MULTIPLICATION LEARNING PROGRAM
Honors Project ID 499
Donna J. Broome
Dr. Ramon L. Avila
August 1980
ID 499
furma J. BrO<Jl're
August 1980
Let us look into a lIDdern mathanatics classroom. The teacher
''Why is 2 + 3 = 3 + 2?"
Unhesitatingly the students reply, ''Because both equal 5."
'fu, " reproves the teacher, "the correct answer is because the
rrutative law of addition holds." Her next question is, ''Why is 9 +
ll?"
Again the students respond at once: "9 and 1 are 10 and 1 rrore
"Wrong," the teacher exclaims. '''The correct answer is that by
definition of 2,
asks,
can2 =
is 11."
the
9 + 2 = 9 + ( 1 + 1 ).
But because of the associative law of addition,
9 + ( 1 + 1) = (9 + 1) + 1.
Now 9 + 1 is 10 by the definition of 10 and 10 + ,_ is 11 by the definition
of 11."
Evidently the class is not doing too well and so the teacher tries
a simpler question, "Is 7 a number?" The students, taken aback by the
simplicity of the question, hardly deem it necessary to answer, but the
sheer habit of obedience causes them to reply affinnatively. The teacher
is aghast. "If I asked you who you who you are, what v;ould you say?"
The students are now wary of replying, but oI'_~ m:>re courageous
youngster does so: "I am Robert Smith."
The teacher looks incredulously and says chidingly, ''You 1reaIl that
you are the name Robert Smith? Of course not. You are a person and your
~ is Robert Smith.
Now let us get back to my original question: Is
7 a nunber? Of course not! It is the name of a number. 5 + 2, 6 + 1,
and 8 - 1 are names for the same numbe~The symbol 7 is a numeral for
the mnber."
The teacher sees that the students do not appreciate the distinction
and so she tries another tack. "Is the number 3 half of the number 8?"
she asks. Then she answers her own question: "Of course not! But the
nureral 3 is half of the numeral 8, the right half."
The students are now bursting to ask, ''What then is a nunber?" However, they are so discouraged by the wrong answers they have given that
they no longer have the heart to voice the question. This is extremely
fortunate for the teacher, for to explain what a rn.tnber really is v;ould
be beyond her capacity and certainly beyond the capacity of the students
to tmderstand it. And so thereafter the students are careful to say that
7 is a nurn2ral, not a mmlber. Just what a rn..nnber is they never find out.
-
--
-2The teacher is not fazed by the pupils' poor answers. She asks, ''How
can we express properly the wlDle rrumbers between 6 and 9?"
''Why,'' one pupil answers, "just 7 and 8."
''fu,'' the teacher replies. "It is the set of numbers which is the
intersection of the set of wlDle numbers larger than 6 and the set of
whole numbers less than 9."
Thus are students taught the use of sets and, presumably, precision.
A teacher thoroughly convinced of the vaunted value of precise
language, and wishing to ask her students, whether a number of lolliIX'Ps
equals a number of girls, phrases the question thus: ''Find out if the
set of 1011iIX'Ps is in one-to-one-corresIXJndence with the set of girls."
Needless to say, she gets no answer frem the students.
Bent but not broken, the teacher asks one tmre question, ''How rruch
is 2 divided bv 4?"
A bright st.-OOents says unhesitatingly, ''Minus 2."
''How did you get that result?" asks the teacher.
''Well, " says the student, "you have taught us that division is repeated
subtraction. I subtracted 4 from 2 and got minus 2."
It vxruld sean that the IX'or children 'M)uld deserve some relaxation
after school, but parents anxious to know what progress their children
are IIEking also query than. One parent asked his eight-year-old child,
"How much is 5 + 3?" The answer he received was that 5 + 3 = 3 + 5 by
the cOIImltative law. Flabbergasted, he rephrased the question: 'r.But how
many apples are 5 apples and 3 apples?"
The child didn't quite understand that "and" means "plus" and so he
asked, "l):) you mean 5 apples plus 3 apples?"
The parent hastened to say yes and waited exr:;ctantly.
"Oh," said the child, "it doesn't matter whether you are talking
about apples, pears or books; 5 + 3 = 3 + 5 in every case."
Another father, concerned about how his yotmg son was getting along
in aritl"lrretic, asked him how he was faring.
''fut so well," the boy replied. 'The teacher keeps talking about
associative, camutative and distributive laws. I just add and get the
right answer, but she doesn't like that. ,,1
And so llirris Kline characterizes the problans of the student trying to learn
mathematics.
