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Winter 2011 Math News Hello Parents and Teachers, One of the foundations of numeracy is being able to solve simple addition, subtraction, multiplication and division questions with ease and understanding. Here are some strategies to improve your child or students fluency and understanding of basic operations.

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STRATEGIES TO SUPPORT COMPUTATIONAL FLUENCY WITH UNDERSTANDING Mental Math and Basic Facts

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Computational Fluency? Computational fluency refers to having efficient and accurate methods for computing. Students exhibit computational fluency when they demonstrate flexibility in the computational methods they choose, understand and can explain these methods, and produce accurate answers efficiently. The computational methods that a student uses should be based on mathematical ideas that the student understands well.

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The acquisition of basic math facts should occur in the following three phases: Phase I: Constructing operational meaning Phase II:Reasoning strategies Phase III:Working toward quick recall There is much overlap and students often can work in multiple stages simultaneously and may move through the stages at different rates. Additionally, it is important to point out that these three phases are critical for each of the four basic operations. However, addition and subtraction may develop concurrently and the same can be said about multiplication and division.

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Phase 1 - What does that mean? Students have to have a deep understanding of what numbers and operations are before being asked to use or memorize fact strategies A student will struggle with memorizing facts if they are asked to do the same questions over and over without understanding the meaning and strategies for completing the operations

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Learning with understanding is more powerful than simply memorizing because the act of organizing improves retention and promotes fluency. EDThoughts 2001 p. 81 What does the research say?

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Phase 2 – The Strategies Strategies are taught only after building understanding of the operations through hands on, pictorial and symbolic practice with operations. Strategies may be discovered by students but other students will need direct instruction with these strategies. PRACTICE MAKES PERFECT – it takes time to learn strategies. Be patient and think of the long term goals to teaching strategies -> Fluency with Understanding Strategies like most math concepts should be taught with manipulatives, pictures and then symbolically

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Building Number Concepts Concrete Manipulatives Pictorial Representation I I I I Abstract Symbols 4 + 4 = 8 2 x 4 = 8 Significant time must be spent working with concrete materials and constructing pictorial representations in order for abstract symbol and operational understanding to occur.

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Addition Strategies Turn Around Facts If you know 5 + 4 = 9, then you know 4 + 5 = 9. When adding, the order doesn’t matter. + = + 5 + 4 = 4 + 5 Commutative Property

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Facts with Zero When adding zero to any number, the sum is the other addend. Examples: 7 + 0 = 7 0 + 5 = 5 Identity Property of Addition

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Count Up (One-/Two- More Than) When an addition problem contains a 1 or 2, we can use this strategy. Start by whispering the greater addend and count on the other addend. Example: 2 + 6 = 8 Start at 6 and count up 7, 8.

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Doubles When an addition problem contains two numbers that are the same we recognize this as a doubles problem. These are memorized facts. You can use visual clues to help you. Example: 4 + 4 = 8

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Near Doubles When an addition problem contains consecutive numbers on a number line, double the smaller addend and add 1. 4 + 5 = 4 + 4 +1 = 8 + 1 = 9

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Decompose (Decomposing is what allows make-ten and near doubles to work.) Break down the addends and add the pieces back together. Example: 11 + 4 = (10 + 1) + 4 = 10 + (1 + 4) = 10 + 5 = 15 Associative Property

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Sums of 10 This group includes all facts with a sum of 10. Picture the Ten Frame when solving. Examples: 7 + 3 = 10 2 + 8 = 10

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Make-Ten (Use the Ten Frame) This strategy works well with at least one addend of 8 or 9. When adding 9, picture a Ten Frame. Take one away from the other addend and move it over in your mind. For 9 + 6 think: 9 in the ten frame means that I need one more to make ten. If I move one from the 6 over, I have 5 left. So I can add 10 + 5 and that equals 15. 9 + 6 has the same sum as 10 + 5 Do the same for 8, except you have 2 open in the Tens Frame.

