Root Riddles Slides Root Riddles
Finding Irrational Numbers on the Line
8th Grade Pre-Algebra
Warm-Up Challenge
5 Minutes
On your Root Hunter Handout, complete the Perfect Square Grid as fast as you can!
Target
1 to 144
Skill
Mental Math
Goal
Accuracy
Go!
\[x^2\]
The Square Root Duel
Watch Carefully
Embedded media
Watch from 3:04: Comparing \(\sqrt{120}\) vs \(\sqrt{121}\)
"One of these is an integer... the other is an imposter."
The "Perfect Square" Rule
Rational
If a number inside a square root is a Perfect Square , the answer is an integer.
\(\sqrt{121} = 11\)
Irrational
If it's NOT a perfect square, the answer is a never-ending decimal.
\(\sqrt{120} \approx 10.954...\)
Pro Tip
"If you can't find a whole number that squares to it, it's IRRATIONAL."
Number Line Challenge
Grab your List of Square Roots . Your mission is to find their "Home" between two perfect integers.
Square Root Lower Perfect Square & Upper Perfect Square
Work in Pairs
20 Minutes
Think-Pair-Share
"How do we know \(\sqrt{120}\) is irrational without using a calculator?"
Classify It
Exit Ticket
\(\sqrt{50}\)
Rational or Irrational? Why?
\(\sqrt{49}\)
Rational or Irrational? Why?
Root Hunter Handout Root Hunter
Finding Irrational Numbers in the Wild
Name:
Date:
Warm-Up: The Perfect Grid
Quickly fill in the perfect squares for each base number.
1²
2²
3²
4²
5²
6²
7²
8²
9²
10²
11²
12²
The Number Line Mapping
For each irrational root, identify the two perfect squares it falls between. Then, find those whole number square roots (the "Home Integers").
Irrational Square Root Between Squares... Between Integers... Approx. Value \(\sqrt{5}\) 4 & 9 2 & 3 \(\sqrt{10}\) \(\sqrt{20}\) \(\sqrt{50}\) \(\sqrt{80}\) \(\sqrt{130}\)
The Visual Plot
Challenge 1: \(\sqrt{10}\)
3
4
Estimate where \(\sqrt{10}\) lands by comparing distance between 9 and 16.
Challenge 2: \(\sqrt{50}\)
7
8
Exit Ticket
1. Is \(\sqrt{50}\) Rational or Irrational?
Explain your reasoning using the "Perfect Square Rule".
2. Is \(\sqrt{49}\) Rational or Irrational?
Explain your reasoning using the "Perfect Square Rule".
Root Reference Cards Hunter's Creed
1 Identify the number inside the radical (\(x\)).
2 Find the nearest Perfect Square smaller than \(x\).
3 Find the nearest Perfect Square larger than \(x\).
4 Take the square roots of those to find the "Home Integers."
Example Hunt: \(\sqrt{18}\)
Smaller Perfect Square 16
Target Number 18
Larger Perfect Square 25
"Since \(\sqrt{16}=4\) and \(\sqrt{25}=5\), then \(\sqrt{18}\) is between 4 and 5."
Glue this reference sheet into your notebook or keep it handy during the challenge!
Target List
Map these roots onto your Number Line Handout.
\(\sqrt{5}\)
\(\sqrt{10}\)
\(\sqrt{20}\)
\(\sqrt{50}\)
\(\sqrt{80}\)
\(\sqrt{130}\)
\(\sqrt{15}\)
\(\sqrt{32}\)
Elite Challenge
Estimate \(\sqrt{150}\). Which two whole numbers is it between? Can you get within 0.1 of the actual value?
Root Guide Teacher Resource Root Guide
Teacher Resource & Answer Key
8TH GRADE PRE-ALGEBRA
Topic: Irrational Roots
Instructional Flow
1
Warm-up (5m)
Focus on speed and precision with squares 1-144. Students must internalize these values to "see" where irrationals fall later.
2
The "Gap" (5m)
Use the video to highlight that most roots aren't integers. \(\sqrt{120}\) is the "almost perfect" example to show irrationality.
3
Challenge (20m)
Students use the Root Hunter Handout and Reference Cards . Encourage them to draw the "Perfect Square Homes" on the line.
Discussion Anchors
"Why is \(\sqrt{4}\) rational but \(\sqrt{3}\) isn't?"
Target Answer: 4 is a perfect square (\(2 \times 2\)). There is no whole number that squares to 3, so its root is a non-repeating, never-ending decimal.
"If \(\sqrt{10}\) is between 3 and 4, is it closer to 3 or 4?"
Target Answer: Closer to 3. Because 10 is much closer to 9 (\(3^2\)) than it is to 16 (\(4^2\)).
Answer Key: Number Line Challenge
Root Between Squares Home Integers Approx. Value \(\sqrt{5}\) 4 & 9 2 & 3 ~2.2 \(\sqrt{10}\) 9 & 16 3 & 4 ~3.2 \(\sqrt{20}\) 16 & 25 4 & 5 ~4.5 \(\sqrt{50}\) 49 & 64 7 & 8 ~7.1 \(\sqrt{80}\) 64 & 81 8 & 9 ~8.9 \(\sqrt{130}\) 121 & 144 11 & 12 ~11.4
Exit Ticket Standards
\(\sqrt{50}\): Irrational
50 is not a perfect square. The nearest perfect squares are 49 and 64. Since no integer squared equals 50, its root is irrational.
\(\sqrt{49}\): Rational
49 is a perfect square (\(7 \times 7 = 49\)). Therefore, \(\sqrt{49} = 7\), which is an integer and a rational number.
