Division Expressions with Fractions and Mixed Numbers
I can divide a fraction by a fraction and divide with mixed numbers by multiplying by the reciprocal.
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🎯 Content Objective / Objetivo de contenido
I can divide a fraction by a fraction and divide with mixed numbers by multiplying by the reciprocal.
Today's Flow
Total pacing: ~45 min · Progress bar at top tracks your place
LAUNCH
⏱ ~10 min
⏱️ 3 MIN · THINK-PAIR-SHARE
Agent Rivera splits a 3/4-pound evidence bag into 1/8-pound portions. Will she get more or fewer than 1 portion? Estimate before solving.
Check for Understanding #1
Teacher: If >30% thumbs down, re-teach with a fresh example before moving on.
Detective Agency Case File
Agent Rivera is dividing a 3/4-pound evidence bag into portions that each weigh 1/8 pound. She needs to know exactly how many portions she can make for testing. Can you solve the case?
Concept Launch
💡 How do I divide one fraction by another fraction?
When you divide a fraction by a fraction, you keep the first fraction, change ÷ to ×, and flip the second fraction (the divisor) to its reciprocal. Then multiply and simplify.
Dividing Fractions Algorithm (Keep, Change, Flip). Formula: a/b ÷ c/d = a/b × d/c. 1. Keep the first fraction unchanged. 2. Change the division symbol (÷) to multiplication (×). 3. Flip the second fraction to its reciprocal (d/c), then multiply across and simplify
Check for Understanding #2
Teacher: If >30% thumbs down, re-teach with a fresh example before moving on.
Your turn — a mixed number this time
VOCABULARY
⏱ ~8 min
| Term / Término | Meaning / Significado | Example / Ejemplo | Visual |
|---|---|---|---|
| Divide Fractions Dividir fracciones |
Multiply the first fraction by the reciprocal of the second fraction. Multiplicar la primera fracción por el recíproco de la segunda fracción. |
2/3 ÷ 1/4 = 2/3 × 4/1 = 8/3 | |
| Dividend Dividendo |
The total number being divided into equal groups (the number inside the division bracket). El número total que se divide en partes iguales (el número dentro de la casilla de división). |
In 3/4 ÷ 1/2, the dividend is 3/4 — the total amount being split | |
| Divisor Divisor |
The number of equal groups you are dividing into (the number outside the division bracket). El número de partes iguales en las que divides (el número fuera de la casilla de división). |
In 3/4 ÷ 1/2, the divisor is 1/2 — the size of each group | |
| Reciprocal Recíproco |
A fraction turned upside down. Una fracción volteada de arriba abajo. |
The reciprocal of 2/3 is 3/2. To divide by 2/3, multiply by 3/2. | |
| Quotient Cociente |
The answer when you divide. La respuesta cuando divides. |
In 3/4 ÷ 1/4 = 3, the quotient is 3 — three 1/4-size pieces fit into 3/4 | |
| Simplify Simplificar |
To write a fraction with smaller numbers that names the same amount, like 2/4 = 1/2. Escribir una fracción con números más pequeños que nombra la misma cantidad, como 2/4 = 1/2. |
8/12 → GCF is 4 → 8÷4 / 12÷4 = 2/3 |
Which Word Fits?
Keep the first fraction, change ÷ to ×, and flip the ___.
Use It In a Sentence
Check for Understanding #3
Teacher: If >30% thumbs down, re-teach with a fresh example before moving on.
Turn & Talk — Launch
Agent Rivera splits a 3/4-pound evidence bag into 1/8-pound portions. Will she get more or fewer than 1 portion? Estimate before solving.
👂 Listen For
Students recognize the divisor 1/8 is smaller than the dividend 3/4, so the quotient is greater than 1.
Extend: Estimate the exact number of portions and explain your reasoning using equivalent fractions before computing.
EXPLORE & PRACTICE
⏱ ~18 min
Visual Modeling Workspace
Use the drawing tray below to annotate the visual model. Teacher: say "Click to reveal" on key steps.
Explore Activity
Use the bar model to see how many 1/12-pound portions fit into 5/6 pound.
✍️ Explore Discourse
Why does 5/6 ÷ 1/12 = 10? How does the bar model show this?
Whiteboard Moment
Show your work clearly. Be ready to explain your thinking to a partner.
Turn & Talk — Explore
Why do we multiply by the reciprocal when we divide one fraction by another?
👂 Listen For
Students explain that flipping the divisor turns division into multiplication, and they simplify the final quotient.
Extend: Show why 3/4 ÷ 1/8 gives the same answer whether you count eighths in 3/4 or multiply by the reciprocal.
Practice Check A
A recipe needs 2/3 cup of sugar. You only have a 1/6-cup scoop. How many scoops do you need?
✍️ Show Your Work
Explain why your answer is correct using today's vocabulary.
Practice Check B
What is the reciprocal of 3/5?
✍️ Show Your Work
Explain why your answer is correct using today's vocabulary.
Expression Simplify
Complete the interactive activity using today's strategy.
✍️ Justify Your Thinking
Lock 5 — Sort each cracked code: is the quotient a WHOLE NUMBER or NOT a whole number?
A classmate turned in the work below. One step has a mistake. Read every step, find it, name it, and fix it.
Choose ONE option to show what you know — then do it in the workspace below.
Use evidence from today's lesson to complete each frame.
Today's key idea is: "Dividing Fractions Algorithm (Keep, Change, Flip). Formula: a/b ÷ c/d = a/b × d/c. 1. Keep the first fraction unchanged. 2. Change the division symbol (÷) to multiplication (×). 3. Flip the second fraction to its reciprocal (d/c), then multiply across and simplify" — and it works because ___.
