Welcome to Pre-Algebra
Pre-Algebra is how arithmetic becomes algebraic thinking: how signed numbers behave, how operation structure controls results, and how quantities relate through equations and ratios. Here you will move from integers and order of operations into fractions and decimals, variables and expressions, solving equations, and proportional reasoning—the core toolkit that supports Algebra I.
Each chapter is built to be used, not skimmed: interact with worked examples step by step, open solutions line by line, and try practice problems with mixed difficulty. Follow the numbered lessons in the sidebar at your own pace, take the Practice Exam when you are ready, and use Exam Remediation to focus on what still needs work.
2. Adding and subtracting integers (same sign vs. different signs)
3. Multiplying and dividing integers (when the answer is + or −)
4. Longer problems — go step by step, left to right
Integers are counting numbers, zero, and their opposites: …, $-3$, $-2$, $-1$, $0$, $1$, $2$, $3$, … On the number line, positives are to the right of zero; negatives are to the left.
Absolute value $|x|$ means “how many steps from $0$?” So $|5|=5$ and $|-5|=5$. It is never negative.
Adding integers: Same sign — add the sizes, keep the sign. Different signs — subtract the sizes, use the sign of the bigger-sized number.
Subtracting integers: $a-b$ is the same as $a+(-b)$: add the opposite.
import numpy as np
import matplotlib.pyplot as plt
from matplotlib.animation import FuncAnimation
# Sequence: start at 0, then apply each op. Each step animates the move.
steps = [('+', 5), ('-', 8), ('+', 2), ('-', 4), ('+', 3)]
# Precompute positions before/after each step for framing.
positions = [0]
for op, n in steps:
positions.append(positions[-1] + (n if op == '+' else -n))
fig, ax = plt.subplots(figsize=(9, 2.6), dpi=72)
ax.set_ylim(-1.2, 1.6)
ax.set_xlim(-8, 8)
ax.axhline(0, color='#3d5a7a', lw=1.5)
for k in range(-8, 9):
ax.plot([k, k], [-0.12, 0.12], color='#3d5a7a', lw=1)
ax.text(k, -0.5, str(k), ha='center', va='top', fontsize=10,
color='#3d5a7a', fontweight='bold' if k == 0 else 'normal')
ax.set_yticks([]); ax.set_xticks([])
for s in ('top', 'right', 'left', 'bottom'):
ax.spines[s].set_visible(False)
marker, = ax.plot([0], [0.35], marker='v', color='#c8963e', markersize=18)
op_txt = ax.text(0, 1.15, 'start at 0', ha='center', fontsize=13,
color='#0f2e4a', fontweight='bold')
# Animate: 12 frames per step (smooth glide) + 4 frames pause.
FRAMES_PER = 12; PAUSE = 4
total_frames = len(steps) * (FRAMES_PER + PAUSE)
def update(i):
step_idx = i // (FRAMES_PER + PAUSE)
within = i % (FRAMES_PER + PAUSE)
if step_idx >= len(steps):
step_idx = len(steps) - 1
within = FRAMES_PER + PAUSE - 1
start = positions[step_idx]
end = positions[step_idx + 1]
op, n = steps[step_idx]
if within < FRAMES_PER:
t = within / FRAMES_PER
cur = start + (end - start) * t
else:
cur = end
marker.set_data([cur], [0.35])
op_txt.set_text(f'{start} {op} {n} = {end} (moving {"right" if op == "+" else "left"})')
return marker, op_txt
anim = FuncAnimation(fig, update, frames=total_frames, interval=70, blit=True)
_mp_html = anim.to_jshtml(default_mode='loop')
plt.close(fig)
How far $x$ is from $0$ on the number line (always zero or positive):
Multiplying and dividing integers: Same sign → positive answer. Different signs → negative answer. A negative times a negative is positive.
1. Absolute value
Why learn this?
How far, not which way. Absolute value is great for distance, temperature size, or “how off” a guess was—without worrying about + or −.
Helpful hints
Think “steps from zero.” $|-a|=|a|$. Absolute value by itself never comes out negative.
Remember
Distance from 0 · $|x|\ge 0$ always · Do work inside $|~|$ first if there is a sum inside.
2. Adding & subtracting integers
Why learn this?
Real life uses + and −. Temperature, height above sea level, and money in a bank account all use gains and losses—just like integer addition and subtraction.
Helpful hints
Same vs. different signs. Same sign → add the sizes, keep the sign. Different signs → subtract the sizes, use the sign of the bigger one. Remember: $a-b=a+(-b)$.
Remember
Hop on the number line · Opposite · Try a tiny example on paper if you’re unsure.
3. Multiplying & dividing integers
Why learn this?
Groups and sharing. Multiplication can mean “groups of”; division can mean “split into equal groups.” The +/− rules work the same for both.
Helpful hints
Count the minus signs. Odd number of negatives → negative answer. Even number → positive. For division: same sign → positive, different signs → negative.
Remember
$(-)(-)=+$ · Zero · Never divide by zero; you can flip pairs of negatives to pluses.
4. Mixed integer expressions
Why learn this?
Problems mix operations. You’ll often see $+$, $-$, $\times$, and $\div$ in one expression—take it one step at a time.
Helpful hints
Go in order. Turn subtraction into adding the opposite. Do × and ÷ before + and − (Lesson 2 has the full PEMDAS story). Work left to right when things are “equal priority.”
Remember
One step at a time · Watch every sign — most mistakes here are sign mistakes.