Transcript Mathematical Relationships

Mathematical Relationships
Addition is the inverse or opposite of subtraction.
5 + 5 = 10 10 – 5 = 5
Multiplication is repeated addition.
5 X 5 = 25
5 + 5 + 5+ 5+ 5 = 25
Division is repeated subtraction and the inverse or opposite
of multiplication.
20
5 = 4
5 x 4 = 20
1 20 – 5 = 15 2 15 - 5 = 10
3 10 – 5 = 5 4 5 - 5 = 0
Division Symbols
• Remember that Math is a language. Therefore
the symbols that we use have meaning. The
following symbols all mean divide by or represent
division.
•
x
y x
x
y
•
y
Hint: All of these division symbols are read as x
divided by y. The variables x and y can represent
any number.
Multiplication concepts
• Represent 4 X 3 as a model
• We read this math sentence as 4 groups of
three.
• Represent 3 X 4 as a model
•
How is this different
from
4 x 3?
Division Concepts
• Represent 12
3 as a model
• We read this math sentences as 12 divided by
three or four groups of three.
• Represent 12
•
from
4 as a model
How is this different
than 12 divided by 3?
Multi-digit Multiplication
• Use a model to show 12 X 4.
• This means 12 groups of 4.
• This could also be a model for 48
4
Traditional method
12
x4
48
We can look at this at 12 groups of 4,
but using the algorithm we first multiply 4
x 2 and get 8 ones. Then we multiply
the 4 x the 1 ten to get for tens. We then
add the 40 or 4 tens plus 8 ones to get 48.
Traditional method
How does the algorithm look if we have multi-digits numbers on
both top and below?
1
48
X 12
96
1
+480
576
What is happening here? We are asking what is 12
groups of 48. The tricky part here is remembering how
this ties back to our models and place value.
Solve
Use the Traditional method and a
model
253
x45
452
x 76
25
x4
Long Division or Magic Seven
• When we are dividing we are asking how many
times one quantity can go into another quantity.
• If the quantity that we are dividing into is larger
than the quantity we are dividing by we end up
with an answer greater than 1.
• If the quantity that we are dividing into is less
that the quantity we are dividing by we end up
with an answer that is less than one (decimal or
fraction).
Long division and Magic 7
• What we are asking is how many 20 are in 1000.
•
50
20 1000
-100
00
- 0
0
50
20 1000
- 400
600
- 400
200
- 100
100
- 100
0
20
20
5
+ 5
50
Subtraction in the division and Magic 7
• Did you notice in both algorithms whether it was
Magic 7 or Long division they involved
subtraction. This shows the relationship between
division and subtraction.
• Did you notice that by taking the answer from a
division problem and multiplying it by the
quantity you divided this you end up with the
number you were dividing into. This explains the
relationships between multiplication and division.
Solve using either Magic 7 or division
.
26 320 .
.
320 26.
What is the difference between these two.
One results in a number less than one and one results
in an answer greater than 1. Why?
The decimal number is always present even if it is not
written it is inferred that it is there.
Practice
45 627