1
Discrete
Math
Grades 6 & 7
Andrew Sundberg
Laporte School
[email protected]
Heather Prestegord
Climax-Shelly Schools
[email protected]
2
Executive Summary
Section One
Title: “Basic Probability & Its Various Representations”
Location in this Document: pg. 6-19
Grade: 6
MN Math Standard(s):
6.4.1.1 – Determine the sample space (set of possible outcomes) for a given experiment and
determine which members of the sample space are related to certain events. Sample space may
be determined by the use of tree diagrams, tables or pictorial representations.
Overview:
In this unit, students will be able to use pictorial representations, tables, and tree
diagrams as they learn about probability. They will be able to see tangible items to see
how to discover various combinations. They will be given time to explore each of these
representations with their peers and share their thinking, which will guide them to
deeper understanding.
3
Executive Summary (continued)
Section Two
(To be used after the algebra unit)
Title: “Vertex-Edge Graphs”
Location in this Document: pg. 20-36
Grade: 7
MN Math Standard(s):
7.2.2.1 – Represent proportional relationships with tables, verbal descriptions, symbols,
equations and graphs; translate from one representation to another. Determine the unit rate
(constant of proportionality or slope) given any of these representations.
Overview:
In this unit, students will make weighted vertex-edge graphs as well as learn to use the
“Nearest Neighbor” and “Greedy” algorithms. They will discuss the benefits of each
algorithm and use these algorithms in the context of careers and mileage, which will
give the students a real-life situation to work with.
Section Three
(Option: T o be used before the algebra unit)
Title: Recursive Equations (Supplement)
Location in this Document: pg. 37-38
Grade: 7
Overview: This section is to be used before our algebra unit, if students need some practice
(or need an introduction) with recursive equations.
4
Sample MCA Question this Unit Covers
5
Table of Contents
Basic Probability Unit………………………………………………………………………………………………………………….6
Probability Pre-Test/Post-Test…………………………………………………………………....................7-8
Pictorial &Table Representations……………………………………………………………………………….9-13
Connected Pictorial Representation………………………………………………………………………….14-15
Tree Diagrams…………………………..……………………………………………………………………………..16-18
Unit Wrap-Up………………………………………………………………………………………………………………..19
Vertex-Edge Graphs Unit……………………………………………………………………………………………………………20
Vertex-Edge Graphs Pre-Test/Post-Test…………….………………………………………................21-22
Weighted Vertex-Edge Graphs………………………………………………………………………………….23-28
“Nearest Neighbor” Algorithm………………………………………………………………………………….29-32
“Greedy” Algorithm..………………..……………………………………………………………………………..33-36
Recursive Supplement…………………………………………………………………………………………………………..37-38
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Title: Basic Probability & Its Various Representations
Mathematics Content: Basic Probability
Objectives: To introduce probability using tree diagrams, tables, & pictorial representations
To use results of tree diagrams to discuss outcomes, and to be able to answer related
questions
Time: Approximately two to three 50 minute class periods
MN Standards:
6.4.1.1 – Determine the sample space (set of possible outcomes) for a given experiment and
determine which members of the sample space are related to certain events. Sample space may
be determined by the use of tree diagrams, tables or pictorial representations.
7
Basic Probabilities
Pre-Test/Post-Test
Name:__________________
Date:___________________
Hour:___________________
Follow the directions for each problem below. Be sure to show all work!
1. Charles can have a snack after school, either a cookie, a piece of cake, or an ice cream cone.
Then, he can play either basketball or football. Create a pictorial representation of all
possibilities. How many possible different options does Charles have? ___________
2. Fran has three sweatshirts (a green one, a yellow one, and an orange) and three pairs of pants
(jeans, khaki pants, & black pants).
Create a table to represent the possible outfits. How many different outfits are there? ________
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3. At a restaurant, Ron can choose a choice of meat, potato, and beverage. For meat, he can
choose chicken or steak. He can choose mashed potatoes, a baked potato, or French fries. He
can choose soda or milk.
How many possibilities are there? __________________
Create a tree diagram to represent the possibilities. Be sure to list the outcomes.
