Understanding Multiplication and Division with Whole Numbers

Understanding Multiplication and Divisionwith Whole Numbers

Adapted from: Guiding Children's Learning of Mathematics by Leonard M. Kennedy (Author),

Steve Tipps (Author), Art Johnson (Author)


Student Goals for Multiplication and Division Understands and explains the effects of addition,

subtraction, and multiplication on whole numbers… and the effects of division on whole numbers including the inverse relationship of multiplication and division.

Selects the appropriate operations to solve specific problems…

Adds, subtracts, multiplies, and divides whole numbers to solve real world problems using appropriate methods of computing, such as mental mathematics, paper and pencil, and calculator. (5th grade, Sunshine State Standards)


Conceptual Understanding Real-life problem-solving stories provide a meaningful

context for multiplication and division Children learn about multiplication and division facts

and rules behind them by modeling and acting out with physical objects by representing the operations with pictures through naming and labeling the results of their

actions


Multiplication – Repeated Addition Joining or combining sets of equal size

Tony has 4 packages of juice boxes. Each package has 3 juice boxes in it..

Jaranda has 5 quarters in her pocket. Kayleigh earns $9 an hour babysitting and worked

11 hours last week.


Multiplication – Geometric Interpretation

Objects are lined up in rows and columns

Paolo arranged the chairs in 4 rows with 5 chairs on each row.

Gina measured the room 9 feet long by 10 feet wide. The ceiling has 14 tiles across and 12 tiles front to back.


Multiplication – Cartesian Product

Combining each object from one set

with each object from another set Molly wanted to know how many school outfits she

had with 5 blouses and 3 pairs of pants.. At her birthday party, Kara had 2 kinds of cake and 3

kinds of ice cream. How many combinations of cake and ice cream could she serve?.


Division – Measurement

Repeated subtraction of the same number

Marisa served 5 cookies to each person at her party. She started with 25 cookies.

Sandy had $93 saved. He spent $8 each week..


Division – Partition or Sharing

Distribute a set as evenly as possible

into a known number of groups

Kim had 19 strawberries to put into 4 boxes. The chef has 25 eggs. How many eggs can each of 8

people have?


Properties of Multiplication and Division Commutative property – order of factors does not

matter Associative property – factor may be paired in any

order Identity element – multiplication or division by 1 does

not change the original number value Distributive property over addition – multiplication

and division may be applied to factors represented as sums


The Meaning of Remainders in Division Introduce remainders as a natural result of division. Discuss the meaning of remainders and the various

ways to handle remainders. Ignore the remainder Round up and adjust the solution Treat the remainder as a fraction

Reread the problem to understand exactly what is wanted.


Strategic Learning of Basic Facts Children invent and learn strategies for the basic facts.

skip counting commutative property (flip flop facts) multiplication by 0, 1 (identity), and 2 division by 1 square numbers, multiplication by 9 associative and distributive property

Children practice facts with the strategies to develop accuracy and reasonable speed.


Guidelines for Practice Accuracy is the primary goal. Speed is secondary. Practice strategies and facts students understand. Keep practice sessions brief – 5 minutes. Use a variety of materials and procedures such as

games, computer activities, and flash cards. Emphasize individual improvement rather than same

goal for all children. Review and maintain basic facts to keep the skill

active.


Transition to Larger Numbers The same situations introduced with numbers 1-100

apply to multiplication and division situations with larger numbers.

Multiplication and division with larger numbers requires understanding of the operations and basic facts understanding place value representation of larger

numbers


Working with Numbers to 1000 and Beyond Problem-solving stories extend learning multiplication

and division with larger numbers by modeling and acting out with base ten models up

to 100, 500, and 1000 by representing the operations with pictures of place

value materials through recording the results of their actions and

pictures with numerals and place value.


Developing Algorithms After actions with materials and recording results,

students invent and/or learn alternative and standard algorithms as step by step procedures for multiplication and division with numbers to 1000 and larger than 1000.

At the same time, students learn when and how to use other computational processes. Estimation Mental computation Technology- calculator and computer


The purpose of computation is to solve problems. Thus, although computation is important in

mathematics and in daily life, our technological age requires us to rethink how

computation is done today.

(NCTM, 1989, p. 44)


Traditional and Alternative Multiplication Algorithms

In multiplication, regrouping and naming is the traditional algorithm. Many children have trouble with order of operation and lining up numbers.

Low stress algorithm start with the largest place value and works left to right.

Transitional and historical algorithms provide students with other approaches to multiplication with larger numbers..


Traditional Transitional Low-stress

74 74 74

x89 x89 x89 666 36 5600

5920 630 320

6586 320 630

5600 36

6586 6586


Traditional and Alternative Division Algorithms

Using trial divisors is the traditional subtraction algorithm.

Ladder alternative algorithm is based on repeated subtraction procedures

It allows children to start the division process with any convenient divisor..

The value of the numbers is maintained throughout the ladder algorithm.


Traditional Ladder Algorithm

23 R 5 23 R 5

37 ) 856 37 ) 856

74 740 20

116 116

111 111 3

5 5


Number Sense, Reasonableness, and Estimation Throughout development of multiplication and division

concepts and procedures, number sense is developed. In problem situations, students ask themselves, “Does

this answer make sense?” Estimation procedures are developed in situations that do

not require exact answers. Front-end estimation Rounding estimation


Estimation for Multiplication

Multiplication Front-End Rounding

74 70 70

x 89 x 80 +90

5600 6300

Students should identify most reasonable answers using high and low estimates.


Estimation for Division

Make an Easy Division Problem

Using Rounding / Reasonableness

____ __20__ __ 22 37) 856 40) 800 40 ) 880

Students should identify reasonable answers through several approximations.


Mental Computation

Alternative algorithm and estimation procedures build confidence in mental computation.

Understanding of multiplication and division of larger numbers is improved when students understand how to work with powers of ten. 20 x 400 = 2 x 4 x 10 x 100 = 8 x 1000 = 800 9000 divided by 30 is 9 divided by 3 x 1000 divided by 10 = 300


Regardless of the particular [computation] method used, students should be able to

explain their method, understand that many methods exist, and see the usefulness of

methods that are efficient, accurate, and general. Students also need to be able to estimate and judge the reasonableness of

results. (NCTM, 2000, p 32)

Multiplication Possibilities

Find as many ways as you can to fill in this problem so the arithmetic is correct:

__ __ __ x _________ 9 6 6

Division Problems

Bring a loaf of sliced bread to class. Choose a loaf that is packaged in a clear wrap so children can count the slices. Ask them to figure how many sandwiches can be made from the loaf.

Division Problems

Tell the class that for a math activity, each student needs eight tiles. A box has two hundred tiles. Have them figure out if everyone in the class can participate in the activity at one time.

Division Problems

Talk with the children about the way paper is packaged. Tell them that a ream contains five hundred sheets. Present this problem: If each student needs 20 sheets to make a recording book, how many books can be made from a ream of paper.

Division Problems

Tell the children about a person with an apple tree who had a giveaway celebration to get rid of fifty extra apples. She offered three apples to each person who asked. How many people could get free apples?

Understanding Multiplication and Division with Whole Numbers

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