"But I taught him about dividing fractions and how to make an
improper fraction and a mixed number."
Real understanding of similarity between a mixed fraction and
improper fraction is ambiguous in this and the next question.
"Sprint" (2 min; introduced in Lesson 10) provided strategic, speeded practice on four measurement interpretation topics: identifying whether fractions are equivalent to 1/2; comparing the value of proper fractions; comparing the value of a proper and an
improper fraction; and identifying whether numbers are proper fractions,
improper fractions, or mixed numbers.
The most striking result was the comparison between items that presented a fraction that was greater than one as an
improper fraction and those presented as mixed numbers (t(27) = 7.902, p < 0.001).
In an
improper fraction the numerator of the fraction is larger than the denominator.
After the conceptual lesson, Rachel was again interviewed, and when asked to convert a mixed number to an
improper fraction, she incorrectly applied a procedure before she corrected herself by drawing a picture.
An
improper fraction is a fraction whose numerator is larger than its denominator, and whose value is greater than a whole unit.
She used the calculator to connect fractions and decimals by computing the exact answer to an
improper fraction. Data reduction yielded the following meta-categories for materials: 1) seatwork/worksheets, 2) group work/ worksheet, 3) dry erase boards, 4) overhead 5) fraction pie-pictorial representations, 6) grid models, 7) student-made pictorial representations, 8) pattern blocks, 9) paper folding models, 10) calculators, 11)fraction strips, 12) tiles, 13) fraction bars, and 14) base ten blocks.
For example, a problem-solving behavior to solve a fraction question such as "adding mixed fractions" could be as follows: converting a mixed number to an
improper fraction, reducing fractions to a common denominator, adding fractions with common denominators, and converting an
improper fraction to a mixed number.
This error is a student-developed modification of the poorly understood short-cut algorithm for renaming a mixed number as an
improper fraction (see Category I example E).
Traditional instruction for such a problem would involve finding a common denominator, adding to create an
improper fraction, then dividing to find wholes, as shown in figure 1.
In the game Finding Fractions, students roll two number cubes to form fractions to fit a particular description, such as "greater than 1/2" or "
improper fraction." A reflective piece asks the students to think about strategies used while playing.