4. For dress-up day, Laura can choose between two pairs of shoes (a white pair of socks and a
black pair of socks) and four pairs of socks (a blue pair, a pink pair, a green pair, and an orange
pair.)
How many different combinations of socks and shoes are there? _______________________
Create a representation, of your choice, to show the possibilities.
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DAY 1 – Pictorial/ Table Representation
Students should take Pre-Test on Previous Page, then spend the rest of the hour on the lesson below.
Materials Needed: 2 text books (such as math & spelling) & 3 fun items (such as a football, basketball,
& a kickball) …(The idea is to have two different items of one category and three different items of
another)
Launch: Have the books and sports equipment available for the students to view. Tell students the
following problem. “Johnny’s parents said he could play a sport with a friend of his after finishing one
subject of homework. He can do his math or his spelling. And then he can play either football,
basketball, or kickball.” Make sure the problem is clarified for any students who might be confused.
Johnny can choose one subject and one sport. “Come up with a guess as to how many different
possibilities Johnny has.” Ask students to share their guess. Maybe have a couple of students share
which subject and sport they would choose.
Explore: Allow students to work in groups to try to discover how many combinations can be made.
Walk around the room to see how students are coming up with their answer. Encourage them to talk
among their group about different ways they might be able to tell how many combinations. If a group
finishes before the other groups, ask them to try to think of another way they could show all of the
possibilities.
Share: Allow each group to show their work on the board and discuss their findings with the class.
Summarize: Inform the students that in this unit, they will be learning about different ways to figure
out, and represent, problems like how many different combinations of subjects and sports there are.
Discuss which of the solving methods they think were useful. If time, ask them if they could name some
situations that might be similar to figuring out how many combinations of subjects and sports. “In what
other situations would you have options like this?”
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Day 2 - Pictorial/ Table Representation (Continued)
Materials: Colored Pencils, 3 different colored shirts, 3 pairs of different colored pants (or shorts)
Launch: Ask the students if they have thought of any scenarios where they many various
combinations… Initiate a discussion about keeping their work in an organized fashion, so as not to
miscount or “lose” any of the outcomes.
Show the shirts and the pants to the students and keep them within their viewing range. Tell the
students it is their job today to figure out how many different outfits could be made.
Explore: Allow students to work in groups to complete the worksheet below. Hopefully, some students
drew pictures of possibilities the day previously. Give students hints if they are not sure what the
question is asking or how to fill out the table.
Share: As students finish, ask them to share their thoughts about pictorial representation & using a
table.
Some discussion questions may include:





How is a pictorial representation helpful?
Through which representation is it easier to make sure all of the possibilities are
accounted for?
What is the downfall to pictorial representation? (Answers may include: Takes a lot of
time, easy to forget some, would not be practical if there were lots of choices)
How is the table helpful?
What is the downfall to table representation? (Answers may include: Can’t “see” the
possibilities….generate a discussion about the following: Can a table be used if there are
more than 2 choices? For example, if we had shirts, pants & shoes)
Summarize: “Today we learned about Pictorial and Table Representation.” Clarify any confusion that
may have developed during the discussion. Emphasize that using a table and draw a picture are two
ways to find the different outcomes. Tomorrow, they will learn about another way.
Homework: To see how students are doing with these representations, assign worksheet
“Pictorial & Table Representations 102”
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Pictorial & Table Representations 101
Name:____________________________
1. How many outfits can be created? Discuss with your group. Sketch the possibilities below.
2. To make sure your group didn’t miss any of the possibilities, fill out the table below to list the
different outfits. List the shirt choices across the top and the pant choices along the side. List
the possibilities in the chart.
3. How many outfits could you make if you only had two shirts? Show your work below.
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Pictorial & Table Representations 102
Name:_________________________
More Practice! Create a table & draw pictures for the following problems.
1. At Jerry’s Ice Cream, students can choose between chocolate, caramel, strawberry, or
butterscotch topping. Then, they can choose either whip cream or a cherry. How many
possibilities are there?
