Download d. division by thousands

ABE/ASE TRANSITION ACADEMY
MASTER TEACHER COLLEGE READINESS PROJECT
This step process activity is geared toward helping students
understand the concept of Division and the relationship to
Multiplication by building onto what each student already know
under Bloom’s Taxonomy Chart:
#1 Knowledge (Observe and recall information,
#2 Comprehension (understand information), and
#3 Application (use of information).
Introduction:
Steps used to solve division problems are made easier by staying in
the | zone |.
In using this best practice math strategy students will be able to
find out how much they recall about basic division skills, in
connection with use of their adding, subtracting, and multiplying
skills.
Goal
To improve students understanding of the step process for solving
division word problems and to develop effective academic habits
for transiting to post-secondary programs by staying in the |zone|.
Background:
This project involves repetition, recognition, and practical
application of those knowledge, comprehension and application
skills needed for dividing with whole numbers. In this activity
students are introduced to the division signs (÷) sign means to set
up a division solution as an (equations). The | dividend | is always
written first - before the division sign. Use of this format allows
students to use a variable to represent the quotient they’re solving
for.
(e.g., 84 ÷ 12 = X).
(
84÷ 12-12=12x)
(
84 =12x)
(Divide both sides of the equation by [12])
(x=7)
This  )¯¯ sign means to set up the division solution as a
(problem). The | dividend | the number to be divided goes inside
the division symbol. The | divisor | the number to divide-by goes
outside the symbol. The | quotient | is written above the
|dividend|. (e.g., this problem can be read as: “93 divided by 3” or
“3 divided into 93”. The answer is still “31”.)
This
(─) sign (the fraction bar) means to divide.
Reminder: (─ ) is also a minus sign use in Subtraction.
This  (/) sign (the slash) means to divide.
THE TASK
In order to help students identify division problems they must
understand the | terms|. Understanding the division terms will help
to actively engage the student with divisional math content and the
incorporation of the repetitive word problem solving step process.
Divisor – A | number | by which the dividend is divided.
Dividend – A number that is divided by | another number |.
Quotient – The | answer | to a division problem.
Remainder – A | left over number | that is smaller than the
divisor after the division problem is completed.
5. Placeholder – A | zero | used to hold a place within the
quotient when the divisor is too large to go into a part of the
dividend.
1.
2.
3.
4.
6. Place Value – The place that a digit occupies in a number is
the | value assigned | to that digit. Therefore each digit has a
place value.
7. Place – The | position | of a digit within a number.
8. Zone/Column – A | vertical line | of digits that all have the
same place value within a problem.
9. Digit – Any single | natural | number. (e.g., 0, 1, 2, 3, 4, 5, 6,
7, 8, 9).
10.
Number – A digit or a group of digits with a
| designated value |. (e.g., 2, 24, 468, 3791, 50,000)
11.
Borrowing – Breaking up tens to make ones, breaking
up hundreds to make tens, breaking up thousands to make
hundreds, breaking up ten-thousands to make thousands, etc.
(Also called regrouping–assigning a |different but equivalent|
place value to a given number).
UNDERSTANTING PLACE VALUE:
What do these four numbers have in common?
1.
2.
3.
4.
9217
7912
1729
2197
Each set of numbers has four-digits, however, the four
different numbers all have the same digits i.e., 1, 2, 7, 9.
The digits are the same, but, each number has a difference
place value. Each digit is in a different place in each number.
The numbering system or place value system tells the value
of each digit in each number. In other words, each digit in
each number has a place value.
Millions
↓
Hundred Thousands
0
↓
Ten Thousands
0
↓
Thousands
0
↓
Hundreds
9
↓
Tens
2
↓
Units or Ones
1
↓
7
THE PROCESS
The Multi-Step Process below is used to Solve Division Problems.
Students must repeat the following step process, listed below, until
the division problem is completed. Student must do each step
operation, in the order listed below, in order to find the quotient,
which is the number of times the divisor will go into the numbered
dividend.
Step # 1 Divide – a place value number by the divisor.
