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]





Concrete actions



à

Symbolic representations

à

Formal operations
on symbols

Figure 1
Activity
1: Equal parts: grades P2
Activity
2: Shading parts: grades 24
Activity
3: Equivalent fractions: grades 45
Activity
4: Folding Fraction strips: grades 25
Activity
5: Measuring Fraction strips: grades 68
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".Don’t 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.
Although the outcomes for this activity
are all at the P2 level, it makes a good starting point at any grade level,
to ensure the concept of fractions as equal parts is understood.
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 13 for sixths.
Although the outcomes for this activity
are all at the 24 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.
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.
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 numbertape.(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.
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.