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2. For breakfast, Jill can choose from the following cereals: Cereal, Cheerios, or Kix. She can also
choose to have a muffin, toast, or a doughnut with her cereal. How many possible choices does
Jill have for breakfast?
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Day 3 - Connected Pictorial Representations
Materials: Smart Board (In the graphics section, find 4 different shirts and 2 different colored pants)
Launch: Ask the students, by looking at the Smart Board, if they can guess how many outfits can be
made.
Explore: Rather than redraw each shirt every time, ask if there’s any easier way of drawing out the
possibilities- where you don’t have to draw each shirt over and over again.
Allow students to work in small groups to see if there’s a simpler way of sketching out the possibilities.
Share: Discuss how each group drew their pictures. Perhaps, show them on the board. Guide students
to lining up the four shirts and then connecting the two pair of pants to each shirt. During the next
lesson, students will learn about tree diagrams, so this should help prepare them for that.
Explore: Allow students to work together on the “Connected Pictorial Representations 103” worksheet
below.
Share: Discuss what students are finding as they draw the “Connected Pictorial Representations.” How
is this easier than drawing out each possibility? Can this representation be used more easily in more
situations?
Summarize: “Today we learned about ‘Connected Pictorial Representations.’ Emphasize organization,
neatness, and using the connecting lines every time. (This will help when they begin tree diagrams
tomorrow.)
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Connected Pictorial Representations 103
Name:_______________________
Create connected pictorial representations to represent the following situations.
1. After dinner, Jenny can choose a dessert (either a cookie, an ice cream cone, or a brownie). She
can also have either a juice box or milk. How many options does Jenny have?
2. Bill had two t-shirts (a red and a yellow one), and four pairs of basketball shorts (a black pair, a
brown pair, a blue pair, and a gray pair). How many different outfits does he have?
3. If Bill, from question #2, has two pairs of tennis shoes to choose from (Nike & Adidas), now how
many possible outfits does he have? (Extend your connected pictorial representation above.)
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Day 4 – Tree Diagrams
Launch: Ask the students if they can think of a way to draw a diagram, in which the choices are still
connected (like yesterday), but they don’t have to draw the pictures. “Remember to keep your diagram
neat and organized.”
Feel free to give them a problem where there are three t-shirts, two pairs of pants, and two pairs of
tennis shoes (or some variation thereof.)
Explore: Allow students to work in groups to discuss this and try to draw a similar diagram without
having to draw the actual pictures.
Share: Discuss how each group drew their pictures. Perhaps, show them on the board.
After students have shared their discoveries, perhaps one group has used letters. Explain that that is
called a tree diagram, and guide students to fill in any missing pieces (such as a missing connector, or a
starting point, etc) Also, discuss & emphasize the outcomes column.
Explore: Allow students to work together on the Beginning Tree Diagrams 104 worksheet below.
Share: As students are finishing up with #1, ask the groups to put #1 on the board so it can be discussed
as a class. Look at each groups tree diagram. Are there missing pieces? Are the outcomes listed? Are
they neatly organized?
Summarize: “Today, we learned about tree diagrams. Tree diagrams are useful to organize our
outcomes.” Emphasize organization, neatness, and using the connecting lines every time. (This will help
when they begin tree diagrams tomorrow.) Have students finish the worksheets for tomorrow.
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Beginning Tree Diagrams 104
Name:_________________________________
Draw a tree diagram for each of the following situations. Remember to list your outcomes! 
1. At Delilah’s Diner, the dinner special is offered for $6.75. Customers may choose a meat choice
(either a hamburger, hot dog, or cheeseburger) and a side choice (either French Fries or salad).
How many meal options does Delilah’s Diner offer?
2. Each student at Stanley Middle School is going to donate a school supply (either a box of pencils,
a box of crayons, or a box of markers) and a toy (either a baseball, a board game, or a doll).
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3. At the end of a community race, runners can choose a bracelet (red, yellow, or green) and either
a ribbon, medal, or certificate. How many choices do the runners have?
