Chapter 2 of Algebra 1: Concepts and Skills, built for a visual learner. The real number line, absolute value as distance, adding and subtracting as jumps, the negative-factor count for multiplying and dividing, the distributive property as an area model, and combining like terms with algebra tiles.
Subject: Algebra 1 · 62 slides · symbolic lesson
Open the interactive version of this deck · Homework for this lesson
Title
Algebra 1 · Chapter 2
The number line, absolute value, signed arithmetic, and the two properties that power all of algebra
Objectives
Chapter 1 gave you letters. This chapter makes sure the numbers underneath them behave, including the negative ones.
Figure (svg): A roadmap of chapter two from the number line through to dividing real numbers
Section
Section 2.1
Concept
The real numbers are all the numbers that fit on a line with no gaps: whole numbers, fractions, decimals, and everything between.
Figure (svg): A number line from minus five to five with negative numbers shaded blue on the left and positive numbers shaded green on the right
opposites — Two numbers the same distance from zero on opposite sides, such as negative three and three. Their sum is always zero.
Picture it
You never have to reason about which of two numbers is larger. You only have to look at which one is further right.
Figure (svg): A number line with minus five and minus two marked, showing minus five sits to the left and is therefore smaller
The rule that feels wrong — that negative five is less than negative two — is obvious once you stop reading the digits and start reading the position.
Prediction
Commit before you reason it out.
\[ -7 \quad \text{versus} \quad -3 \]
Predict first
Which number is greater, negative seven or negative three?
Correct: Negative three is greater, because it sits further to the right on the number line.
Why: The digits pull you the wrong way here: seven is bigger than three, so negative seven feels bigger too. But negative seven is seven units to the left of zero and negative three is only three units left, so negative three is the one further right and therefore the larger number. A thermometer helps: minus three degrees is warmer than minus seven.
Ranking
Order these five numbers from smallest to largest. Position on the line decides, not the size of the digits.
Put in order
Why: Reading left to right along the line: negative eight is furthest left, then negative six, then negative one point five, then zero, then two. Among the negatives the ordering feels reversed because the bigger the digits the further left you are, which is exactly the habit this exercise is built to retrain.
Real world
Negative numbers are not an exam invention. Three places you already meet them:
Discussion prompt
For temperature, bank balances and elevation, say what zero means in each case and what a negative value means.
Hint: In each case, what event or place was picked as the zero?
Answer:
Temperature: zero is the freezing point of water, and negative means colder than that.
Bank balance: zero is owing nothing, and negative means you owe money.
Elevation: zero is sea level, and negative means below it.
In every case zero is a chosen reference point rather than nothing at all — which is exactly what the origin on a number line is.
Section
Section 2.2
Concept
The absolute value of a number is how far it sits from zero, with the direction thrown away.
Figure (svg): A number line showing that both minus four and positive four sit four units away from zero, with two curved distance arrows
\[ |-4| = 4 \qquad |4| = 4 \qquad |0| = 0 \]
Worked example
Evaluate the expression below.
\[ |-6| + |2 - 9| \]
Treat each pair of bars as a grouping symbol
Why: Absolute value bars sit on the top rung of the ladder with brackets, so whatever is inside them is finished before the bars are removed.
\[ |-6| + |-7| \]
Replace each absolute value by its distance
Why: Distance is never negative, so both bars come off as positive numbers regardless of the sign inside.
\[ 6 + 7 \]
Add
Why: Only ordinary addition is left.
\[ = 13 \]
Figure (svg): A number line marking minus six and minus seven with their distances from zero shown as six units and seven units
Verify: recompute the inside of the second bars
Why: Two minus nine is negative seven, and the distance from zero to negative seven is seven. Six plus seven is thirteen, so the answer stands.
Error analysis
A student produced the line below. Something went wrong.
Annotate
On: \( |3 - 8| \;\overset{?}{=}\; |3| - |8| = 3 - 8 = -5 \)
\[ |3 - 8| = |-5| = 5 \]
Edge cases
Absolute value has two special cases worth pinning down now.
Discussion prompt
What is the absolute value of zero, and can the absolute value of any number ever be negative? Explain using distance.
Hint: How far is zero from zero?
Answer:
The absolute value of zero is zero: it sits no distance at all from itself.
No absolute value can ever be negative, because a distance is a length and lengths do not run backwards. That single fact rules out whole families of wrong answers later — if you ever solve an absolute-value equation and get a negative on the far side, there is no solution and you can stop.