In his book, Why Johnny Can't Add:
The Failure of the New Math, he
contends that hundreds of thousands of problems exist in millions of kids.
The
problans, while varied directly with the rrumber of students, are nonetheless that,
problans, and to each and every student, each is as :i.mp:::>rtant and as inSUrtIDuntable
as any other obstacle in the road to learning.
In any public school classrCXJffi, these students bring not only their successes
with mathematics and numbers, but their failures, as well.
If their background
should be unlucky enough to include a large dose of this type of teaching, one
-
-3can rrore easily understand why students have problems with math.
Yet we, as
math teachers in junior and senior high schools are not only expected to deal
with students wh:> have these problems, but we are expected to help them overcane
the problems and to learn mathem:ttics, algebra, geometry or other fonns of higher
math.
Perhaps it is time that
s~
attention focused itself on the slow student--
the one for whom each homework exercise is a seperate little Hell through which
he rrust go without the aid of a guide.
In short, the student who has problffllS
deserves some help.
Obviously, in a limited situation such as a Senior Honors' Proj ect at Ball
State University in M..mcie, Ind., to attempt to deal with the problems of all slow
students in mathem:ttics
-
be-
~uld"both
futile and foolish.
focused on those students who lack fundamental facts
Instead, this project has
~nd
skills as they relate
to single digit rrultiplication. This project is a unit of instruction.
be used for any of the following instances:
individual
~rk
It may
for a student in a
''nonnal" junior or senior high classroom who gives evidence that he or she lacks
basic multiplication facts, group
~rk
with students who overall demonstrate a
lack of such skills, raraiiation for students in upper elementary grades who have,
for one reason or another, failed to master basic multiplication skills and for
those students in the lower elementary grades who demmstrate readiness to learn
the rrultiplication facts.
The instructional unit involves several facets:
cognitive through use of flash
cards and feedback, through use of visually-read questions and taped answers, through
~s"\e0.J
the use of pre and post tests designed specifically to test mastry of the infoI1M.tion
presented in the unit, and psychorrotor, through the use of physical obj ects to
- --
count representing the act of multiplication.
Through these
tv.Q
domains, the stu-
dent will be exposed several times and through several different :rredia to the information being conveyed.
-
-4Students learn best through rrultiple instruction techniques, as Suydam and
Dessart support in the National Cmmcil of Teachers of Mathematics m:mograph,
"Classroan Ideas fran Research on Computational Skill~," p: p. 10-11. 2
They con-
tend that by teaching students to IIultiply by using a lattice method, students
were able to answer problens faster and with greater accuracy in a tim=d, IIl.lltidigit, multiplication drill than students taught by mJre conventional methods.
Another example Suydam and Dessart cite is the difference in perfonnance by students taught by intuitive (word problem examples) rather than formal, abstract
facts, also known as "conventional" methods.
Based on these two studies, the
evidence for varying the rrethod of both teaching and learning is so strong that
it carmet be overlooked, particularly when dealing with students whose backgrotmds
demonstrate a lack of mathematics success.
In establishing a multiple method theory for teaching multiplication, Charles
H. D'Augustine of Ohio University lists a pair of "readiness experiencesllthat
students of nultiplication must have had before they can honestly expect to master
the multiplication skills.
These skills are COtmting and addition, and both nust,
logically, precede a student's perfonnance of the mJre complex skill, multiplication? In this instructional tmit, the preliminary matter of COtmting is dealt
with by the student's using sticks, either for COtmting or for rrultiplication work.
These sticks are grouped by nu.ltiplication fact and are grouped into cups by their
total number.
The student clarifies his tmderstandin3 both through the COtmting
as well as through cmmting the groups and detennining how many sticks are in each
pack.
D'Augustine shows a mJdel of the same concept, alghough his differs with
ccnplexity and with grouping as this one.From this obviously basic cornerstone
,
carnes the theoretical or abstract knowledge one needs to begin multiplication.
-
-5We also know that changing presentation media assists the teacher of the
slow student in finding a greater degree of success in helping his or her students.
In the instructional tmit developed, the different media include the
pre and post tests, the slides, the tape recorder, the flash cards, the sticks
for physical manipulation and the written material for students to read.
Even
for the student who has a deficeincy in reading, the tmit IDuld be able to at
least provide for his or her needs, as well as for those who read on either a
slightly higher or lower than average grade level. 4 This type of presentation
also provides for the student a change of pace in the presentation of material.
