Establishing Number Sense Through Alternate Strategies
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Establishing Number Sense Through Alternate Strategies
5th GRADE DIVISION
Created by Christine M. Nobley
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STANDARDS
GPS 2011-2012
M5N3b. Explain the process of division
including situations where divisors are whole
numbers and decimals.
CCGPS 2012-2013MCC5.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.
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“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)
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DEALING OUT (SHARING)by one
30 5
I I I I I
I I I I I I I I I I
I I II I I
I I II I I
I I II I I
I I II I I
I I II I I
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DEALING OUT By multiples
222
222
222
222
222 30 5
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REPEATED ADDITION• Uses existing addition skills • Helps with multiplication skills• Limited view of division
0 3 6 9 12 15 18
18 3
3 + 3 + 3 + 3 + 3 + 3
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REPEATED SUBTRACTION• Uses existing subtraction skills• Most kids not as proficient in subtraction
18 3
18 15 12 9 6 3 0
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• Divide 6 into 8, subtract, get 2, bring down 4?• 6 X 10= 60 6 X 4= 24 14
• Half of 84 is 42. 42 divided by 6 = 7. 7 X 2= 14.
• ½ of 84 = 42 ½ of 6= 3 42 3 = 14 so 84 6 = 14
84 6
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COUNTING ON• Uses knowledge of addition or subtraction• Large numbers time consuming
54
6 12 18 24 30 36 42
48 60 66 72 78 84
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HALF THEN DOUBLE• Can only use if dividend is even
24 412 4 = 3 Double that = 6
108 654 6 = 9 Double that = 18
8662 414331 41 Clearly, this strategy is not helpful.
91 7 Strategy not helpful because 91 is odd
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PROPORTIONAL REASONING100 50 = 250 25 = 2
10 5 = 22 1 = 2
48 12 = 424 6 = 4
8 2 = 44 1 = 4
288 16144 872 4
36 2 = 18
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SINGLE DIGIT ARRAYS• Shows how to build up to the dividend• Uses existing multiplication knowledge
3007 3007
70 70 70 70 147
10 10 10 10 2
6 left over
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3 DIGIT BY 2 DIGIT ARRAY884 26
26 260 260 260 52 52
10 10 10 2 2
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MULTIPLYING UP• Builds on strengths in multiplication• Relates multiplication to division
15 x 10 = 15015 x 10 = 15015 x 10 = 15015 x 10 = 15015 x 10 = 15015 x 10 = 15015 x 4 = 60
15 x 20 = 30015 x 20 = 30015 x 20 = 30015 x 2 = 3015 x 2 = 30
15 x 30 = 45015 x 30 = 45015 x 4 = 60
960 15
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DISTRIBUTIVE PROPERTY
132 12(120 + 12) 12120 12 12 12
10 + 1 = 11
65 5
(30 + 30 + 5) 56 + 6 + 1 = 13
645 20(600 + 40 + 5) 30 + 2 = 32 R 5
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PARTIAL QUOTIENTS• Uses friendly
multiplication facts
• x 2 • x 5• x 10
18 468- 180 10
288
- 180 10 108- 90 5 18- 18 1 0
Answer = 26
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PARTIAL QUOTIENT
• http://www.youtube.com/watch?v=qWstA8EZr2w
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STANDARD ALGORITHM 1775 885 -5 38 -35 35 -35 0
Divide
Multiply
Subtract
Bring Down
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REVIEW OF DIVISION STRATEGIES
1. Dealing Out/Sharing2. Repeated Addition3. Repeated Subtraction4. Counting Up5. Half then Double6. Proportional Reasoning7. Rectangular Arrays8. Multiplying Up9. Distributive Property10. Partial Quotients
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“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).
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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
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HOW CAN I PRACTICE?• 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/221-
230/Day08.pdf• http://teachmath.openschoolnetwork.ca/wordpress/grade-5/division/distributive-property-f
or-division/• http://www.youtube.com/watch?v=4o_5PUZ4wvU• http://eternaltreasures.hubpages.com/hub/MATHEMATICS-Shortcuts-in-Multiplication-Divisi
on-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
Web Slice Gallery.url