Thinking Skills
Children should be led to make their
own investigations, and to draw upon
their own inferences. They should be
told as little as possible which can
produce unlimited learning potential.
Herbert Spencer
Intellectual Moral and Physical
1864
Research Based Curriculum



Mathematics is more meaningful when it is rooted in real
life contexts and situations, and when children are given
the opportunity to become actively involved in learning.
Children begin school with more mathematical
knowledge and intuition than previously believed.
Teachers, and their ability to provide excellent
instruction, are the key factors in the success of any
program.
Think Algebraically
Math could be spark curiosity!
Is there anything interesting about
addition and subtraction
sentences?
Write two number sentences…
4+2=6
3+1=4
7 + 3 = 10
To 2nd graders: see if you can find some that don’t
work!
How does this work?
Math could be fascinating!
Is there anything more exciting
than memorizing multiplication
facts?
What helps people memorize?
Something memorable!
Let’s multiply 53 x 47

“OK, 53 is near 50”
47
2500
48
49
50
about 50
51
52
53
…OK, 47 is also near 50”
 Actually, they are both 3 units away!

To do…
53
 47
I think…
50  50 (well, 5  5 and …)… 2500
Minus 3  3
–9
2491
But nobody cares if kids can
multiply 47  53 mentally!
What do we care about, then?
50  50 (well, 5  5 and place value)
 Keeping 2500 in mind while thinking 3  3
 Subtracting 2500 – 9
 Finding the pattern
 Describing the pattern

Algebraic thinking
Algebraic language
Science
59=
(7 – 2)  (7 + 2) = 7  7 – 2  2
n
n–d
= 49 – 4
= 45
n+d
(n – d)  (n + d) = n  n – d  d
We also care about thinking!

Kids feel smart!
Teachers feel smart!
 Practice.

It matters!
Gives practice. Helps me memorize, because it’s memorable!

Something new.
Foreshadows algebra. In fact, kids record it with algebraic language!

And something to wonder about:
How does it work?
What could mathematics be
like?
It could be surprising!
Surprise! You’re good at algebra!
A number trick
Think of a number.
 Add 3.
 Double the result.
 Subtract 4.
 Divide the result by 2.
 Subtract the number
you first thought of.
 Your answer is 1!

How did it work?
Think of a number.
 Add 3.
 Double the result.
 Subtract 4.
 Divide the result by 2.
 Subtract the number
you first thought of.
 Your answer is 1!

How did it work?
Think of a number.
 Add 3.
 Double the result.
 Subtract 4.
 Divide the result by 2.
 Subtract the number
you first thought of.
 Your answer is 1!

Why a number trick?
Why bags?
Computational practice, but much more
 Notation helps them understand the trick.

invent new tricks.

undo the trick.
 But most important, the idea that
notation/representation is powerful!

Who Am I?
I. I am even
II. All of my digits < 5
3rd grade detectives!
I.
IV. I am less than 400
V. Exactly two of my digits
are the same.
I am even.
II.
All of my digits < 5
III.
h+t+u=9
IV.
I am less than 400.
V.
III. h + t + u = 9
Exactly two of my
digits are the same.
ht u
h
t
u
1 4 4
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
432
342
234
324
144
414
Representing ideas and
processes
Bags and letters can represent numbers.
 We need also to represent…

— multiplication
 processes — the multiplication algorithm
 ideas
Combinations
Four skirts and three shirts: how many
outfits?
Five flavors of ice cream and four toppings:
how many sundaes? (one scoop, one topping)
How many 2-block towers can you make
from four differently-colored Lego
blocks?
Lesson Components
Math Messages
 Alternative Algorithms
 Mental Math and Reflection
 Explorations
 Games

Arithmetic Tricks

Multiply by 11

Take the original number and imagine a space between
the two digits:
52 x 11

5_2

Now add the two numbers together and put them in the
middle:

5_(5+2)_2

That is it – you have the answer: 572.
Arithmetic Tricks

If you need to square a 2 digit number ending in
5, you can do so very easily with this trick.

Multiply the first digit by itself + 1, and put 25 on
the end. That is all!

25 x 25 = (2 x (2+1)) & 25

2 x 3 & 25
6 & 25
625
Arithmetic Tricks

Multiply by 5

Take any number, then divide it by 2 (in other
words, halve the number). If the result is whole,
add a 0 at the end. If it is not, ignore the
remainder and add a 5 at the end. It works
everytime:
2682 x 5 = (2682 / 2) & 5 or 0
5887 x 5
2682 / 2 = 1341 (whole
number so add 0)
2943.5 (fractional number
(ignore remainder, add 5)
13410
29435
Arithmetic Tricks

Divide by 5

Dividing a large number by five is actually very simple.
All you do is multiply by 2 and move the decimal point:

195 / 5
Step1:
Step2:
195 * 2 = 390
Move the decimal: 39.0
or just 39
Step 1:
Step2:
2978 * 2 = 5956
595.6
= 39

2978 / 5
= 595.6
Puzzle

Suppose you have a list of
numbers from zero to one
hundred. How quickly can you
add them all up without using
a calculator?

HINT: There is a swift way to add these
numbers. Think about how the numbers
at the opposite ends of the list relate to
each other.
Putting It Together Solution

The list contains fifty pairs of numbers that add
to 100
(100+0, 99+1, 98+2, 97+3, etc.)
with the number 50 as an unpaired leftover:
50 X 100 + 50 = 5,050

Challenge:
Using four 4's and any operations, try
to write equations to produce the
values 0 to 10.
Example:
0 = 44 – 44
1=?
10 = ?
FOUR 4’s Puzzle Solution
0 (4+4) – (4+4)
6 ((4+4)/4)+4
1 (4+4–4)/4
7 (4+4) – (4/4)
44/44
44/4–4
2 (4*4)/(4+4)
8 (4*4) – (4+4)
3 (4+4+4)/4
4+4+4–4
(4*4–4)/4
4 (4–4)*4+4
5 (4*4+4)/4
9 (4/4)+4+4
10 (44–4)/4
try to write equations
to produce the values
0 to 100.