Today I work the students on making connections between the mathematical procedures used to convert mixed numbers to improper fractions and improper fractions to mixed numbers.
It is important for students to really know how these procedures connect to the models because this develops a conceptual understanding, rather than memorization.
As a warm-up, I ask students to refer back to their math journals to an entry they wrote about an encounter with mixed numbers or improper fractions. Then, make a model of the numbers they recorded in that journal.
I will come back to these models at then end of class when the students work on their ticket out.
The focus of today is to help students make connections between models and procedures. I make this clear to the students right at the start of the lesson. Participation and asking questions is encouraged (this is always encouraged, but I state it explicitly today because many students may think it is "easy" complete the procedures, but not ask about the connections).
I ask all students to model 6 and 3/10 on their paper while one student does the same on the board. I choose a student to model on the board who has begun to progress out of needing a model. This student draws each of the wholes and then writes "10" in side, rather than make 10 pieces and shade them in. He is able to explain that the 10 represents the 10 - 10ths in each whole. I choose him to model on the board, because through his approach, the connection to multiplication of the the whole and the denominator is mades more clear.
This student explains that he used 6 wholes with 10 parts in each so he knew that was 60 tens. While he is explaining, I write 60/10 on the board.
Then, he points to the 3/10 from the part of the next whole that is shaded. He said that once he knew there were 60 he added the other 3 to get 63 tenths.
I write 60/10 + 3/10 = 63/10 to help students connect their knowledge of adding fractions to converting fractions to mixed numbers. I make this more explicit by writing 10/10 + 10/10 + 10/10 + 10/10 + 10/10 + 10/10 = 60/10.
Then, I ask students probing questions to consider as a group.
• Why did I write 10/10 6 times in this equation?
• Where did the 10 in the denominator come from?
Next, I ask students to revisit this model (6 wholes broken into 10 parts). Is there another way to determine the total number of parts in these 6 wholes without repeated addition?
Allow students to turn and talk and then share out. Students will notice that they can multiply 6 x 10 to get 60 tenths.
I then point out the short cut (they way their parents do it) multiply the denominator by the whole number, then add the numerator, and keep the denominator the same.
I am careful to comment on how confusing/ meaningless the trick is if you don't know why it works.
Then, I go back through the procedure again, connecting it to the model.
I use gradual release to have students work through the connections between models and procedures at this point. I ask students to work in their groups to model 5 and 5/6, and then try applying a procedure.
While students are working, I move around the room and ask them questions:
• Why did you multiply 5 x 5? Where does this appear in the model?
• When you multiplied 5 x 5 to get 25, what does this mean?
• Why did you add 4/5 to 25/5?
• Can you show this procedure using addition instead?
Then, I rely on interactive modeling to move through the conversion as a group, constantly connecting the algorithm and the model.
Students work in pairs to convert 3 more mixed numbers into improper fractions.
While they are working, I post the same questions I asked during the guided practice on the board as "something to think about"
Learning to convert mixed numbers to improper fractions peaks curiosity in students about working the other way.
Once students have had an opportunity to practice converting mixed numbers, I call them back together for more guided practice, this time about improper fractions.
I use the fraction 63/10 to work backwards from the original mixed number that we started with.
This time, I call attention to the fraction bar, and ask students to turn and talk about what a fraction bar represents. A fraction bar is a sign for division, at the beginning of the year we learned that division can be written 3 ways. Fraction bars was one of them.
I bring the model from the beginning of class back up on the board. As I divide 63 by 10, and get 6 wholes - I connect this to the model. Then, continue dividing to get a reminder 3. I ask students how this 3 connects to the model.
Finally, I ask the students, "What size pieces are these 3 that are left over? How do you know? Where can you see this in the model? where can you find this information in the division problem?
Note: Converting from improper fractions is more familiar to the students, because we have spent so much time with interpreting remainders in division. Therefore, I have students practice one more problem as a small group before allowing them to have more "mixed independent practice".
Students work in pairs to continue to practice converting between mixed numbers and improper fractions. Focusing on the connections between the models and the procedures.
Students return to the improper fraction or mixed number they worked with at the beginning of class. In their journals, they convert the number using models and a procedure. If there is time, they explain how the model and the procedure are connected.