3rd grade to Percents. At each step we will look at several examples of what
I see as truly important to be part of our students understanding.
3rd Grade: In this grade we look at reducing fractions. What we are doing is factoring out values of 1. EXAMPLE: 6 ÷ 15 OR 6/15. Is there a number that is a FACTOR of both 6 and 15.
Lets see!
Lets see!
3 x 2 = 6
3 x 5 = 15 thus 3 is a factor of both!
We factor out 3/3 =1 and we are left with 2/5
THUS 6/15 = 2/5
HOW DO WE KNOW WE ARE RIGHT!
6 x 5 = 2 x 15
30 = 30 we are correct!
Not only do we first see the power of factoring, but we also see how we
know that we have done our work correct, because the cross products are =.
The only reason a proportion is correct. Our students also see that 1 times
any other number does not change the value of that number, a powerful
concept.
Our students have written their first proportion 6 is to 15 as 2 is to 5.
And they know it is true by multiplying on the diagonals.
EXAMPLE: Reduce 4 ÷ 12 or 4/12. As above what is the largest factor of 4 and 12.
4 x 1 = 4
4 x 3 = 12 thus 4 is a factor of both 4 and 12
We factor out 4/4 = 1 and we are left with 1/3
THUS 4/12 = 1/3
4 is to 12 as 1 is to 3 and we can check that we are correct
4 x 3 = 12 x 1
12 = 12 We are correct!
We can also know when we have made an error.
EXAMPLE: Reduce 12 ÷ 20 or 12/20. Now we look for our factor.
4 x 3 = 12
4 x 5 = 20
12/20 = 4/5 But 20 x 4 ≠ 12 x 5
80 ≠ 60 oops!
12/20 = 3/5
3 x 20 = 12 x 5
60 = 60 we are correct!
Our next stop is in the 4th and 5th grade as we add and subtract fractions
and start doing unit pricing.
4TH AND 5TH GRADE
UNIT PRICING OR UNIT COST: It was not long ago that I saw 2
different company's cans of peas in a grocery store. The cans held the same volume
of peas, and were exactly the same size.
The first had sale! 3 cans for a dollar
The second read $.25 per can
The ones that had the sale were almost sold out, while the others were almost
untouched. I asked the grocer if they had filled the $.25 group up. He said
no, but that they were equal in numbers when they opened in the morning.
Lets think
3/$1.00 = 1/$.33⅓ HOWEVER 1/$.25 = 4/$1.00
So, we're either unable, or unwilling to do some thinking that would save us
money, or get us another can of peas.
When we buy many of the same thing for a certain cost, it is vital that we
understand how much it costs for one of those items.
Buy teaching PROPORTIONS, we get a generation that thinks in
PROPORTIONS and becomes better consumers.
1. 10 pounds of potatoes cost $6.30. How much
does 1 pound cost?
10/$6.30 = 1/c or $6.30/10 = c/1
1 x $6.30 = 10 x c $.63 = c
$6.30 ÷ 10 = c
$.63 = c
Fractions: Now for the fun stuff, adding and subtracting fractions.
The rules are simple. In order to ADD OR SUBTRACT fractions
we must have a COMMON DENOMINATOR.
That's the number on the bottom the one on top is the NUMERATOR
Example: ⅓ + ⅓ = ⅔ It is like 1 apple + 1 apple = 2 apples
The denominator stays the same.
The student must learn to read the problem
ONE–THIRD + ONE–THIRD = TWO–THIRDS
Easy doings when the denominators are the same, but what is if they are not?
½ + ⅓ = ?
We need to fix those denominators. To do it we build PROPORTIONS
½ = x/6
+ ⅓ = y/6
Remember our rule of proportions, "the cross products must be equal"
2 ⋄ x = 1 ⋄ 6 and 3 ⋄ y = 1 ⋄ 6
x = 3 and y = 2
3/6 + 2/6 = 5/6
How do I find the common denominator?
It is the smallest number that both of the numbers go into equally.
Example: 3/4 = a/common denominator
+ 1/6 = b/common denominator
4 x 6 = 24 so 24 could be the common denominator, but 4 and 6 go into 12
and we want the smallest number. So we will chose 12.
3 x 12 = 36 which means 4 x a must = 36, a = 9
1 x 12 = 12 which means 6 x b must = 12, b = 2
Thus 9/12 + 2/12 = 11/12!
Example: 7/8 = a/common denominator
– 1/3 = g/common denominator
Again 8 x 3 = 24 but in this case 24 is the lowest common denominator.
7 x 24 = 168 and so must 8 x a = 168. However we can make it easier
and with this process have a way of solving proportions.
7/8 = a/24 divide 24 by 8 to get 3, then 3 x 7 = 21
1/3 = g/24 divide 24 by 3 to get 8, then 8 x 1 = 8
Thus 21/24 – 8/24 = 13/24
Converting fractions to Percents:
This is now a natural conversion as a
percent is part of the number 100
Consider the fraction: 3/5
3/5 = g/100
Remember above, 5 x g = 3 x 100, or 5 goes into 100, 20
times, and 20 x 3 = 60. g = 60%
g is part of 100, thus g is a percent.
g= 60%
We have solved a proportion and at the same time converted a fraction
to a percent. This basic law of proportions that says that the product of
the diagonals must be equal has taken us from reducing fractions to the
concept of percents. We have only scratched the surface of its use in
Math. However, this fundamental idea needs to be stressed year after
year.