Write the quotient in the |zone| above the number place.
Step # 2 Multiply – the quotient by the divisor.
Write the product beneath the numbered place value.
Step # 3 Subtract (– minus sign) the product from the dividend,
and then,
Step # 3.5 Compare the Reminder,
Step # 4 Bring Down - the Next Number/Digit from the
dividend and write it next to the subtracted remainder. Repeat the
process of divide (÷), multiply (x), subtract (-), bring down,
divide (÷), multiply (x), subtract (-), bring down, until there are
no more dividend |place valve| numbers to bring down or the
remainder is zero (0) or less than the divisor.
WORD PROBLEM PROCESS ANALYSIS
I. IDENTIFY THE STEP PROCESS
1. Understand the question being asked.
a. This process will fine tune the students’ problemsolving technique to make them a more efficient
learner.
2. Determine necessary data needed to answer the
question being asked.
a. This process will help students to identify
discardable data through vocabulary recognition,
prior knowledge, and background diversity learning
skills.
3. Select an operation and develop a strategy for solving
the presented problem based upon the data determined
to be necessary to help solve the problem. (Discard).
a. This process will improve reading comprehension,
teach vocabulary usage and promote problemsolving techniques.
b. This process will also help to coordinate the
generalized skills learned in class to the students’
daily life skills.
4. Set-up the problem for resolution and/or solve for an
actual outcome.
Step # 5
Check Answer to Division Problems:
(a)Multiply the |divisor| and the |quotient|.
(b) Then add the |remainder| if there is one.
(c) The results should equal the |original dividend|.
POINTERS ON USING PROBLEM SOLVING STRATEGIES
1. Act it out.
a. Relate problems and vocabulary to specific skills and
background.
b. Apply problems to real life situations.
2. Make a table, chart, model or picture.
a. Teach concepts with concrete examples.
b. Use manipulatives (e.g., blocks, buttons, money, algebra
tiles, etc., to promote creativity outside of reliance on
worksheets) and make learning enjoyable.
3. Find a pattern.
a. Realizing differing methods, systems and concepts which
may or may not be universal.
b. Learning by trial and error.
4. Working actual problems
a. Develops thinking through use of traditional direct
instructions, paper, pen and pencil drills, and repetitive
practice instructions.
b. Kinesthetic learners (These learners learn best by doing.)
c. Use of a transitional teaching process (i.e., auditory,
visual, and kinesthetic) designed to relax students,
promote enjoyment of the learning experience, help raise
self-esteem, and blend self-awareness of students through
increased participation in the strategic teaching process.
A. DIVISION BY UNITS of ONES- 1-digit number in the
dividend ÷ by a 1-digit number in the divisor. (Equation
set-up, Horizontal Division).
How to divide a one-digit number in the dividend by a one-digit
number in the divisor.
Step # 1
Divide 8 ÷ 4 = 2. Divide the divisor (4) into the
dividend (8). Put the quotient (2) above the dividend (8).
Step # 2
Multiply 2 X 4 = 8. Multiply the quotient (2)
times (x) the divisor (4) = the product (8). Put the product (8)
directly under the dividend (8).
Step # 3
Subtract 8 – 8 = 0. Subtract the product (8) from
the dividend (8) = the remainder (0). Draw double lines under
the remainder (0) to show completion of the division problem or
equation.
B. DIVISION BY TENS
2-digit number in the dividend ÷ by a 1-digit number in the
divisor. (Equation set-up, Horizontal Division).
Step # 1
Divide 46 ÷ 2 = 23. Divide the divisor (2) into the
first place dividend (4). Put the quotient (2) above the first place
dividend (4).
Step # 2
Multiply 2 X 2 = 4. Multiply the quotient (2)
times (x) the divisor (2) = the product (4). Put the product (4)
directly under the first place dividend (4).
Step # 3
Subtract 4 – 4 = 0. Subtract the product (4) from
the dividend (4) = the remainder (0).
Step # 4
Compare the remainder (0) to the divisor (2). (The
remainder should always be less than the divisor.
Step # 5
Bring down the second place digit in the dividend
(↓) and repeat Steps # 1, # 2, # 3, # 4, and # 5 as needed. Draw
double lines under the remainder (0) to show completion of the
division problem or equation.
How to divide a two-digit number by a one-digit number
(e.g. 42 ÷ 7) (Problem set-up, Vertical Division).

