Whole Number divided by a Fraction

Over the next few weeks we are going to dive deep into dividing fractions!  I can honestly say I have been teaching fractions for years and while I did understand why the multiplicative inverse worked, I never fully understood what happened when two fractions got divided.  Now, after looking at fraction division using manipulatives and visuals, I can finally say I get it!  It is amazing how much visuals really help in deep understanding.  Today we will be focusing on a whole number divided by a fraction.  I will be using Fraction Tiles which can be purchased here, or you can print and laminate a set of fractions tiles for free here.  

The first problem we will look at is 1 ÷ 1/8.  Or we could say, “How many one-eighth pieces are in 1 whole?”  Have students use their fraction tiles to justify their answer.  Students will see that there are 8 one-eighths pieces in 1 whole.

Try several similar problems using the fraction tiles and have students justify their answer.  For example, 2 ÷ 1/4, 1 ÷ 1/10.

What about 2 ÷ 2/3?  “How many 2/3 pieces are in 2?”  Have students represent 2 and then line up their third pieces.  Students should see how it takes 3 two-third pieces to make 2.

Then ask, “Why is our answer getting so much bigger?”  Have a class discussion on the fact that since the piece we are dividing by is smaller that the dividend, it can fit inside many times.  It is essential that students understand that when you divide a whole number by a fraction, your answer gets bigger.

What happens if the division doesn’t work out evenly?  For example, 2 ÷ 3/4.  In this case we have to look at the number of pieces that are left out.  We are able to make two whole groups with two-fourth pieces left out.  The two-fourths left out is two thirds of what we are dividing by.  Therefore, our answer would be 2 and 2/3.

Once students have a strong understanding, they can solve problems on a worksheet by drawing a model to get their answer.  I would not rush to multiplying by the reciprocal until a deep understanding is formed.

I hope you enjoy this lesson with your students.  I would love your feedback.

~MN

Comparing Fractions and Decimals

Students sometimes have a difficult time understanding how fractions and decimals are related and how they can be compared.  As always, providing a visual is essential for student understanding.  Once students have a strong understanding on how to compare fractions, you can add decimals to the mix.

Begin by having students put together their fraction tiles.  Give students 2 fractions to compare using one of the 5 strategies for Comparing Fractions.  For example, 2/5 and 2/8.  Have a discussion on how 2/5 is larger because both fractions have 2 pieces and fifths are larger than eighths.

Ask, how would the decimal 0.3 compare to those two fractions?  Students should realize that 0.3 is 3/10 and therefore would fall between the two fractions.

Finally, ask students “where would the decimal 0.35 fall?”  Discuss with students who this decimal is half way between 0.3 and 0.4.

This is a common misconception for students and providing that visual is a great way to strengthen student understanding of fractions and decimals.  Have students try several similar problems to strengthen their understanding of how decimals and fractions compare.

What strategies do you have for comparing fractions and decimals?