Establishing Number Sense Through
Alternate Strategies
th
5
GRADE DIVISION
Created by Christine M. Nobley
STANDARDS
GPS 2011-2012
CCGPS 2012-2013
M5N3b. Explain the
process of division
including situations where
divisors are whole
numbers and decimals.
MCC5.NBT.6 Find the whole
number quotients of whole
numbers with up to 4 digit
dividends and 2 digit divisors
using strategies based on place
value, the properties of
operations, and/or the
relationship between
multiplication and division.
“Students with good number sense
can think and reason flexibly with
numbers, use numbers to solve
problems, spot unreasonable
answers, understand how numbers
can be taken apart and put together
in different ways, see connections
among operations, figure mentally,
and make reasonable estimates”.
Burns, M. (2007)
DEALING OUT (SHARING)
by one
30
5
I
I
I
I
I
I I
I I
I I
I I
I I
I I I
I I I
I I I
I I I
I I I
I I I
I I I
I I I
I I I
I I I
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
DEALING OUT
By multiples
30 5
REPEATED ADDITION
• Uses existing addition skills
• Helps with multiplication skills
• Limited view of division
0
3
6
9
18 3
12
3+3+3+3+3+3
15
18
REPEATED SUBTRACTION
• Uses existing subtraction skills
• Most kids not as proficient in subtraction
18
18
15
12
3
9
6
3
0
84 6
• Divide 6 into 8, subtract, get 2, bring down 4?
• 6 X 10= 60 6 X 4= 24 14
• ½ of 84 = 42 ½ of 6= 3
42 3 = 14
so
84 6 = 14
• Half of 84 is 42. 42 divided by 6 = 7.
7 X 2= 14.
COUNTING ON
• Uses knowledge of addition or subtraction
• Large numbers time consuming
6
12
18
24
30
36
42
48
54
60
66
72
78
84
HALF THEN DOUBLE
• Can only use if dividend is even
24 4
12 4 = 3
Double that = 6
108 6
54 6 = 9
Double that = 18
8662 41
4331 41
Clearly, this strategy is not helpful.
91
Strategy not helpful because 91 is odd
7
PROPORTIONAL REASONING
100 50 = 2
50 25 = 2
10 5 = 2
2 1=2
48
24
8
4
12 = 4
6=4
288 16
144 8
72 4
36 2 = 18
2=4
1=4
SINGLE DIGIT ARRAYS
• Shows how to build up to the dividend
• Uses existing multiplication knowledge
7
300
7
7 300
10
10
10
10
2
70
70
70
70
14
6 left over
3 DIGIT BY 2 DIGIT ARRAY
884
26
26
10
10
10
2 2
260
260
260
52 52
MULTIPLYING UP
• Builds on strengths in multiplication
• Relates multiplication to division
960
15 x 10 = 150
15 x 10 = 150
15 x 10 = 150
15 x 10 = 150
15 x 10 = 150
15 x 10 = 150
15 x 4 = 60
15
15 x 20 = 300
15 x 20 = 300
15 x 20 = 300
15 x 2 = 30
15 x 2 = 30
15 x 30 = 450
15 x 30 = 450
15 x 4 = 60
DISTRIBUTIVE PROPERTY
65
(30 + 30 + 5)
6 + 6 + 1
5
645
5
20
(600 + 40 + 5)
30 + 2 = 32
= 13
132
12
(120 + 12)
120 12
12
12
10
+
12
1
= 11
R5
PARTIAL QUOTIENTS
• Uses friendly
multiplication
facts
• x2
• x5
• x 10
18 468
- 180
288
- 180
108
- 90
18
- 18
0
10
10
5
1
Answer = 26
PARTIAL QUOTIENT
• http://www.youtube.com/watch?v=qWstA8EZ
r2w
STANDARD ALGORITHM
177
5 885
-5
38
-35
35
-35
0
Divide
Multiply
Subtract
Bring Down
REVIEW OF DIVISION
STRATEGIES
1. Dealing Out/Sharing
2. Repeated Addition
3. Repeated Subtraction
4. Counting Up
5. Half then Double
6. Proportional Reasoning
7. Rectangular Arrays
8. Multiplying Up
9. Distributive Property
10. Partial Quotients
“By abandoning the rote
teaching of algorithms we
are not asking children to
learn less, we are asking
them to learn more”
(Fosnot & Dolk, 2001,
p.102).
References
Burns, M. (2007). About teaching mathematics: a k-8 resource (3rd ed.). Sausalito, Ca:
Scholastic, Inc.
Chapin, S., & Johnson, A. (2000). Math matters: grades k-6. Sausalito, Ca: Math Solutions
Publications.
Fosnot, C. T., & Dolk, M. (2001). Mathematicians at work: Constructing multiplication and
division. Portsmouth, NH: Heinemann.
Hickendorff, M., van Putten, C., Verhelst, N., & Heiser, W. (2010). Individual differences in
strategy use on division problems: mental versus written computation. Journal of
Educational Psychology, 102(2), 438-452.
Mulligan, J., & Mitchelmore, M. (1997). Young children's intuitive models of multiplication and
division. Journal for Research in Mathematics Education, 28(3), 309-331.
Partial quotients method for division. (August 26, 2009). Retrieved June 21, 2012 from
http://www.youtube.com/watch?v=qWstA8EZr2w
HOW CAN I PRACTICE?
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http://www.sophia.org/partial-quotients-algorithm-tutorial
http://www.slideshare.net/guestb30cd4/partial-quotients
http://www.williston.k12.sc.us/kees/KEES%20Curriculum/MAPLessonPlans/Num&Oper/221230/Day08.pdf
http://teachmath.openschoolnetwork.ca/wordpress/grade-5/division/distributive-propertyfor-division/
http://www.youtube.com/watch?v=4o_5PUZ4wvU
http://eternaltreasures.hubpages.com/hub/MATHEMATICS-Shortcuts-in-MultiplicationDivision-Addition-Subtraction-secrets-in-mental-math-calculation-speed
http://www.teachertube.com/viewVideo.php?title=Partial_Quotients_Division&video_id=20
032 http://www.youtube.com/watch?v=By9Kwoz8oT4 5
http://www.youtube.com/watch?v=340UK0aox1I
http://www.youtube.com/watch?v=FJ-P5c_bZlc
http://www.youtube.com/watch?v=hCKd3C4P6Uk