Cube3D 立方体数理Maths · Primary school
Why can different fractions name the same amount?
A fraction counts equal parts of a chosen whole. The denominator tells us how many equal parts make that whole; the numerator counts how many are shaded. If we split each part again without adding or removing any shaded area, the numbers change but the fraction still names the same amount. This investigation keeps every bar equally long and every disk equally large.
Repartition the whole without changing the shaded amount
Start with 1/2. Divide each half into three equal smaller parts. There are now 2 × 3 = 6 parts in the whole, and the shaded half contains 1 × 3 = 3 of them. That gives 1/2 = 3/6. The extra dividing lines do not make the shaded region larger. In the experiment, change the split factor while leaving A’s numerator and denominator fixed; the bar and disk show the same boundary before and after repartitioning.
The same reasoning gives n/d = (n × k)/(d × k) for a positive integer k. For example, set A to 2/3 and the split factor to 4: two shaded thirds become eight shaded twelfths, so 2/3 = 8/12. Increasing only the numerator would shade more; increasing only the denominator while keeping the numerator fixed would count fewer of the original-sized amounts. Both numbers must be scaled by the same factor.
A common denominator makes the parts comparable
To compare 2/3 and 3/4, keep the wholes the same size and divide both into twelfths. Each third becomes four twelfths, so 2/3 = 8/12. Each quarter becomes three twelfths, so 3/4 = 9/12. Because eight and nine now count parts of the same size, 8/12 is smaller than 9/12, and therefore 2/3 < 3/4.
Try this directly: set A to 2/3 and B to 3/4, then enable “Show a common denominator”. The model uses the least common multiple of the original denominators; it is 12 here. It does not simply compare the numerators 2 and 3 while ignoring their different units. The independent split factor for A illustrates equivalent fractions; the comparison panel chooses its own common denominator from A and B.
Check the boundaries and test a tempting shortcut
Choose “Zero shaded parts”. Zero fourths, zero sixths and zero twelfths all represent zero of an equal-sized whole. Choose “One whole”: 4/4 and 6/6 both shade the complete whole. These cases also obey the scaling rule. Reset restores A = 1/2, split factor 3 and B = 2/3.
Now choose “What if both numbers go up by 1?” Adding one to both numbers changes 1/2 into 2/3, which is not equal. The common-denominator bars show 3/6 and 4/6. One extra sixth has appeared. Multiplying both numbers by the same positive integer is a different operation from adding the same number to both.
Check your understanding
A is 3/4. Predict the fraction after splitting each quarter into three. Then compare 3/4 with B = 5/6 using the common-denominator bars. Which is larger?
Show the explanation
The equivalent fraction is 9/12. For the comparison, 5/6 = 10/12, so 3/4 = 9/12 < 10/12 = 5/6. The common unit is one twelfth of the same-sized whole.
What the model assumes
This is an exact mathematical area model for fractions from 0 to 1, with original denominators 1–12 and positive integer split factors 1–6. It does not model improper or negative fractions. The dividing strokes are visual guides, not area measurements. Comparisons use integer cross-products, so they do not depend on rounded decimal displays. The interactive lab requires Family Plus or Teacher access; this learning note is public.