Radical Rivals Worksheet
8th Grade Math Lab
RADICAL RIVALS
Estimating & Comparing Non-Perfect Square Roots
Manual 1.0A
Name:
Date:
Period:
THE BATTLEFIELD: CLASSIFYING NUMBERS
Rational Numbers \\( \frac{a}{b} \\) Form
Any number that can be written as a fraction of two integers. Their decimal representations either terminate (stop) or repeat.
\\( 0.75 \\) (\\(\frac{3}{4}\\)) \\( -5 \\) \\( 0.33... \\) \\(\sqrt{25}\\) (=5)
Irrational Numbers Non-Fractional
Numbers that cannot be written as fractions. Their decimals are infinite and never repeat. Non-perfect square roots are always irrational.
\\( \pi \\) \\( \sqrt{2} \\) (1.41...) \\(\sqrt{12}\\) \\( 1.101... \\)
THE ESTIMATION PLAYBOOK (NEAREST TENTH)
1 Squeeze
Find the perfect squares directly above and below your target radical.
2 Target
Decide which integer boundary the radical is closer to, and guess a tenth.
3 Verify
Multiply to test. Select the tenth whose square is closest to the target.
WORKED EXAMPLE Estimate \\(\sqrt{12}\\) to the nearest tenth
Step 1: Squeeze
\\(9 < 12 < 16\\)
\\(3 < \sqrt{12} < 4\\)
So, \\(\sqrt{12}\\) is between 3 and 4.
Step 2: Target
12 is almost exactly halfway between 9 and 16.
Let's try \\(3.5\\).
Step 3: Verify
\\(3.5^2 = 12.25\\) (diff 0.25)
\\(3.4^2 = 11.56\\) (diff 0.44)
\\(\sqrt{12} \approx 3.5\\)
GUIDED DRILLS: RUN THE PLAYBOOK
DRILL 01: \\(\sqrt{19}\\)
Step 1: Squeeze
< 19 <
< \\(\sqrt{19}\\) <
Step 2 & 3: Target & Verify
Scratchpad: Test your square options below.
Final Answer:
\\(\sqrt{19} \approx\\)
DRILL 02: \\(\sqrt{42}\\)
Step 1: Squeeze
< 42 <
< \\(\sqrt{42}\\) <
Step 2 & 3: Target & Verify
Scratchpad: Test your square options below.
Final Answer:
\\(\sqrt{42} \approx\\)
Unit 2: Real Number System Page 1 of 2
Lab Activities
APPLICATIONS & SHOWDOWNS
RADICAL RIVALS // PART 2
ROUND 1: CLASSIFICATION CHALLENGE
Check the correct classification box for each real number. If a radical can be simplified to an integer, it is rational!
| Number | Rational | Irrational | Justify Your Answer |
|---|
| \\(\sqrt{15}\\) | | | |
| | | |
|
|
| \\(\sqrt{64}\\) |
|
|
|
| \\(0.375\\) |
|
|
|
| \\(-\frac{22}{7}\\) |
|
|
|
| \\(\pi\\) |
|
|
|
ROUND 2: NUMBER LINE TARGET PRACTICE
Approximate each point's value to the nearest tenth, then plot and label points A, B, and C on the number line.
Point A: \\(\sqrt{7}\\)
Squeeze: ____ < 7 < ____
Tenth: ____
Check: ____ \\(\times\\) ____ = ____
A \\(\approx\\)
Point B: \\(\sqrt{18}\\)
Squeeze: ____ < 18 < ____
Tenth: ____
Check: ____ \\(\times\\) ____ = ____
B \\(\approx\\)
Point C: \\(\sqrt{32}\\)
Squeeze: ____ < 32 < ____
Tenth: ____
Check: ____ \\(\times\\) ____ = ____
C \\(\approx\\)
2 3 4 5 6
ROUND 3: THE INEQUALITY ARENA
Compare the values using \\(<\\), \\(>\\), or \\(=\\). Show the decimal approximation you used for your comparison in the box below each pair!
\\(\sqrt{11}\\)
\\(3.3\\)
\\(\approx\\) _______________ \\(=\\) \\(3.30\\)
\\(4.5\\)
\\(\sqrt{21}\\)
\\(=\\) \\(4.50\\) \\(\approx\\) _______________
\\(\sqrt{5}\\)
\\(2\frac{1}{4}\\)
\\(\approx\\) _______________ \\(=\\) _______________
\\(\frac{19}{3}\\)
\\(\sqrt{38}\\)
\\(\approx\\) _______________ \\(\approx\\) _______________
Unit 2: Real Number System Page 2 of 2
Radical Rivals Answer Key
8th Grade Math Lab (Teacher Edition)
RADICAL RIVALS
★ COMPLETE ANSWER KEY & TEACHER GUIDE ★
Verified Key
Name:
TEACHER ANSWER KEY
Date:
EXAM PERIOD
Period:
ALL
THE BATTLEFIELD: CLASSIFYING NUMBERS
Rational Numbers \\( \frac{a}{b} \\) Form
Any number that can be written as a fraction of two integers. Their decimal representations either terminate (stop) or repeat.
\\( 0.75 \\) (\\(\frac{3}{4}\\)) \\( -5 \\) \\( 0.33... \\) \\(\sqrt{25}\\) (=5)
Irrational Numbers Non-Fractional
Numbers that cannot be written as fractions. Their decimals are infinite and never repeat. Non-perfect square roots are always irrational.
\\( \pi \\) \\( \sqrt{2} \\) (1.41...) \\(\sqrt{12}\\) \\( 1.101... \\)
THE ESTIMATION PLAYBOOK (NEAREST TENTH)
1 Squeeze
Find the perfect squares directly above and below your target radical.
2 Target
Decide which integer boundary the radical is closer to, and guess a tenth.
3 Verify
Multiply to test. Select the tenth whose square is closest to the target.
WORKED EXAMPLE Estimate \\(\sqrt{12}\\) to the nearest tenth
Step 1: Squeeze
\\(9 < 12 < 16\\)
\\(3 < \sqrt{12} < 4\\)
So, \\(\sqrt{12}\\) is between 3 and 4.
Step 2: Target
12 is almost exactly halfway between 9 and 16.
Let's try \\(3.5\\).
Step 3: Verify
\\(3.5^2 = 12.25\\) (diff 0.25)
\\(3.4^2 = 11.56\\) (diff 0.44)
\\(\sqrt{12} \approx 3.5\\)
GUIDED DRILLS: RUN THE PLAYBOOK
DRILL 01: \\(\sqrt{19}\\)
Step 1: Squeeze
16
< 19 <
25
4
< \\(\sqrt{19}\\) <
5
Step 2 & 3: Target & Verify
Scratchpad: (19 is closer to 16, test 4.3 and 4.4)
4.4² = 19.36
(diff 0.36)
4.3² = 18.49
(diff 0.51)
Final Answer:
\\(\sqrt{19} \approx\\)
4.4
DRILL 02: \\(\sqrt{42}\\)
Step 1: Squeeze
36
< 42 <
49
6
< \\(\sqrt{42}\\) <
7
Step 2 & 3: Target & Verify