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Day 5:
Answer any questions students may have before taking the post-test
Post-Test (Same as Pre-Test)
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Title: Vertex-Edge Graphs Unit
Mathematics Content: Vertex-Edge Weighted Graphs, Vertex-Edge Graphs using the “Nearest
Neighbor” method, Vertex-Edge Graphs using the “Greedy” method
Objectives: To introduce various Vertex-Edge Graphs (as listed in the “Mathematics Content”)
To use results of Vertex-Edge Graphs to learn about creating efficient routes between cities
Time: Approximately Eight 50-minute class periods
MN Standards:
7.2.2.1 – Represent proportional relationships with tables, verbal descriptions, symbols, equations and
graphs; translate from one representation to another. Determine the unit rate (constant of
proportionality or slope) given any of these representations.
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Vertex-Edge Graphs
Pre-Test/Post-Test
Name:__________________
Date:___________________
Hour:___________________
Use the scenario below to answer the following questions.
Carlos is a salesman. He lives in Philadelphia and has meetings throughout the next two
weeks in Minneapolis, New York City, Chicago, and Washington D.C. He wants to begin
and end his week at home, in Philadelphia. He can schedule his meetings in any order he
chooses. The following chart shows the miles between each city.
CITIES
Chicago
Chicago
Minneapolis
New York City
Philadelphia
Washington,
D.C.
Minneapolis
New York
City
Philadelphia
Washington,
D.C.
---------
354
711
664
544
354
--------
1015
982
931
711
1015
-------
80
203
664
982
80
------
123
544
931
203
123
-----
1. Make a weighted graph using the cities as vertices and the distances between cities as edges.
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2. Use the “Nearest Neighbor” method to find a route for Carlos. Calculate his total mileage.
3. Use the “Greedy” method to determine Carlos’ route. Explain your route and calculate his total
mileage.
4. Which route seemed more efficient? Explain.
5. Bonus! Is there a more efficient method for this route than either of those two methods?
Explain. (Feel free to use the back of your test to show your calculations and to explain.)
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DAY 1
Students should take Pre-Test on Previous Page, then spend the rest of the hour beginning the lesson
below.
Launch: “If you were in a band and were going to put up fliers to advertise for your upcoming show in
various cities around, how would you go about planning your route?”
“How many miles is it from here to ________?” “What are some ways that we can find the distance
between two cities?” (Perhaps this discussion will lead into talking about mileage tables)
Explore: Continue with exploring the launch question. “Is there a way we could sketch out a graph that
would show the distances between three cities?”
Give them the following information (or feel free to actually use cities close to your school):



From city A to city B is 15 miles.
From city A to city C is 12 miles.
From city B to city C is 24 miles.
Allow students to see if they can make some kind of graph representation. After a few minutes,
suggest that they make three “vertices” (dots), one to represent each city.
Share: Let them share their findings, but guide them to creating a weighted vertex-edge graph. (A
weighted vertex-edge graph could be similar to the one seen on the “Weighted Vertex-Edge Graphs 2”
sheet. That diagram just needs the miles listed along each edge (line.)
Summarize: Clarify to the students about what a weighted vertex-edge graph is. Emphasize the city
locations/dots as being “vertices”, the lines between as the “edges,” and the “weighted” part comes
from being the mileage listed along the edges of the graph.
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DAY 2
Launch: “Who can tell me what a ‘weighted vertex-edge’ graph is?” Listen for responses and make
clarifications. “Today, we get to create a couple more ‘weighted vertex-edge’ graphs. Why would we
use them? How would using a ‘weighted vertex-edge’ graph simplify our advertising route?”
Explore: Have the students work together in small groups on the “Weighted Vertex-Edge Graphs 1”
worksheet below.
Share: Allow each group to draw one (or both, if time) of their graphs of the board. Have each group
explain their graph. Discuss any differences that may exist among the graphs.
Summarize: Again, clarify to the students about what a weighted vertex-edge graph is. Emphasize the
city locations/dots as being “vertices”, the lines between as the “edges,” and the “weighted” part comes
from being the mileage listed along the edges of the graph.
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Weighted Vertex-Edge Graphs 1
Name:_________________________
Create a weighted vertex-edge graph, given the following distances between cities. (Assume the
numbers in the table are the distances between the two cities.)
1.
