Using visuals is essential for deep mathematical understanding. Today we are going to explore what happens when two fractions are divided.
First, we are going to look at 1/3 ÷ 1/4. Have students represent both fraction tiles. “How many times can 1/4 fit in 1/3?” Have students ponder that questions and discuss with a partner an estimate.
Students should have a sense that since 1/3 is larger than 1/4, it can fit in 1 time with a little bit left over. How can we find out what fraction is left over? Let’s explore! If you line up the twelfth piece, you can see that there is 1/12 left over. But how does this relate to what we are dividing by, the 1/4 piece. If you convert fourths into twelfths, you can see that 1/12 is 1/3 of 1/4. Therefore, 1/3 ÷ 1/4 is 1 and 1/3.
Let’s look at the reverse, 1/4 ÷ 1/3. “How many times can 1/3 fit in 1/4?”
This time, the divisor is larger than the dividend. Students should see that 1/3 can’t fit into 1/4, because 1/3 is larger then 1/4. Therefore, our answer must be less than 1. We have to determine what fraction 1/4 is of 1/3. If we use our twelfth pieces again, we can see that 1/4 = 3/12 and 1/3 = 4/12. “What fraction of 4/12 is 3/12?” It is 3/4 of 4/12. Therefore, 1/4 ÷ 1/3 is 3/4.
Try some more similar examples with your students using the fraction tiles. You will see that they start to really develop that deep mathematical understanding that will stay with them forever.
CCSS: 6.NS.A.1 and 7.NS.A.2
Check for understanding: When you divided a smaller fraction by a bigger fraction, does your answer get bigger or smaller? Explain. When you divided a bigger fraction by a smaller fraction, does your answer get bigger or smaller? Explain.