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During a PD day, some teachers were looking at Page 5 in 4th grade NF Progression (the discussion of equivalent fractions).
The teachers understood how looking at equivalent fractions as partitioning a 2/3 piece into 4 times as many parts of 4 times as many pieces would generate an equivalent fraction of 8/12. Then they asked how one would explain equivalence when beginning with 8/12 and wanting to “simplify” it as 2/3.
While we shared that the goal of this discussion was to see that 8/12 can be renamed as 2/3 using a numberline or area model, we are not sure how one would explain to students that 2/3 is one-fourth as many pieces of one-fourth as many parts–given their limited experience with division of whole numbers. We figured that this discussion must belong in a different grade, but we were unable to determine exactly what grade that would be??
Our initial thoughts:
Grade 5 where division of fractions begins (NF)?
Grade 6 where complex fractions are introduced (NS)?
Grade 6 where division and multiplication of fractions is concluded (NS)?
Any help you can provide would be greatly appreciated!
Thanks for the question!
Fraction equivalence in both directions (either with the numerator and denominator getting larger or smaller) falls into forth grade. Since in third grade there is a standard which says students will be fluent in multiplication and division within 100, students in forth grade will have the foundation to move in either direction for equivalence.
The progression encourages students to think of equivalence by dividing each piece to see a multiple of the total number of pieces as well as a multiple of the selected pieces. We can see equivalence moving from larger numerators and denominators to smaller in much the same way by merging the same number of pieces together so that they are still equal in size. So 8/12 would merge four pieces together into three equal groups to form 2/3.
Check the Illustrative Mathematics Website starting tomorrow (11/8) for an illustration of the fractions progression with 8 modules with videos describing this concept! Links will be on the left hand side.
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