As earlier indicated, the unit can either be used in a small group or in a
individual setting.
This provides for tmre personalized instruction which is,
according to Sam Duker in Individualized Instruction in Mathematics, the rrost
effective way to instruct the slower student as well as the average student.
I
Also, the utility of having a unit that can be used equally well with individuals
J
alrrost any situation.
,
and with small groups carmot be tmderstated by the public school teacher in
Specifically, by the design of the instructional unit,
I
f
the student has as an option, continuing on a specific section or tmving through
I
the entire packet to the next leaming activity.
I
I
In brief, then, the learning packet or instructional tmit consists of the
I
pretest, followed by the physical review, followed by individual flashcard or
screen drill, followed by a IDrksheet and concluded with a post test for each of
two divisions of the multiplication facts.
The subtmit of multiplication facts
is :imnediately made reI event with a short section of verbal application or "story"
problems which bnmediately precedes the final post test for the unit.
-
The instructional unit is specifically designed so that the student will have
no difficulty y;orking through the problans, provided the facts have first been
mastered, a task provided for in the design of the unit.
It also allows students
-6to segregate their individual learning problans fran the remainder of the basic
nultiplication facts.
Finally, the
~diate
feedback for the whole unit is the
verbal problem unit which applies the facts previously learned.
By utilizing sllch an individual approach, one
~uld
rope that the teacher
who, at the beginning of this paper, was about to be tarred and feathered by
her cotIIll.lIlity will no longerhave to fear for her life; she has a no-nonesense
guide about how to teach multiplication to the slower junior and senior high
school student.
,-
ENDOOTES
~y
Johnny Can't Add: The Failure of the Nevv Math,
York, pp. 1-3, St. Martin's Press.
furris Kline, 1973, New
2Classroom Ideas fran Research on ~utational Skills, M. N. Saydam and D. J.
Dessart, 1976, Reston, VA, p. p. 0:1, National Council of Teachers of Mathematics.
3Multiple ~trods of Teaching Mathanatics in the Eleneltary School, C. H. D'Augustine,
1973, p. p. 133-150, Harper and Roe, Nevv York.
4Individualized Instruction in Mathema.tics, S. Ducker, 1972, Scarecrow Press,
MetuChen, NJ, p. p. 17-74.
BIBLICCRAPHY
D'Augustine, Charles H. Mlltiple Methods of Tea~ Mathematics in the
Elementary School, Harper & Roe,New York, 19 .
Thlker, Sam, Individualized Instruction in Mathematics, Scarecrow Publishing Co.,
Metuchen, NJ, 1972.
Hirschi, L. Edwin, Buil~ Mathematical Concegts in Grades Kindergarten Through
Eight, Internationa TextbOOk Company, 197 .
M:>rris Kline, Why Johrmy Canlt Add:
Press, 1973.
The Failure of the New Math, St. MartinIs
Ginsburg, Herbert, Qri.ldren's Arithmetic:
Company, New York, 1977.
the learning process, D. Van N::>sttand
Suydam, Marilyn and IXmald Dessart, Classroom Ideas from Research on
tational
Skills, Natiot13.l Cotn1til of Teachers 0 Mathematics, Reston, VA, 19 6.
,
.....
suw
Donna J. Hoile
INDIVIDUAL LEARNING PACKET
ON
MU1T IP1ICATION
This individual learning packet is intended as one means of helping first
year
h1gE_.~c):lOol stu4~ts
who do not understand multiplication.
It is pa:rt of
what will be a four-part remedial unit for individual student,use.
r chose the individual learning packet because I've found when students
reach high school, they are frequently hesitant to accept help with any of
the four basic processes.
This packet will help those students by giving them
a non-threatening learning situation in which to learn at least survival-level
skills in multiplication.
The overall goal of the packet is to enable a student who is unable to
multiply ..t9 ..p,t leal?t master mul tiplica~lon tables from 1 through 9 and to use
.
.-~
that knowledge in story or applied problems.
-~."..
The secondary objective is to
help the student who is weak in multiplication skills to develop those skills,
again, through drill and application.
The packet is intended to build both
knowledge and skills.
The selection of media for the packet was based on the learning-by-doing
concept.
With the groups of sticks, students can actually see, feel and count
to understand the "how" and "why" questions.
Flash cards provide students with
individual tests and immediate feed back, considering answers are on the reverse
sides of the cards.
Through the pre- and post-tests and practice problems, the student will be
able to determine where (s)he needs to concentrate more work.