Matching
Absolute value bars are grouping symbols. Read the inside first, then take the distance.
Match the pairs
Why: The second one is the case worth staring at: the bars produce nine as always, and the minus sign sits outside them and is applied afterwards. So a negative answer is possible when a minus sign is outside the bars, but never from the bars themselves.
Definition probe
Absolute value means distance from zero. Sort each statement by whether it can ever happen.
Sort into buckets
Possible, or impossible?
Section
Section 2.3
Concept
Adding a positive number jumps right. Adding a negative number jumps left. That is the entire model.
Figure (svg): A number line showing a jump of plus five from zero to five followed by a jump of minus eight landing on minus three
\[ 5 + (-8) = -3 \]
Pattern
Look at these four sums before generalising.
| sum | result | which distance was bigger |
|---|---|---|
| 9 + (-4) | 5 | the positive one |
| 4 + (-9) | -5 | the negative one |
| -6 + 2 | -4 | the negative one |
| -2 + 6 | 4 | the positive one |
Predict first
When you add a positive and a negative number, what decides the sign of the answer?
Correct: The sign of whichever number is further from zero — the bigger jump wins.
Why: The two jumps go in opposite directions, so they partly cancel and whatever is left over points the way the longer jump pointed. Order of writing has nothing to do with it, which is why nine plus negative four and negative four plus nine both give five.
Worked example
Find the sum below.
\[ -12 + 5 \]
Compare the two distances from zero
Why: Twelve is further from zero than five, so the leftward jump is the longer one and the answer will land left of zero.
Subtract the smaller distance from the larger
Why: The jumps oppose each other, so only the difference survives.
\[ 12 - 5 = 7 \]
Attach the sign of the longer jump
Why: The longer jump was leftward, which is negative.
\[ -12 + 5 = -7 \]
Figure (svg): A number line showing a jump of twelve to the left from zero and then five back to the right, ending at minus seven
Verify: check with a real-world reading
Why: Owing twelve dollars and then earning five leaves you owing seven, which is negative seven. The picture, the arithmetic, and the situation all agree.
Sorting
Decide the sign of each sum without computing the exact value.
Sort into buckets
Sort each sum by the sign of its answer.
Picture it
Every addition of signed numbers is the same picture: two arrows laid end to end.
Figure (svg): Three number lines showing same-direction jumps piling up and opposite-direction jumps partly cancelling
Notice you never needed a rule at all. The rule is just a description of what the arrows do.
Two truths and a lie
Three claims about adding signed numbers. Knock out the false one.
Eliminate the wrong options
Which statement about adding signed numbers is false?
Survives elimination: C
Why: Keep the false statement, which is C. A positive plus a negative can land anywhere: nine plus negative four is positive five, four plus negative nine is negative five, and four plus negative four is exactly zero. The sign of the answer is decided by which jump was longer, not by the presence of a negative.
Section
Section 2.4
Concept
To subtract a number, add its opposite. This one rewrite removes the need for a second set of rules.
Figure (svg): Two number lines stacked: subtracting three and adding negative three both land on the same point
\[ a - b = a + (-b) \]
Worked example
Simplify the expression below — the case that trips everyone.
\[ 4 - (-6) \]
Rewrite the subtraction as adding the opposite
Why: The opposite of negative six is positive six, so the whole expression becomes an addition.
\[ 4 + 6 \]
Add
Why: Both jumps now go right, so their lengths simply pile up.
\[ = 10 \]
Figure (svg): A number line showing a jump of four right then a further six right, landing on ten, illustrating subtracting a negative
Verify: read it as a temperature change
Why: Going from negative six degrees up to four degrees is a rise of ten degrees, which is exactly the difference the expression asked for.
Trap
Simplify seven minus negative two.
Treat the two signs as one subtraction and ignore the second
Why: The brackets look decorative, so the minus in front of the two seems to be the only one doing work.
\[ 7 - (-2) \;\to\; 7 - 2 = 5 \]
Five is wrong, and nothing about the working looks unusual, which is why this error survives so long.
Simplify the same expression by rewriting it before computing anything.
Replace subtract with add the opposite, then simplify the sign
Why: Subtracting negative two means adding positive two. Do the rewrite on paper rather than in your head, and the two signs cannot merge by accident.
\[ 7 - (-2) = 7 + 2 = 9 \]
The habit that saves you: never simplify two signs mentally, always rewrite the line first.
