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Fraction Equivalence

Alignments to Content Standards: 4.NF.C.5


Explain why $\frac{6}{10} = \frac{60}{100}$. Draw a picture to illustrate your explanation.

IM Commentary

The explanation below is a teacher-level explanation. Students may not articulate every detail, but the basic idea for a case like the one shown here is that when you have equivalent fractions, you have just cut the pieces that represent the fraction into more but smaller pieces. Explaining fraction equivalences at higher grades can be a bit more involved (e.g. $\frac68 = \frac{9}{12}$), but it can always be framed as subdividing the same quantity in different ways.

Note that there is a subtlety with the idea of "equal pieces." It is easiest to show this with a drawing where the "pieces" are represented by congruent figures; later students will deal with figures that are not congruent but have equal area. The figures shown in the solution show congruent pieces, avoiding this complexity.


The picture below shows a square divided into ten equal pieces. The square represents 1. Since six of them are shaded, the shaded part represents $\frac{6}{10}$.


If we divide each of the ten pieces into ten smaller, equal-sized pieces, the square is divided into $10 \times 10 = 100$ equal-sized pieces, and each small piece represents $\frac{1}{100}$.


The six shaded pieces are now each divided into ten pieces as well, so there are $6 \times 10$ shaded pieces. The shaded area represents $\frac{60}{100}$.

Since the area of the square that is shaded hasn’t changed, the two fractions represent the same amount, so the two fractions are equal.

Anthony says:

over 3 years

Starting equivalent fractions using the base of ten is crucial. Without the understanding of this task going further 3/12 = 6/24 would never be understood. Far too often this lesson is often done quickly with little mastery. Using rods and units is a must when teacher fraction equivalence. Stressing breaking the 'whole' into equal size pieces doesn't change the whole at all is why we can compare and say 6/10 = 60/100. If the whole were not the same size one could not compare them. This task is very suitable when introducing equivalence, especially when they are asked to include an illustration.

Mike says:

over 5 years

This is where my class struggled the most trying to understand how to find different equivalent fractions on their own without a visual representation. We camped out on this for a while.

Laurie Flood says:

about 5 years

I have a PDF that appears as though it would help students make connections between the manipulatives and computation, and I also found a MARS lesson I found that would be helpful in summative assessment. I am a graduate student and former teacher working on a Grades 5 - 8 mathematics certification. I would love to speak with you and see how these or other things you have used have worked to help students make this connection. You can contact me via information at the bottom of this website: http://satreadingwriting.wordpress.com/about/

Kristin says:

over 5 years

Did they make progress on it eventually? Can you share any strategies you used to help them make sense of these kinds of problems?