Since this is
designed to be used one student at a time, several students each day might use
the packet; therefore, all materials are easily cleaned.
To use the packet, the teacher will announce its availability.
Following
class work with multiplication, the teacher will be able to refer individual
students to the packet for remedial work.
Effectiveness of the packet will be shown by stUdents' better performance
of classwork dealing with multiplication. The packe~ contains several opportunities for the student to evaluate their performance as they proceed through the
material; however, the mastery of the packet material only, with no carry-over,
is not a desirable outcome.
In regard to the evaluation of the project, my objective was to develop a
format for a four-part packet by way of the multiplication skills.
----------------------,-------------------------
I think I've
·.
Page 2
met that objective.
As for the packet's operational objectives, the only
accurate way to determine their effectiveness is to use it in a classroom
setting; this I have not yet accomplished.
Creation of the project was difficult because I had no model to follow.
From my research, I've found the project to be theoretically sound, though.
Based on this quarter of the four arithmetic functions, I think I will be able
to develop lndi vidual leaming packets wi t:1 j;;,uch less difficulty.
PURPOSE:
This individual learning packet has been designed to help you develop
your skill in multiplication. You need to keep in mind that you will be working on your own. checking your own progress and that you will be able to
look at the answers at any time; however, if you look at the answers before
doing the work or if you skip steps. that may seem impractical, you will not
be learning all you can.
If you have any questions, please ask your teacher.
GENERAL PROClt:nURE:
1.
The procedure to use to answer any question in this packet is to write
the correct answer in the blank (unless told otherwise) with the special
pen included in the packet.
2.
To check your answers after you have finished, take out the appropriate
answer sheet and place it on top of your problem sheet. The correct
answers will be either to the right of your answers or below them. If
you have missed any answers, mark the correct answer on your problem
sheet. After you finish correcting your answers, review the problems
you answered incorrectly.
3.
After reviewing any missed problems, take a damp cloth and wipe off
your problem sheet.
4.
When you finish with a page, please replace it into the correct envelope
in the correct order so that the next person will have a ready-to-use
packet.
5.
Now, to get a general idea of what to do, please take out the "Flow
Chart" from this envelope.
6.
Begin working through the packet by removing the "Instruction Sheetl l
from this envelope.
INSTRUCTION SHEET I
1.
Remove Pretest I and answer the questions.
2.
Check your answers with Answer Sheet-Pretest I (See # 2 on this
envelope) .
3.
Did you miss more than two problems on the pretest? If not, please
remove Pretest II and answer those questions. Go to #5.
4.
If you missed more than two problems on Pretest I, take envelope II
and follow the instructions on the outside.
5.
Check your answers with Answer Sheet-Pretest II (See # 2 on this
envelope) .
6.
Did you miss more than five answers on Pretest II? If not, please
proceed to envelope IV.
7.
If you missed more than five answers on Pretest II, please proceed
to envelope III.
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _oooo-_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
~
\\0
*-__________n_ _
-_·_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _~
=
4
X 0
:3
x2 =
2 X 0
=
:3
x1 =
=
O.X 0 ..
2 X
:3
x :3
•
1 X
:3 ..
2 Xl"
oX
1 -
4X4=
:3
X 4 •
2 X 4 -
4 X 2 =
2 X 2 ..
4 Xl"
oX
o x :3
..
1 X 2 •
2 ..
1 X 4 •
1 X 1 =
:3
x0
4 =
1 X 0
:3
=
x :3
=
..
4
oX
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-----------------,----------------------------
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PRETEST
9 x 8
=- -
--
3
x
9
=- -
Ox8= _ _
5
x 1
=- -
=- -
8 x 0
=- -
7
=--
6 x 1
=- -
8 x 1
=--
1 x 9
=- -
8 ":.
--
9xO":. _ _
7xl'::. _ _
4x9= _ _
7x3":.
--
5x4-:.
--
9x7'::. _ _
5
8x2= _ _
5x3-:.
--
7x5= _ _
5 x 8 = --
3x6-:.