Fill the middle
Complete the middle step of this subtraction.
Fill in the blanks
-5 - 9 \;=\; -5 + (-9) \;=\; -14
Why: Subtracting nine means adding negative nine. Now both jumps go left, so their lengths add together to fourteen units left of zero, giving negative fourteen. Students who expect subtraction to make things smaller in size are surprised here, but the two leftward jumps genuinely pile up.
Explain it to yourself
You keep turning subtraction into addition. Say why that is allowed rather than just useful.
Discussion prompt
Why does taking away six give the same result as adding negative six?
Hint: Draw the arrow for each phrase and compare them.
Answer:
Because moving six units left is one single motion, and there are two English sentences for it: take away six, or add negative six. The arrow on the number line is identical, so the results must be identical.
The payoff is that you only ever need one set of sign rules. Every subtraction becomes an addition, and addition you already understand as two jumps.
Analogy
Subtracting a negative shows up in ordinary life more than you would think. Pair each situation with what it does to the total.
Match the pairs
Why: Removing something bad is the everyday meaning of subtracting a negative, and in every case it moves you in the positive direction. If the algebra ever feels arbitrary, translate it into forgiven debts and it becomes obvious again.
Section
Section 2.5
Concept
The sign of a product depends on how many negative factors there are, and nothing else.
Figure (svg): A two by two grid of multiplication sign rules showing same signs give a positive product and different signs give a negative product
An even count of negatives gives a positive answer; an odd count gives a negative one.
Worked example
Simplify the product below.
\[ (-2)(3)(-4) \]
Count the negative factors
Why: There are two of them, and two is even, so the final answer will be positive. Deciding the sign first means the arithmetic cannot corrupt it.
Multiply the sizes, ignoring signs for a moment
Why: Two times three is six, and six times four is twenty-four.
\[ 2 \cdot 3 \cdot 4 = 24 \]
Attach the sign you already decided
Why: Two negatives cancel, so the product is positive twenty-four.
\[ (-2)(3)(-4) = 24 \]
Figure (svg): Three factors shown as chips with the two negative ones paired off and cancelling to leave a positive product
Verify: multiply in a different order
Why: Three times negative four is negative twelve, and negative two times negative twelve is positive twenty-four. A different route reaches the same answer, so the sign was handled correctly.
Tweak it
Change the number of negative factors and watch the answer jump across zero.
Parameter explorer
Slide the multiplier through zero. What happens to the output as it crosses?
\[ y = {a}x \]
Prediction
Do not multiply. Just count.
\[ (-1)(-2)(-3)(-4)(-5) \]
Predict first
Is the product positive or negative?
Correct: Negative — there are five negative factors, and five is odd.
\[ (-1)(-2)(-3)(-4)(-5) = -120 \]
Why: Every pair of negatives cancels to a positive. Five negatives make two complete pairs with one left over, and that leftover negative decides the sign of the whole product. The size happens to be 120, so the answer is negative 120, but you knew the sign before doing any arithmetic at all.
Notation
Three expressions that look alike and are not. Read where each minus sign is actually attached.
Annotate
On: \( -3^2 \qquad (-3)^2 \qquad -(3^2) \)
Brackets are not decoration here — they change the answer by a sign.
Section
Section 2.6
Concept
The distributive property says a factor outside a bracket multiplies every term inside it.
Figure (svg): An area model of three times the quantity x plus four, split into a three by x rectangle and a three by four rectangle
\[ a(b + c) = ab + ac \]
Worked example
Expand the expression below — the version where signs get lost.
\[ -3(2x - 5) \]
Multiply the outside factor by the first term
Why: Negative three times two x is negative six x. Carry the sign with the factor rather than leaving it behind.
\[ -6x \]
Multiply the outside factor by the second term, sign included
Why: The second term is negative five, and negative three times negative five is positive fifteen. Two negatives make a positive here.
\[ -6x + 15 \]
Figure (svg): An area model of minus three times the quantity two x minus five, with both products labelled and the second one positive
Verify: substitute x equals two into both forms
Why: The original gives negative three times the quantity four minus five, which is negative three times negative one, which is three. The expanded form gives negative twelve plus fifteen, which is also three. They agree, so the expansion is right.
Trap
Expand four times the quantity x plus three.
