Establishing Number Sense Through Alternate Strategies

5th GRADE DIVISION

Created by Christine M. Nobley

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.

“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 II I I

I I II I I

I I II I I

I I II I I

I I II I I

DEALING OUT By multiples

222

222

222

222

222 30 5

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

REPEATED SUBTRACTION• Uses existing subtraction skills• Most kids not as proficient in subtraction

18 3

18 15 12 9 6 3 0

• 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

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

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

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

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

3 DIGIT BY 2 DIGIT ARRAY884 26

26 260 260 260 52 52

10 10 10 2 2

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

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

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

PARTIAL QUOTIENT

• http://www.youtube.com/watch?v=qWstA8EZr2w

STANDARD ALGORITHM 1775 885 -5 38 -35 35 -35 0

Divide

Multiply

Subtract

Bring Down

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

“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?• 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