Not only do we first see the power of factoring, but we also see how we
know that we have done our work correct, because the cross products are =.
The only reason a proportion is correct. Our students also see that 1 times
any other number does not change the value of that number, a powerful
concept.
Our students have written their first proportion 6 is to 15 as 2 is to 5.
And they know it is true by multiplying on the diagonals.
EXAMPLE: Reduce 4 ÷ 12 or 4/12. As above what is the largest factor of 4 and 12.
4 x 1 = 4
4 x 3 = 12 thus 4 is a factor of both 4 and 12
We factor out 4/4 = 1 and we are left with 1/3
THUS 4/12 = 1/3
4 is to 12 as 1 is to 3 and we can check that we are correct
4 x 3 = 12 x 1
12 = 12 We are correct!
We can also know when we have made an error.
EXAMPLE: Reduce 12 ÷ 20 or 12/20. Now we look for our factor.
4 x 3 = 12
4 x 5 = 20
12/20 = 4/5 But 20 x 4 ≠ 12 x 5
80 ≠ 60 oops!
12/20 = 3/5
3 x 20 = 12 x 5
60 = 60 we are correct!
Our next stop is in the 4th and 5th grade as we add and subtract fractions
and start doing unit pricing.
4TH AND 5TH GRADE
UNIT PRICING OR UNIT COST: It was not long ago that I saw 2
different company's cans of peas in a grocery store. The cans held the same volume
of peas, and were exactly the same size.
The first had sale! 3 cans for a dollar
The second read $.25 per can
The ones that had the sale were almost sold out, while the others were almost
untouched. I asked the grocer if they had filled the $.25 group up. He said
no, but that they were equal in numbers when they opened in the morning.
Lets think
3/$1.00 = 1/$.33⅓ HOWEVER 1/$.25 = 4/$1.00
So, we're either unable, or unwilling to do some thinking that would save us
money, or get us another can of peas.
When we buy many of the same thing for a certain cost, it is vital that we
understand how much it costs for one of those items.
Buy teaching PROPORTIONS, we get a generation that thinks in
PROPORTIONS and becomes better consumers.
1. 10 pounds of potatoes cost $6.30. How much
does 1 pound cost?
10/$6.30 = 1/c or $6.30/10 = c/1
1 x $6.30 = 10 x c $.63 = c
$6.30 ÷ 10 = c
$.63 = c
Fractions: Now for the fun stuff, adding and subtracting fractions.
The rules are simple. In order to ADD OR SUBTRACT fractions
we must have a COMMON DENOMINATOR.
That's the number on the bottom the one on top is the NUMERATOR
Example: ⅓ + ⅓ = ⅔ It is like 1 apple + 1 apple = 2 apples
The denominator stays the same.
The student must learn to read the problem
ONE–THIRD + ONE–THIRD = TWO–THIRDS
Easy doings when the denominators are the same, but what is if they are not?
½ + ⅓ = ?
We need to fix those denominators. To do it we build PROPORTIONS
½ = x/6
+ ⅓ = y/6
Remember our rule of proportions, "the cross products must be equal"
2 ⋄ x = 1 ⋄ 6 and 3 ⋄ y = 1 ⋄ 6
x = 3 and y = 2
3/6 + 2/6 = 5/6
How do I find the common denominator?
It is the smallest number that both of the numbers go into equally.
Example: 3/4 = a/common denominator
+ 1/6 = b/common denominator
4 x 6 = 24 so 24 could be the common denominator, but 4 and 6 go into 12
and we want the smallest number. So we will chose 12.
3 x 12 = 36 which means 4 x a must = 36, a = 9
1 x 12 = 12 which means 6 x b must = 12, b = 2
Thus 9/12 + 2/12 = 11/12!
Example: 7/8 = a/common denominator
– 1/3 = g/common denominator
Again 8 x 3 = 24 but in this case 24 is the lowest common denominator.
7 x 24 = 168 and so must 8 x a = 168. However we can make it easier
and with this process have a way of solving proportions.
7/8 = a/24 divide 24 by 8 to get 3, then 3 x 7 = 21
1/3 = g/24 divide 24 by 3 to get 8, then 8 x 1 = 8
Thus 21/24 – 8/24 = 13/24
Converting fractions to Percents:
This is now a natural conversion as a
percent is part of the number 100
Consider the fraction: 3/5
3/5 = g/100
Remember above, 5 x g = 3 x 100, or 5 goes into 100, 20
times, and 20 x 3 = 60. g = 60%
g is part of 100, thus g is a percent.
g= 60%
We have solved a proportion and at the same time converted a fraction
to a percent. This basic law of proportions that says that the product of
the diagonals must be equal has taken us from reducing fractions to the
concept of percents. We have only scratched the surface of its use in
Math. However, this fundamental idea needs to be stressed year after
year.