--
6 x 2
=--
5 x 2
=--
7
.l.L X
x
5=
7 x 8
=- -
3x7-:' _ _
6 x 6
o
9
=- -
8x3= _ _
9x5= _ _
8 x 8
=- -
2 x 9 ':
--
2 x 6
=--
2x8= _ _
9
3
= --
1 x 7
=- -
o x 5 ::;
--
6 x 4
=--
9x2= _ _
9 x 9
=- -
--
5 x 6 ':
--
1 x 8
=- -
4x5= _ _
4 x 7 =
--
9
=--
6 x 9 ':
--
3x8= _ _
9x6= _ _
7 x 9=
--
5x0
=- -
8 x 4
=- -
1x5= _ _
8x 7
7 x 4
=--
6 x 0 =
--
9x4= _ _
ox
5
9
=- -
5 x 7 = --
8x6= _ _
6 x 3
=- -
6 x 7
=- -
3x5= _ _
7 x 0
= --
2 x 5
= --
o
x
7 x 6 =
d x
x
x 6
=--
x
x
=- -
9 x 1
=- -
--
8 x 5
=--
6 x 5
=- -
7 x 2
= --
1 x 6
=- -
6
8
=- -
2 x 7
=- -
7 =
4x6= _ _
x
I
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INSTRUCTIONS FOR ENVELOPE II
1.
Procedure to use sticks to solve multiplication problems
1.
Remove containers I, 2, 3. and 4 from the box.
2.
Since 0 x any number equals zero, there are no sticks. Therefore,
zero times any number, or any number times zero equals zero.
3.
Work through the following example:
Problem: 4 x 3 =
Take container 3 a-n~d'--r-emove four bundles of sticks
Count the number of sticks you have
III III III III = 12.
This means 4 x 3 = 12.
II.
4.
Complete all problems on the Stick Problem sheet in this way.
5.
Check your answers.
6.
Review mistakes.
Procedure to use flash cards
1.
III.
IV.
Take flash cards from box A. Place them so the answers are facing
away from you. Go through each card. Try to figure out the answer,
then ehcek yourself by turning the card over. Continue until you have
gone through all the cards at least twice.
Practice Problems
1.
Answer the problems on Practice Problem sheet
2.
Check your answers
3.
Review mistakes
Remove the post test and answer the problems.
1.
Check your answers. If you missed more than two, return to Step I
at the top of this sheet.
2. If you didn't miss more than two, proceed to envelope 3.
STICK PROBLEMS
Jxl= _ _
2xJ"' _ _
1 x J =
--
Oxl- _ _
2x4= _ _
4 x 1 =
--
lx2"' _ _
lx4"' _ _
2xO= _ _
ox
--
J x 2
=- -
4xJ= _ _
lxl= __
2 x 1
= --
JxO= _ _
Jx4- ___
4 x 4
=--
2x2= _ _
4 x 2 :::
--
o
= --
o x 4 :::
ox
--
lxO= _ _
4 x 0
=- -
--
0 ...
2 '"
JxJ= _ _
__
~~._.
~_T_~~·~'
_______________,__________________
x J
I
,~\'-<,,'-I..JQ-' '::3nee\
-1~
5\ \C~ -~lJ\t'\(,\'- cs
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f
3
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k
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C
c
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,
,
10 '
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o h
r
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PRACTICE PROBLEMS
lx4= _ _
1 x 2
=- -
4 x 1 =
2x4= _ _
ox
= --
1 x 3
=- -
2:x:3= _ _
3xl= _ _
ox
=- -
ox 3
=
--
2x2= _____
3x4= _ _
2 x 1
= --
4.x3= _ _
ox
0
=--
2 x 0
=- -
3x3= _ _
1 x 0
=- -
1
2
--
Ox4= _ _
4 x 2
=- -
4x4= _ _
3xO= _ _
1 x 1
=- -
3 x 2
=- -
4 x 0
= --
4 x 1
=- -
2 x 3
=--
o
=- -
ox 1 =
o
=- -
x 3
1 x 0 =
--
4 x 4
=
x 0
3 x 2= _ _
= --
1 x 3 =
ox
2
=- -
3x4= _ _
4 x 3 =
3 x 3
=- -
4 x 2 =
--
1 x 1 =
1 x 4
= --
2x4= _ _
3 x 1 =
2 x 2
=- -
2 x 1
=--
2 x 0 =
Ox4= _ _
3
=--
4 x 0
1 x 2
x 0
=
I~
o
.()
,,8
J{P
o
()
l
3
••.• _. (J
t
L~;····~~.',' 0
POST TEST
4xO-::' _ _
3x2'::. _ _
lxl::' _ _
3
--
4x4': _ _
4x2::
Ox4'::. _ _
lxO= _ _
3x):: _ _
x 0 -=-
--
2 x 0
1:
--
ox
0
=--
4xJ= _ _
2 x 1
=- -
ox
1
=--
3x4= _ _
2 x 2
= --
o
x 3
=- -
o
3xl= _ _
2 x 4
=- -
1 x 4
=- -
2x3= _ _
4 x 1
=--
x 2
1 x
J
=- =
--
lx2= _ _
- I
a
:'1
1'1,.
