Development of fractions, decimals, percent, ratio, proportion, their operations, and connections

Development of Fractions

What are fractions?

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

• Fractions are unsolved division problems. 3/5, three divided by five.
• Fractions are equal parts of a whole. 3/5, three equal parts of a whole ...
• Fractions are multiplication and division. 3/5 are a whole divided into fifths and one-fifth multiplied three times.

Young learners assign a fractional value to a part, without systematic reference of the part's values. For example, they may associate one-half to any part of a whole.

Before young learners  conserve area they explore different shapes visually and determine, which are identical and equivalent with visual comparisons. Identical shapes are more accurately compared than those, which are not. The harder it is to compare shapes or parts of shapes by fitting them together visually, then the less accurate they will be. However, many experiences are necessary for students to develop accurate operations to verify or refute visual evidence.

For example: Experiences to spatially determine how two identical shapes can fit into a larger shape. A square can be made with identical rectangles (each 1/2 of the square), two identical triangles (each 1/2 of the square). Concretely at first, and then perceptually manipulating pattern blocks, cutting paper, or folding paper to explore size relationships and how to communicate those relationships.

Discovering how many objects it takes to complete a larger shape entirely, are beginning experiences for young learners to identify the number of cubes required to cover a certain area, which are later used to understand and communicate fractional parts. The number of smaller objects to cover a larger area, becomes the part and the larger area, becomes the whole. Thus, initial experiences with whole numbers, can develop into applications with fractions. One of two tiles can represent 1/2, one of three tiles ... 1/3, & one of four ... 1/4. Where the relationship of one object in two is 1/2 of the two, or group, and these concrete representations help beginning learners to develop their spatial abilities to understand fractions. However, there is much development that is necessary to interchange fractional ideas and link them to the fundamental idea of a fraction as division.

These activities also help learners develop  spatial abilities to rotate and flip shapes and objects to identify fractional parts in more complicated representations.

Learners need many experiences representing fractions in multiple ways: concrete objects, models, pictures, diagrams, words (written and oral), act out, symbols with multiple objects. See instructional ideas, activities, & assessment to develop understanding of fractional values

As learners explore shapes as parts of other shapes, fractional relationships, or parts of a whole are developed and can be communicated as divisions.

Comparing fractions

Learners initially consider the larger of the numbers (numerator or denominator) in a fraction to determine the larger fractions. Later they will develop accurate algorithms as comparisons become more complex. Algorithms such as:

1. If the denominators are the same number, then the denominators are equal, they will compare numerators. For example: 2/3 compared to 1/3. Thirds are the same and two is twice one, therefore 2/3 > 1/3. Learners who do this, may or may not understand the fractional parts are also related to the same or equivalent whole or unit.
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.

Eventually the idea that: The whole matters because fractions are part whole relationships, becomes a solid foundation for comparison.

Since fractions are counter intuitive, in several ways, it is helpful to frequently ask what is one (the unit) and to check their understanding of different values by asking how close the fractional numbers, they are working with, are to land mark numbers, (0, 1/2, 1, or other landmarks).

Addition and subtraction of fractions

Add and subtract fractions of equal parts:

Samples:

• A city block is about 1/12 of a mile. Chris walks three blocks to store, two blocks to the post office and five blocks back home. Home much of a mile did Chris walk? Represent it on a diagram and use it to explain.
• Chris comes to supper and finds that there are three pieces of a pizza, that was cut into four equal parts, left. If Chris takes two, how much of the pizza did Chris take and how much was left? (Both right with little hesitation super, both right or close to being right close, one wrong by a lot or both wrong not yet.
• Use of fractions in real life: Give me three ways you use fractions in your life? Without hesitation super, with hesitation and much thinking close, not able to not yet.

Suggestions for the introduction of addition and subraction of fractional numbers.

• Have students represent problems in multiple ways: concrete objects, models, pictures, diagrams, words (written and oral), act out, symbols.
• Start with concrete or visual representations.
• Introduce problems with real life stories.
• Start with problems that have common denominators.
  
• Create and give problems that include all the different addition and subtraction syntax (Join, separate, part-part-whole, compare or equalize.) Example: Join, with missing end or result unknown. Sandra ate 1/4 of a pie and Jimmy ate 1/4. How much was eaten all together?
• Include problems with mixed numbers along with problems that have common denominators.
• Move to problems with different denominators by doing problems similar to using measuring cups and pour 1/2 cup colored water into a measuring cup with a 1/4 cup oil and ask how much together. Discuss with models, pictures... and use problems from all groups of the different types of addition and subtraction.
• Introduce multiples and ratio table for students to use to find common multiples.
• Introduce mixed problems that have numbers and fractions with common and uncommon denominators.

Equivalent fractions

Use concrete, diagrams, and number lines to represent fractional values for learners to convince themselves fractional values relationships.

Algorithms for learners to develop:

Use multiples of one (2/2, 3/3, 4/4, ...) to divide or multiply to see if numbers are equivalent. Remember one time anything is the anything.

• Is 1/2 equivalent to 2/4, check by multiplying by one (2/2)
  • 1/2 * 2/2 = 2/4;
• Or Is 1/2 equivalent to 2/4, check by dividing by one (2/2)
  • 2/4 / 2/2 = 1/2;

Invert one fraction, multiple, & reduce. Can check to see if fractions are equivalent by knowing:
if a/b is equal to c/d, (equivalent fractions), then (a*d)/b = c?

• Is 1/2 equivalent to 2/4, check
  • (1 * 4)/ 2
  • 4 / 2 equals 2,
  • ∴ 2 = 2 & 1/2 is equivalent to 2/4
• Is 3/5 equivalent to 6/10,
• (3 * 10)/ 5
• 30 / 5 is 6 ∴ yes

Multiplication & division of fractions

Learners operate on fractional numbers with the same strategies as those used with different types of multiplication and division of whole numbers.

As learners become familiar with different ways multiplication and division operate to describe actions and compare numbers they are discovering how numbers are related with these operations. Different models can be used to illustrate and better develop these relationships when solving problem, for example: Cuisenaire models to develop four different types of multiplication of fractions.

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 (distribute, partition, divide). 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 partitive (distribute, partition, divide) since there is only part of a group (the floor).

Partitive

Three fourths, 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 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.
Partitive 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)

 

Quotitive and Measurement

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.

• A pie is cut into four equal pieces. One piece or 1/4 of the pie is removed. There are three equal pieces of one-four left. Three pieces are three times one piece, 3/4 = 3 * 1/4.

Quotative

Three fourths, or three out of four parts of the same one whole are shaded. This 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.
Quotitive 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/quotitive (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: Quotitive

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.

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 1/2 of the whole was so the 1/3 of the whole is twice it or 2/3.

1. Therefore, 3 1/3 divided into halves will result in 6 2/3 halves.
2. 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.