Folding fractions

The following set of activities is intended to develop the concepts of fractions as equal parts of a whole, equivalent fractions, numerators as kinds of fractions, denominators as counts of fractions, comparing fractions, ordering fractions, and fractions as measures. As with any mathematical activity these should be proceeded by a discussion of what the students already know about fractions.On the basis of this discussion it may be possible to abbreviate or skip some of these activities.
The underlying teaching model for these activities is a progression from concrete actions, to pictorial representations, to symbolic representations, to formal actions on symbols (see Figure 1). This progression should proceed at a pace appropriate to the students' understanding.Not all grade levels need to complete the progression to formal operations.

 
 

Folding paper and shading parts
Drawing pictures of folded and shaded paper
Using numbers to represent shaded parts of folded paper
Operating on numbers representing a fraction to arrive at different representations for that fraction
[Folding a piece of paper into four parts and shading three of them]
¾
Concrete actions 
à
Pictorial representations
à
Symbolic representations
à
Formal operations on symbols

Figure 1

Activity 1: Equal parts: grades P-2

Activity 2: Shading parts: grades 2-4

Activity 3: Equivalent fractions: grades 4-5

Activity 4: Folding Fraction strips: grades 2-5

Activity 5: Measuring Fraction strips: grades 6-8

Materials

Activities 1-3: Lots of 8½by 11 paper, blank on one side.Memos from administration work well.
Activities 4-5: Adding machine tapes cut in one metre strips.
It is assumed the students have journals for math in which they record important ideas, problems they are working on, and personal reflections on learning mathematics.



Activity 1: Equal parts: grades P-2

1. Give everyone a sheet of paper and ask them to fold it in half.There are many possibilities, but the three most likely are:

If everyone does it the same way, do it differently and ask if your way is also a half.

2.Ask how two different foldings, that result in different shapes (long rectangles, short rectangles, triangles) can all be a half.Lead a discussion of this question, drawing attention especially to children's comments related to dividing equally, two equal parts, or two parts the same.

3.Have them record the different foldings by drawing them in their math journal with the labels "half" and "½".Donít worry about explaining the parts of ½ yet.

4. (Grade 2+ only) Repeat this process for thirds, fourths, sixths, and eighths.At this level you might ask what the number under the one in a fraction (e.g., the 4 in ¼) represents.You hope someone says something like "The number of equal parts."Have them record that a fraction is one equal part of something, that a half is one of two equal parts, that a third is one of three equal parts, etc. in their math journals.

Outcomes: 

P-A7, 
1-A8, 
2-A4

Although the outcomes for this activity are all at the P-2 level, it makes a good starting point at any grade level, to ensure the concept of fractions as equal parts is understood.

Activity 2: Shading parts: grades 2-4

1. Have everyone fold a sheet of paper into four equal parts.They can reuse their folding from the previous activity.Have them shade one side of the folded up paper and unfold it.They should get something like:

2.Remind them that ¼ and "one fourth" represent one of four equal parts.Ask them to shade two of the four equal parts.Ask them what that would be called and how you would write it in symbols.Have them draw the picture and the symbolic representation in their journals.

3.Repeat 2 for three parts and four parts.

4.Repeat 1-3 for sixths.

Outcomes: 

2-A4, 
3-A3, 
4-A1

Although the outcomes for this activity are all at the 2-4 level, it makes a good bridge for higher grades going from activity 1 to the other activities to ensure the concepts of numerators as kinds of fractions and denominators as counts of fractions are understood.

Activity 3: Equivalent fractions: grades 4-5

1. Have everyone fold a sheet of paper into four equal parts and shade two of them. They can reuse their folding from the previous activity.Ask them what other fraction is the same.Ask them to explain why.The criteria for a good explanation should be that the other students understand it.One possible explanation would use shading like this:

2.Ask them what other fractions are the same as ½.

3.Ask them what fractions are the same as 2/3.

4.Ask them what they notice about the fractions that are the same as ½.You hope they see something like the denominators are always double the numerators.

5.Ask them to find more fractions that are the same as ½ based on this pattern.

6.Repeat 5 for 2/3, ¾ and other fractions chosen by the students.

7.Have them summarise in their journals the relationship between equivalent fractions.

Outcomes: 

4-A6, B6, 
5-A3, C8



Activity 4: Folding Fraction strips: grades 2-5

1. Have everyone fold a one meter long strip of adding machine tape into two equal parts. Ask them to mark one end "0" (zero).If they start there how much of the strip will they have travelled along when they get to the fold?(we hope someone says "one half").Have them mark "½" at that fold.How much of the strip will they have travelled along when they get to the end?(We hope they say "all of it" or "the whole thing" or better yet "2/2").Have them mark "1" at that end.
2.Have them fold the tape in fourths.How much of the strip will they have travelled along when they get to the first new fold?Mark it "¼".Where would 2/4 be?(We hope they say "the middle" or "the half".Mark it "2/4".How much of the strip will they have travelled along when they get to the remaining new fold?Mark it "¾".

3.Repeat 2 for thirds, sixths, and eighths.

4.Ask them to tell you a fraction bigger than ½. Ask for more than one answer.Ask how they know.Relate this to being further along their number-tape.(If they have a number line in the room you might make the connection that their number tape is a close up of the space between 0 and 1 on the number line. )

5.Have them challenge each other to name factions that are bigger than, smaller than, or the same as some fraction.

Outcomes: 

2-A4, 
3-A3, 
4-A1, A5, A6, 
5-A3, A10, C8

 


Activity 5: Measuring Fraction strips: grades 6-8

1. Have everyone measure their strip and mark the length in centimetres (100) at the 1 end. Have everyone measure how far the half fold is from the 0 end.(50 cm.)
2.Ask how much 100 percent is, and how much 50 percent is.They will probably know they are 1 and ½.Point out the connection between the length and the percent.Ask why this happened.

3.Have them mark the percent for the other fractions on their tape.

4.Ask them to mark off tenths and the multiples of 10% on their tape.

5.Ask them what 0.1 is in words (we hope someone says "one tenth" instead of "zero point one")Relate this to the fraction "one tenth" and 10%.

6.Have them add the decimals 0.1, 0.2, 0.3, etc. to their tape.

7.Ask them to put the decimals for other fractions on their tape.

8.Have them summarise in theirjournals the relationship between fractions, percents, and decimals.

Outcomes: 

6-A2, A9
7-A7
8-A8 



[Fractions P-8] [Teaching Elementary Math]
[Copyright -- David A Reid]  [School of Education] [Acadia University]