Fractions, Decimals, Percent, Ratio, and Proportion Development

Fraction to decimal to percent connections

Develop connections like:

What are fractions?

A fraction is a relationship of parts to a whole expressed as division and multiplication with a numerator and denominator.

Comparing fractions

If fractions are relationships of parts to a whole expressed with multiplication and division, then how do we compare fractions?

They must be compared by using the same whole. Here are three ways to compare fractions:

  1. If the denominators are the same number, then the denominators are equal and the numerators can be compared to the same whole. For example: 2/3 compared to 1/3. Thirds are the same and two is twice one, therefore 2/3 > 1/3.
  2. If the numerators are the same numbers, then the numerators are equal and the denominators can be compared to a whole.
    • Example 1: 2/3 compared to 2/5. The numerators are the same so compare thirds and fifths to a whole. Thirds are more than fifths, therefore, 2/3 > 2/5.
    • Example 2: Compare 1/3 to 1/4 to 1/2. The numerators are all the same so when the denominators are compared to a whole, fourths are smaller than thirds, and both are smaller than halfs. Therefore, 1/4 < 1/3 < 1/2.
  3. Neither the numerators or denominators are the same. 2/3 & 4/5. They must be compared using the same whole so make the numerators or the denominators the same so they can be compared.
    • Example 1: Compare 2/3 with 4/5. Make the numerators the same, 4/6 & 4/5 and compare the denominators to the same whole. Fifths are more than sixths. Therefore, 4/5 > 4/6.
    • Example 2: Compare 2/3 with 4/5. Make the denominators the same, 10/15 & 12/15 and compare numerators. Ten fifteenths are less than twelve fifteenths. Therefore, 12/15 > 10/15.

The whole matters because fractions are relations.

Equivalent fractions

Use multiples of one to divide or multiply

Chck equivalent fractions by using a/b = c/d, therefore ad/b = c?

Connecting multiplication and division to fractions.

Partative

3/4, or three divided among or partitioned into four groups. (3/4 = 3/4). The equation seems like its saying the same thing so let's make it a more concrete. Three sandwiches shared with four people. Each person gets 3/4 of a sandwich.

Three dollars shared with four people. Each person gets three quarters or 75 cents.

Review with whole numbers.
Partative is thought of as division where a set is put into a certain number of groups.
For example 24 cookies are to be wrapped and put into lunch sacks for a team of eight. How many cookies can be wrapped for each player? (24 / 8 = 3) or (24 / ? = 8)
Or if a set a is put into b groups, it results in c number in each group.
c, is the quotient of a and b. (a / b = c ).
(Quotient is the answer to a division problem)

If divison is introduced as fair sharing it is more likely seen as division and division with multiplication when 3/4 is understood as 3 * 1/4.

 

Quotative and Measurement

Quotative

3/4, or three out of four parts of the same one/whole are shaded. Can be thought of as both multiplication and division. One whole divided into four parts with one-fourth shaded three times. Three-fourths put into one group.

Review with whole numbers.
Quotative is thought of division where a set is put into groups.
For example 12 sandwiches are packaged 3 sandwiches to a bag, resulting in 4 bags.
Or Set a is put into groups the size of b, it results in a certain number of groups, c of size b being formed.
c, is the quotient of a and b. (a / b = c )

 

Measurement

When fractions are experienced as measurement/quotative (shaded parts of a whole) the idea of division and multiplication can be lost.

Review with whole numbers.
Measurement is thought of division where a value or set is used as a unit of measure to measure another group.
If you want teams of five, and you have 20 kids, then how many teams will be needed? Or where set a is measured by the size of b, it results in a certain number of groups, c of size b.
c, is the quotient of a and b. (a / b = c )

Sample problems

3 1/3 / 1/2 = 6 2/3

Solution 1: Quotative

This procedure is easy to memorize. However, it is much harder to model and explain how it works.

