Radical Remodel Slides RADICAL REMODEL
Multiplying Expressions with the Distributive Property
Building Blocks
Recall the Distributive Property:
\(a(b + c) = ab + ac\)
The Radical Rule:
Numbers outside stay outside. Numbers inside stay inside.
\(3 \cdot \sqrt{5} = 3\sqrt{5}\)
\(\sqrt{2} \cdot \sqrt{7} = \sqrt{14}\)
\(2\sqrt{3} \cdot 5\sqrt{2} = 10\sqrt{6}\)
Monomial Remodel
Example 1
\(3(\sqrt{5} + 2)\)
Step 1: Distribute to 1st term
\(3 \cdot \sqrt{5} = 3\sqrt{5}\)
Step 2: Distribute to 2nd term
\(3 \cdot 2 = 6\)
Final Layout: \(3\sqrt{5} + 6\)
Matching Foundations
Example 2
\(\sqrt{2}(\sqrt{6} + 3)\)
1st Distribute: \(\sqrt{2} \cdot \sqrt{6}\) \(\sqrt{12}\)
2nd Distribute: \(\sqrt{2} \cdot 3\) \(3\sqrt{2}\)
WAIT! INSPECTION REQUIRED!
\(\sqrt{12}\) can be simplified to \(2\sqrt{3}\)
Final: \(2\sqrt{3} + 3\sqrt{2}\)
Large Scale Project
The FOIL Method
First
Outside
Inside
Last
\((4 + \sqrt{3})(2 - \sqrt{3})\)
F: \(4 \cdot 2\) \(8\)
O: \(4 \cdot -\sqrt{3}\) \(-4\sqrt{3}\)
I: \(\sqrt{3} \cdot 2\) \(2\sqrt{3}\)
L: \(\sqrt{3} \cdot -\sqrt{3}\) \(-3\)
Combine like terms: \(8 - 3 - 4\sqrt{3} + 2\sqrt{3}\)
\(5 - 2\sqrt{3}\)
Crew Challenge
Simplify the following construction project:
\(\sqrt{6}(3\sqrt{2} - 5)\)
Step 1: Distribute
\(3\sqrt{12} - 5\sqrt{6}\)
Step 2: Simplify
\(3(2\sqrt{3}) - 5\sqrt{6}\)
\(6\sqrt{3} - 5\sqrt{6}\)
Project Summary
Inside x Inside
Numbers under the radical stay under the radical.
Outside x Outside
Coefficients stay outside as coefficients.
Post-Production
Always simplify your radicals after multiplying!
Radical Remodel Worksheet Radical Remodel
Job Site: Multiplying Radical Expressions
Foreman:
Shift Date:
PHASE 1
Monomial Foundations
Distribute the term outside the parentheses. Remember: Numbers outside stay outside, numbers inside stay inside!
\(3(\sqrt{2} + 5)\)
\(\sqrt{5}(\sqrt{3} - 6)\)
\(2\sqrt{7}(\sqrt{2} + 4)\)
\(\sqrt{6}(\sqrt{6} - 3)\)
PHASE 2
Binomial Remodel (FOIL)
Expand the expressions using FOIL. Don't forget to combine like terms and simplify all radicals in your final blueprints!
\((3 + \sqrt{5})(2 + \sqrt{5})\)
\((5 - \sqrt{2})(4 + \sqrt{2})\)
PHASE 3
Advanced Inspections
\((2\sqrt{3} + 1)(\sqrt{3} - 4)\)
Workspace Area:
\((\sqrt{10} + 2)(\sqrt{10} - 2)\)
Workspace Area:
Inspector's Final Challenge
A rectangular building site has dimensions of \((4 + \sqrt{2})\) meters by \((3\sqrt{2} - 1)\) meters. Find the total area of the site in square meters. Give your answer in simplest radical form.