Multiply the four by the x and copy the rest across
Why: The four visibly touches the x, so it is easy to believe its job is finished after that first product.
\[ 4(x + 3) \;\to\; 4x + 3 \]
Test it with x equal to one: the original is four times four, which is sixteen, but this gives seven.
Expand the same expression by drawing the arrows before writing anything.
Multiply the outside factor by every term inside, one arrow each
Why: Two terms inside means two arrows and two products. Drawing them makes a missing one visible.
\[ 4(x + 3) = 4x + 12 \]
Test it with x equal to one: four plus twelve is sixteen, matching the original.
Reverse engineer
Here is the finished expansion. Reconstruct what was outside the bracket.
Fill in the blanks
7(x + 3) \;=\; 7x + 21
Why: Both terms share a factor of seven, so seven is what was outside. Dividing each term by seven recovers the bracket: 7x over 7 is x, and 21 over 7 is 3. Reading the distributive property backwards like this is exactly what factoring means, and it is the whole of Chapter 10.
Pattern
This procedure never changes, however ugly the numbers get.
The check at the end is not optional politeness — it catches every sign error in a single line of arithmetic.
Picture it
The distributive property is a statement about area, and this is the picture behind it.
Figure (svg): One rectangle shown whole with width x plus four, then the same rectangle cut into two pieces of area three x and twelve
Any time you doubt an expansion, draw the rectangle. It settles the question without any rules.
Sorting
Sort each expansion. Check the second term especially — that is where signs die.
Sort into buckets
Which expansions are correct?
Section
Section 2.7
Concept
Like terms have exactly the same variable part. Only like terms may be combined, and combining them just adds their coefficients.
Figure (svg): Algebra tiles sorted into two piles: three long x tiles in one pile and five small unit tiles in another, showing they cannot merge
coefficient — The number multiplying the variable part of a term. In five x the coefficient is five, and in x the coefficient is an invisible one.
Worked example
Simplify the expression below.
\[ 7x + 4 - 3x + 9 \]
Group the terms by shape
Why: Move terms around with their signs attached. The sign in front of a term belongs to that term and travels with it.
\[ (7x - 3x) + (4 + 9) \]
Add the coefficients within each group
Why: Seven x minus three x is four x — the x itself is not touched, only how many of them there are.
\[ 4x + 13 \]
Figure (svg): Algebra tiles showing seven x tiles with three removed leaving four, alongside four units plus nine units making thirteen
Verify: substitute x equals two into both forms
Why: The original gives fourteen plus four minus six plus nine, which is twenty-one. The simplified form gives eight plus thirteen, which is also twenty-one, so nothing was lost.
Discrimination
Sort each pair by whether the two terms may be combined. Do not combine them.
Sort into buckets
Which pairs are like terms?
Explain it
A younger student asks why three x plus four cannot just be seven x.
Discussion prompt
Explain it in two sentences using an everyday comparison, no algebra words.
Hint: Try counting two different objects.
Answer:
Something like: three bags plus four apples is not seven of anything, because bags and apples are different things — you can only add up things that are the same kind.
Then make it concrete: if x is 10, then 3x plus 4 is 34, but 7x would be 70. Trying it on one number is the fastest way to convince anyone the shortcut is wrong.
Check
Two chapter skills in one question. Paper first.
Check your understanding
Simplify 5(2x - 3) + 4x.
Answer: A
Why: Distributing the five gives 10x minus 15. Adding the 4x combines with the 10x because both are x terms, giving 14x minus 15. The minus 15 has no like partner, so it stays put.
Error analysis
This simplification has gone wrong twice. Find both before reading the notes.
Annotate
On: \( 6x - 2(x + 4) \;\overset{?}{=}\; 6x - 2x + 8 \;=\; 4x + 8 \)
\[ 6x - 2(x + 4) = 6x - 2x - 8 = 4x - 8 \]
Faded example
Same skill, fewer supports. Fill in what is missing.
Fill in the blanks
3(2x - 1) + 5x \;=\; 6x - 3 + 5x \;=\; 11x - 3
Why: Distributing the three gives six x minus three. The five x then combines with the six x to give eleven x, while the minus three has no like partner and simply comes along unchanged. The commonest slip is combining the constant with an x term, which would wrongly give something like eight x.
Section
Section 2.8
Concept
Dividing is multiplying by the reciprocal, so the sign rule carries over unchanged: same signs give positive, different signs give negative.
\[ \frac{-24}{6} = -4 \qquad \frac{-24}{-6} = 4 \]
Figure (svg): Two division statements: zero divided by five equals zero is allowed, while five divided by zero is crossed out as undefined
Counterexample
A classmate says: dividing always makes a number smaller.
