WillieWheelMan AIMS

W illie the Wheel Man
Common Core State
Standards for Mathematics*
• Make sense of problems and persevere in
solving them. (MP.1)
• Reason abstractly and quantitatively. (MP.2)
• Model with mathematics. (MP.4)
• Attend to precision. (MP.6)
• Look for and make use of structure. (MP.7)
• Look for and express regularity in repeated
reasoning. (MP.8)
• Write and interpret numerical expressions.
(5.OA.A)
• Analyze patterns and relationships. (5.OA.B)
• Graph points on the coordinate plane to
solve real-world and mathematical problems.
(5.G.A)
You Need
Wheel-shaped pasta (see Before You Begin 1)
Student pages
Colored pencils, 4 different colors
Straight edge, such as a ruler
Before You Begin
1. One box of rotelle pasta (wagon wheel
shape) wheel-shaped cereal, or other
circular cereal/candy will provide plenty
of manipulatives, as students will find that
the pattern is easy to discern and will soon
discontinue its use.
5th GR MATHEMATICAL PRACTICES
Patterns and Relationships
Students will use tables and graphs to analyze
the relationship between various vehicles and
the number of wheels they need.
Do This
1. Ask students how many wheels his/her
bicycle has. Discuss how many wheels
are on a bike and how many bicycle
wheels they would have if every member
in their family had a bicycle. Record these
responses on the board. Have the class
play a guessing game of “In the family.”
For example: If a student says his or her
family would need 10 wheels, the class
would respond by saying that there are five
in the family, because five bicycles have
10 wheels. Record this as an ordered pair,
(5,10). Do several of these examples, leaving the ordered pairs randomly recorded on
the board.
2. Tell the class that you have a friend, Willie
the Wheel Man, and that he deals in wheels.
Explain that his customers tell him what
kind of vehicle they are building and how
many vehicles they want to build, and he
figures out how many wheels they will need.
3. Show the students the wheel-shaped pasta
and explain that they may use it to help
them model the bicycles.
4. Distribute the pasta and first student page.
Explain that when Willie gets the information from a customer, he makes a t-table.
Ask students to fill in the t-table with the
number of wheels needed for the various
numbers of bicycles, using the pasta as
necessary.
1
5. Once the table has been filled out, ask
students how they determined the number
of wheels. [E.g., used the pasta to model
the bikes and counted, skip counted (2,
4, 6, …) the wheels per bicycle, multiplied
the number of bicycles by two to find the
number of wheels, etc.] Emphasize the
importance of understanding multiple
methods to solve a problem. (MP.4, MP.7)
6. Ask the students to share the patterns they
see in the t-table. Have them look at the
horizontal pattern and decide as a group
what rule they used to find the number of
wheels on a certain number of bikes and
write the rule using the letter B on the first
student page. Share out as a class. [Any
number of bikes times two is the number
of wheels needed, 2B.] Invite the students
to use the expression to answer how many
wheels are needed for 27 bicycles, 50
bicycles, etc. (MP.1, MP.2, MP.4)
7. Distribute the second student page and
colored pencils. Inform students that they
are going to look at the information in a
different way by building a graph on a
coordinate plane. On the first student page,
help students write each row in the t-table
as an ordered pair. On the second page,
draw students’ attention to the two axes.
The horizontal axis is labeled number of
vehicles because they will be graphing lines
for three types of vehicles—a bicycles, tricycles, and wagons. Now have them graph
© 2012 AIMS Education Foundation
each ordered pair on the coordinate plane
with one of their colored pencils. Assist as
needed. Ask: From the graph, how can you
tell how many wheels are on four bicycles?
…five bicycles? Which model is easier for
you to find the number of wheels needed?
Why? (MP.4, MP.6)
8. Ask groups: How can you use the graph to
determine the number of wheels on seven
bicycles? After groups answer the question,
have a few groups share answers. If necessary, steer students to using their straight
edge to extend the line. Ask: How many
wheels do you need for eight bicycles?
…10 bicycles? (MP.6)
9. Tell the class that now they are going
to apply their new skills, looking at the
number of wheels they need to order from
Willie for tricycles. Students will fill out a
t-table, write the ordered pairs, find the
rule, and add the line to the graph on the
student page in a different color. Be sure
they color the key.
10. Ask: How does the tricycle line compare to
the bicycle line? [It’s above the bicycle line.
Both are straight.]
