virginia birch mfnerc numeracy specialist

Virginia BirchMFNERC Numeracy Specialist

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

Report Card

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 2013K - Gr 8 Manitoba Curriculum Framework of Outcomes 2013

• 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

• One of the Math Competencies for the Grade 3 and Grade 7 Math Assessment

• (see report templates)

Mental Math…what is it

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

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.

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…

Counting On – Grade 1Concept: 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 + 5think 5 + 1 + 1 + 1 is 8;think 5, 6, 7, 8

Practising the strategy…

2+7=9+3=3+12=

Counting Back – Grade 1

Concept: Subtraction

Meaning: Students begin with the minuend and count back to find the difference.

Example: for 6 – 2think 6 – 1 – 1 is 4; think 6, 5, 4

Practising the Strategy

9-2=10-3=7-1=

Using One More – Gr 1, 2Concept: 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


5+6=6+7=8+9=

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

Practising the strategy

7+6=9+8=6+5=

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 107 + ____ is 10; so 23 + ____ is 30


28 +___ = 3015 +___=2016 + ___= 40

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


3+3=__ , so ___-3=___

9+9=__ , so ___-9=___

8+8=__ , so ___-8=___

Using Addition to Subtract – Gr 1, 2, 3

Concept: SubtractionMeaning: This is a form of part-part-whole representation. Thinking of addition as: part + part = wholeThinking of subtraction as: whole – part = part

Example: for 12 – 5think 5 + ____ = 12so 12 – 5 is 7


13-7= ? , think 7 +___= 1311-6= ? , think 6 +___=1115-7= ? , think 7+___= 15

The Zero Property of Addition – Gr 2


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

Using Doubles – Gr 2, 3Concept: Addition, SubtractionMeaning: Students learn doubles, and use this to extend facts: using doubles doubles plus one (or two) doubles minus one (or two) Example:

for 5 + 7think 6 + 6 is 12;for 5 + 7think 5 + 5 + 2 is 12for 5 + 7 think 7 + 7 – 2 is 12


for 6+ 8,think doubles

for 6 + 8, think double 6 plus two

for 6 + 8, think double 8 minus two

Building on Known Doubles – Gr 2, 3


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


3+4=7+6=8+9=

Adding from Left to Right – Gr 3

Concept: Addition

Meaning: Using place value understanding to add 2-digit numerals.

Example: for 25 + 33think 20 + 30 and 5 + 3 is 50 + 8 or 58


17+22=26+21=45+34=

Making 10 – Gr 3Concept: 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 + 5think 8 + 2 + 3 is10 + 3 or 13


8+7=7+9=5+7=

Compensation – Gr 3Concept: 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 + 33think 25 + 35 – 2 is 60 – 2 or 58


47+22= 18+15=39+17=

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 as3 + 4

Compatible Numbers – Gr 3, 4


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


4+7=9+8=7+8=

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

for 12 ÷3 think


5x5=3x4=4x2=

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 as5 x 4

Skip Counting – Gr 3Concept: 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


2x5=3x4=5x4=

Zero Property of Multiplication– Gr 4


Meaning: Multiplying a factor by zero will always result in zero.

Example: 30 x 0 is 00 x 15 is 0

Multiplicative Identity– Gr 4Concept: 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 1221 ÷ 1 is 21

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

4 x 7 =6 x 8=8 x 7=9 x 8=


Doubling or Halving– Gr 4, 5


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 28for 48 ÷ 6,

think the double of 24 ÷ 6 is 8

8 x 4 =think double 8 x 2 = ___

32 ÷ 4 =think double 16 ÷ 4 = ____


Using the Pattern for 9s- Gr 4


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

4 x 9 = 365 x 9 = 456 x 9 = 547 x 9 = 638 x 9 = 72


Repeated Doubling– Gr 4, 5


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

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 816 + 16 = 32 so,4 X 8 = 32

8 X 8 is double 4 X 832 + 32 = 64 so,8 X 8 = 64


Using multiplication to divide - Gr 4

Concept: DivisionMeaning: This is a form of part-part-whole representation. Thinking of addition as: part x part = wholeThinking of subtraction as: whole ÷ part = partExample: for 35 ÷ 7

think 7 x ____ = 35• so 35 ÷ 7 is 5

36 ÷ 6 =Think 6 x ___ = 36 42 ÷ 7 = Think 7 x ___ = 42


Distributive property – Gr 4, 5

Concept: MultiplicationMeaning: 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 154think 2 x 100 plus 2 x 50 plus 2 x 4

is 200 + 100 + 8 or 308

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

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

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

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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