Your Child's Math Education
www.nctm.org
Why does the math that my child brings home look
different from the math I remember?
Yesterday — Focus on Memorization
The mathematics you remember firom years ago may have focused on memorized
facts and methods for solving problems. In the past, teachers thought students
were good in mathematics i f they could "do math" quickly even i f they didn't
understand what they were doing. Students are still being taught the same skills
you learned,, but they learn them with understanding.
Today — Focus on Understanding
Today, more emphasis is placed on thinking and understanding. No matter
how well your child can do calculations, this ability is not very useful i f he
' or she doesn't understand thera or know how or when to use particular math
skills. National and intemationai studies have shown that students have made
steady improvement in math since 1990 when the National Council of Teachers
of Mathematics began advocating standards-based mathematics education
and learning with understanding. For example, results for the 2005 National
Assessment of Educational Progress (NAEP, the Nation's Report Card) show
that 80 percent of fourth graders and 69 percent of eighth graders performed at
or above the basic level in math, compared to only 50 percent and 52 percent in
1990.
Active Learners
Because society has changed, the math that students need to know has also
changed. Instead of worksheets filled with problems calling only for numerical
calculations, your child may be bringing home problems that relate to real life,
such as working with salaries and the cost of living and Ufe expectancy, and
making decisions based on comparisons. Because technology is used in so
many difTerent ways-today, students need to be able to reason about problems
and explain mathematics. Real leaming is more than just a student listening to
a teacher. Think about your own leaming experiences. You probably remember
those times when you actively participated in a learning activity much more than
when you just hstened to and watched the teacher
NATIONAL COUNCIL OF
NCTM
TEACHERS OF MATHEMATICS
Your Child's Math Education
www.nctm.org
Yesterday's Classroom
Today's Classroom
Straight rows of desks
Groups of desks
Teachers showing students how to solve a problem
while the students copied the process from the board
Students working together and discussing how to solve a
problem, with guidance from teachers
Students repeating given rules and memorizing what
to do in operations on numbers
Students applying the mathematics they know to develop;:
new skiils as they examine and question solutions to
problems, including real-world situations
Today's students are solving problems that they might come across in their everyday lives. They are learning many
of the same sldlls you leamed—and more. For example, think about how you might have learned to divide fractions
in school. To solve 3 ^ 1/4, you were probably taught to "invert and multiply," like this: 3/1 x 4/1 = 12/1. Did you
understand why you were inverting or what the answer meant?
Today's students might relate this problem to a real-life situation such as the following:
Jason has 3 pounds of hamburger and wants to make patties that weigh VA ofa pound apiece. How
many patties can he make? They begin by drawing a model ofthe problem, 3 1/4
Suppose that each circle represents 1 (one unit, or one whole—in this case, a pound of hamburger).
The shaded sUce in the first circle represents 1/4 (one patty).
How many one-fourths are in 3 wholes? We can see that twelve of the one-fourth pieces, four "slices" per circle,
will fill all three of the circles.
Putting the problem in a real-life context helps students make a picture of it in their minds, and this can help them
later in understanding why the process of mverting and multiplying works.
The goal of mathematics education today is to develop a lifelong understanding that is useful both at home and in
the workplace. Whatever your chUd chooses to do in life, having a strong understanding of mathematics will open
doors to a productive fiiture.
NATIONAL COUNCIL OF
NCTM
TEACHERS OF MATHEMATICS
Addition Algorithms
Combining Tens and Ones
453
+ 359.
700 (400+300)
100 (50 + 50)
12 (9 + 3)
812
Incremental
267 + 353
267 + 300 => 567 + 3 => 570
570 + 30 => 600 + 20 => 620
Compensation
67 add 3
+ 24 subtract 3
70
=- (+) 21
91
123 subtract 3 =
38 add 3
=
120
(+) j j .
161
Subtraction Algoritlims
Combining Tens and Ones
362
-_227
100 (300-200)
40 (60 - 20)
-=5_ (2-7)
135
Incremental
227 +|lQOh> 327 - { 3 >> 330 -lf30>> 360
360 {2\=> 362
135 answer
362 - 200 => 162- 20 => 142 - 2 => 140
140-5 --=> 135
Compensation
223
119
addl
addl
224
= (-)i20
104
223
114
subtract 4
subtract 4
=
219
=
(r}110
109
Multiplication Algorithms
Area Model
Partial Products
60
50x60
50x7
3x60
3x7
67
x53
3000
350
180
+ 21
3551
50
3
50x60 = 3000
3 x 6 0 = 180
3000 + 350+ 180 + 21 =3551
Division Algorithm
12
10
_3.
13
Answer: 13 with remainder of 2
7
5 0 x 7 = 350
3 x 7 = 21
!
What can I do to help
Addition Strategies
One-MoreThan and
Two-MoreThan Facts
What? Facts that have an
addend of 1 or 2.
