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Fairfield Public Schools Elementary School Mathematics A Guide to Multiplication & Division Fairfield Public Schools 2011-2012

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“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 Fairfield Public Schools 2011-2012

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

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

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

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How would you record your thinking? 39 x 4 6 3 15 4 x 9 = 36 carry the 3 4 x 3 = 12 add the 3 = 15 Actually, it is 4 x 9 = 36 carry the 3 tens 4 x 3 tens = 12 tens add the 3 tens = 15 tens Digit-Oriented Right to Left Fairfield Public Schools 2011-2012

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Another way to record thinking 39 x 4 3 OR + 120 39 x 4 120 + 36 156 3 6 4 x 9 = 36 & write the 3 tens Then 4 x 3 tens = 120 Number-Oriented Left to Right Right to Left Fairfield Public Schools 2011-2012

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Another way to record thinking OR re-write the problem horizontally and decompose 39 according to place value. 4 x (30 + 9) Fairfield Public Schools 2011-2012

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Connecting an alternative strategy to algebraic properties 4 x 39 = ? 4 x 30 + 9 = 120 + 36 = 156 ( ) Distributive Property of multiplication over addition Fairfield Public Schools 2011-2012

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4 x 39 = ? 4 x 40 - 1 = 160 - 4 = 156 ( ) Distributive Property of multiplication over subtraction Connecting an alternative strategy to algebraic properties Fairfield Public Schools 2011-2012

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25 x 49 = ? How would you record your thinking? Try another problem and solve it mentally: Fairfield Public Schools 2011-2012

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Did you think… Money? (4 quarters) = $1.00 (or 100) 8 quarters = 200, 12 quarters = 300, 16 quarters = 400… 48 quarters = 1200 + 1 more quarter is 49 quarters = 1225, Fairfield Public Schools 2011-2012

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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 Fairfield Public Schools 2011-2012

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OR did 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 Property Distributive Property Fairfield Public Schools 2011-2012 10 x 5 x 25 =

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OR did you think… (4) 49 x 25 5 Fairfield Public Schools 2011-2012

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

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OR did you think… (4) 49 x 25 245 0 Fairfield Public Schools 2011-2012

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OR did you think… (1) (4) 49 x 25 245 80 Fairfield Public Schools 2011-2012

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

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OR did you think… (1) (4) 49 x 25 245 + 980 1225 Fairfield Public Schools 2011-2012

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

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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 decide… Fairfield Public Schools 2011-2012

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

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How would you record your thinking? 1 -12 3 6 36 3 12 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. Digit Oriented Fairfield Public Schools 2011-2012

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OR did you think… 120 10 36 - 3 13 10 x 12 = 120 120 from 156 leaves 36 3 x 12 = 36 Number Oriented Fairfield Public Schools 2011-2012

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

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Area Model Closed Array 3 rows of 4 or, 3 groups of 4 3 x 4 = 12 3 4 Open Array Fairfield Public Schools 2011-2012

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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

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Multiplication with Partial Products 3 x (5 + 4) 3 54 + 12 = 27 15 9 Or (3 x 5)+(3 x 4) The Distributive property of multiplication over addition Fairfield Public Schools 2011-2012

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12 13 100 10 + 2 10 + 3 12 x 13 =(10 + 2) x (10 + 3) = 10 x 10 = 10 x 3 = 30 2 x 3 = 2 x 10 = 6 20 Fairfield Public Schools 2011-2012

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Division Partial Quotients (Partial Factors) 918 9 3 3 +3= 6 Fairfield Public Schools 2011-2012

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Division Make it friendly to solve mentally 12 156 12036 10 + 3 = 13 Fairfield Public Schools 2011-2012

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OR 12 156 12 1 + 1= 13 144 Fairfield Public Schools 2011-2012 OR Record it:

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OR use your understanding of 5 x 12… 12 156 + 3+ 5 5 36 60 = 13 Fairfield Public Schools 2011-2012 Which can be recorded as:

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Differences between invented strategies and the traditional algorithm Invented Flexible Left to right Number-oriented Traditional Rigid Right to left Digit oriented Fairfield Public Schools 2011-2012

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Advantages of traditional algorithms Better for calculations with large numbers One procedure for all problems within a given operation Fairfield Public Schools 2011-2012

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Advantages of invented strategies Built on understanding - easier to make sense Builds understanding with estimation Reinforces place value concepts Much easier to perform mentally Avoid system errors Fairfield Public Schools 2011-2012

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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

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“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) Fairfield Public Schools 2011-2012

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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 gamesmath games Include them in everyday activities and highlight the mathematics – e.g. cooking, measuring, shopping, traveling, building… Delay the “standard algorithm” until your child understands why it works. Fairfield Public Schools 2011-2012

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More ways……Primary Suggestions Number 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 you’ve 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? Fairfield Public Schools 2011-2012

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More ways……Primary Suggestions Make 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 activities Practice the facts: play card games like “Double War” where it’s 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, it’s 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! Fairfield Public Schools 2011-2012

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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. Fairfield Public Schools 2011-2012

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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.FPS Fun Facts Fluency Tell time using analog clocks Do jigsaw puzzles and thinking games like Mastermind and Othello Read 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 Fairfield Public Schools 2011-2012

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