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Virginia Birch MFNERC Numeracy Specialist

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Overview Mental Math… what is it? Grade 1 to 3 Strategies Indicators from the Provincial Report Card

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Mental Math… what is it Conceptual strategies that enhance flexible thinking and number sense and number skills (critical numeracy) calculating mentally without the use of external memory aids. provides a cornerstone for all estimation processes offering a variety of alternate algorithms and non-standard techniques for finding answers K - Gr 8 Manitoba Curriculum Framework of Outcomes 2013

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developing mental math skills and recalling math facts automatically Facts become automatic for students through repeated exposure and practice. When facts are automatic, students are no longer using strategies to retrieve them from memory. Mental Math… what is it K - Gr 8 Manitoba Curriculum Framework of Outcomes 2013

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One of the Math Competencies for the Grade 3 and Grade 7 Math Assessment (see report templates) Mental Math… what is it

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Mental Math: Should not be timed. Students differ on the amount of time they need to process concepts. Can be done daily approximately five minutes for a daily math routine. Can be practiced with math games or learning centers where students can practice the strategies. Create a Mental Math Bulletin Board Create an Estimation Bulletin Board

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The development of mental math strategies is greatly enhanced by sharing and discussion. Students should be given the freedom to adapt, combine, and invent their own strategies.

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Math Tools you can use… Ten Frames Base 10 Blocks 100 – Chart Number Line Array Cards Dominoes Finger Patterns Linking cubes Going from concrete to pictorial to abstract…

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Counting On – Grade 1 Concept: Addition Meaning: Students begin with a number and count on to get the sum. Students should begin to recognize that beginning with the larger of the two addends is generally most efficient. Example: for 3 + 5 think 5 + 1 + 1 + 1 is 8; think 5, 6, 7, 8

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Practising the strategy… 2+7= 9+3= 3+12=

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Counting Back – Grade 1 Concept: Subtraction Meaning: Students begin with the minuend and count back to find the difference. Example: for 6 – 2 think 6 – 1 – 1 is 4; think 6, 5, 4

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Practising the Strategy 9-2= 10-3= 7-1=

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Using One More – Gr 1, 2 Concept: Addition Meaning: Starting from a known fact and adding one more. Example: for 8 + 5 if you know 8 + 4 is 12 and one more is 13

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Practising the Strategy 5+6= 6+7= 8+9=

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Using One Less - Gr 1, 2 Concept: Addition Meaning: Starting from a known fact and taking one away. Example: for 8 + 6 if you know 8 + 7 is 15 and one less is 14

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Practising the strategy 7+6= 9+8= 6+5=

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Making 10 – Gr 1, 2 Concept: Addition, Subtraction Meaning: Students use combinations that add up to ten and can extend this to multiples of ten in later grades. Example: 4 + ____ is 10 7 + ____ is 10; so 23 + ____ is 30

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Practising the strategy 28 +___ = 30 15 +___=20 16 + ___= 40

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Starting from Known Doubles – Gr 1 Concept: Addition, Subtraction Meaning: Students need to work to know their doubles facts. Example: 2 + 2 is 4 and 4 – 2 is 2

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Practising the strategy 3+3=__,so ___-3=___ 9+9=__,so ___-9=___ 8+8=__,so ___-8=___

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Using Addition to Subtract – Gr 1, 2, 3 Concept: Subtraction Meaning: This is a form of part-part-whole representation. Thinking of addition as: part + part = whole Thinking of subtraction as: whole – part = part Example: for 12 – 5 think 5 + ____ = 12 so 12 – 5 is 7

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Practising the Strategy 13-7= ?, think 7 +___= 13 11-6= ?, think 6 +___=11 15-7= ?, think 7+___= 15

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The Zero Property of Addition – Gr 2 Concept: Addition, Subtraction Meaning: Knowing that adding 0 to an addend does not change its value, and taking 0 from a minuend does not change the value. Example: 0 + 5 = 5; 11 – 0 = 11

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Using Doubles – Gr 2, 3 Concept: Addition, Subtraction Meaning: Students learn doubles, and use this to extend facts: using doubles doubles plus one (or two) doubles minus one (or two) Example: for 5 + 7 think 6 + 6 is 12; for 5 + 7 think 5 + 5 + 2 is 12 for 5 + 7 think 7 + 7 – 2 is 12

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Practising the strategy for 6+ 8,think doubles for 6 + 8, think double 6 plus two for 6 + 8, think double 8 minus two

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Building on Known Doubles – Gr 2, 3 Concept: Addition, Subtraction Meaning: Students learn doubles, and use this to extend facts. Example: for 7 + 8 think 7 + 7 is 14 so 7 + 8 is 14 + 1 is 15

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Practising the Strategy 3+4= 7+6= 8+9=

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Adding from Left to Right – Gr 3 Concept: Addition Meaning: Using place value understanding to add 2-digit numerals. Example:for 25 + 33 think 20 + 30 and 5 + 3 is 50 + 8 or 58

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Practising the Strategy 17+22= 26+21= 45+34=

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Making 10 – Gr 3 Concept: Addition, Subtraction Meaning: Students use combinations that add up to ten to calculate other math facts and can extend this to multiples of ten in later grades. Example: for 8 + 5 think 8 + 2 + 3 is 10 + 3 or 13

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Practising the strategy 8+7= 7+9= 5+7=

