Understanding Multiplication and Division with Whole Numbers
description
Transcript of 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)
Understanding Multiplication and Divisionwith Whole Numbers
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)
Understanding Multiplication and Divisionwith Whole Numbers
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
Understanding Multiplication and Divisionwith Whole Numbers
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.
Understanding Multiplication and Divisionwith Whole Numbers
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.
Understanding Multiplication and Divisionwith Whole Numbers
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?.
Understanding Multiplication and Divisionwith Whole Numbers
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..
Understanding Multiplication and Divisionwith Whole Numbers
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?
Understanding Multiplication and Divisionwith Whole Numbers
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
Understanding Multiplication and Divisionwith Whole Numbers
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.
Understanding Multiplication and Divisionwith Whole Numbers
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.
Understanding Multiplication and Divisionwith Whole Numbers
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.
Understanding Multiplication and Divisionwith Whole Numbers
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
Understanding Multiplication and Divisionwith Whole 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.
Understanding Multiplication and Divisionwith Whole Numbers
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
Understanding Multiplication and Divisionwith Whole Numbers
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)
Understanding Multiplication and Divisionwith Whole Numbers
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..
Understanding Multiplication and Divisionwith Whole Numbers
Traditional Transitional Low-stress
74 74 74
x89 x89 x89 666 36 5600
5920 630 320
6586 320 630
5600 36
6586 6586
Understanding Multiplication and Divisionwith Whole Numbers
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.
Understanding Multiplication and Divisionwith Whole Numbers
Traditional Ladder Algorithm
23 R 5 23 R 5
37 ) 856 37 ) 856
74 740 20
116 116
111 111 3
5 5
Understanding Multiplication and Divisionwith Whole Numbers
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
Understanding Multiplication and Divisionwith Whole Numbers
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.
Understanding Multiplication and Divisionwith Whole Numbers
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.
Understanding Multiplication and Divisionwith Whole Numbers
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
Understanding Multiplication and Divisionwith Whole Numbers
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?