Cities
Grand Forks
Fargo
Detroit Lakes
Grand Forks
--------81
131
Fargo
Detroit Lakes
131
46
-------
81
-------46
2.
Cities
Bemidji
Mankato
Duluth
St. Paul
Bemidji
--------296
152
230
Mankato
296
-------235
88
Duluth
152
235
------150
St. Paul
230
88
150
------
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DAY 3
PART 1
Launch: “What if you were given a ‘weighted edge graph’…could you create the table?” What would be
some advantages or disadvantages of using a table verses a ‘weighted edge graph’?
Explore: Have the students work together in small groups on the “Weighted Vertex-Edge Graphs 2”
worksheet below.
Share: Allow groups to check with other groups as they fill in the rest of the Weighted Graph & fill in the
table. Discuss as a class how this is different from what they did yesterday.
Summarize: Again, clarify to the students about what a weighted vertex-edge graph is. Emphasize the
city locations/dots as being “vertices”, the lines between as the “edges,” and the “weighted” part comes
from being the mileage listed along the edges of the graph.
Part 2
Launch: “Remember the Bicycle Algebra Unit? Could we create a ‘Weighted Vertex-Edge Graph’ to
represent the cities we used in that unit? ”
Explore: Have the students continue working together in small groups – this time on the “Weighted
Vertex-Edge Graphs 3” worksheet below.
Share: Allow groups to share their “Weighted Vertex-Edge Graph” on the board.
Possible Discussion Questions:



Why would we want to make a weighted graph? What can we learn? How are they
useful?
Could we create a ‘weighted vertex-edge graph’ for any number of cities? Would
drawing them get easier or worse if we add more cities?
Why would we want each city to be represented by just one vertex? (Perhaps refer to a
few days ago, when students might have represented a city by a few different vertices
(or dots)).
Summarize: Again, clarify to the students about what a weighted vertex-edge graph is. Emphasize the
city locations/dots as being “vertices”, the lines between as the “edges,” and the “weighted” part comes
from being the mileage listed along the edges of the graph. Provide clarifications students may need
from the earlier discussion.
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Weighted Vertex-Edge Graphs 2
Name:_________________________
Given the following Vertex-Edge Graph for the fictitious cities of Appleville, Bananaville, Cherryville, &
Dateville, fill in the miles on the graph, then complete the table. (The graph is not drawn to scale.)
From Appleville to Bananaville it is 125 miles, from Bananaville to Cherryville it is 173 miles, from
Cherryville to Dateville it is 81 miles, from Appleville to Dateville it is 68 miles, from Appleville to
Cherryville it is 243 miles, & from Bananaville to Dateville it is 209 miles.
B
A
D
C
Cities
-------------------------
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Weighted Vertex-Edge Graphs 3
Name:_________________________
Using the information from the bicycle algebra unit, create a distance table and weighted
vertex-edge graph. Be sure to include all of the following cities: LaPorte, Grand Rapids,
Duluth, Silver Bay, Grand Marias, and Grand Portage.
Cities
------------------------------------
Weighted Vertex Edge Graph:
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DAY 4 – The “Nearest Neighbor” Graphing Method
Launch: “What are some of the cities around here?” (List them on the board as students call them off.)
After the list includes most of the neighboring cities, ask the students, “which city would be considered
our nearest neighbor?” (If there are a couple of close cities, perhaps use an online map service to find
out which city is closest.)
Explore: “Now let’s say we have three of these towns.” Include the “nearest neighbor,” and then add
the city you’re in to the list of the three towns. (A total of four cities.)
As a class, create a vertex-edge graph for the four cities. Continue to use an online map service to find
the distances.
Then have the class draw just the vertices of a graph. (Draw one also on the board, and fill in the labels
for each of the vertices.)
Give the students the following problem:
“Let’s say Greg is a traveling salesman. He lives here in _______ so he is going to start the day
here and end the day here. To try to save gas money, he is always going to travel to the nearest
town. We’ll call it the ‘nearest neighbor.’” So from here, he is going to travel to _______.