Radical Remodel Answer Key Inspector's Answer Key
PROJECT: RADICAL REMODEL
Status
APPROVED
Phase 1: Monomial Foundations
\(3(\sqrt{2} + 5)\)
\(3\sqrt{2} + 15\)
\(\sqrt{5}(\sqrt{3} - 6)\)
\(\sqrt{15} - 6\sqrt{5}\)
\(2\sqrt{7}(\sqrt{2} + 4)\)
\(2\sqrt{14} + 8\sqrt{7}\)
\(\sqrt{6}(\sqrt{6} - 3)\)
Intermediate: \(6 - 3\sqrt{6}\)
\(6 - 3\sqrt{6}\)
Phase 2: Binomial Remodel
\((3 + \sqrt{5})(2 + \sqrt{5})\)
FOIL: \(6 + 3\sqrt{5} + 2\sqrt{5} + 5\)
\(11 + 5\sqrt{5}\)
\((5 - \sqrt{2})(4 + \sqrt{2})\)
FOIL: \(20 + 5\sqrt{2} - 4\sqrt{2} - 2\)
\(18 + \sqrt{2}\)
Phase 3 & Inspector's Challenge
\((2\sqrt{3} + 1)(\sqrt{3} - 4)\)
FOIL: \(2(3) - 8\sqrt{3} + \sqrt{3} - 4 \rightarrow 6 - 7\sqrt{3} - 4\)
\(2 - 7\sqrt{3}\)
\((\sqrt{10} + 2)(\sqrt{10} - 2)\)
FOIL: \(10 - 2\sqrt{10} + 2\sqrt{10} - 4\)
\(6\)
Final Challenge Solution:
Area = \((4 + \sqrt{2})(3\sqrt{2} - 1)\)
F: \(4 \cdot 3\sqrt{2} = 12\sqrt{2}\)
O: \(4 \cdot -1 = -4\)
I: \(\sqrt{2} \cdot 3\sqrt{2} = 3(2) = 6\)
L: \(\sqrt{2} \cdot -1 = -\sqrt{2}\)
Combine: \(12\sqrt{2} - 4 + 6 - \sqrt{2}\)
\(2 + 11\sqrt{2}\) m\(^2\)
Radical Remodel Teacher Guide Project Manager's Handbook
Lesson Guide: Radical Remodel
Project Type
Algebraic Construction
Project Goal
Master the distribution of radical expressions through monomials and binomials (FOIL).
Timeline
45-60 Minutes
Materials
Slides, Worksheets, Calculators (optional), "Final Inspection" Exit Tickets.
The Construction Blueprint
0-10 min
Foundational Reinforcement
Review the Distributive Property using integers. Introduce the "Radical Rule" (Inside x Inside, Outside x Outside) via Slide 2.
10-25 min
Structure Building
Walk through monomial and binomial examples (Slides 3-5). Emphasize that "Inspection" (simplification) is required at the end of every project.
25-50 min
Job Site Practice
Students work through the "Radical Remodel Worksheet". Circulate the room to check for "building code violations" (common errors).
50-60 min
Final Inspection
Administer the Exit Ticket. Review one tricky problem from Phase 3 together.
Safety Briefing: Common Violations
Illegal Mergers
"Adding an outside number to an inside number."
Wrong: \(2 + \sqrt{3} = \sqrt{5}\)
Correction: These are not like terms!
Forgotten Inspections
"Leaving \(\sqrt{12}\) as a final answer."
Fix: Always check for perfect square factors in the product.
Lost Negatives
"Dropping the sign during FOIL."
Tip: Treat the radical like a variable (\(x\)) for signs.
Radical Square
"\(\sqrt{x} \cdot \sqrt{x} = \sqrt{x^2} = x\)"
Remind students: Squaring a radical cancels it out!
Radical Remodel Exit Ticket Final Inspection
Exit Ticket: Radical Remodel
Crew Member:
Complete this final task to clear the job site for the day. Show all distributive steps.
1
\(2\sqrt{3}(\sqrt{6} + 4)\)
2
\((\sqrt{5} + 2)(\sqrt{5} - 3)\)
Self-Inspection: I simplified my radicals.
Final Inspection
Exit Ticket: Radical Remodel
Crew Member:
Complete this final task to clear the job site for the day. Show all distributive steps.
1
\(2\sqrt{3}(\sqrt{6} + 4)\)
2
\((\sqrt{5} + 2)(\sqrt{5} - 3)\)
Self-Inspection: I simplified my radicals.
