T EACHING F OR AND V IA P ROBLEM S OLVING : T HE C ASE OF M ULTIPLICATION AND D IVISION David McKillop, Consultant Pearson Education Canada

W HAT IS MEANT BY T EACHING F OR P ROBLEM S OLVING

T EACHING F OR P ROBLEM S OLVING Mathematics concepts and procedures in school mathematics are used in the world. These applications should be front and centre in the study of any concept or procedure, NOT treated as an afterthought. For example, all units on addition and subtraction should centre on the different structures of story problems and their solutions.

P OWER OF R ELEVANT C ONTEXT

A S TEACHERS, WE CANNOT OVER ESTIMATE THE ROLE WORD PROBLEMS PLAY FOR MOST STUDENTS TO SEE THE RELEVANCE OF A MATH CONCEPT, AND THE VERY ESSENCE OF THE CONCEPT.

M ULTIPLICATION S TORY P ROBLEMS There were 4 plates with 5 cookies on each plate. How many cookies were there altogether? Equal Groups Story Problems

M ULTIPLICATION S TORY P ROBLEMS There are four rows of 5 cadets marching in the parade. How cadets are marching? Array Story Problems

M ULTIPLICATION S TORY P ROBLEMS You rode 5 km each day for 4 days. How many kilometres did you ride altogether? Day 1Day 2Day 3Day 4 Rate Story Problems

M ULTIPLICATION S TORY P ROBLEMS You have 5 comic books. Your friend has four times as many comics as you do. How many does your friend have? Yours Your Friend’s Comparison Story Problems

M ULTIPLICATION S TORY P ROBLEMS You have a choice of four different cones and five different flavours of ice cream. How many different choices do you have for an ice cream cone? Flavour 1 Flavour 2 Flavour 3 Flavour 4 Flavour 5 Cone AXXXXX Cone BXXXXX Cone CXXXXX Cone DXXXXX Combinations Story Problems

M ULTIPLICATION S TORY P ROBLEMS A rectangle has a length of 5 cm and a width of 4 cm. What is the area of this rectangle? 4 cm 5 cm Product of Measures Story Problems

M ULTIPLICATION S TORY P ROBLEMS How many centimetres are there is 6 metres? How many millimetres are there in 8 cm? How many grams are there in 4 kilograms? Special Rate Problems: conversions of units of measurement.

L ANGUAGE OF M ULTIPLICATION Four groups of five Four sets of five Four rows of five Four fives Four times five The product of four and five Four times as many as five

C ONCEPTUAL U NDERSTANDING : C ONNECTIONS A MONG M ULTIPLE R EPRESENTATIONS OF C ONCEPTS Contexts Concrete Models Pictures Symbols Language

Understanding can be defined as a measure of the quality and quantity of connections that an idea has with existing ideas. Understanding depends on the existence of appropriate ideas and on the creation of new connections. …Understanding is never an all-or-nothing proposition. (John A. Van de Walle)

T RADITIONAL A PPROACH : O NE D OOR TO A M ATH C ONCEPT

N EW A PPROACHES : M ANY D OORS TO A M ATH C ONCEPT

W RITE A NUMBER SENTENCE FOR EACH PROBLEM There are 18 students in the gym class. They form 3 teams. How many students are on each team? There are 18 students in the gym class. They form teams of 3 members. How many teams are formed?

T WO I NTERPRETATIONS OF A D IVISOR Divisor as the the number of groups: There are 18 students in the gym class. They form 3 teams. How many students are on each team? Divisor as the the number in each group: There are 18 students in the gym class. They form teams of 3 members. How many teams are formed? What is the interpretation of the 6 in each case?

W HY CAN THERE BE TWO INTERPRETATIONS OF THE D IVISOR AND Q UOTIENT ? Write two division sentences related to each multiplication sentence:

C REATE TWO DIVISION WORD PROBLEMS RELATED TO THIS MULTIPLICATION WORD PROBLEM There were 4 plates with 5 cookies on each plate. How many cookies were there altogether? There were 20 cookies on 4 plates. If each plate had the same number of cookies, how many cookies were on each plate? There were 20 cookies placed on plates, 5 to a plate. How many plates were needed?

C REATE TWO DIVISION WORD PROBLEMS RELATED TO THIS MULTIPLICATION WORD PROBLEM There are four rows of 5 cadets marching in the parade. How cadets are marching? Twenty cadets are marching in four rows in the parade. How many cadets are in each of the rows? Twenty cadets were marching in rows of 5 in the parade. How many rows of cadets are there?

C REATE TWO DIVISION WORD PROBLEMS RELATED TO THIS MULTIPLICATION WORD PROBLEM You rode 5 km each day for 4 days. How many kilometres did you ride altogether? You rode 20 kilometres in 4 days. If you travelled the same distance each day, how far did you ride each day? You rode a total of 20 kilometres, riding 5 km each day. How many days did you ride?

C REATE TWO DIVISION WORD PROBLEMS RELATED TO THIS MULTIPLICATION WORD PROBLEM You have 5 comic books. Your friend has four times as many comics as you do. How many does your friend have? You have 5 comic books and your friend has 20 comic books. How many times as many comic books does your friend have than you? Your friend has 20 comic books. This is 4 times as many comic books as you have. How many comic books do you have?

W HAT IS MEANT BY T EACHING V IA P ROBLEM S OLVING

NCTM P RINCIPLES AND S TANDARDS, 2000 Solving problems is not only a goal of learning mathematics but also a major means of doing so…

T EACHING V IA P ROBLEM S OLVING Van de Walle’s Three-Part Lesson Planning: Teachers: Before, During, and After Students: Explore (Show and Share) Connect Practice (Reflect)

T EACHING V IA P ROBLEM S OLVING Teach in ways that students tell you what you used to tell them!

T EACHING V IA P ROBLEM S OLVING Each learner constructs meaning in his/her own unique way. A learner, therefore, must be actively interacting with the idea. That is, he/she must be thinking (inducing, deducing, reasoning, sequencing, classifying, finding patterns and relationships). It is the quality of this interaction that is crucial to the development of understanding.

T EACHING V IA P ROBLEM S OLVING What does it look like in units on multiplication and division at all grade levels?

M OVING F ROM S ITUATION TO O PERATION Situation A. Represent with a model using role play, materials, diagrams, or numbers B. Solve by counting Represent the model and solutions with a number sentence Situation Represent with numbers and symbols (operate) Solve using a computation (compute)

U SING COUNTERS, MODEL THIS PROBLEM AND SOLVE IT Marcy noticed there were 3 rows of 14 soldiers marching in the Remembrance Day parade. How many soldiers were marching?

U SING COUNTERS, MAKE AN ARRAY FOR 3 X 14 3 x 14 = 3 x x 4 = = 42 Distributive Property

H OW MANY SQUARE CENTIMETRES DO YOU SEE ? 10 x 10 = x 7 = 70 3 x 10 = 303 x 7 = x