Transcript + 1 - Dalton State College

Addition and
Subtraction
Do not train children to learning by
force and harshness, but direct them to
it by what amuses their minds, so that
you may be better able to discover with
accuracy the peculiar bent of the
genius of each.
Plato
Arithmetic Today

Arithmetic has generally been learned
through basic algorithms, but it has great
potential through problem solving
techniques.
Current Traditional Algorithm
Addition
1
47
+28
75
“7 + 8 = 15. Put down the 5 and
carry the 1. 4 + 2 + 1 = 7”
Subtraction
7 13
83
- 37
46
“I can’t do 3 – 7. So I borrow from
the 8 and make it a 7. The 3
becomes 13. 13 – 7 = 6.
7 – 3 = 4.”
Expanded Column Method
Number Line Method
Add on Tens, Then Add Ones
46 + 38
46 + 30 = 76
76 + 8 = 76 + 4 + 4
76 + 4 = 80
80 + 4 = 84
Partitioning Using Tens Method
Nice Numbers Method
Lattice Method

First arrange the numbers in a columnlike fashion.

Next, create squares directly under
each column of numbers.

Then split each box diagonally from the
bottom-left corner to the top-right
corner. This is called the lattice.

Now add down the columns and place
the sum in the respective box, making
the tens place in the upper box and the
ones place in the lower box.

Lastly, add the diagonals,
carrying when necessary.
Counting Down Using Tens Method
Partitioning Using Tens Method
Nice Numbers Method
The Counting-Up Method
The Counting-Up Method
Nines Complement
827
- 259
→
→
827
740
+ 1
1568
568
(nines complement)
(to get the ten's complement)
(Drop the leading digit)
Benefits of Alternative Algorithms
 Place value concepts are enhanced
 They are built on student understanding
 Students make fewer errors
Suggestions for Using/Teaching
Traditional Algorithms



We are not saying that the traditional algorithms
are bad.
The problems occur when they are introduced
too early, before students have developed
adequate number concepts and place value
concepts to fully understand the algorithm.
Then they become isolated processes that stop
students from thinking.
Integers

Integers can be easily approached by
thinking in regards of basic
addition/subtraction and determining its
position on the number line
 Is the final result positive or negative?
Integer Addition Rules
 Rule #1 – If the signs are the same, pretend
the signs aren’t there. Add the numbers and then
put the sign of the addends in front of your
answer.
9 + 5 = 14
-9 + -5 = -14
Integer Addition Rules
 Rule #2 – If the signs are different pretend the
signs aren’t there. Subtract the smaller from the
larger one and put the sign of the one with the
larger absolute value in front of your answer.
-9 + 5 =
9 - 5 = 4 Answer = - 4
Larger abs. value
One Way to Add Integers Is
With a Number Line
When the number is positive, count to the right.
When the number is negative, count to the left.
-
+
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
One Way to Add Integers Is
With a Number Line
+3 + -5 = -2
+
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
-
One Way to Add Integers Is
With a Number Line
-3 + +7 = +4
-
-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6
+
Adding Integers with Tiles
We can model integer addition with tiles.
 Represent -2 with the fewest number of
tiles


Represent +5 with the fewest number of
tiles.
ADDING INTEGERS

What number is represented by combining
the 2 groups of tiles?
+3

Write the number sentence that is
illustrated.
-2 + +5 = +3
ADDING INTEGERS

Use your red and yellow tiles to find each sum.
-2 + -3 = ?
+
= -5
ADDING INTEGERS
-6
+
+2
+
-3
+
+
=?
= -4
+4
=?
=
+1
SUBTRACTING INTEGERS


We often think of subtraction as a “take away”
operation.
Which diagram could be used to compute
+3 - +5 = ?
+3
+3
SUBTRACTING INTEGERS
SUBTRACTING INTEGERS

Use your red and yellow tiles to model each
subtraction problem.
-2
- -4 = ?
SUBTRACTING INTEGERS
Now you can
take away 4 red
tiles.
2 yellow tiles are
left, so the
answer is…
-2
-
-4
=
+2
SUBTRACTING INTEGERS
Work this problem.
+3
-
-5
=?
SUBTRACTING INTEGERS
• Add enough red and yellow
pairs so you can take away 5
red tiles.
• Take away 5 red tiles, you
have 8 yellow tiles left.
+3
- -5 = + 8
Why is adding fractions a difficult
concept for students to grasp?


