Here is an activity for middle school students that borrows a familiar number sense routine to have students think deeply about operations with fractions.
Have students sit in a circle. Tell them that they are going to count around the circle by 3/8. Ask them to predict what number will be the last one said after everyone has had a turn, what number will be reached halfway around the circle, a quarter way, etc. Also ask, if we were to continue counting around the circle again and again, whether students believe the number 30 will be called out. How about 40? If so, how many numbers do we have to go through until we get to them? If not, why not? Record the predictions and then begin counting, with each student taking a turn. Record the numbers on the board as students say their numbers. Also have students express any answers greater than 1 as whole or mixed numbers (don’t simplify the fractions – patterns will be easier to see if the numbers are left in terms of eighths).
Students will count and record as follows: 0, 3/8, 6/8, 1 1/8, 1 4/8, 1 7/8, 2 2/8, 2 5/8, 3, 3 3/8, 3 6/8, etc.
Go over the predictions and ask students to justify their thinking/reasoning and why they think they were right or wrong. Discuss any patterns observed, as well as the connection to multiplying by fractions (what will be the last number called?) and dividing fractions (will we say 30 – in other words, does 3/8 divide into 30 a whole number of times? What does the quotient tell us?)
Repeat with other fractions, recording and discussing patterns. Students should begin to notice connections between operations and between fractions and whole numbers. They might even notice some patterns – like the 3/8 sequence going up by 3 for every 8 terms – that will help them connect to ratio, rate, and proportional reasoning.