(What they decided was the nearest neighbor). Then at that spot, he is going to look to the next
nearest neighbor. Remember he will not be traveling home until the end of the day. And he will
want to visit the other cities only once.”
Have students work on this problem in groups. (They should connect the vertices as they form
the route Greg should follow.)
Share: Let them share their findings. If students go to the wrong city, ask them “From ____, is ____ the
nearest neighbor?”
Possible Discussion Questions:


Using the “Nearest Neighbor” method, is there more than one correct answer?
Why would people use the “Nearest Neighbor” method? (trying to save gas, not have to
drive as much)
Summarize: Clarify to the students that this is called the “Nearest Neighbor” method and that it can be
used to find routes between cities.
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DAY 5 – The “Nearest Neighbor” Graphing Method (continued)
Launch: “If you were in Fargo, ND, which city would be considered your ‘nearest neighbor’?”
Explore: Have students work in groups on the “Nearest Neighbor 1” worksheet below.
Share: Allow each group to draw one (or both, if time) of their graphs of the board. Have each group
explain their graph. Discuss any differences that may exist among the graphs. Again, ask if they went to
the “Nearest Neighbor” after each stop.
Possible Discussion Questions (two are repeated from yesterday and can be expanded on):



Using the “Nearest Neighbor” method, is there more than one correct answer?
Why would people use the “Nearest Neighbor” method? (trying to save gas, not have to
drive as much)
Do you think the “Nearest Neighbor” method will always give you the shortest mileage?
Why or why not? (Perhaps not when there are four or more cities because the final
route ends up being quite long.) Will it always work when there are three total cities?
Summarize: “Nearest Neighbor” is one method to graph routes. From whatever city you are at, the goal
is to get to the next shortest city. The “Nearest Neighbor” may or may not give the shortest route.
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Routes Using the “Nearest Neighbor” Method 1
Name:_________________________
Leonard is a traveling salesman. He lives in Grand Forks and needs to travel Fargo and Detroit Lakes.
He wants to begin and end his day at home, in Grand Forks. Use the “nearest neighbor” method to
determine his route. Also, calculate his total mileage.
1.
Cities
Grand Forks
Fargo
Detroit Lakes
Grand Forks
--------81
131
Fargo
81
-------46
Detroit Lakes
131
46
-------
32
Lynnette is a family counselor. She lives in Duluth & has meetings this week in Bemidji, Mankato, and
St. Paul. She wants to begin and end her week at home, in Duluth. She can schedule her meetings in
any order she chooses. Use the “nearest neighbor” method to determine her route. Also, calculate her
total mileage.
2.
Cities
Bemidji
Mankato
Duluth
St. Paul
Bemidji
--------296
152
230
Mankato
296
-------235
88
Duluth
152
235
------150
St. Paul
230
88
150
------
33
DAY 6 – “Greedy Method”
Launch: “We’ve been talking lately about finding possible routes for travel. What if you were a traveling
salesman and your boss asked you to hit the four towns we talked about yesterday? He also told you
that you could have a thousand dollar bonus if you found the shortest route to travel.”
Explore: Have the students work together in small groups. Use the four cities from yesterday’s list and
see if students can come up with a smaller total mileage.
Share: Allow groups to share their thought processes (even if they didn’t get a better mileage). I
If someone mentions wanting to utilize all of the “least distances,” allow that to lead into the
introduction of the next method, the “Greedy Method.” Otherwise, look at the smallest distances
between the four cities, and tell the class “I want to be Greedy, so I want to use the least mileage
possible.”
Connect the two cities with the shortest distance, then the two with the next shortest distance, etc, but
do not enclose a circuit (route) until all of the cities have been included.
If there is time, give the students a similar problem and have them find a route using the “Greedy”
method.
Discussion Questions:



How is the “Greedy” method different than the “Nearest Neighbor?”
Which one gave us lower mileage? Will it always work that way?
Were our routes different? Will they always be different/the same?
Summarize: “Today we learned about the ‘Greedy’ method for graphing routes.” Emphasize
connecting the cities with the smallest distance first, then the next smallest, then the next, etc, without
enclosing the circuit (route) until the entire route is complete.