Place the divisor (7) before the division bracket
and place the dividend (42) under it.


7)42



Examine the first digit of the dividend(4). It is
smaller than 7 so it can't be divided by 7 to
produce a whole number. Next take the first
two digits of the dividend (42) and determine
how many 7's it contains. In this case 42 holds
six sevens (6*7=42). Place the 6 above the
division bracket.

6
7)42



Multiply the 6 by 7 and place the result (42)
below the 42 of the dividend.



6
7)42
42
0
Draw a line under the 42 and subtract it from
42(42-42=0). Since the result is 0 the division is
finished and 6 is the answer/quotient.
C. DIVISION BY HUNDREDS
3-digit number in the dividend ÷ by a 1digit number in the
divisor. (Equation set-up, Horizontal Division)..
Step # 1
Divide 755 ÷ 5 = 151. Divide the divisor (5) into
the first place dividend (7). Put the quotient (1) above the first
place dividend (7).
Step # 2
Multiply 5 X 1 = 5. Multiply the quotient (1)
times (x) the divisor (5) = the product (5). Put the product (5)
directly under the first place dividend (7).
Step # 3
Subtract 7 – 5 = 2. Subtract the product (5) from
the dividend (7) = the remainder (2).
Step # 3.5
Compare the remainder (2) to the divisor (5). (The
remainder should always be less than the divisor.
Step # 4
Bring down the second place digit in the dividend
(↓) (5) and repeat Steps # 1, # 2, # 3, # 4, and #5 as needed.
Draw double lines under the remainder (0) to show completion
of the division problem.
How to divide a three-digit number by a one-digit
number (e.g. 416 ÷ 7). (Problem set-up, Vertical
Division).

Place the divisor before the division bracket
and place the dividend (416) under it.



7)416


Examine the first digit of the dividend(4). It is
smaller than 7 so it can't be divided by 7 to
produce a whole number. Next take the first
two digits of the dividend (41) and determine
how many 7's it contains. In this case 41 holds
five sevens (5*7=35) but not six (6*7=42). Place
the 5 above the division bracket.



5
7)416


Multiply the 5 by 7 and place the result (35)
below the 41 of the dividend.




5
7)416
35


Draw a line under the 35 and subtract it from
41 (41-35=6). Bring down the 6 from the 416
and place it to the right of the other 6.

5

7)416

35

66
Divide 66 by 7 and place that answer above the
division bracket to the right of the five.







59
7)416
35
66


Multiply the 9 of the quotient by the divisor (7)
to get 63 and place this below the 66. Subtract
63 from 66 to give an answer of 3. The number
3 is called the remainder and indicates that
there were three left over.







59 R 3
7)416
35
66
63
3

D. DIVISION BY THOUSANDS
4-digit number in the dividend ÷ by a 1-digit number in the
divisor. (Equation set-up, Horizontal Division).
Step # 1
Divide 9608 ÷ 8 = 1201. Divide the divisor (8)
into the first place dividend (9). Put the quotient (1) above the
first place dividend (9).
Step # 2
Multiply 8 X 1 = 8. Multiply the quotient (1)
times (x) the divisor (8) = the product (8). Put the product (8)
directly under the first place dividend (9).
Step # 3
Subtract 9 – 8 = 1. Subtract the product (8) from
the dividend (9) = the remainder (1).
Step # 3.5
Compare the remainder (1) to the divisor (8). (The
remainder should always be less than the divisor.
Step # 4
Bring down the second place digit in the dividend
(↓) (6) and repeat Steps # 1, # 2, # 3, # 4, and # 5 as needed.
Draw double lines under the remainder (0) to show completion
of the division problem or equation.
How to divide a four-digit number by a 0ne-digit
number (e.g. 4138 ÷ 7)(Problem set-up, Vertical
Division).

