Sample Analogous Tasks
Multiplicative Structures
Multiplicative Structure and Analogous Tasks
Multiplication and Analogous Tasks
Dr. Roger Fischer
EMAT Project Facilitator
Montana State University
October 14, 2016
References
Sample Analogous Tasks
Multiplicative Structures
Multiplicative Structure and Analogous Tasks
O VERVIEW
Sample Analogous Tasks
Multiplicative Structures
Multiplicative Structure and Analogous Tasks
References
Sample Analogous Tasks
Multiplicative Structures
Multiplicative Structure and Analogous Tasks
References
A NALOGOUS P ROBLEM S AMPLE FROM THE GROUP
1. Kate financed her kitchen renovation. The cost of the
renovation was $1200. She put $300 down, then paid 8%
interest on the remainder. What was the total cost of the
renovation?
2. Hurley has 6 cups of bread dough. He uses 2 cups of the
dough to make rolls. He lets the rest double in size before
baking it into two bread loaves. What is the total volume
of dough?
Sample Analogous Tasks
Multiplicative Structures
Multiplicative Structure and Analogous Tasks
References
A NALOGOUS P ROBLEM S AMPLE FROM THE GROUP
1. Kate financed her kitchen renovation. The cost of the
renovation was $1200. She put $300 down, then paid 8%
interest on the remainder. What was the total cost of the
renovation?
1200 − 300 = 900
900 × 0.08 = 72
1200 + 72 = $1272
2. Hurley has 6 cups of bread dough. He uses 2 cups of the
dough to make rolls. He lets the rest double in size before
baking it into two bread loaves. What is the total volume
of dough?
6−2=4
4×2=8
8 + 2 = 10 cups
Sample Analogous Tasks
Multiplicative Structures
Multiplicative Structure and Analogous Tasks
M EANING OF M ULTIPLICATION
What does multiplication mean? Please share any relevant
thoughts you have in the chat window.
References
Sample Analogous Tasks
Multiplicative Structures
Multiplicative Structure and Analogous Tasks
D EFINITION FROM B ECKMANN
If A and B are nonnegative numbers, then
A × B or A · B
which we read as “A times B,” means the total number of
objects in A groups if there are B objects in each group.
Beckmann (2012, p. 140)
References
Sample Analogous Tasks
A CTIVITY
Multiplicative Structures
Multiplicative Structure and Analogous Tasks
References
Sample Analogous Tasks
Multiplicative Structures
Multiplicative Structure and Analogous Tasks
R ECTANGULAR A RRAYS
View the rows as the groups. Then try it with the columns.
References
Sample Analogous Tasks
Multiplicative Structures
Multiplicative Structure and Analogous Tasks
References
T RANSFERRING THE A NALOGOUS C ONCEPT
The similar structure here is a multiplicative one.
A groups ×
B objects
= A × B objects.
group
What are the groups and objects in the following problem?
I
A restaurant serves cheese sandwiches that are made from
a piece of bread and a piece of cheese. There are three types
of bread to choose from: wheat, white, and rye, and there
are 4 types of cheese to choose from: cheddar, provolone,
Swiss, and American. How many types of cheese
sandwiches can the restaurant make with these choices?
Sample Analogous Tasks
Multiplicative Structures
Multiplicative Structure and Analogous Tasks
T RANSFERRING THE A NALOGOUS C ONCEPT
The similar structure here is a multiplicative one.
A groups ×
B objects
= A × B objects.
group
What are the groups and objects in the following problem?
I
Robert and Samuel each have a hermit crab collection.
Robert has 3 times as many hermit crabs as Samuel.
Samuel has 7 hermit crabs. How many hermit crabs does
Robert have?
References
Sample Analogous Tasks
Multiplicative Structures
Multiplicative Structure and Analogous Tasks
S UMMARY OF M ULTIPLICATIVE S TRUCTURES
I
Array
I
Ordered Pair
I
Mult. Comparison
References
Sample Analogous Tasks
Multiplicative Structures
Multiplicative Structure and Analogous Tasks
S UMMARY OF M ULTIPLICATIVE S TRUCTURES
I
Array problems
I
I
I
Ordered pair problems
I
I
I
The number of rows (or the columns) is the first factor.
The number of objects in each row (or columns) is the
second factor.
The number of first choices is the first factor.
The number of pairs per choice is the second factor.
Multiplicative comparison problems
I
I
The multiplier used for comparison is the first factor.
The number being compared to is the second factor.
References
Sample Analogous Tasks
Multiplicative Structures
Multiplicative Structure and Analogous Tasks
References
F INDING M ULTIPLICATIVE S TRUCTURE
Analyze this problem for multiplicative structure. What are the
objects and the groups? What type of multiplication problem is
this?
Anita Bath drives her 1977 Ford Pinto with wood
paneling for 8 hours at a speed of r miles per hour.
How far does she drive?
Sample Analogous Tasks
Multiplicative Structures
Multiplicative Structure and Analogous Tasks
I DENTIFYING A NALOGOUS TASKS
Is this problem analogous to the one with Anita Bath?