11. Ask: Using your experience with the patterns from bicycles and tricycles, where on
the graph do you think the line for wagons
will be? What patterns will help you make
your predictions? Justify your answer.
(MP.3, MP.6, MP.8)
12. Now have students fill out the t-table, write
the ordered pairs, find the rule, and add
the line to the graph on the second student
page in a different color for wagons.
5th GR MATHEMATICAL PRACTICES
Ask These
1. How many wheels will be needed for 15
bicycles? …for 13 tricycles? Describe two
ways you could find out. [Students can use
the expression or the graph to answer the
question. If students use the graph, they
will need to make it larger.]
2. If someone orders 24 wheels, how many
bicycles could they make? [12] How many
tricycles? [8] How many wagons? [6] How
do you know? [When I find 24 wheels on
the number of wheels axis, I follow the line
until it intersects with the line graph of the
bicycle, tricycle, or wagon. Then I use the
coordinates to give me the answer to
how many bicycles, tricycles, or wagons
I could make.]
Journal Prompt
Make a table, write the ordered pairs, and
graph the possibilities if you needed to find the
number of wheels for a unicycle.
Digging Deeper
Have students design a vehicle with multiple
wheels. On a blank piece of paper, have students write a prediction of where the line will be
on the graph. Then, have them draw a t-table,
find the rule, and add the line to the graph on
the student page in a different color.
* © Copyright 2010. National Governors Association
Center for Best Practices and Council of Chief State
School Officers. All rights reserved.
3. How is the t-table helpful? …ordered pairs?
…rule? …graph? [The t-table helps us to
see numerical patterns. Ordered pairs help
us to make our graph. The rule helps me
represent the math procedure I use to find
any number of wheels for a certain vehicle.
The graph is a visual representation that
allows me to see patterns and make
generalizations.]
4. Explain in your own words how you can
use a graph to know how many wheels are
on three wagons. …on eight bicycles.
5. Which method did you find easiest for
determining the number of wheels on a
vehicle: using the rule, using the graph,
adding wheels, or skip counting? Explain
why you thought it was the easiest.
2
© 2012 AIMS Education Foundation
W illie the Wheel Man
Bicycles
1. Fill in the t-table column Number
of Wheels.
2. Look for the horizontal pattern.
3. Write the rule.
4. Use the information from the
t-table to write ordered pairs.
Number of
Bicycles
Number of
Wheels
Ordered
Pairs
0
0
(0,0)
1
(
,
)
2
(
,
)
3
(
,
)
4
(
,
)
5
(
,
)
Rule:
Tricycles
Wagons
Number of
Tricycles
Number of
Wheels
Ordered
Pairs
Number of
Wagons
Number of
Wheels
Ordered
Pairs
0
0
(0,0)
0
0
(0,0)
1
(
,
)
1
(
,
)
2
(
,
)
2
(
,
)
3
(
,
)
3
(
,
)
4
(
,
)
4
(
,
)
5
(
,
)
5
(
,
)
Rule:
5th GR MATHEMATICAL PRACTICES
Rule:
3
© 2012 AIMS Education Foundation
W illie the Wheel Man
1. Graph the data.
2. Color the graph key to correspond to the graph.
Number of Wheels
W
Willie
Key
Bicycles
Tricycles
Wagons
30
29
28
27
26
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
Number of Vehicles
5th GR MATHEMATICAL PRACTICES
4
© 2012 AIMS Education Foundation
W illie the Wheel Man
CON
N
EC
Ask These
T I NG
AR
LE
NI
N
G
1. How many wheels will be needed for 15
bicycles? …for 13 tricycles? Describe two
ways you could find out.
2. If someone orders 24 wheels, how many bicycles
could they make? How many tricycles? How many
wagons? How do you know?
3. How is the t-table helpful? …ordered pairs? …rule?
…graph?
5th GR MATHEMATICAL PRACTICES
5
© 2012 AIMS Education Foundation
W illie the Wheel Man
CON
N
EC
T I NG
AR
LE
NI
N
G
4. Explain in your own words how you can
use a graph to know how many wheels
are on three wagons. …on eight bicycles.
5. Which method did you find easiest for determining
the number of wheels on a vehicle: using the rule,
using the graph, adding wheels, or skip counting?
Explain why you thought it was the easiest.
5th GR MATHEMATICAL PRACTICES
6
© 2012 AIMS Education Foundation