Download Mental Math and Basic Facts

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.
Mental Math and Basic Facts
STRATEGIES TO SUPPORT COMPUTATIONAL
FLUENCY WITH UNDERSTANDING
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.
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.
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
What does the research say?
Learning with understanding is more
powerful than simply memorizing
because the act of organizing
improves retention and promotes
fluency.
EDThoughts 2001 p. 81
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
Building Number Concepts
Concrete
Manipulatives
Pictorial
Representation
IIII
Abstract
Symbols
4+4=8
2x4=8
IIII
Significant time must be spent
working with concrete materials
and constructing pictorial representations
in order for abstract symbol and operational understanding to occur.
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
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
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.
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
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
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
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
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.
Making 10 Examples
We can find the answer to this fact by
making ten.
5
+ 6
Place the larger number in a ten
frame.
5
+ 6
Use part of the other number to
fill the ten frame.
5
+ 6
Then you can look at what is left
outside the ten-frame and tell what
the answer is.
5
1
+ 6 = 10 +
= 11
We can find the answer to this
fact by making ten.
8
+ 4
Place the larger number in the
ten-frame.
8
+ 4
Fill the ten-frame with part of the
other number.
8
+ 4
What is left outside tells you the
answer. You have 10 and 2 more.
8
12
+ 4 =
Strategies To Support Fact Learning
Subtraction
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.
L Count Up Strategy - This strategy works best
when subtracting two numbers that are close
together. Ex. 11 - 8 = 3 …….(9, 10, 11)
Strategies To Support Fact Learning
Subtraction
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
2
6
7
8
9
10
11
3
12 13
14
Strategies To Support Fact Learning
Subtraction
L Subtracting 9 from a teenager!
When subtracting nine from a teenager number, simply add the digits of the
teenager!
14
17
13
-9
-9
-9
5
8
4
Strategies To Support Fact Learning
LShow and discuss patterns!
9-8=1
8-7=1
7-6=1
6-5=1
5-4=1
4-3=1
3-2=1
2-1=1
Strategies To Support Fact Learning
Subtraction
L Fact Families Strategy - Can be used with all subtraction facts.
3-2=1
3
3-1=2
2+1=3
1+2=3
1
2
Strategies To Support Fact Learning
L Write Fact Families!
(3, 8, 11)
3 + 8 = 11 11 - 8 = 3
8 + 3 = 11 11 - 3 = 8
Think of a related addition
fact.
12
?
- 7 =
? + 7 =
12
Subtract from Ten
To find 15 - 8, start with
15.
15 is 10 &
5
To find 15 - 8, take 8 from the
ten. You can see that 7 is left.
15 - 8 = 7
To find 12 7, start
with 1.
12 is 10 &
2
To find 12 - 7, take 7 from the
ten. You can see that 5 is left.
12 - 7 = 5
Remember
that 13 is 10
and 3.
13
-8
10 + 3
-8
2+3
Take 8 from
the 10.
Combine this
with the other
3.
Remember
that 13 is 10
and 3.
13
-8
5
10 + 3
-8
2+3
Take 8 from
the 10.
Combine this
with the other
3.
13 - 8 =
5
Take 8 from
the 10.
12
-8
Combine this
with the other
2.
Altogether,
what is the
answer?
Multiplication Facts
are easiest to learn when...
 You find patterns.
 You use rhymes.
 You use stories.
 You relate them to what you
already know.
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
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
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
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
Five’s Pattern - Step 2
If you are multiplying an odd number, drop th
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
Five’s Pattern - Step 3
If you are multiplying an even number, just ad
a zero
5 x 2 =
5 x 6 =
5 x 4 =
5 x 8 =
 5 x 10 =
1 0
3 0
2 0
4 0
50
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
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= 9
+
3= 9
+2 =
9
+
9= 9
+
8= 9
+
4= 9
+
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
Six’s Pattern (even #’s only)
Step #2 The number you are
multiplying by 6 is the 2nd #.
6 x 4 = 2
2
4
6 x 6 = 3
6
6 x 8 = 4
8
6 x 2 = 1
Strategies To Support Fact Learning
L Stick Drawing! Count the dots!
4 x 3 = 12
Strategies To Support Fact Learning
Multiplication
L Zero Rule-
zero
any number multiplied by zero is
5x0=0
L One Rule- the product is the other number
6x1=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
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
L
Six Rule - think five groups of the number plus one more group.
4 x 5 = 20
5 x 5 = 25
6 x 7 = 5 x 7 + 7 = 42
Strategies To Support Fact Learning
L
L
Seven Rule- memorize two facts:
Brain hook…..56 is 7 x 8…Think 5..6..7..8
7 x 7 = 49 and 7 x 8 = 56
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
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
Strategies To Support Fact Learning
L Make Connections Between Multiplication
and Division
What facts
do you see?
3 x 5 = 15
15  3 = 5
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.