How? Use Join or partpart whole problems in
which one of the addends
is a 1 or a 2.
Example: Beth saw 8 cats
run into the yard. Then
two more cats came in
a f t e r . How many cats did
Beth see in ail?
Zeros
What? Facts that have 0
as one of the addends.
How? Use word problems
involving zero. Have
students draw a picture
that shows two parts with
one part empty. Often,
students have the
misconception that
answers to addition are
always bigger.
Example: James has 2
marbles. His friend has 0
marbles. How many
marbles do they have?
0
1
2
3
4
5
6
7
8
9
+1
+1
+1
+1
+1
+1
+1
+1
+1
+1
0+2
1+2
2+2
3 +2
4 +2
5 +2
6+2
7 +2
8 +2
9 +2
0
1
2
3
4
5
6
7
8
9
+0
+0
+0
+0
+0
+0
+0
+0
+0
+0
1 +0
1 +1
1 +2
1 +3
1 +4
1+5
1 +6
1 +7
1+8
1 +9
2+0
2+1
2+2
2+3
2+4
2+5
2+6
2+7
2+8
2+9
0
0
0
0
0
0
0
0
0
0
+0
+1
+2
+3
+4
+5
+6
+7
+8
+9
Addition Strategies
Doubles
0 +0
1 +1
2+2
3+3
4+4
5+5
6 +6
7 +7
8+8
9 +9
What? Facts that have
two of the same addends.
How? These are relatively
easy f o r students to learn.
Have students make
picture cards f o r doubles.
Example: Taylor and
Ashton each found 5 rocks
on the playground. How
many did they f i n d
together?
Near-Doubles
What? Facts include all
combinations where one
addend is one more than
the other.
How? Double the smaller
number and add one. Be
sure students know the
doubles before focusing
on this strategy. Practice
using word problems.
Example: 5 + 6 = (5 + 5 ) + 1
0 +1
1 +2
2+3
3+4
4+5
5+6
6+7
7+8
8 +9
1 +0
2 +1
3 +2
4+3
5 +4
6+5
7 +6
8+7
9 +8
Addition
Strategies
Facts
e 01
Description
mimmmim
Make Ten
What? Facts that have at
least one addend of 8 or 9.
How? Build on the 8 or 9
up to 10 and then add on
t h e rest.
Exannple: For 6 + 8, start
with 8, then 2 more makes
10, and that leaves 4 more
f o r 14.
The Last Six
What? The remaining 12
facts.
How? Spend several days
with word problems where
these facts are the
addends.
Example Strategies that
could be used:
Make Ten
Counting On
Ten Frame Facts
Turn-around F a c t s
Commutative Property:
no difference.;
tlie result is the sume.
2.1=1.2
2 +8
3 +8
4 +8
5 +8
6+8
7 +8
8 +8
9 +8
1 +9
2 +9
3 +9
4 +9
5 +9
6 +9
7 +9
8 +9
9 +9
8 +2
8+3
8+4
8 +5
8 +6
8 +7
8 +8
8 +9
9 +1
9 +2
9+3
9+4
9+5
9+6
9+7
9 +8
9 +9
3+5
3 +6
3+7
4 +6
4 +7
5 +7
5+3
6+3
7+3
6+4
7 +4
7+5
12 X 12
Fact Strategies
l-GCtS
Turn-around F a c t s
.
Type OT FQCT
Descriptior.
Commutative Toperty:
Changing the order of the fad|-ors makes no difference;
the result is 1 he same.
8X3 = 3
Twos
(Doubles)
What? Facts that have 2
as a factor.
How? These are equivalent
to the addition doubles. 2
X 7 is double 7 and the
same is true of 7 X 2.
Example: 6 X 2 = 6 + 6
Fives
What? Facts that have 5
as a factor.
How? Practice counting by
5s using arrays with rows
of 5. Relate to the minute
hand on the clock.
Example: 6 X 5 is 6 rows
of 5. When t h e minute
hand points t o the 6 it is
30 because t h e clock is in
increments o f 5.
2X0
2X1
2X2
2X3
2X4
2X5
2X6
2X7
2X8
2X9
2X10
2X11
2X12
0X2
1X2
2X2
3X2
4X2
5X2
6X2
7X2
8X2
9X2
10X2
11 X 2
12X2
5X0
5X1
5X2
5X3
5X4
5X5
5X6
5X7
5X8
5X9
5X10
5X11
5X12
0X5
I X5
2X5
3X5
4X5
5X5
6X5
7X5
8X5
9X5
10X5
II X 5
12X5
12 X 12 F a c t Strategies
Turn-around F a c t s
Zeros and
Ones
What? Facts that have a t '
least one factor with 0 or
1.
How? Avoid rules like,
"Any number multiplies by
zero is zero." Students
might confuse rules they
may have learned f o r
addition.
Instead, use Story
problems to develop the
concepts behind these
facts.
