Demonstration 1 of 4
Count steps, land on a place
Where does a fraction live, and how do you find it without a formula?
Each curved hop is one step of size 1/b. The dot is where the count of hops stops. The numbers above the line are the marks; the hops are the spaces between them.
\[a/b\]
\[1/4 + 1/4 + 1/4 = 3/4\]
In a/b, the denominator b says how many equal steps make one whole, so each step is 1/b. The numerator a counts how many of those steps you take from zero.
Predict first. Keep 4 steps and switch the step size from fourths to eighths. Will the dot land past 1, at 1, or before 1?
Choose an example
Constructed example: the chapter's quarter-marked number line, with eighths added.
Calculated values
- Step size (denominator)
- 1/4
- Steps taken (numerator)
- 3
- Location
- 3/4
- Same place by division
- 3 / 4 = 0.75
- Marks from 0 to here, counting both ends
- 4
Each step is 1/4 of the whole. 3 steps: 1/4 + 1/4 + 1/4 = 3/4. Division agrees: 3 / 4 = 0.75. Count the 3 spaces, not the marks: 3 spaces use 4 marks including 0. The place is between 0 and 1.
Use the idea
On a measuring cup or a ruler, find the size of one small step first, then count steps from zero, not marks.
Where the conclusion applies
The whole from 0 to 1 is split into equal steps. If the steps are unequal, counting them no longer gives a fraction of the whole.
Check your understanding: On a line marked in sixths, where do 5 steps from zero end, and how many marks have you passed, counting 0?
Chapter 9 source: section "A quantity is also a location". Demonstration C09-D01.
Demonstration 2 of 4
Seven fourths passes one
What happens when the count of fourths is bigger than 4?
Shade fourths in order. Every time four are shaded, one strip is a complete whole. Whatever is shaded in the next strip is the fractional part of the mixed number.
\[4/4 = 1\]
\[\frac{11}{4}=\frac84+\frac34=2\frac34\]
11/4 means eleven copies of 1/4, and 7/4 means seven. 4/4 is one whole. The mixed number 2 3/4 means 2 + 3/4: the space stands for plus, not times.
Predict first. Choose 8 fourths. Will any fourths be left over after the complete wholes?
Choose an example
Constructed example: the chapter's 7/4 and 11/4 examples, with 4/4 and 8/4 as boundaries.
Calculated values
- Fourths counted
- 7
- Complete wholes
- 7 = 1 x 4 + 3, so 1
- Fourths left over
- 3
- Mixed number
- 1 3/4
- Back to fourths
- 1 x 4 + 3 = 7
7/4 = 4/4 + 3/4 = 1 3/4. 3 fourths remain beyond 1, so the place sits between 1 and 2. Reverse it: 1 x 4 + 3 = 7 fourths, the same count you started with.
Use the idea
When a recipe or gauge reads in small units past a whole, group the units into wholes first, then read the leftover.
Where the conclusion applies
All strips are the same length, so every 4/4 is the same whole. If the strips differed in length, two of them would not add to 2.
Check your understanding: Write 9/4 as a mixed number and check it by turning it back into fourths?
Chapter 9 source: section "Seven fourths passes one". Demonstration C09-D02.
Demonstration 3 of 4
One ribbon, two correct names
How can the same piece be one fourth and one half at once?
The ribbon sits between two number lines. The top counts ribbons; the bottom counts meters. The dashed line marks one physical spot that gets a name on each line.
\[2\div4=\frac12\]
The whole ribbon is one whole on the top line. One meter is one whole on the bottom line. 2 / 4 is the length of one piece in meters.
Predict first. With 3 pieces shaded, what does the meters line read at the dashed line?
Choose an example
Constructed example: the chapter's 2-meter ribbon, with a 3-meter ribbon for comparison.
Calculated values
- Length of one piece
- 2 / 4 = 1/2 meter
- Fraction of the ribbon
- 3/4
- Length in meters
- 3 x 1/2 = 1 1/2
- Pieces not shaded
- 1 (1/2 m)
Each piece is 1/4 of the ribbon, and its length is 2 / 4 = 1/2 meter. 3 pieces are 3/4 of the ribbon and measure 3 x 1/2 = 3/2 = 1 1/2 meters. Check: 1 1/2 + 1/2 = 2 meters. Same endpoint, two correct names, because the wholes differ.
Use the idea
Whenever a fraction could refer to two wholes, write the unit beside it: 3/4 of the ribbon, or 1 1/2 meters.
Where the conclusion applies
The four pieces are equal. The ribbon's length is a stated value, not a measurement of anything real. Without a unit, "one half" cannot be checked.
Check your understanding: A 3-meter ribbon is cut into 4 equal pieces. How long are 2 pieces, and what fraction of the ribbon are they?
Chapter 9 source: section "Worked example: a whole can contain several measurement units". Demonstration C09-D03.
Demonstration 4 of 4
One place, more than one name
Why is 4/8 the same place as 1/2 when both numbers changed?
Both strips are the same whole. Cutting every part into n multiplies the count of parts and the count of shaded parts by n, so the shaded length stays put.
\[a/b = (a × n)/(b × n)\]
\[1/2 = (1 × 4)/(2 × 4) = 4/8\]
a/b is the starting fraction. n is a positive whole number: every old part is cut into n equal smaller parts.
Predict first. Start from 3/4 and cut every part into 2. What is the new name, and does the endpoint move?
Choose an example
Constructed example: the chapter's 1/2 = 4/8 and 3/4 = 6/8 renamings, with other factors.
Calculated values
- Starting name
- 1/2
- Factor n
- 4
- New name
- (1 x 4)/(2 x 4) = 4/8
- Part size
- 1/2 becomes 1/8
- Endpoint as a decimal
- 1 / 2 = 0.50 and 4 / 8 = 0.50
(1 x 4)/(2 x 4) = 4/8. There are 4 times as many parts, each 1/4 as big, so the count and the size change together, and the endpoint stays at 0.50 of the whole. Reverse it by dividing both by 4: (4 / 4)/(8 / 4) = 1/2. No piece was added or removed.
Use the idea
To compare 3/4 with 5/8, rename 3/4 with n = 2 as 6/8. Now both count eighths, and 6 eighths reach farther than 5.
Where the conclusion applies
The same whole, the same factor on top and bottom. Adding the same number to top and bottom is not renaming: 2/3 and 4/5 are different places.
Check your understanding: What factor n turns 2/3 into 6/9, and does the place move?
Chapter 9 source: section "The same place can have more than one name". Demonstration C09-D04.