12
o
- 3
...._---."..-," ' - - - - - - , - - - - -
"--~----
_ .._--
INSTRUCTIONS FOR ENVELOPE III
1. Procedure to use sticks to solve multiplication problems
1.
Remove containers 1- 9 from the box.
2.
Since 0 x any number equals zero, there are no sticks. Therefore,
zero times any number, or any number times zero equals zero.
3.
Work through the following example
Problem: 5 x 4 =
Take container 4 a-n~d~r-emove five bundles of sticks
Count the number of sticks you have
IIII lIIT lIIl IIII lIII = 20
This means 5 x 4 = 20.
II.
4.
Complete all problems on the Stick Problem sheet in this way
5.
Check your answers.
6.
Review mistakes
Procedure to use flash cards
1.
III.
Take flash cards from box B. Place them so the answers are facing
away from you. Go through each card. Try to figure out the answer,
then check yourself by turning the card over. Continue until you have
gone through all the cards at least twice.
Practice Problems
1.
Answer the problems on Practice Problem sheet
2.
Check your answers
3. Review mistakes
IV.
Remove the post test and answer the problems.
1.
Check your answers. If you missed more than five, return to Step I
at the top of this sheet.
2. If you didn It miss more than five, proceed to envelope 4.
----------------------------------.----
STICK PROBID1S
7x8= _ _
5x1= _ _
o
x 8
=--
1x9:: _ _
4 x 5
=--
9xl= _ _
lx6= _ _
7
x 1
= --
9
5
=- -
1 x 5
=- -
4x8= _ _
5 x 3
= --
9xO= _ _
ox 5
=- -
5 x 0
=- -
5x7= _ _
2 x 5
=- -
5x9= _ _
6 x 9
=--
1 x 7
=- -
3x7= _ _
8 x 2
=--
Ox6= _ _
3x8= _ _
7xl= _ _
7x 2
=--
8 x 7
=- -
9 x 9
=--
8 x 1
=- -
5x5= _ _
7 x 0 =
--
6 x 7
=- -
7 x 4
=--
5 x 6
=- -
8x8= _ _
3x6= _ _
7x3= _ _
8 x 6
=- -
6 x 4 =
x
6x6= _ _
--
=_
4 x 6
=--
8 x 5
=--
9 x 6
=--
9 x 3
=- -
x 7
=- -
8xO= _ _
8 x 9
=--
7 x 0
=- -
6 x 0
=- -
8 x 9
=- -
2 x 9
=- -
7 x 8
=- -
5 x 4
=- -
3
5
=- -
lx8= _ _
6x2= _ _
2 x 7
=- -
6x 5
=- -
7 x 9
=- -
9x2= _ _
6xl= _ _
4x9= _ _
3
9
=- -
6 x 3
=- -
8x4= _ _
7 x 6
=- -
8x3= _ _
5 x 8
=- -
o
x 9
=- -
7
x
5
9 x 4
=- -
4 x 7 =
--
x
x
2x6= _ _
9 x 7
=- -
6x8= _ _
o
x 7 =
=--
5 x 2
=--
3x 7
=--
9
x
2
x
8
8
--
=- -
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PRACTICE PROBLEl1S
7x8= _ _
9x8= _ _
8 x
9
=- -
3x9:: _ _
7
x 0 ::
--
=- -
3x7= _ _
8 x 0
=--
4 x 9
=- -
.5
x ~ ::
--
Ox8= _ _
5x2= _ _
7
7
= --
6xl= _ _
1 x 9
=- -
2x8= _ _
9 x 3
=--
9 x 2
4 x 5
=- -
4 x 7
= --
9 x 6
=--
7 x 9 ::
--
8 x 7 =
--
9 x 1
=- -
ox7
=- -
8 x 5
=- -
6 x 5 ::
--
7
x 2 ::
--
2 x 7
=- -
7
x 1
=- -
6
=--
6 x 6
=- -
lx8= _ _
3 x 8
=--
5
x 1
1x6= _ _
6x8= _ _
x
4x6= _ _
=--
=- -
8 x 1
=- -
9
=- -
x
9
7 x 1
=- -
9 x 7
9
=- -
2x6= _ _
6x4= _ _
9 x 4 =
--
8
=--
3 x 5
=- -
o
x 6
=- -
ox
--
7x3= _ _
5 x 4
=- -
8 x 2
=- -
5x 8
=--
3x6= _ _
7 x 8 ::
--
3x 7
= --
--
8 x 3
=--
8x8= _ _
2 x 9 ::