Discussion prompt
Find a division that makes the number bigger, and say what the classmate is really thinking of.
Hint: What happens if the divisor is a fraction less than one?
Answer:
Dividing eight by one half gives sixteen, which is larger than eight. Dividing by any number between zero and one makes the result grow.
\[ 8 \div \tfrac{1}{2} = 16 \]
The classmate is thinking of dividing by numbers bigger than one, which is the only case where the result shrinks. It is a good example of a rule learned from a narrow diet of examples.
Worked example
Simplify the expression below.
\[ \frac{-3(8) + 6}{-3} \]
Finish the numerator first
Why: The fraction bar is a grouping symbol, so the whole top must collapse to one number before dividing.
\[ \frac{-24 + 6}{-3} = \frac{-18}{-3} \]
Count the negative signs in the division
Why: There are two, which is even, so the answer is positive.
Divide the sizes
Why: Eighteen divided by three is six.
\[ = 6 \]
Figure (svg): A fraction bar shown as a grouping symbol, with the numerator collapsing to minus eighteen before the division happens
Verify: multiply the answer back by the divisor
Why: Six times negative three is negative eighteen, which is the numerator we computed, so the division was carried out correctly.
Step zero
Look at the expression below and plan the route. Do not compute anything yet.
\[ \frac{-4(6) + 10}{-7} \]
Discussion prompt
List the moves in the order they must happen, and say which sign decisions you can make in advance.
Hint: Which grouping symbol outranks the division?
Answer:
Plan: finish the numerator first because the fraction bar groups it — multiply, then add. Only then divide.
Sign decided in advance: the numerator will come out negative and the divisor is negative, so that is two negatives and the final answer will be positive. Knowing the sign before computing means an arithmetic slip cannot flip it without you noticing.
Running the plan gives negative twenty-four plus ten, which is negative fourteen, divided by negative seven, which is positive two.
Comparison
Fill in the blanks. Once this table is complete you have all of signed arithmetic on one page.
Comparison matrix
| operation | what to do with the signs | worked case |
|---|---|---|
| adding same signs | add the sizes, keep the sign | -3 + (-5) = -8 |
| adding different signs | subtract sizes, keep the sign of the larger | -9 + 4 = -5 |
| subtracting | add the opposite, then use the addition rules | 6 - (-2) = 8 |
| multiplying or dividing | count the negatives: even gives positive | (-4)(-5) = 20 |
Invariant
Watch an expression being simplified and name the thing that never changes.
Step through it
One quantity is identical in every frame. Which one, and why does that matter?
Simplifying changes the appearance of an expression and never its value. That invariant is what makes every step checkable.
Commit first
Answer, then rate your confidence.
\[ \text{Simplify } -2(4 - 7x) + 3x \]
Predict first
What does the expression simplify to?
Correct: 17x - 8
\[ -2(4 - 7x) + 3x = -8 + 14x + 3x = 17x - 8 \]
Why: Distributing negative two gives negative eight plus fourteen x, because negative two times negative seven x is positive fourteen x. Adding the three x gives seventeen x, and the constant is negative eight. The commonest error is losing the double negative and getting negative fourteen x instead.
Exit ticket
The most useful thirty seconds of the chapter.
Predict first
Which of these still feels shakiest?
Correct: Whatever you picked is the right answer — it is the thing to drill first.
Why: Every one of these four shows up in every remaining chapter, so a wobble here becomes a wobble everywhere. Naming it now means you can practise ten of that one type rather than a scattered mixture that never quite fixes anything.
Connect it up
Draw it once and the rules stop feeling like eight separate things.
Draw it
Draw a map with the number line at the centre. Attach: absolute value, adding, subtracting, multiplying, dividing, distributing, like terms. On each arrow, write the one sentence that connects it back to the line.
If subtraction is not connected to addition on your map, redraw that arrow — it is the most useful link in the chapter.
Recap
Signed arithmetic is the machinery every later chapter runs on, and you now have all of it.
| if you remember one thing | it should be |
|---|---|
| about the line | further right is always larger, even among negatives |
| about subtraction | rewrite it as adding the opposite before you compute |
| about signs in products | count the negatives; even cancels, odd survives |
| about simplifying | the value never changes, only the appearance |
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