Place the divisor before the division bracket
and place the dividend (4138) under it.



7)4138


Examine the first digit of the dividend(4). It is
smaller than 7 so can't be divided by 7 to
produce a whole number. Next take the first
two digits of the dividend (41) and determine
how many 7's it contains. In this case 41 holds
five sevens (5*7=35) but not six (6*7=42). Place
the 5 above the division bracket.



5
7)4138


Multiply the 5 by 7 and place the result (35)
below the 41 of the dividend.



5
7)4138

35


Draw a line under the 35 and subtract it from
41 (41-35=6). Bring down the 3 from the 4138
and place it to the right of the 6.





5
7)4138
35
63


Divide 63 by 7 and place that answer above the
division bracket and to the right of the two.





59
7)4138
35
63


Multiply the 9 of the quotient by the divisor (7)
to get 63 and place this below the 63 under the
dividend. Subtract 63 from 63 to give an
answer of 0. Bring down the 8 from the
dividend 4138 and place it next to the 0







59
7)4138
35
63
63
08


Divide 08 by 7 and place that answer (1) above
the division bracket and to the right of the
four.








591
7)4138
35
63
63
08
Multiply the 1 of the quotient by the divisor (7)
to get 07 and place this below the 08 under the
dividend. Subtract 07 from 08 to give an
answer of 1.








591
7)4138
35
63
63
08
07
1


There are no more digits in the dividend to
bring down so the 1 is a remainder. The final
answer could be written in several ways.
591 remainder 1 or sometimes 591r1 or as a
mixed number 591 1/ 7
THE EVALUATION
A. Student performance will be evaluated by establishment of
basic weakness found in use of the other mathematical
operation processes (i.e., addition, subtraction,
multiplication).
B. The evaluation should allow for a quick and easy
determination of those students who can be grouped together
for instructional purposes.
C. The evaluation will provide immediate feedback to prevent
practicing incorrect methods by teaching students to stay
within the | zone |.
D. If students respond incorrectly to more than twenty (20%)
percent of the division problem process, begin remediation
before proceeding to the next higher skill level instructions.
THE CONCLUSION
The student should have strengthened his understanding of:
1. what a divisor is (the number that the dividend will be
divided by).
2. what a dividend is (the number that is being divided).
3. what a quotient is (the number of times the divisor will go
into the dividend).
4. what a remainder is (a number that is less than the divisor
and is too small to be divided by the divisor to form a whole
number).
Staying within the |zone| helps to eliminate directional and
placement errors:
1. Directional Errors  Where the computation is correct, but
the steps are performed in the wrong |zone| or order.
2. Placement Errors  Where the computation is accurate, but
the numbers are written in the wrong |zone|, therefore the
quotient or mathematical operation will be incorrect.
The student will be able to recognize and write division problems
in a Horizontal and vertical format.
1. using variables in setting up division equation.
2. Using vertical notation in changing a horizontal equation to a
vertical top to bottom division problem.
The student will be able to create visual drawings and incorporate
symbols to help solve word problems, thereby making
comprehension of the math question easier.
THE CREDIT RESOURCES
1. ABOUT.COM: Mathematics
(a) Long Division Practice Problems.
2. AA [email protected]
(a). Dividing by Repeated Subtraction  Practice
Exercises.
(b). Multiplication and Division Relationship 
Practice Exercises.
(c). Division and Multiplication Relationship 
Practice Exercises.
3. ENRIGHT DIAGNOSTIC INVENTORY OF BASIC
ARITHMETIC SKILLS by Brian E. Enright
4. CONTEMPORARY’S NUMBER POWER
by Jerry Howett
5. CONTEMPORARY’S COMPLETE G.E.D.
by McGraw-Hill Publisher.
6. MATH.COM  Basic Math
(a) Multiplying and Dividing Integers.