--
1 x 7
= --
o x 5 :: - -
7 x 6
= --
5x 6
=- -
8
x
--
6 x 9
=- -
--
7 x 4
=- -
6
.x. 0
=- -
5x 9
=- -
=--
6 x 7
=--
7
x 0 ::
5
=- -
x
5
lC
5 =
--
4x8= _ _
5 x 3
=--
9x 0 =
= --
9 ::
5 x 0
=- -
8 x 4 ::
5 x 7
=- -
6 x 3
7
x
x
5
6
x 2
9 ::
_ _ _ _ _ _ _ _ _ _ _ _
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--
7x3= _ _
8x2= _ _
5x8= _ _
7 x 8
=- -
ox9
=- -
8x8= _ _
1 x 7
=- -
7x6= _ _
8 x 9
=- -
5
=- -
7x4= _ _
5 x 9
=- -
6 x 3 =
7 x 0
=- -
5x4= _ _
5x 3
=--
3x
-6 =- -
3 x 7 =
4 x 8 =
x 0
9 x 0 =
--
--
3
=- -
2 x 9
=- -
ox 5
=- -
5 x 6
=- -
6 x 9
=- -
8 x 4
=- -
6
0
=- -
5 x 7
=- -
6 x 7 = --
2 x 5
=- -
7 x 1
=- -
9 x 7
=- -
7 x 5
=--
6 x 2
8
x
J(
=--
6x6= _ _
3x8= _ _
2x6= _ _
6x4= _ _
lx8= _ _
9 x 4
=--
8x6= _ _
3
x
5
=- -
o
6
=- -
--
5 x 5
=- -
ox 7
=- -
5
x 2
=--
7 x 7
=- -
8xl= _ _
2 x 8
=--
4 x 5
=- -
9 x 6
=- -
8 x 7
=- -
ox8
=- -
=- 6 x 5 =- -
1
=- -
4 x 6
=- -
9x 8
=- -
3 x 9
=- -
5 x 1 =
--
8 x 0
=- -
6 x 1
=- -
9x5= _ _
1 x
5
=--
-x9=
9 x 2
x
6
x
9 x 3= _ _
9 x 9
=- -
4 x 7
=- -
7 x 9
=- -
=- -
7 x 2
=--
6
8
=--
2 x 7
=--
.-------.-----------------.----~ ..
- .. - . - - . -
1 x 9 =
9 x 1
--
=- -
8 x 5
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-
INSTRUCTIONS FOR ENVELOPE IV (STORY PROBLEMS)
1.
Remove the proce dures sheet and read it twice.
2.
Remove the example problem and read through it twice.
3.
Using the above procedures, work through the Practice Story Problems.
4.
Check your answers.
5.
Review your mistakes.
6.
If you feel confident working with story problems, remove the
Individual Learning Packet Post Test.
7.
ON A SEP ARATE SHEET OF PAPER, write the answers to all 64
questions.
8.
Submit your answers to the Individual Learning Packet Post Test to
your teacher.
--
.1
PROCEDURE TO FOLLOW TO SOLVE A STORY PROBLEM
~"Mi"'."
1.
read through the problem
2.
find the question asked
).
find the information given pertaining to the question
4.
set up an equation
5.
write a multiplication equation
6.
write the answer with the correct title
AN EXAMPLE OF SOLVING A STORY PROBLEM
1.
rea.d through the problem
Frank helped his uncle plant some small pine trees.
Frank planted 3 rows of trees. He put 8 trees in each
row. How many trees did he plant?
2.
find the question asked
How many trees did he plant?
3.
find the information given pertaining to the question
planted 3 rows of trees & put 8 in each row
4.
set up an equation
3 rows x 8 trees in each row
5.
write a multiplication equation
3 x 8
6.
=
?
write the answer with the correct title
24 trees
--------__-_,______
-------"-~.
--_ ..--
PRACTICE STORY
1.
Jill went t.o the store with her mat. Jill bought 9 pieces of bubble gum.
She paid 2 cents for each piece of bubble gum. How many cents did she
spend'?
Answer
Multiplication Equation
2.
