Download Mental Math Strategies #2

Virginia Birch
MFNERC 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
KK -- Gr
Gr 88 Manitoba
Manitoba Curriculum
Curriculum
Framework
Framework of
of Outcomes
Outcomes 2013
2013
Mental Math…
what is it
• 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.
K - Gr 8 Manitoba Curriculum
Framework of Outcomes 2013
Mental Math…
what is it
• One of the Math Competencies for
the Grade 3 and Grade 7 Math
Assessment
• (see report templates)
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…
•
•
•
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Ten Frames
Base 10 Blocks
100 – Chart
Number Line
•
•
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Array Cards
Dominoes
Finger Patterns
Linking cubes
Going from concrete to pictorial to abstract…
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
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 – 2
think 6 – 1 – 1 is 4;
think 6, 5, 4
Practising the Strategy
9-2=
10-3=
7-1=
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
Practising the Strategy
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 10
7 + ____ is 10;
so 23 + ____ is 30
Practising the strategy
28 +___ = 30
15 +___=20
16 + ___= 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
Practising the strategy
3+3=__ ,
so ___-3=___
9+9=__ ,
so ___-9=___
8+8=__ ,
so ___-8=___
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
Practising the Strategy
13-7= ? , think 7 +___= 13
11-6= ? , think 6 +___=11
15-7= ? , think 7+___= 15
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
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
Practising the strategy
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
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
Practising the Strategy
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 + 33
think 20 + 30 and 5 + 3 is 50 + 8
or 58
Practising the Strategy
17+22=
26+21=
45+34=
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
Practising the strategy
8+7=
7+9=
5+7=
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
Practising the strategy
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 as
3+4
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
Practising the Strategy
4+7=
9+8=
7+8=
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



Practising the strategy
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 as
5x4
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
Practising the Strategy
2x5=
3x4=
5x4=
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
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
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
Practising the Strategy
4x7=
6 x 8=
8 x 7=
9 x 8=
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
Practising the Strategy
8x 4=
think double 8 x 2 = ___
32 ÷ 4 =
think double 16 ÷ 4 = ____
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
Practising the Strategy
4x9=
5x9=
6x9=
7x9=
8x9=
36
45
54
63
72
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
Practising the Strategy
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
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
Practising the Strategy
36 ÷ 6 =
Think 6 x ___ = 36
42 ÷ 7 =
Think 7 x ___ = 42
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
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
(a + b) x (c + d) = ac + ad + bc + bd
13 x 12
= (10 + 3) x (10 + 2)
= (10 x 10) + (10 x 2)
+ (3 x 10) + (3 x 2)
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