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DAY 7 – “Greedy Method” (continued)
Launch: “How many of you think you could use the ‘greedy method’ to earn that $1000 from your boss
by finding the shortest route today?” (refer to yesterday’s launch)
Explore: Have the students work together in small groups on the “Routes Using the ‘Greedy’ Method 1”
worksheet below. (The data is the same as when they used the “Nearest Neighbor” method.)

If groups finish early, ask them to compare the mileage they had using the Greedy
Method with the mileage they had using the Nearest Neighbor. Which one got the
better mileage? Why?
Share: Allow groups to share their graphs on the board and explain their thought processes.



When would people prefer the “Nearest Neighbor” method? (Perhaps they enjoy
driving longer distances later in the day or they want to see as many clients in the
morning. Perhaps they would do the reverse route if they wanted to see more people in
the evening (when people get home from work.))
When is the “Greedy” Method beneficial?
Which method do you think is used more often? Why?
Summarize: “Today we continued to learn about the “Greedy” method for graphing routes.” Again,
emphasize connecting the cities with the smallest distance first, then the next smallest, then the next,
etc, without enclosing the circuit (route) until the entire route is complete.
35
Routes Using the “Greedy ” Method 1
Name:_________________________
Leonard is a traveling salesman. He lives in Grand Forks and needs to travel Fargo and Detroit Lakes.
He wants to begin and end his day at home, in Grand Forks. Use the “Greedy” method to determine
his route. Explain your route and calculate his total mileage.
1.
Cities
Grand Forks
Fargo
Detroit Lakes
Grand Forks
--------81
131
Fargo
81
-------46
Detroit Lakes
131
46
-------
36
Lynnette is a family counselor. She lives in Duluth & has meetings this week in Bemidji, Mankato, and
St. Paul. She wants to begin and end her week at home, in Duluth. She can schedule her meetings in
any order she chooses. Use the “Greedy” method to determine her route. Explain your route and
calculate her total mileage.
2.
Cities
Bemidji
Mankato
Duluth
St. Paul
Bemidji
--------296
152
230
Mankato
296
-------235
88
Duluth
152
235
------150
St. Paul
230
88
150
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37
Recursive Supplement (To be used prior to Algebra unit if recursion is new to your class)
Launch: “How many of you would like to be on a game show?” (Perhaps allow the students to share
what game show they would like to be on.)
Explore: Create an “x-y chart” (see page 38) on the board, with “Questions Correct” as the heading on
the left column and “Number of Prizes” on the right Column.
“On this game show, if you get one question correct, you get one prize. If you get two questions correct,
you get two prizes. If you get three questions correct, you get four prizes. If you get four questions
correct, you get eight prizes.”
Fill in a few more values on the table. (The number of prizes doubles each time, but we want the
students to discover that.)
Allow students to work on seeing the pattern with their group. Encourage them to look at the right side
of the equation, and following the pattern, create an equation that uses the words “Next” and “Now.”
Share: Allow groups to explain their thought processes. And guide students to the recursive equation
“Next = 2(Now).”
Explore: Give students a few more tables to work on, having them develop the recursive equation. You
may use the worksheet “Recursive Formulas,” or give them similar tables and save the worksheet for
homework or a quiz.
Share: Have students share how they develop answers, showing their work on the board.
Discussion Questions:


What is your strategy for forming the recursive equation?
Why would we want to create a recursive equation?
Summarize: “Today we learned about recursive equations.” Be sure to stress that the words “Next” and
Now” should be in every equation, and their equation should work for each value on the table.
38
Recursive Formulas
Name:_________________________
Write the recursive formula based on the following tables. Fill in any values in the table that were left
open for you.
2.
1.
Hours
Amount ($)
H
M
1
2
1
3
2
8
2
5
3
14
3
7
4
20
4
9
5
5
11
6
Recursive Formula:
Recursive Formula:
__________________________
__________________________
3.
4.
X
Y
Hours
Amount ($)
0
2
0
1
1
6
1
3
2
18
2
6
3
54
3
9
4
Recursive Formula:
5
Recursive Formula:
__________________________
_______________________