Although children learn addition of whole numbers with
ease, addition of fractions — though conceptually the
same as addition of whole numbers — is much harder.
It requires knowledge of fraction equivalencies.

To add two fractions, you have to know that they
must be thought of in terms of like units.

We take this for granted when we add whole
numbers: 3 + 5 is really 3 ones + 5 ones

— but not when we add fractions: 3 halves + 5
fourths is, for purposes of addition, 6 fourths + 5
fourths.
Let’s Eat Pizza
The pizza is currently 8 pieces
What if I wanted to eat one
eighth of the pizza?
One fourth of the pizza?
One sixteenth of the pizza?
One twelfth of the pizza?
Addition of Fractions

The objects must be of the same type

We combine bundles with bundles and sticks
with sticks.

Addition means combining objects in two or
more sets

In fractions, we can only combine pieces of
the same size

In other words, the denominators must be the
same
Addition of Fractions
Example:
1 3

8 8
+
Click to see animation
= ?
Addition of Fractions
Example:
1 3

8 8
+
=
Addition of Fractions
Example:
1 3

8 8
+
The answer is
(1  3) 4

8
8
=
which can be simplified to
1
2
Addition of Fractions with
equal denominators
More examples
2 1
 
5 5
3
5
6 7
13
3
 
 1
10 10
10
10
6 8
14
 
15 15
15
Addition of Fractions
With different denominators
In this case, we need to first convert them into equivalent fraction with
the same denominator.
Example:
1 2

3 5
An easy choice for a common denominator is 3×5 = 15
1 1 5 5


3 3  5 15
Therefore,
2 23 6


5 5  3 15
1 2 5
6 11
 


3 5 15 15 15
Addition of Fractions
With different denominators
•
When the denominators are bigger, we
need to find the least common
denominator by factoring.
•
If you do not know prime factorization yet,
you have to multiply the two denominators
together.
More Exercises:
3 1 = 3 2 1 = 6 1
 
 
 
4 8
4 2 8
8 8
=
6 1 7

8
8
3 2 = 3 7 2  5
 


5 7
5 7 7  5
=
21 10


35 35
5 4 = 5 9 4 6
 


6 9
69 9 6
=
45 24


54 54
=
=
21  10 31

35
35
45  24 69
15

1
54
54
54
Subtraction of Fractions

Subtraction means taking objects away


Objects must be of the same type

we can only take away apples from a
group of apples
In fractions, we can only take away
pieces of the same size. In other
words, the denominators must be the
same.
Subtraction of Fractions
Example:
equal denominators
11
3

12
12
This means to take away
11
12
11 3 11  3
8 2
 


12 12
12
12 3
3
12
Subtraction of Fractions
More examples:
15 7
15  7
8
1
 


16 16
16
16 2
6 4
69 47
54 28
54  28
26
 





7 9
79 9 7
63 63
63
63
7 11
7  23 11 10
7  23  11 10
161  110





10 23
10  23 23 10
10  23
230
51

230
Did you get all the answers right?
Adding/Subtracting
Fraction Simplification
3 - 1
8
8
8
=
2
1
= 8 = 4
ab
a b


Fraction Addition/Subtraction
c c
c
Adding/Subtracting
11  22 4
11
2 11
11
8
=
+
+
+
=
???
 4
28
7 28
7
28
28
28  7
19
11
+
8
=
=
28
28
ab
a b Denominator
Common
= ??????


Fraction Addition/Subtraction
c
c
c
28
Fractions: Steps for Success
1.
2.
3.
Know the fraction rules and how to apply
them
Show your work and write out each step
Check your work for errors or careless
mistakes