
H
J
Use this space and the back to write your story and fill in tables.
Time Distance Time Distance
0 0
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
9 9
10 10

Activity – Two Brothers Race Name:
1Standard: SS12.3.1a; 12.3.1b; DK2; DK3 Period:
http://media.mivu.org/mvu_pd/a4a/resources/applets/2race.html
1. 2.
3. 4.
5. 6.

7. 8.
9. 10.
11. 12.

1. Circle the appropriate responses
Ryan
Ryan

Ryan
Ryan

6. Describe what happened in the race
Ryan
Ryan

Ryan
Ryan

9. Sketch a graph for the race.
Use a solid line for Ryan, dotted line for the other racer.
10. Sketch a graph for the race

11. Sketch a graph for the race.
12. Sketch a graph for the race

WKS – Journey to the Bus Stop Name:
1AStandard: Period:
Journey to the Bus Stop
Every morning Tom walks along a straight road from his home to a bus stop, a distance of 160 meters.
The graph shows his journey on one particular day.
1. Describe what may have happened. You should include details like how fast he walked.
A).____________________________________________________________________
B).____________________________________________________________________
C).___________________________________________________________________
D).___________________________________________________________________
2. Are all sections of the graph realistic? Fully explain your answer.
__________________________________________________________________________________
__________________________________________________________________________________
__________________________________________________________________________________
__________________________________________________________________________________
A
B
C
D

WKS – The Four Sister Race Name:
1BStandard: Period:
The Four Sister Race
One day four sisters, Alberta, Betsy, Carla, and Dora had a 1000 meter race. The graph below is a sketch describing what
happened that day. You are the race announcer. Write a description of what happened during the race. Since it is a
rough sketch you do not need to measure anything accurately, just tell what happened.
1000
10
Distance
(meters)
Time (minutes)
Alberta
Betsy
Carla
Dora

Activity – Mountain Climbers Name:
2Standard: Period:
Carlos was climbing a mountain that Vijay was descending when they met. Carlos left at 8 am from an
altitude of 350 feet and gained 100 ft/hr. Vijay left at 8 am from an altitude of 1850 feet and lost 150 ft/hr.
1. Complete the following table:
Time of
Day
Carlos’
Altitude
Vijay’s
Altitude
2. What is Carlos’ constant rate of change (speed)?
3. What is Vijay’s constant rate of change (speed)?
4. What is the y-intercept (start) for Carlos?
5. What is the y-intercept (start) for Vijay?
6. What time did they meet?
7. What altitude did they meet?
8. Graph
(Titles, Labels, Units, etc.)

Notes – Rates of Change & Slope Name:
2Standard: Period:
Find the rate of change for each situation – Use correct units!!
1. 2.
Gallons Hours
3. 4.
Days Lunches Purchased
Definition
________________________ allows you to see the relationship between two quantities that are changing. If one
quantity depends on the other, then the following is true.
𝑟𝑎𝑡𝑒 𝑜𝑓 𝑐ℎ𝑎𝑛𝑔𝑒 =
𝑣𝑒𝑟𝑡𝑖𝑐𝑎𝑙 𝑐ℎ𝑎𝑛𝑔𝑒
ℎ𝑜𝑟𝑖𝑧𝑜𝑛𝑡𝑎𝑙 𝑐ℎ𝑎𝑛𝑔𝑒
=
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒
𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑖𝑛𝑑𝑒𝑝𝑒𝑛𝑑𝑒𝑛𝑡 𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒
Using the 2 pieces of information given, determine the units that would be used in expressing the Rate of Change. You
do NOT need to find the rate of change.
5. 16 carrots, 6 people 6. 2 people, 10 slices of pizza
7. 500 texts, $50 8. 35 mph, 1.25 hours
Miles
Miles
MessagesSent
$$remaining

Find the rate of change for each situation using the information given.
9. After walking for 75 minutes you have gone 5.1 miles. At 110 minutes, you have travelled 7.7 miles.
10. In a set amount of time, 3 people can eat 67 Oreos
cookies. In the same amount of time, 5 people can
eat 98 cookies.
11. You place 10 grams on a spring and it stretches 8cm.
When you place 35 grams on the spring it stretches
28cm.
Find the rate of change between each of the following pairs of points.
12. (2, 5) & (3, 7) 13. (-1, 6) & (3, -5)
14. (-3, -2) & (4, -8) 15. (3, 5) & (7, 7)
16.
X Y
1 7
2 10
3 13
17.
X Y
-2 12
4 0
8 -6

Rate of Change Lesson Plan

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