  1. Divide 3 1/3 into groups of 1/2. How many groups?
  2. Three whole in half to make 6,
  3. The 1/3 is less than 1/2.
  4. So the answer is between 6 and 7.
  5. How do we find how much 1/3 is of 1/2?
  6. Draw a unit of one, divide it into thirds and mark 1/3 and 2/3.
  7. Then divide the same unit of one in half and mark 1/2.
  8. Notice that the 1/2 of the whole divides the middle third in half. Which tells us that piece is 1/3 of the 1/2 of the whole. This gets tricky so make sure you argree with it because you are going to need to use it soon.
  9. This next part is the tricky part. We want to know how much of the 1/2 of the whole does the 1/3 of the whole represent? Well we just saw what 1/3 of to 1/2 of the whole was so the 1/3 of the whole is twice it or 2/3.
  10. Therefore, 3 1/3 divided into halves will result in 6 2/3 halves.
  11. Another way of thinking about it is the thirds of 1/2 are 1/6 of one whole. Therefore, 2/6 of one = 1/3 but the 2/6 of one is 2/3 of the 1/2. This is fairly confusing, but is easier seen with diagrams.

Solution 2: Measurement

Or how many 1/2 pieces are in 3 1/3? This way takes a 1/2 piece and measures the 3 1/3. Again pretty easy for the 3 (six pieces of 1/2). But when the 1/2 is placed beside the left over 1/3 it becomes harder to see. However, a picture as described above should show that the 1/2 would measure 2/3 of the 1/2.

Solve this one both ways:

3 1/3 / 1/3 (=10)

Sample problem

2 2/3 / 1/4

  1. Ask, how many fourths in two two-thirds? (10).
  2. Can measure with 1/4. Four fourths in each one. For a total of 8., but how many fourths in two-thirds?

Solve with decimals

  1. 2/3 = .66 2/3, and .66 2/3 = .25 + .25 + .16 2/3, and 2/3 = 1/4 +1/4 + 1/6, because .16 2/3 is 2/3 of .25, because .08 1/3 + .08 1/3 + .08 1/3 = .25,
  2. But what is it of one? (.16 2/3) / 1 or (16 2/3) / 100 = 1/6, or how many 1/4 in 1/6 is the same as 1/6 / 1/4 = 2/3. Or 1/4 (4) + 1/4 (4) + 1/6 (4) = 1 + 1 + 2/3.

The answer (2 2/3) is in relation to 1/4 of an inch (first whole to consider), but there is also the inch (second whole to consider).

Another way to think of it is 2/3 / 1/4, find a common multiple /denominator of three and four and get, 8/12 / 3/12, then ask how many 3/12 (1/4) in 8/12 (2/3), 3/12 + 3/12 + 2/12, these three numbers represent the one whole to consider 2/3 put into 2 piles of 1/4 (3/12) and a leftover pile 2/12, To find the value in relationship to the second whole (one) we multiply all three by 4.

4 * 3/12 + 4 * 3/12 + 4* 2/12, and get 12/12 + 12/12 + 8/12, that is 2 2/3.

 

Multiplication.

As in division of fractions, multiplication or fractions requires that two wholes be consider for each problem: A relation of a relation.

Example

1/6 of 1/2, (1/12)

  1. Asks what is 1/6 of one-half
  2. Need to consider one whole to take one-half of.
  3. Need to consider 1/2 as one whole to take 1/6 of.
  4. Then finally need to consider 1/12 as part of one whole that was used to start.

 

Decimals and percent

Decimals and percentage are specific equivalents of fractional relations. Decimals are fractional base-ten equivalents using place value. Percents are relationships based on a one-hundred-part whole.

In addition and subtraction of fractions there is one whole that needs to be considered.

Distributive and quotative division models each are likely to bring different ideas.

Sample problem

A group of six people have five candy bars and want to share them equally. How much would each person get?

a ratio of candy bars to children. Cut each candy bar into six pieces and pass them out to five people (distributed). Each person gets five pieces or 5 x 1/6 or 5/6.