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Making 10 Examples We can find the answer to this fact by making ten. 5 + 6

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Place the larger number in a ten frame. 5 + 6

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Use part of the other number to fill the ten frame. 5 + 6

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Then you can look at what is left outside the ten-frame and tell what the answer is. 5 + 6 = 10 + 1 = 11

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We can find the answer to this fact by making ten. 8 + 4

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Place the larger number in the ten-frame. 8 + 4

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Fill the ten-frame with part of the other number. 8 + 4

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What is left outside tells you the answer. You have 10 and 2 more. 8 + 4 = 12

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Strategies To Support Fact Learning Back L Count Back Strategy - This strategy works best when subtracting 0, 1, 2 or 3. Ex. 12 - 3, start from 12 and count back three numbers, 11, 10, 9. Up L Count Up Strategy - This strategy works best when subtracting two numbers that are close together. Ex. 11 - 8 = 3 …….(9, 10, 11 ) Subtraction

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Strategies To Support Fact Learning L Distance From Ten Strategy - When the number ten lies between the two numbers of the subtraction fact, find the distance from ten for each of the numbers, then add their distances together. This strategy works best when both numbers are close to ten. Ex. 13 - 8 = 2 + 3 = 5 Subtraction 6 7 8 9 10 11 12 13 14 23

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Strategies To Support Fact Learning L Subtracting 9 from a teenager! Subtraction When subtracting nine from a teenager number, simply add the digits of the teenager! 141713 -9-9-9 584

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Strategies To Support Fact Learning 9 - 8 = 1 8 - 7 = 1 7 - 6 = 1 6 - 5 = 1 5 - 4 = 1 4 - 3 = 1 3 - 2 = 1 2 - 1 = 1 LShow and discuss patterns!

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Strategies To Support Fact Learning L Fact Families Strategy - Can be used with all subtraction facts. Subtraction 3 12 3 - 2 = 1 3 - 1 = 2 2 + 1 = 3 1 + 2 = 3

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Strategies To Support Fact Learning L Write Fact Families! (3, 8, 11) 3 + 8 = 1111 - 8 = 3 8 + 3 = 1111 - 3 = 8

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Think of a related addition fact. 12 - 7 = ? ? + 7 = 12

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Subtract from Ten To find 15 - 8, start with 15. 15 is 10 & 5

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To find 15 - 8, take 8 from the ten. You can see that 7 is left. 15 - 8 = 7

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To find 12 - 7, start with 1. 12 is 10 & 2

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To find 12 - 7, take 7 from the ten. You can see that 5 is left. 12 - 7 = 5

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13 Remember that 13 is 10 and 3. Take 8 from the 10. Combine this with the other 3. 10 + 3 -8-8 2 + 3 -8

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13 Remember that 13 is 10 and 3. Take 8 from the 10. Combine this with the other 3. 13 - 8 = 5 -8-8 10 + 3 -8-8 2 + 35

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Take 8 from the 10. Combine this with the other 2. 12 - 8 Altogether, what is the answer?

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Multiplication Facts are easiest to learn when... You find patterns. You use rhymes. You use stories. You relate them to what you already know.

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Zero Pattern 0 times any number is 0 0 x 3 = 0 0 x 7 = 0 0 x 4 = 0 0 x 1 = 0 0 x 0 = 0 0 x 9 = 0

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One’s Pattern 1 times any number is the same number 1 x 3 = 3 1 x 7 = 7 1 x 4 = 4 1 x 1 = 1 1 x 2 = 2 1 x 9 = 9

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Two’s Pattern 2 times any number is that number doubled 2 x 3 = 6 2 x 7 = 14 2 x 4 = 8 2 x 1 = 2 2 x 2 = 4 2 x 9 = 18

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Five’s Pattern - Step 1 Cut the number you are multiplying in half 5 x 3 = 1.5 5 x 7 = 3.5 5 x 8 = 4 5 x 1 =.5 5 x 2 = 1 5 x 6 = 3

