Understanding Multiplication and Divisionwith Whole Numbers Adapted from: Guiding Children's Learning of Mathematicsby 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 x89x89x89 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 74740 20 116 116 111111 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 89x 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?