Fairfield Public Schools Elementary School Mathematics A Guide to Multiplication & Division

Fairfield Public SchoolsElementary School MathematicsA Guide toMultiplication & Division

Fairfield Public Schools 2011-2012


Our experience, discussions, and review of the literature have convinced us that school mathematics demands substantial change

- Adding it Up National Research Council Center for Education



Mathematics in the 21st CenturyIt is no longer sufficient to just learn arithmetic.Todays students need to be mathematical thinkers and problem solvers.Fairfield Public Schools 2011-2012


How does the standard algorithm work?The following is a sampling of different ways of thinking about how to solve basic computation problems.Consider thisFairfield Public Schools 2011-2012


How would you mentally calculate? 4 x 39 = ?Think about how you would solve this problem before you continue.Fairfield Public Schools 2011-2012


How would you record your thinking? 39 x 4

63154 x 9 = 36 carry the 34 x 3 = 12 add the 3 = 15Actually, it is4 x 9 = 36 carry the 3 tens4 x 3 tens = 12 tens add the 3 tens = 15 tensDigit-Oriented Right to LeftFairfield Public Schools 2011-2012


Another way to record thinking 39 x 4

3 OR+ 120 39 x 4120+ 36 15615636 4 x 9 = 36 & write the 3 tensThen 4 x 3 tens = 120Number-Oriented Left to RightRight to LeftFairfield Public Schools 2011-2012


Another way to record thinkingORre-write the problem horizontally and decompose 39 according to place value.

4 x (30 + 9)Fairfield Public Schools 2011-2012


Connecting an alternative strategy to algebraic properties 4 x 39 = ?

4 x 30 + 9 = 120 + 36 = 156

( ) Distributive Property of multiplication over additionFairfield Public Schools 2011-2012


4 x 39 = ?

4 x 40 - 1 = 160 - 4 = 156

( ) Distributive Property of multiplication over subtractionConnecting an alternative strategy to algebraic propertiesFairfield Public Schools 2011-2012


25 x 49 = ?

How would you record your thinking?Try another problem and solve it mentally:Fairfield Public Schools 2011-2012


Did you thinkMoney? (4 quarters) = $1.00 (or 100)8 quarters = 200, 12 quarters = 300, 16 quarters = 40048 quarters = 1200 + 1 more quarter is 49 quarters = 1225,Fairfield Public Schools 2011-2012


OR did you think?25 x 49(20 + 5) x 49 = (20 x 49) + (5 x 49) (2 x 49 x 10) + (5 x (40 + 9)) = (98 x 10) + (200 + 45) =(980 + 200) + 45 = 1225



ORdid you think

49 x 25 is (50 1) x 25 or,

5 x 25 = 125 and 10 x 125 = 1250

(50 1) x 25 is one less group of 25 from 1250 (50 x 25) (1 x 25) = 1225( )Associative PropertyDistributive PropertyFairfield Public Schools 2011-201210 x 5 x 25 =


OR did you think

(4) 49 x 25 5Fairfield Public Schools 2011-2012


OR did you think

(4) 49 x 25 245Fairfield Public Schools 2011-2012


OR did you think

(4) 49 x 25 245 0Fairfield Public Schools 2011-2012


OR did you think (1) (4) 49 x 25 245 80



OR did you think (1) (4) 49 x 25 245+ 980 Fairfield Public Schools 2011-2012


OR did you think (1) (4) 49 x 25 245+ 980 1225



Did you have another approach?Which way is most efficient for you?Fairfield Public Schools 2011-2012


Compare your strategy with someone else.Which approach is the correct approach?

Why does the traditional algorithm in the US work?

When might you use alternative strategies for computation?

Look at the numbers, then decideFairfield Public Schools 2011-2012


DivisionHow would you divide

156 12 = ?Solve it mentally and think about how you approached the problem.Fairfield Public Schools 2011-2012


How would you record your thinking?1-12 3636 312 goes into 15 once.Put down the 12 and subtract from 15 to get 3.Bring down the 6 to make 36. 12 goes into 36 three times.DigitOrientedFairfield Public Schools 2011-2012


OR did you think1201036-31310 x 12 = 120120 from 156 leaves 363 x 12 = 36Number OrientedFairfield Public Schools 2011-2012


How do models support thinking?The following examples use the array model to represent thinking.Fairfield Public Schools 2011-2012


Area ModelClosed Array

3 rows of 4 or, 3 groups of 43 x 4 = 1234Open ArrayFairfield Public Schools 2011-2012


Algebraic properties are used with whole numbers in the elementary grades.