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Five’s Pattern - Step 2 If you are multiplying an odd number, drop the decimal point 5 x 3 = 5 x 7 = 5 x 5 = 5 x 1 = 5 x 9 = 1.5 3.5 2.5. 5 4.5

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Five’s Pattern - Step 3 If you are multiplying an even number, just add a zero 5 x 2 = 5 x 6 = 5 x 4 = 5 x 8 = 5 x 10 = 1 0 3 0 2 0 0 50 4

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Nine’s Pattern - Step 1 Subtract 1 from number you are multiplying 9 x 3 = 2 9 x 7 = 6 9 x 8 = 7 9 x 1 = 0 9 x 2 = 1 9 x 6 = 5

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Nine’s Pattern - Step 2 Add a 2nd number that totals nine with the 1st number 9 x 3 = 2 9 x 7 = 6 9 x 8 = 7 9 x 1 = 0 9 x 2 = 1 9 x 6 = 5 7 3 2 9 8 4 + + + + + + = = = = = = 9 9 9 9 9 9

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Six’s Pattern (even #’s only) Step #1 Cut the number you are multiplying in half. 6 x 2 = 1 6 x 4 = 2 6 x 6 = 3 6 x 8 = 4

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Six’s Pattern (even #’s only) Step #2 The number you are multiplying by 6 is the 2nd #. 6 x 2 = 1 6 x 4 = 2 6 x 6 = 3 6 x 8 = 4 2 4 6 8

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Strategies To Support Fact Learning L Stick Drawing! Count the dots! 4 x 3 = 12

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Strategies To Support Fact Learning Multiplication L Zero Rule- any number multiplied by zero is zero 5 x 0 = 0 L One Rule- the product is the other number 6 x 1 = 6 L Two Rule - add the number to itself 8 x 2 = 8 + 8 = 16 L Three Rule- double the number, then add the number again 7 x 3 = 14 + 7 = 21

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Strategies To Support Fact Learning Multiplication L Four Rule- double the number twice. 6 x 4 = 12 + 12 = 24 Silly Saying (You’ve got to be 16 to drive a 4 x 4) L Five Rule- count by fives. Products will end with 0 or 5.3 x 5 = 15 4 x 5 = 20 5 x 5 = 25 L Six Rule - think five groups of the number plus one more group.6 x 7 = 5 x 7 + 7 = 42

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Strategies To Support Fact Learning L Seven Rule- memorize two facts: 7 x 7 = 49 and 7 x 8 = 56 Brain hook…..56 is 7 x 8…Think 5..6..7..8 L Eight Rule- Memorize one fact: 8 x 8 = 64 Silly Saying (I ate and I ate until I got sick on the floor!) 8 8 6 4

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Strategies To Support Fact Learning Multiplication L Nine Rule- subtract 1 from the number you are multiplying with nine. Then think….What should I add to that number to equal 9? 7 x 9 …One less than 7 is 6. 6 + ? = 9 6 + 3 = 9 so the product is 63 L Ten Rule- put a zero on the number you are multiplying by. 9 x 10 = 90

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Strategies To Support Fact Learning L Make Connections Between Multiplication and Division What facts do you see? 3 x 5 = 15 15 3 = 5

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Phase 3 – Working Towards Quick Recall Students must know all the facts or strategies to find all the facts before emphasis on speed is placed Quick recall of math facts is usually defined as the ability to solve a basic number computation in a few seconds or less (without resorting to inefficient methods like counting).. Phase III is simply about increasing a student’s quick recall. Spending more time in Phase III will not lead to quick recall of unknown facts. Successfully addressing Phases I and II will significantly decrease the need for prolonged work in Phase III. This involves: increasing the speed in which the student selects and applies a strategy for solving the problem and using highly organized and planned practice for the purpose of devoting facts to memory. Phase III may include practice (and/or drill) with specific groups of facts through fact cards, games, paper/pencil practice, and the use of technology. Once students acquire the ability to quickly recall the facts, Phase III focuses on maintenance of the facts.

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