Neft Teacher · Learn It

Unit 6 · Standard 6.NOS.1

Division Expressions with Fractions and Whole Numbers

Lesson 6-1 · Learn It — how the math works

Why we're learning this

Agent Reyes finds a 3-foot strip of tape at the crime scene. The lab needs the tape cut into 1/4-foot sections for analysis. How many sections can be made? The answer cracks the code to the locked evidence vault.

✱ Math Workbench

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What does it mean to divide by a fraction?

Dividing by a fraction asks one question: how many of that small piece fit inside the whole amount? For 3 ÷ 1/4, we ask how many 1/4-foot pieces fit into 3 feet.

Picture itSee what the math looks like before you read about it

Agent Reyes has a 3-foot strip of evidence tape. For the lab she needs to cut it into sections that are each 1/4 foot long.
Where you see this in real lifeAgent Reyes has a 3-foot strip of evidence tape. For the lab she needs to cut it into sections that are each 1/4 foot long.

Watch me solve itI do — read one step at a time

The problem looks like this
A bar model. 3 whole bars, each labelled 1, sit above a bar of exactly the same length cut into 12 equal pieces, each one fourth of a whole. It shows that 12 pieces of size 1/4 fit inside 3 wholes, so 3 divided by 1/4 is 12.3 ÷ 1/4 = 123 wholes111cut into pieces of 1/4 foot1/41/41/41/41/41/41/41/41/41/41/41/4Count them: 12 pieces of 1/4 fit inside 3 wholes.
  1. I want to cut a 3-foot strip of tape into 1/4-foot pieces, so I write 3 ÷ 1/4. It asks: how many 1/4-foot pieces fit in 3 feet?3 ÷ 1/4
  2. Each foot holds 4 quarter-foot pieces, so 3 feet hold 3 × 4 = 12 pieces.3 × 4 = 12
  3. That is why the shortcut works: dividing by 1/4 gives the same answer as multiplying by its , 4/1. So 3 ÷ 1/4 = 3/1 × 4/1 = 12. (Some people call this Keep, Change, Flip.)3 ÷ 1/4 = 3/1 × 4/1 = 12
  4. Check by multiplying: 12 pieces × 1/4 foot = 12/4 = 3 feet. So 3 ÷ 1/4 = 12 pieces.
  5. One more move to know: mixed number → improper fraction: multiply the bottom by the whole number, add the top, keep the same bottom. 3 1/2 → (2 × 3) + 1 = 7 → 7/2.(2 × 3) + 1 = 7

Watch out — mistakes students make here

  • Taking 1/4 of 5 is multiplication. Division by a fraction asks 'how many pieces fit inside.'
  • Subtraction is not the operation here. Division by 1/4 asks how many of those small pieces fit.
  • Addition combines lengths. To find how many books fit, divide the shelf length by one book's width.

Try it with meWe do — you answer first, then check

Same steps, new numbers.

  1. Let's try 2 ÷ 1/3. The bar model counted 6 thirds inside 2 wholes — now let's see why.2 ÷ 1/3
  2. Each whole holds 3 thirds, so 2 wholes hold 2 × 3 = 6 thirds. That is 2/1 × 3/1, multiplying by the of 1/3.2 × 3 = 6
  3. Another way to see it: write 2 as 6/3. Six thirds ÷ one third = 6, just like 6 apples ÷ 1 apple = 6.
  4. Check by multiplying: 6 × 1/3 = 6/3 = 2. So 2 ÷ 1/3 = 6 — exactly what we counted on the bars.6 × 1/3 = 6/3 = 2

On your ownYou do — solve this one by yourself

Before you start, remember:

  • Now try 1/2 ÷ 4. This one asks how much is in ONE group: half a pan of brownies is shared equally by 4 people.
  • Cut the half into 4 equal parts. Each part is 1/8 of the whole pan, because 8 of those parts would fill the whole pan.
  • The shortcut agrees: dividing by 4 is the same as multiplying by its , 1/4, so 1/2 × 1/4 = 1/8.
  • Check by multiplying: 4 × 1/8 = 4/8 = 1/2.
  • One more: the top number does not have to be a 1. For 3/4 ÷ 2, share 3/4 into 2 equal parts: 3/4 × 1/2 = 3/8, and 2 × 3/8 = 6/8 = 3/4 checks it.

Solve: A shelf is 5 feet long. Books are each 1/4 foot wide. How many books fit on the shelf? Which expression represents this?

My answer

Stuck? Open one hint at a time.

Hint 1

Spot the roles: 5 feet is the whole shelf, and 1/4 foot is the width of one book.

Hint 2

Counting how many books fit means total ÷ piece size — and the count should be bigger than 5, since books are thin.

Hint 3

Set it up as 5 ÷ 1/4, put a 1 under the 5, and flip the divisor to test which choice matches.

Check my answer
✅ Answer: 5 ÷ 1/4 = 20 books
The expression is 5 ÷ 1/4. Dividing by a fraction is the same as multiplying by its reciprocal: 5/1 × 4/1 = 20 books. Check by multiplying: 20 × 1/4 = 20/4 = 5 feet.

Turn & TalkSay it out loud with a partner

Agent Reyes cuts a 3-foot strip of tape into 1/4-foot sections. Before calculating, will the number of sections be more or less than 3? Why?

Try starting with:

  • The number of sections will be ___ than 3 because ___.El número de secciones será ___ que 3 porque ___.
  • Each 1/4-foot piece is ___ than one whole foot, so ___.Cada pieza de 1/4 de pie es ___ que un pie entero, así que ___.
Use these wordsdividenddivisorunit fraction