Because Divide Fractions means ___, but a tricky part is ___, so I have to ___.
A common mistake with Divide Fractions is ___. It happens because ___, and the fix is ___.
I can prove my answer is correct by ___, using Dividend to check my work.
✍️ TWR · WRITE 3 SENTENCES · 7 MIN
Dividing Fractions Algorithm (Keep, Change, Flip). Formula: a/b ÷ c/d = a/b × d/c. 1. Keep the first fraction unchanged. 2. Change the division symbol (÷) to multiplication (×). 3. Flip the second fraction to its reciprocal (d/c), then multiply across and simplify because ___
Dividing Fractions Algorithm (Keep, Change, Flip). Formula: a/b ÷ c/d = a/b × d/c. 1. Keep the first fraction unchanged. 2. Change the division symbol (÷) to multiplication (×). 3. Flip the second fraction to its reciprocal (d/c), then multiply across and simplify but ___
Dividing Fractions Algorithm (Keep, Change, Flip). Formula: a/b ÷ c/d = a/b × d/c. 1. Keep the first fraction unchanged. 2. Change the division symbol (÷) to multiplication (×). 3. Flip the second fraction to its reciprocal (d/c), then multiply across and simplify so ___
🌱 TWR · GROW THE KERNEL · 6 MIN
Answer these to add detail
Sentence starters (tap to use)
Student Workspace
Use the bar model to see how many 1/12-pound portions fit into 5/6 pound.
| Column A | Column B |
|---|---|
✏️ Sketch Your Strategy
Differentiation Paths
Step-by-step with a worked model and sentence frames.
What is the reciprocal of 3/5?
Core practice aligned to the standard.
Extension with error analysis or multi-step reasoning.
Partner Activity
Work with your partner on the practice problems at your differentiation path level. Explain each step using math vocabulary.
Check for Understanding #4
Teacher: If >30% thumbs down, re-teach with a fresh example before moving on.
Real-World Connection
🌍 Math in the Wild
A carpenter has a board that is 5/6 of a foot long. She needs to cut pieces that are each 1/12 of a foot. She wants to know how many pieces she can cut from the board.
✍️ Connection Reasoning
How does dividing fractions help the carpenter?
The carpenter can cut ___ pieces because 5/6 ÷ 1/12 = ___.
Turn & Talk — Connect
How could dividing fractions help you share food or materials fairly?
👂 Listen For
Students give a fair-share scenario and identify which fraction is the dividend and which is the divisor.
Extend: A classmate flipped the dividend instead of the divisor. Critique this error and explain the correct step.
Summarize: Division Expressions with Fractions and Mixed Numbers
Dividing Fractions Algorithm (Keep, Change, Flip). Formula: a/b ÷ c/d = a/b × d/c. 1. Keep the first fraction unchanged. 2. Change the division symbol (÷) to multiplication (×). 3. Flip the second fraction to its reciprocal (d/c), then multiply across and simplify
In your own words:
CLOSURE & REFLECT
⏱ ~8 min
Today I learned that ___ because ___.
One thing I am still not sure about is ___.
What is 5/6 ÷ 1/12?
Bonus Exit Check
What is 1/2 ÷ 1/4?
✍️ Show Your Work
Explain why your answer is correct using today's vocabulary.
Reflection & Self-Assessment
Continue Learning
Launch the Full Interactive Activity
Students continue practice in the HTML lesson engine with auto-check, hints, and differentiation.
Family Connection
Share tonight's family homework and discuss one vocabulary word at home.
Open Family Homework ↗Teacher Notes
⏱️ Pacing Guide
- Launch & vocab: 12 min
- I Do / We Do / You Do: 15 min
- Explore & practice: 15 min
- Connect & closure: 8 min
Total: ~45 min
🎯 Listen For · Common Errors
• Students recognize the divisor 1/8 is smaller than the dividend 3/4, so the quotient is greater than 1.
• Students explain that flipping the divisor turns division into multiplication, and they simplify the final quotient.
• Students give a fair-share scenario and identify which fraction is the dividend and which is the divisor.
• Listen for students naming a specific strategy tied to 6.NOS.1 — not just "I multiplied." They should connect steps to the key idea.
Common mistake: A common mistake in Divide Fractions is flipping the wrong fraction: students flip the dividend (the first fraction) instead of the divisor (the second fraction). For example, solving 3/4 ÷ 5/8 by flipping 3/4 to 4/3 and keeping 5/8 gives 4/3 × 5/8 = 20/24 = 5/6 — but only the divisor gets flipped. Keep 3/4, change ÷ to ×, and flip only the divisor 5/8 to its reciprocal 8/5: 3/4 × 8/5 = 24/20 = 6/5. Before you submit, check that the fraction you kept (the dividend) is still facing the same way it started.
Answer Key (Teacher Appendix)
Hide this slide during presentation or move to the end of your copy.
✓ Practice 1: 4 scoops — 2/3 ÷ 1/6 = 2/3 × 6/1 = 12/3 = 4 scoops.
✓ Practice 2: 5/3 — To find the reciprocal, flip the fraction. The reciprocal of 3/5 is 5/3.
✓ Practice 3: 2 — 1/2 ÷ 1/4 = 1/2 × 4/1 = 4/2 = 2. Two 1/4-size pieces fit into 1/2.
✓ Practice 4: 6 — 3/4 ÷ 1/8 = 3/4 × 8/1 = 24/4 = 6. Six 1/8-size pieces fit into 3/4.
✓ Exit ticket: 10 — 5/6 ÷ 1/12 = 10. Using the reciprocal, 5/6 × 12/1 = 60/6 = 10. Since 5/6 = 10/12, ten 1/12-foot pieces fit into the board.