PROB~lS
Tony went to the neighborhood carnival with his friend Tim. Tony rode on
3 different rides. He rode each ride 4 times. How many times did he ride
all together?
Multiplication Equation
Answer
3. Brad plays softball every Saturday.
He played in 7 games.
Brad hit 4 runs every time he played.
How many runs did he hit?
Multiplication Equation
4.
Answer
Cindy likes to play house with her friends. Cindy has 6 dolls.
has 5 dresses. How many doll dresses does Cindy have?
Multiplication Equation
Every doll
Answer
/
5. Elizabeth saves StaJllpS.
pages.
She puts 8 stamps an each page.
How many stamps has she saved'?
Multiplication Equation
6.
J full
Answer
John collects marbles. He puts 5 marbles in each ~.
How many marbles does he have?
Multiplication Equation
She has
Answer
He has 9 bags.
PRACTICE STORY PROBLEMS (CCNTINUED)
7.
Marmie went to the post office for her mom. Marmie bought
stamp cost 8 cents. How much did Marmie spend?
Multiplication Equation
8.
Each
Answer
Answer
Kathy has a lot of jewelry. She keeps 4 rings in each jewelry box.
has 2 jewelry boxes. How many rings does Kathy have?
Multiplication Equation
10.
stamps.
Mark helped. his grandfather count how many cows he had. His grandfather
had 9 trucks. He put 9 cows in each truck. How many cows did Mark's
grandfather have?
Multiplicat.ion Equation
9.
7
She
Answer
Robert helped his father-in-law trim Christmas trees. He trimmed 3 rows
of trees. Each row had 9 trees in it. How many trees did Robert trim?
Multiplication Equation
Answer
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INDIVIDUAL LEARN ING PACKET POST TEST
1.
9 X 4" _ _
22.
9 X 9 .. _ _
43.
4 X 9" _ _
2.
6 X 0 ...
--
23.
9 X 2 ... _ _
4ih
OX3- _ _
3.
7 x 4'" _ _
24.
6 X 4 ... _ _
45.
6 Xl"
4.
1 X 2'" _ _
25.
0 X
46.
1 X 0 ..
5.
9x
6.
3
7.
8 X 7 ..
8.
1 X
9.
5 ..
--
III
__
26.
1 X 7" _ _
47.
7,x 7
&:
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__
27.
9 X 3 .. _ _
48.
6 X 6
m
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718- _ _
29.
2 X 8 ,. _ _
50.
ox
8 X 4'" _
30.
1 Xl"
--
51.
5
10.
5 X 0 .. _ _
31.
2 X 6 .. _ _
52.
3 X 6 - __
11.
1 X 4'" _ _
)2.
0 X 8 ..
--
53.
1 X
3 .. _ _
12.
7 X 9" _ _
33.
2 X 4'" _ _
54.
0 X
7"" _ _
13.
9 X 6 .. _ _
)4.
8 X 5'" _ _
55.
2 X
3" _ _
143X8- _ _
35.
8 X 8'" _ _
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15.
8 X 9" _ _
36.
9 X 5" _ _
57.
5 X 3 ... _ _
16.
4 X 5'" _ _
37.
0 X
9 .. _ _
58.
0 X 2 ... _ _
17.
1 X 8'" _ _
38.
2 X2
59.
2 X
7
III
18.
5 X 6 ... _ _
--
39.
3
X
9" _ _
60.
4 X 6
OR
19.
4 X 4 ... _ _
40.
5
X
5" _ _
61.
6 X 8 ...
20.
7
.. _ _
41.
9 X 7:11 _ _
21.
3 X 4'" _ _
42.
7 X 3" _ _
62.
Charlie decided to put 3 rows of tomatoe plants in his garden. He put 6 plants
in each rOlf. How many tomatoe plants did Charlie plant?
Multiplication Equation
Answer _ _ _ _ _ _ _ _ _ _ __
63.
Beth wanted soml~ extra money, so she took 9 empty pop bottles to the store. She
got 5 cents for each bottle. How much money did she receive at the store?
Multiplication .fi::quat1on
Answer _ _ _ _ _ _ _ _ _ _ __
64.
Toni keeps his I~iniature car collection in a special box. There are 5 rows of
cars in the box with 8 cars in each row. How many cars does Toni have?
Multiplication Equation
Answer _ _ _ _ _ _ _ _ _ _ __
..
1
5 - __
x6
-
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4"
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--
0 ..
X 2 ..
X 5 ... _ _
---