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Compensation – Gr 3 Concept: Addition, Subtraction Meaning: Using other known math facts and compensating. For example, adding 2 to an addend and taking 2 away from the sum. Example: for 25 + 33 think 25 + 35 – 2 is 60 – 2 or 58

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Practising the strategy 47+22= 18+15= 39+17=

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Commutative Property – Gr 3 Concept: Addition Meaning: Switching the order of the two numbers being added will not affect the sum. Example: 4 + 3 is the same as 3 + 4

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Compatible Numbers – Gr 3, 4 Concept: Addition, Subtraction Meaning: Compatible numbers are friendly numbers (often associated with compatible numbers to 5 or 10). Example: for 4 + 3 students may think 4 + 1 is 5 and 2 more makes 7

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Practising the Strategy 4+7= 9+8= 7+8=

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Array – Gr 3 Concept: Multiplication, Division Meaning: Using an ordered arrangement to show multiplication or division (similar to area). Example: for 3 x 4 think for 12 ÷ 3 think

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Practising the strategy 5x5= 3x4= 4x2=

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Commutative Property – Gr 3 Concept: Multiplication Meaning: Switching the order or the two numbers being multiplied will not affect the product. Example: 4 x 5 is the same as 5 x 4

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Skip Counting – Gr 3 Concept: Multiplication Meaning: Using the concept of multiplication as a series of equal grouping to determine a product. Example: for 4 x 2 think 2, 4, 6, 8 so 4 x 2 is 8

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Practising the Strategy 2x5= 3x4= 5x4=

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Zero Property of Multiplication– Gr 4 Concept: Multiplication Meaning: Multiplying a factor by zero will always result in zero. Example: 30 x 0 is 0 0 x 15 is 0

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Multiplicative Identity– Gr 4 Concept: Multiplication Meaning: Multiplying a factor by one will not change its value. Dividing a dividend by one will not change its value. Example: 1 x 12 is 12 21 ÷ 1 is 21

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Skip-Counting from a Known Fact– Gr 4, 5 Concept: Multiplication, Division Meaning: Similar to the counting on strategy for addition. Using a known fact and skip counting forward or backward to determine the answer. Example: for 3 x 8 think 3 x 5 is 15 and skip count by threes 15, 18, 21, 24

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4 x 7 = 6 x 8= 8 x 7= 9 x 8= Practising the Strategy

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Doubling or Halving– Gr 4, 5 Concept: Multiplication, Division Meaning: Using known facts and doubling or halving them to determine the answer. Example: for 7 x 4, think the double of 7 x 2 is 28 for 48 ÷ 6, think the double of 24 ÷ 6 is 8

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8 x 4 = think double 8 x 2 = ___ 32 ÷ 4 = think double 16 ÷ 4 = ____ Practising the Strategy

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Using the Pattern for 9s- Gr 4 Concept: Multiplication, Division Meaning: Knowing the first digit of the answer is one less than the non-nine factor and the sum of the product’s digits is nine. Example: for 7 x 9 think one less than 7 is 6 and 6 plus 3 is nine, so 7 x 9 is 63

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4 x 9 = 36 5 x 9 = 45 6 x 9 = 54 7 x 9 = 63 8 x 9 = 72 Practising the Strategy

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Repeated Doubling– Gr 4, 5 Concept: Multiplication Meaning: Continually doubling to get to an answer. Example: for 3 x 8, think 3 x 2 is 6, 6 x 2 is 12, 12 x 2 is 24

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To ﬁnd 8 X 8, ﬁrst ﬁnd 2 X 8, then double, then double again. 2 X 8 = 16 4 X 8 is double 2 X 8 16 + 16 = 32 so, 4 X 8 = 32 8 X 8 is double 4 X 8 32 + 32 = 64 so, 8 X 8 = 64 Practising the Strategy

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Using multiplication to divide - Gr 4 Concept: Division Meaning: This is a form of part-part-whole representation. Thinking of addition as: part x part = whole Thinking of subtraction as: whole ÷ part = part Example: for 35 ÷ 7 think 7 x ____ = 35 so 35 ÷ 7 is 5

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36 ÷ 6 = Think 6 x ___ = 36 42 ÷ 7 = Think 7 x ___ = 42 Practising the Strategy

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Distributive property – Gr 4, 5 Concept: Multiplication Meaning: In arithmetic or algebra, when you distribute a factor across the brackets: a x (b + c) = a x b + a x c (a + b) x (c + d) = ac + ad + bc + bd Example: for 2 x 154 think 2 x 100 plus 2 x 50 plus 2 x 4 is 200 + 100 + 8 or 308

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Place a straw between two columns. What does it now show? a x (b + c) = a x b + a x c Record it as 3 x 7 = 3 x 2 + 3 x 5

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13 x 12 = (10 + 3) x (10 + 2) = (10 x 10) + (10 x 2) + (3 x 10) + (3 x 2) (a + b) x (c + d) = ac + ad + bc + bd

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How do you assess for mental math strategies? Use checklists Observe strategies students are using through games Make anecdotal notes while conversing with a student Collect student sample work

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Indicators from the Provincial Report Card (Gr 1 to 8) Determines an answer using multiple mental math strategies Applies mental math strategies that are efficient, accurate, and flexible Makes a reasonable estimate of value or quantity using benchmarks and referents. Uses estimation to make mathematical judgments in daily life. Provincial Report Card Document Pg. 43

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