The shortfin Mako is known to have the fastest speed
of all sharks. The sailfish has the fastest speed of all
fish. The top speed recorded for a sailfish is
approximately 25 mph faster than the fastest speed of
a Mako shark. A sailfish can swim 14 mi in the same
time that a Mako shark can swim 9 mi. Find the speed
of each sea animal. – Bittinger (2015)
References
Sample Analogous Tasks
Multiplicative Structures
Multiplicative Structure and Analogous Tasks
References
ATTENDING TO THE M EANING OF Q UANTITIES
9 miles =
r miles
(t hours)
1 hour
14 miles =
(r + 25) miles
(t hours)
1 hour
Sample Analogous Tasks
Multiplicative Structures
Multiplicative Structure and Analogous Tasks
References
ATTENDING TO THE M EANING OF Q UANTITIES
9 miles =
r miles
(t hours)
1 hour
14 miles =
(r + 25) miles
(t hours)
1 hour
9 miles =
r miles
hours)
(t 1
hour
14 miles =
(r + 25) miles
(t hours)
1
hour
Sample Analogous Tasks
Multiplicative Structures
Multiplicative Structure and Analogous Tasks
References
ATTENDING TO THE M EANING OF Q UANTITIES
9 miles =
r miles
(t hours)
1 hour
14 miles =
(r + 25) miles
(t hours)
1 hour
9 miles =
r miles
hours)
(t hour
1
14 miles =
(r + 25) miles
(t hours)
hour
1
9 miles
r miles
=t
14 miles
(r+25) miles
=t
Sample Analogous Tasks
Multiplicative Structures
Multiplicative Structure and Analogous Tasks
References
ATTENDING TO THE M EANING OF Q UANTITIES
9 miles =
r miles
(t hours)
1 hour
14 miles =
(r + 25) miles
(t hours)
1 hour
9 miles =
r miles
hours)
(t hour
1
14 miles =
(r + 25) miles
(t hours)
hour
1
9 miles
r miles
=t
14 miles
(r+25) miles
=t
9
miles
r
miles
=t
14 miles
(r+25) miles
=t
Sample Analogous Tasks
Multiplicative Structures
Multiplicative Structure and Analogous Tasks
References
ATTENDING TO THE M EANING OF Q UANTITIES
9 miles =
r miles
1 hour (t
hours)
14 miles =
r miles
1 hour ((t
+ 25) hours)
14 miles = r1 miles
hours)
((t + 25) hour
hours)
9 miles = r1 miles
(t hour
9 miles
r miles
=t
14 miles
(r+25) miles
=t
9
miles
r
miles
=t
14 miles
(r+25) miles
=t
9
r
=
14
r+25
Sample Analogous Tasks
Multiplicative Structures
Multiplicative Structure and Analogous Tasks
I DENTIFYING A NALOGOUS TASKS
Is this problem analogous to either of the previous problems?
A new crew of painters takes two times as long to
paint a small apartment as an experienced crew.
Together, both crews can paint the apartment in 4
hours. How many hours does it take the new crew to
paint the apartment?
hours for the new crew to
It takes
paint the apartment. – EdReady Montana (2016)
References
Sample Analogous Tasks
Multiplicative Structures
Multiplicative Structure and Analogous Tasks
A UBIQUITOUS MATHEMATICAL STRUCTURE
W = rt
W
t=
r
W
r=
t
W represents some amount of work completed
r represents the rate at which that unit of work is completed
t represents the amount of time spent working
For W = d, we have the familiar d = rt relation.
References
Sample Analogous Tasks
Multiplicative Structures
Multiplicative Structure and Analogous Tasks
A N I NVITATION
For our next meeting:
1. Find two mathematical objects that have similar
mathematical structures.
2. For each mathematical object, construct a word problem
whose solution would involve the use of said
mathematical object.
3. Send me a digital copy of your tasks within two weeks.
4. I will typeset the problems and include them in future
sessions for discussion and feedback.
5. All solution authors will remain anonymous.
References
Sample Analogous Tasks
Multiplicative Structures
Multiplicative Structure and Analogous Tasks
MEA-MFT C ONFERENCE
I will be presenting at the MEA-MFT conference next Friday.
I Analogous Tasks throughout the K-12 Curriculum
I
I
HHS Room 37 11AM
The Pythagorean Theorem: Much more than squares
I
HHS Room 34 12PM
References
Sample Analogous Tasks
Multiplicative Structures
Multiplicative Structure and Analogous Tasks
References
J UST IN CASE . . .
Analyze this problem for multiplicative structure. What are the
objects and the groups? What type of multiplication problem is
this?
A computer has 128 gigabytes of memory. One
gigabyte is 1 × 109 bytes. A floppy disk, used for
storage by computers in the 1970s, holds about 80
kilobytes. There are 1000 bytes in a kilobyte. How
many kilobytes does a modern computer have? –
Illustrative Mathematics (2016)
Sample Analogous Tasks
Multiplicative Structures
Multiplicative Structure and Analogous Tasks
T HANKS FOR COMING !
Dr. Roger Fischer
Mathematics Instructor
Gallatin College – Montana State University
201 Hamilton Hall
Bozeman, MT 59717
[email protected]
References
Sample Analogous Tasks
Multiplicative Structures
Multiplicative Structure and Analogous Tasks
References
R EFERENCES I
Beckmann, S. (2012). Mathematics for elementary teachers with
activity manual. Boston, MA: Addison-Wesley.
EdReady Montana. (2016, June). Knowledge check: Solving
rational equations and applications. Retrieved June 10, from
https://mtdademo.edready.org/
studyPlan?groupId=17705&assessmentId=
24615&unitId=245848&from=assessments
Illustrative Mathematics. (2016, June). 8.EE: Choosing
approrpiate units. Retrieved June 15, 2016, from
https://www.illustrativemathematics.org/
content-standards/tasks/1981
Bittinger, M. L. (2015). Introductory algebra. Boston, MA:
Pearson.