0X0
1 xo
2X0
3X0
4X0
5X0
6X0
7X0
8X0
9X0
10X0
11 XO
12X0
0X0
0X1
0X2
0X3
0X4
0X5
0X6
0X7
0X8
0X90X10
0X11
0X12
1 XO
1X1
1X2
1 X3
1X4
1 X5
1X6
1 X7
1X8
1X9
1 XlO
1 X 11
1 X12
0X1
1X1
2X1
3X1
4X1
5X1
6X1
7X1
8X1
9X1
10X1
11 X I
12X1
Example: I f I have 5
groups of 0, how many do
I have?
Nines
What? Facts that have 9
as a factor.
How? Since 9 is close to
10, students can take one
set away. Another way is
that the tens digit of the
product is always one less
than the other factor.
Example: 7 X 9 is the
same as 7 X 10 less one
set of 7, or 70 - 7. 9 X 7 =
63, 6 is one less than the
7 in the tens place and the
digits equal nine (6 + 3 =
9).
9X0
9X1
9X2
9X3
9X4
9X5
9X6
9X7
9X8
9X9
9X10
9X11
9X12
0X9
I X9
2X9
3X9
4X9
5X9
6X9
7X9
8X9
9X9
10X9
II X 9
12X9
-act Strategies
Facts
Type o f Fad-
Tens
Description
What? Facts that have 10
as a factor.
How? Count by tens using
arrays.
Example: 10 rows of 10 is
100.
Elevens
What? Facts that have 11
as a factor.
How? Use 10 and add one
more set.
Example: 11 X 12 = 122
(11 X 10) + (11 X 1) = 110 +
12
Twelves
What? Facts that have 12
as a factor.
How? Use 10 and add two
more sets.
Example: 1 2 X 9 = 108
(10 X 9) + (2 X 9) = 90 + 18
Turn-around Facts
Commutative i^roperiy:
Changing the order of the facj-ors makes no difference;
X L
IX •
J
he same.
the result is •
10X0
10X1
10X2
10X3
10X4
10X5
10X6
10X7
10X8
10X9
10X 10
10X11
10X12
0X10
1 XlO
2 X 10
3X10
4X10
5X10
6X10
7X10
8X10
9X10
10X10
11 X 10
12X10
11 XO
11 X I
11 X 2
11 X 3
11 X 4
11 X 5
11 X 6
11 X 7
11 X 8
11 X 9
11X10
11 X I I
11 X 1 2
0X11
1 X 11
2X11
3X11
4 X 11
5X11
6X11
7X11
8X11
9X11
10X11
11 X I I
12X11
12X0
12X1
12X2
12X3
12X4
12X5
12X6
12X7
12X8
12X9
12X 10
12X11
12X12
0X12
1 X12
2X12
3X12
4 X 12
5X12
6 X 12
7X12
8X12
9X12
10X 12
11 X 1 2
12X12
12 X 12 F a c t S t r a t e g i e s
Turn-around F a c t s
Type of F a c t
Threes,
fours, sixes,
sevens and
eights
(The helping
facts)
What? These are the 25
facts l e f t a f t e r the other
strategies are learned.
Actually, oniy 15 facts
remain to master because
20 of them consist of
pairs of turnarounds.
How? They can be learned
by relating each t o an
already known f a c t or
"helping fact."
3X3
3X4
3X6
3X7
3X8
4X4
4X6
4X7
4X8
6X6
6X7
6X8
7X7
7X8
8X8
4X3
6X3
6X4
7X3
7X4
7X6
8X3
8X6
8X4
8X7
Examples:
Threes: Double and one
more set-For example, 7 X
3, Double 7 Is 14. One
more 7 is 21.
Fours: Double and double
again-For exampie, 6 X 4 ,
Double 6 Is 12. Double
again is 24.
Facts with one even
factor: Double & HalvingFor example, 6 X 8, 3
times 8 is 24. Double 24 Is
48.
Any f a c t : Add one more
set. For example, 7 X 6, 5
sevens are 35. One more 7
Is 42.
*Squares are facts with the same digit as both factors (5 X 5). All of these facts are addressed within another
strategy An additional strategy would be to build arrays of these numbers and notice that they are all equal rows
and equal columns, hence the name, squares.
Online Resources for Mathematics
Includes several online
educational games from a monthly
column in the NCTM News
Bulletin.
http ://www. nctm. org/resou rces/content. aspx?id=:2147483782
Online resources for teaching math
that consists of activities, lessons and
more web links.
http://illuminations.nctm.org/
National Library of Virtual Manlpylatives
The National Library of Virtual Manipulatives (NVLM) is an interactive, webbased library of virtual manipulatives and/or concept tutorials for K-12.
http://nlvm.usu.edu/
Ever wonder about how mathematics, science,
technology and engineering is connected to real world
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http://www.thefutureschannel.com/about.php