Partitive take five candy bars and cut three in half. If each person takes half there will be two pieces left. Take the two pieces and cut each into thirds. So each person gets 1/2 + 1/3 or 5/6.

Sample problem

I used 2/3 of a can of paint to cover 1/2 of the porch floor. How much paint will be needed for the whole floor?

It is a ratio of 2/3 can to 1/2 floor. It is hard to see that it is distributive since there is only part of a group (the floor).

Measurement quotative division

Sample problem

How many bags of cookies can be make with 12 cookies and three bags?

Shade three of four parts and label. Starting with a quotative model is not the place to start. It does not emphasize the whole and the relationship to the whole is missed.

Sample problem

John is baking a cake and only has a 1/2-cup measuring tool. The recipe calls for 2/3 of a cup of flour. How many times should he fill the 1/2-cup?

This is quotative, not just because it involves measurement. It asks how many 1/2 cups fit in 2/3rds a cup. This is more difficult for children to understand and is usually done by making common denominators.

Eventually students should link the two ideas and see they are the same. Five bars divided by six kids is 5 x 1/6; The 5/6 mark on a strip is also 5 x 1/6. That shows how multiplication and division are related. Fractions are multiplication and division.

Sample problem

Time on a clock. Mark a clock with fourths, thirds, fifths, sixths, and tenths. Try adding different fractions. 1/3 + 1/4 is how many minutes plus how many minutes? 1/3 (20 minutes) + 1/4(15 minutes) = (35 minutes) 7/12.

Or 1 1/4 - 2/3; 1:15 back 40 minutes to 35 or 7/12.

Sample problem

Chris told Pat that s/he was .8 of the way to their goal of each saving a certain amount of money Pat said that s/he had only .5 of what Chris had. What part of their goal did Pat have?

Arrays make a ten by ten grid and shade .4 of .8. Can also show that 1/2 * 4/5 is the same by making a two by ten array.

Algorithms vs. number sense

6/16 * 8/18 (I would switch the numbers and reduce (1/2 * 1/3 = 1/6.)

Chinese teachers would decompose the numbers to expanded notation and multiply the parts like 6 * 8 * 1/16 * 1/18. American teachers teach the algorithm (multiply top numbers and bottom numbers) without conceptual understanding and treat errors as procedural errors of digit manipulation.

Use arrays to multiply. 3/5 * 5/6 Show in arrays three ways. 3x5 and 5x3 showing 3 * 1/5 with 5 * 1/6 shaded to show four different parts of problem and whole. Then switch the directions of the two numbers and repeat. Then show the answer (15) in relation to the whole (30).

Write problems as 2 * 3 * 1/3 * 1/5 for 2/3 * 3/5 …

 

Use distributive property

Problem

3 1/2 * 14;

  1. Think of the problem as (3 + 1/2) * 14; then
  2. (3 * 14) + (1/2 * 14);
  3. 42 + 7;
  4. 49

see sequences

Arrang dots in a triangular pattern with

Three dots in the top row ... for a total of 24.

Can do different multiplication facts and fractions 1/8, 3/24, 24/8, 1/4, 1/6, 6 * 4, 8 * 3,…3/8, It’s like the three is one and the eight is the whole (8 groups of 3 made from the 24). It's really cool how multiplication and division helps us with fractions. (It's the relationship.)

May want to support younger students with grouping of numbers more. E.g. placing four 8-dot dominoes in a rectangular pattern for the same kind of problem (4 * 8)

Or three groups of pennies in clusters of six (die pattern).

Using time and the clock for problems

Problems

  1. My son suggested that a good way to exercise is to jog for 1/3 of an hour and walk for 1/4 of an hour. How much time is that altogether?
  2. Or I could walk for 1/3, jog 1/2, and walk for 1/3. How much time?
  3. Or 1/4, 1/3, 1/4…

Write as reduced fraction and fraction with denominator of 60, could make ratio table.

Clock chart or pie chart.

Again 1/2 + 1/6 = 2/3, 30/60 + 10/60 = 40/60 = 4/6…

Would the clock be useful for sevenths? Make a list of fractions for which the clock would be useful.

