DIRECTIONS: Write R if the number can be written as a fraction of two whole numbers, I if it cannot.
Rational: one whole number over another. Its decimal stops
or repeats (0.25, 0.333…). Irrational: the decimal never stops and never repeats, like
√2 = 1.41421356…
Part A — R or I?
1.
0.101001000100001… (one more zero each time)
2.
√7
3.
√121
4.
√9
5.
0.35
6.
π
7.
√2
8.
4⁄3
9.
22⁄7
10.
√18
11.
−13
12.
0.333333…
When the Fractions Run Out — continued
Grade 8 · Number · Page 2 of 2
√16 = 4, rational. √17 = 4.1231056…, irrational.
Look under the root sign: is it a perfect square?
Part B — Rational or irrational? If it is rational, write what it equals.
13.
√36=
14.
√12=
15.
√3=
16.
√10=
17.
√13=
18.
√49=
Part C — Explain it.
Ella says √25 must be irrational, “because it has a square root sign on it.” Is she right? What is √25, and what should she have looked at instead?
22⁄7 is often used in place of π. Is 22⁄7 rational or irrational? Is π? And are the two numbers equal — how would you check?
A calculator shows √2 = 1.414213562. Square that number by hand. Does it come out as exactly 2? What does that tell you about what the screen was showing you?
Show your work
Daily Language Review — Set 1
Grade 8 · Mixed Review · Page 1 of 2
NameDate
DIRECTIONS: Fix every mistake and write the sentence right on the lines. Number 1 is done.
running the concession stand at halftime were the worst job i ever had... and my sister got paid moreDone for you
Fixed:Running the concession stand at halftime was the worst job I ever had, and my sister got paid more.
the whole locker room was cleaned by us before coach even asked and it took an hour
if i was you i would just tell mom about the grade before she sees it herself
walking home from practice my phone died and i had to guess the way
my brother said he would help — but then he just sat there and watched the whole thing get built by me
Set 1, continued
Grade 8 · Mixed Review · Page 2 of 2
i told her i would try and finish the poster before class but i forgot it at home
first you should mix the paint then you add the water and then the brush should be rinsed between coats
the vending machine was smashed sometime over the weekend and nobody in the office know who did it
staying up late and then blaming the wifi for my grade are not a strategy according to mom
if you want to make the team you have to show up early — practice is run by coach at six and excuses are not accepted by him
Inside a Kind of Piece
Grade 8 · Atoms · Page 1 of 3
NameDate
DIRECTIONS: There are about ninety kinds of atom. What makes one kind different from another? Read, then answer.
An atom has a tiny, heavy center of protons and neutrons,
with electrons around it. The number of protons decides the kind: 1 is hydrogen, 6 carbon, 79 gold.
#
If the center holds…
Which kind is it?
How do you know?
1.
1 proton
2.
6 protons
3.
6 protons and 8 neutrons instead of 6
4.
8 protons
Row 3 is the interesting one. Say what changed and what did not, and why the kind stayed the same.
How Anybody Knows
Grade 8 · Atoms · Page 2 of 3
DIRECTIONS: Nobody has seen inside an atom. Read the 1909 experiment, then answer.
Fast, heavy particles were fired at thin gold foil. Nearly all went
straight through. About one in eight thousand came back.
What does "nearly all of them went straight through" tell you about an atom?
What does the tiny number that bounced straight back tell you? Say why "small" and "heavy" both follow.
science-11-grade6 sheet 30 works out the inside of a planet from earthquake waves that never arrive. Say what the two investigations have in common.
Mostly Nothing
Grade 8 · Atoms · Page 3 of 3
DIRECTIONS: If an atom's center were a marble in the middle of a football field, the atom would reach the stands.
So a table is almost entirely empty space. Your hand stops because the
electrons of one atom push hard on the electrons of the next.
A friend says solids feel solid because their atoms are packed with no gaps. Using this page, say what is wrong and what actually holds your hand up.
Almost all of an atom's mass is in that tiny center. Explain how something so small can be nearly all of the weight.
Say what you would have concluded in 1909 if every single particle had gone straight through with none coming back.
The Square Number Maze
Grade 8 · Games · Page 1 of 2
NameDate
DIRECTIONS: Step only on perfect squares (a whole-number square root). Go across or down.
The wrong ones are within 3 of a square. Know your squares to 22².
START42↓
√25✗
34
√49
83
80
16
12
363
440
255
227
212
170
199
287
257
√25
√256
172
146
147
398
12
√400
122
326
327
98
170
√361
122
192
22
182
FINISH256
Done for you42 = 16 ✓ → 16 = 4 × 4 ✓. √25 = 5 ✗ not that way. Carry on from there.
The Square Number Maze — Questions
Grade 8 · Games · Page 2 of 2
Write the squares you stepped on, each with its root.
42 (4) → 16 (4) →
Which was the largest square number on your route, and what is its root?
Pick a square you could NOT step on. Name the two perfect squares it sits between.
A right-angled triangle has short sides of 5 and 12. The square on the long side is 169. How long is it, and how did the maze help?
Square Roots and Cube Roots
Grade 8 · Number · Page 1 of 2
NameDate
DIRECTIONS: Fill every gap in both tables, then work out Part C.
√144 = 12, because 12 × 12 = 144.
3√125 = 5, because 5 × 5 × 5 = 125.
Part A — The perfect squares. Fill every gap.
n
n²
n
n²
n
n²
1
6
36
11
2
4
7
49
12
3
8
13
4
16
9
14
196
5
10
15
225
Part B — The perfect cubes.
n
1
2
3
4
5
n³
64
125
Part C — Evaluate.
3.
√100 =
4.
√64 =
5.
√25 =
6.
3√−64 =
7.
3√8 =
Square Roots and Cube Roots — continued
Grade 8 · Number · Page 2 of 2
x² = 49 has two answers, 7 and −7. But √49
means only the positive one: 7.
Part D — Solve. Give every answer the equation has.
8.
x² = 196 x = or x =
9.
x² = 225 x = or x =
10.
x² = 121 x = or x =
11.
x³ = −27 x =
Part E — Explain it.
Priya solves x² = 81 and writes x = 9. She is not wrong, but she is not finished. What is missing — and why does √81 = 9 have only the one answer?
In Part C you worked out 3√−64, and it is a perfectly ordinary number. But √−64 is not a number at all. What is the difference between cubing and squaring that causes that?