Flexible strategies foster a greater sense of number, equivalence, estimation, and the use of algebraic properties.Fairfield Public Schools 2011-2012


Multiplication with Partial Products

3 x (5 + 4)354+ 12 = 27159Or(3 x 5)+(3 x 4)The Distributive property of multiplication over additionFairfield Public Schools 2011-2012


121310010

+

210 + 312 x 13 = (10 + 2) x (10 + 3) = 10 x 10 = 10 x 3 = 302 x 3 = 2 x 10 = 620Fairfield Public Schools 2011-2012


DivisionPartial Quotients (Partial Factors)918933 +3= 6Fairfield Public Schools 2011-2012


DivisionMake it friendly to solve mentally121561203610 + 3= 13Fairfield Public Schools 2011-2012


OR12156121+ 1= 13144Fairfield Public Schools 2011-2012ORRecord it:


OR use your understanding of 5 x 1212156+ 3+ 55366060= 13Fairfield Public Schools 2011-2012Which can be recorded as:


Differences between invented strategies and the traditional algorithmInventedFlexible

Left to right

Number-orientedTraditionalRigid

Right to left

Digit oriented



Advantages of traditional algorithmsBetter for calculations with large numbers

One procedure for all problems within a given operationFairfield Public Schools 2011-2012


Advantages of invented strategiesBuilt on understanding - easier to make sense

Builds understanding with estimation

Reinforces place value concepts

Much easier to perform mentally

Avoid system errors



The numbers and problem often lend themselves to different strategies.If you wanted to buy 3 rolls of wrapping paper on sale for $1.99, how would you calculate your answer?Did you use the standard algorithm or did you use an alternative strategy?Is one strategy more correct than another?Fairfield Public Schools 2011-2012


The single most important principle for improving the teaching of mathematics is to allow the subject of mathematics to be problematic for students. (Hiebert et al., 1996)It is important that students solve problems not to apply mathematics but to learn new mathematics. When students engage in well chosen problem based tasks and focus on the solution methods, what results is new understanding of the mathematics embedded in the task. (Van de Walle and Lovin, 2006)



Ways you can support your child:Ask your child How they solved a problem and how do they know if they are correct? (prove it!)Play math gamesInclude them in everyday activities and highlight the mathematics e.g. cooking, measuring, shopping, traveling, buildingDelay the standard algorithm until your child understands why it works.Fairfield Public Schools 2011-2012


More waysPrimary SuggestionsNumber a paper from 1-12 for each player. Roll two dice and either add or subtract the numbers on your dice. Cross out that number on your paper. Then let your partner roll. Take turns rolling the dice, and adding and subtracting. The first person with all of the numbers crossed out is the winner.Race to $1.00 Each player needs 10 pennies, 9 dimes, $1.00 and a die. Take turns rolling the die to collect that many pennies. When youve collected ten pennies, trade them in for a dime. Continue until someone reaches a dollar. Variation: Race to Zero: Play the same way as above but start with a $1.00, and roll to subtract. The first person to reach zero wins.Play Store: Give your child 1 quarter, 3 dimes, 4 nickels, and 9 pennies. Tag a selection of small toys with prices under $0.84. Take turns counting out the correct coins to buy different items at the store. Ask: Which costs the most? The least?



More waysPrimary SuggestionsMake Pattern Pasta Jewelry: Dye pasta with a little rubbing alcohol and a drop of food coloring and shake in a zip lock gallon bag. Practice counting by twos, fives and tens. Think of different ways to sort the pasta (color, shape, texture). Finally, create a necklace by stringing the pasta in a repeating pattern.Involve counting and numbers in everyday activitiesPractice the facts: play card games like Double War where its just like War but each player throws down two cards and adds them both. The player with the highest sum, gets that hand. If there are equal sums, its Double War! Try subtracting instead and the player with the lowest difference gets the cards.Cook together and focus on measuring, counting, estimating and keeping track of time.Sort laundry, silverware, toys, shells on the beach, etc. to improve classification skills. Talk about the rules for sorting.Tell time using analog clocks.Countdown to big events like holidays, or birthdays. Use a calendar and count back how many days left until the big day!



Intermediate Suggestions

How much is your name worth? Assign each letter in the alphabet a monetary value (e.g., a = $1.00, b = $2.00 or a = 25, b = 50, c = 75, etc.). Whose name is the most expensive in your family? The least expensive?Keep a record or log of how many hours you and your family watch television each week. At that rate, figure how much in a month? The summer? Compare this data to how much time you spend running and playing in the fresh air.Use store flyers to write the grocery list with a given amount of money. Plan schedules using elapsed time. Create a schedule if you go away on vacation- plan the activities for your family and estimate how long each activity will take. Create an expense budget for your family.Decide which route to take to a destination relative to time and distance (using a map and not the GPS!).Save and budget money.



Intermediate Suggestions

Practice your addition, subtraction, multiplication and division facts by playing games like Double War with addition, subtraction and multiplication. See the FPS Fun Facts Fluency packet for more ideas.Tell time using analog clocksDo jigsaw puzzles and thinking games like Mastermind and OthelloRead the newspaper or magazines and find examples of large numbers and numbers represented by percents, decimals or fractions. Make a list of all the numbers you found and put them in order from least to greatest.Create a game (board game, dice game, logic game) of your own! Write up the directions and bring it to school to share with your class.Whenever you see or hear a number, try and think of at least one different way to express it numerically. For example: Your mom tells you that you have 10 minutes before your swim lesson. How many ways can you express the number 10? 2 x 5, (30 6) x 2, or (99 90) + 1



What if we record our thinking differently?*What if we record our thinking differently?**Or 50 x (10 + 10 + 5) = 500 + 500 + 250 = 1250 25 = 1225

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Fairfield Public Schools Elementary School Mathematics A Guide to Multiplication & Division