Prove It Worksheet Prove It! Visual Proofs
Fraction Division Investigation
Student Lab Sheet
Researcher Name
Date
Step 1: The Warm-Up
Draw a bar model (tape diagram) to represent the fraction 4/7. Label your parts clearly.
Step 2: The Visual Proof
The Problem: \( \frac{4}{7} \div \frac{3}{7} = ? \)
Thinking Tip: This question is asking: "How many groups of 3/7 fit inside 4/7?"
1. Draw 4/7 again below. Use the grid to make it precise.
2. Now, circle or bracket groups of 3/7 within your model above.
How many full groups did you find? How much of a group is left over?
Full Groups
Fraction of a Group Left Over
Step 3: Strategy Comparison
The "Video" Way (Algorithm)
Apply "Keep, Change, Flip" to solve:
The "Visual" Way (Proof)
Write the answer you found from your circles and brackets on the first page.
The Big Discovery
Look at your visual model. Why is the answer 4/3 (or \( 1 \frac{1}{3} \))? Explain how you can see both the "4" and the "3" from the final answer in your drawing of groups.
"Math is not about following rules; it's about proving truths."
PRV-IT-01
Reciprocal Revolution Slides Mathematics Investigation
Reciprocal Revolution
Breaking the code of "Keep, Change, Flip"
Grade 5 Advanced
45 Minutes
Warm-Up
On your Prove It Worksheet, locate the Step 1 box.
\( \frac{4}{7} \)
Draw a tape diagram that represents this fraction. Label each part clearly.
Time limit: 5 minutes
The "How" Guide
https://youtu.be/Qhh0W1SL32I
Embedded media
Watch closely. Pay attention to the "Keep, Change, Flip" algorithm.
Quick Discussion
The Procedure
The video showed us how to get the answer quickly.
Is this math magic?
KEEP • CHANGE • FLIP
Our Mission
"But why does flipping a fraction and multiplying work?"
We aren't just calculators. We are investigators.
The Visual Proof
Can you prove this result without using the algorithm?
\( \frac{4}{7} \div \frac{3}{7} = \frac{4}{3} \)
Work with a partner to show how many groups of 3/7 fit into 4/7 on your worksheet.
The Big Connection
In your drawing:
The 4 represented the total pieces you had.
The 3 represented the size of the group you were making.
\( \frac{4}{3} \)
The visual and the algorithm tell the same story!
Reflection Journal
"How does the visual model connect to the math we did in the video?"
Write your thoughts
Sketch a diagram
Reflection Journal Prompts INVESTIGATOR'S LOG
Reciprocal Revolution: Reflection Journal
Subject: FRAC-DIV-05
Lead Researcher
Date of Investigation
01
Connecting the Models
Think back to your tape diagram of 4/7 ÷ 3/7. You found that you could make one full group and then had one third of a group left over, resulting in 4/3.
How does the number of pieces you started with (the 4) and the size of the group you were making (the 3) show up in the algorithm from the video?
02
The "Reciprocal" Secret
If you had to explain to a 4th grader why we "flip" the second fraction (the divisor) when dividing, what would you tell them based on your visual proof today? Hint: Think about what happens to the pieces when we look for "how many groups."
03
Efficiency vs. Proof
The video mentioned that "Keep, Change, Flip" is a fast way to get the answer. When is it better to use the algorithm, and when is it better to draw a visual model?
Use Algorithm When...
Use Visuals When...
Case Closed: Conceptual Mastery Achieved
Reciprocal Revolution Anchor Chart The Reciprocal Revolution
From "How" to "Why" in Fraction Division
The Procedure
The Algorithm
K
KEEP The first fraction exactly as it is.
C
CHANGE Division to Multiplication.
F
FLIP Find the Reciprocal of the divisor.
The Concept
The Visual Proof
"How many groups of 3/7 fit into 4/7?"
1. Start with 4/7
2. Group by 3/7
1 Full Group
1/3 of a group
Result: \( 1 \frac{1}{3} \) or \( \frac{4}{3} \)
WHY?
The Conceptual Bridge
When we FLIP the divisor, we are changing the problem from "pieces of a whole" to "pieces in a group." The numerator (4) is the number of pieces we have, and the denominator (3) is now the number of pieces needed for one full group.
Reciprocal The result of flipping a fraction's numerator and denominator.
Divisor The number we are dividing by (how big our group is).
Quotient The result of division (how many groups we found).
Teacher Investigation Guide Teacher Facilitation Guide
Lesson: Reciprocal Revolution
LEVEL: 5TH GRADE ADVANCED
Learning Objective
Students will move beyond the "Keep-Change-Flip" mnemonic to discover the conceptual proof of fraction division. By the end of the lesson, students will be able to explain why the reciprocal of the divisor represents the number of pieces required to form a "new whole group."
Standards Alignment
CCSS.MATH.CONTENT.5.NF.B.7: Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.MP.3: Construct viable arguments and critique the reasoning of others.
Materials Needed
Prove It Worksheet
Investigation Slides
Reflection Journal
Anchor Chart (Display)
Instructional Flow
05 Minutes
Warm-Up: The Baseline
Students draw 4/7 on a tape diagram. Observation: Ensure students are dividing the whole into seven equal parts and shading four. This is the foundation for the whole lesson.
10 Minutes
Video Investigation
Use the "Strategic Pauses" mentioned in the slides:
0:15: Pause and ask for predictions of the multiplication expression.0:48: Ask for the GCF of 12 and 18 to simplify.1:23: Discuss efficiency vs. brute-force multiplication (4 × 21).
20 Minutes
The "Prove It" Challenge
Students prove \( \frac{4}{7} \div \frac{3}{7} \).
Key Concept: If you have 4 pieces (each size 1/7) and you need 3 pieces to make a "group," you can make 1 full group with 1 piece left over. Since you need 3 pieces for a group, that 1 piece is 1/3 of a group. Total = 4/3.
Answer Key & Teacher Prompts
Visual Proof Result
\( \frac{4}{7} \div \frac{3}{7} = \frac{4}{3} \) or \( 1 \frac{1}{3} \)
Check for: Students circling 3 units of 1/7 as "one group." The remaining 1 unit of 1/7 should be labeled as "1/3 of a group" because the group size is 3.
Critical Thinking Prompt
"Look at the algorithm result (4/3). Where is the 4 in your picture? Where is the 3 in your picture?"
Desired Answer: The 4 is the total shaded parts we have. The 3 is how many parts make up the new 'group' we are dividing by.