Lines

1/4 + 1/5 Lets say that we are going on an imaginary trip. What would be a good distance to use? 100 Draw a line and put 0 and 100 at the ends. Why 100? I thought of 1/4 as 25 and 1/5 as 20. Put on double number line (fractions and whole numbers). How can we add it? 25/100 + 20/100 = 45/100.

I would make the track 20 miles. Again make double number line and put both sets of fractions on both lines. 1/4, 9/20,… 25/100, 45/100,…

What about 10? 2.5 is 1/4 because 5 is 1/2, 2 is 1/5 because 2 * 5 = 10, therefore 2.5/10 + 2/10 = 4.5/10. Put on line with 9/20 and 45/100

2/4 + 1/5

Do with 100, 10, 20,

Open arrays

Are arrays too easily constructed procedurally so that students do not see the conceptual? A person really has to think to see the fractional relationship (an array within an array).

1/3 * 1/4 (array 3 x 4)

2/3 * 1/4 (use same array, is it twice the other? Why?)

2/3 * 3/4 (How does this compare with the two preceding?)

4/3 * 3/2 Make a 3 x 4 array 2 x 3 would be the whole…

Might try 1/2 of 4/3 first, then 2/2 * 4/3, then 3/2 * 4/3

Then do these and transition to swapping numerators and denominators

1/5 * 1/7

3/5 * 4/7

Swapping numerators and denominators

4/5 * 3/7

Or 4 * 3 * 1/5 * 1/7

If only an array is used students will notice that the inside array and the outside array are the same, only rotated 90 degrees, and see that multiplication is commutative (doesn't matter if multiply 1/2 * 2/3 or 2/2 * 2/3) but not understand multiplication of fractions conceptually.

3/8 * 4/9

5/6 * 3/5

4/5 * 5/8

Use to find when swapping is a useful strategy.

Getting rid of a fraction or decimal or percent

Multiplication of fractions

Division

Remember this earlier problem? See quotative and measurement above.

  1. 3 1/3 / 1/2 (multiply both factors by two or double each factor)
  2. 6 2/3 / 2/2
  3. 6 2/3 / 1
  4. 6 2/3

Percentage

Chris wanted to buy a leather coat for $350. Mom said she would help but Chris would have to pay 80%. How much would Chris have to pay?

 

Developing strategies for computation with decimals

All strategies for whole numbers can work with decimals.

Friendly numbers

71.87 + 28.2 = 72.07 + 28 (compensation subtract .2 and add .2)

71.87 + 28.2 = 71.07 + 29 (compensation subtract .8 and add .8)

71.87 + 28.2 = 72 + 28 + .07

Using money

.20 * 9 = (5 * .20 = 1.00 and 4 * .20 = .80) 1.80

.20 * 9 = (.20 * 10 - .20) 1.80

.25 * 9

16 * .25 = 4

How many students would know that a 4x4 array of quarters is 4 dollars?

Using fractions and decimals interchangeably

75 * 80 = 3/4 * 80 * 100 ( because 75 was treated as 75/100, divide 100 need to multiply 100)

1/4 * 80

.25 * 80

25 * 80

1/2 * 60

.5 * 60

.50 * 60

.50 * .60

 

Sample problems

Try to use different strategies to solve these:

1/3 * 1/4
2/3 * 1/4
2/3 * 3/4


1/5 * 1/7
3/5 * 4/7
4/5 * 3/7


3/8 * 4/9
5/6 * 3/5
4/5 * 5/8


6 * 10
12 * 5


24 * 2 1/2
8 * 30
16 * 15


32 * 7 1/2
64 * 3 3/4
18 * 5 1/2


9 * 11


4 1/2 * 22


14 * 3 1/2


4 / 1/2
8 / 1
16 / 1/4
32 / 1/2
64 / 1


5 1/2 / 1/3
16 1/2 / 1
2 1/2 / 1/5

 

Dr. Robert Sweetland's notes
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