Chapter 1 of Algebra 1: Concepts and Skills, built for a visual learner. Variables as labelled boxes, powers as factor counts, the order-of-operations ladder, checking solutions on a balance scale, translating English into symbols, the four-step problem solving plan, tables and graphs, and a first look at functions as machines.
Subject: Algebra 1 · 62 slides · symbolic lesson
Open the interactive version of this deck · Homework for this lesson
Title
Algebra 1 · Chapter 1
Letters that hold numbers, powers, order of operations, and your first functions
Objectives
This chapter is the doorway into algebra. Everything later stands on these eight ideas.
Figure (svg): A roadmap of the chapter showing eight stops from variables through to functions
Section
Section 1.1
Concept
A variable is a letter that stands for a number you have not pinned down yet.
Figure (svg): Three labelled boxes: a box marked n holding the number 7, a box marked w holding 12, and an empty box marked x with a question mark
variable expression — A recipe built from numbers, variables, and operations — such as five times n. It has no equals sign, so it is not a question, it is a value waiting to be computed.
Picture it
Before the rules, get the picture. A letter is not a mystery; it is a placeholder you can refill.
Figure (svg): One recipe card reading five times n, with three different numbers dropped into the n slot giving three different results
That is the whole trade: you give up a fixed number and gain a rule that covers every number.
Worked example
Evaluate five times n when n is three.
\[ 5n \quad \text{when } n = 3 \]
Replace every n with 3
Why: Substitution: the letter was only ever a stand-in for the number, so swapping it back changes nothing about the value.
\[ 5 \cdot 3 \]
Figure (svg): The expression 5n with the letter n being replaced by the number 3, giving 5 times 3 equals 15
Multiply
Why: Once no letters remain, this is ordinary arithmetic and the order of operations takes over.
\[ = 15 \]
Verify: count five groups of three by hand
Why: Five threes are 3, 6, 9, 12, 15 — the same 15, reached without any algebra, so the substitution was done correctly.
Explain it to yourself
You just replaced a letter with a number and kept going as if nothing happened. Say why that is legal.
Discussion prompt
In your own words: why does replacing n with 3 not change the value of the expression?
Hint: What did the letter n actually mean before you substituted?
Answer:
Because the letter was never a different kind of object — it was a name for that number the whole time. Writing 5n when n is 3 and writing 5 times 3 are two spellings of one quantity.
This is the idea every later chapter leans on: you may always swap a name for the thing it names.
Concept
In algebra the multiplication sign is usually dropped, because it looks too much like the letter x.
\[ 5 \times n \;=\; 5 \cdot n \;=\; 5n \]
A number written directly in front of a letter always means multiply. This is the single most common piece of notation in the whole course.
Figure (svg): Three ways of writing five times n, with the compact form 5n highlighted as the one used from now on
Matching
Notation is only useful if it reads the same to everyone. Pair each expression with the sentence it stands for.
Match the pairs
Why: Two of these look almost identical in English but are not the same number: seven minus a number is not the same as a number minus seven. Order matters for subtraction and division, and does not matter for addition and multiplication.
Section
Section 1.2
Concept
A power is repeated multiplication written short. The small raised number says how many copies of the base are multiplied.
\[ 2^5 = 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 = 32 \]
Figure (svg): The power two to the fifth expanded into five twos being multiplied, with the exponent five circled and an arrow to the count of factors
base and exponent — In two to the fifth, 2 is the base (the thing being multiplied) and 5 is the exponent (how many of them there are).
Prediction
Commit before you compute. Powers do not behave like multiplication, and this is where that first bites.
Predict first
Which is larger, two to the fifth power or five to the second power?
Correct: Two to the fifth is much larger: it is 32, while five to the second is 25.
\[ 2^5 = 32 \qquad 5^2 = 25 \]
Why: Swapping the base and the exponent almost never gives the same answer, because they do completely different jobs — the base is what gets multiplied and the exponent is how many times. Two to the fifth is 32 and five squared is 25, so the smaller base with the bigger exponent wins here.
Worked example
Evaluate the square of x when x is six, then evaluate two to the fourth power.
\[ x^2 \quad \text{when } x = 6 \]
Substitute 6 for x
Why: Replace the name with the number, then the exponent is an ordinary counting instruction.
\[ 6^2 = 6 \cdot 6 = 36 \]
Now expand two to the fourth
Why: Write out the factors before multiplying; expanding first is what stops the exponent being treated as a multiplier.
\[ 2^4 = 2 \cdot 2 \cdot 2 \cdot 2 = 16 \]
Figure (svg): A bar chart of the powers of two from two to the first through two to the fifth, growing from 2 to 32
Verify: re-count the factors in each expansion
Why: Six squared shows exactly two sixes and gives 36; two to the fourth shows exactly four twos and gives 16. The factor counts match the exponents, so neither was misread.
Error analysis
A student wrote the line below. Something in it is wrong. Find it before reading on.
Annotate
On: \( 3x^2 \;\overset{?}{=}\; (3x)^2 = 9x^2 \)
An exponent grabs only the symbol immediately to its left, unless brackets tell it to grab more.
Sorting
Sort each expression by what the exponent is actually attached to.
Sort into buckets
Drag each expression into the right bucket.
Section
Section 1.3
Concept
Without a shared order, the same written expression would mean different things to different people. So mathematics fixes one.
Figure (svg): A four rung ladder showing the order of operations from grouping symbols at the top down to add and subtract at the bottom
Two of these rungs say left to right, and those are the two that people forget.
Pattern
Before the machine walks it: what does the expression below come out as?
\[ 3 + 2 \cdot 4^2 \]
Predict first
What is the value of three plus two times four squared?
Correct: 35
Why: Powers come before multiplication, so four squared is 16 first. Then multiplication before addition gives 2 times 16, which is 32, and finally 3 plus 32 is 35. Working strictly left to right instead would give 80, which is the commonest wrong answer here.
Worked example
Evaluate the expression using the ladder, and watch which rung fires each time.
\[ 3 + 2 \cdot 4^2 \]
Rung 2: powers first
Why: There are no brackets, so the highest rung with anything on it is powers. Four squared is 16.
\[ 3 + 2 \cdot 16 \]
Rung 3: multiply
Why: Multiplication outranks addition, so the 2 and the 16 join before the 3 gets involved.
\[ 3 + 32 \]
Rung 4: add
Why: Only addition is left, and there is nothing to its left competing with it.
\[ = 35 \]
Step through it
Step through the same expression and watch one rung fire per frame.
Verify: re-read the ladder from the top
Why: Grouping: none. Powers: done first. Multiply: done second. Add: done last. Every rung fired in order and nothing was skipped, so 35 stands.
Ranking
Here is a messier expression. Rank the four moves in the order they must happen.
\[ (6 - 2)^2 \div 4 + 5 \]
Put in order
Why: Grouping symbols always fire first, which turns the bracket into 4. Powers come next, giving 16. Then divide, giving 4, and only then add 5 for a final answer of 9. Notice that division fires before addition even though it is written further right.
Trap
Evaluate this expression the way it reads out loud, one operation at a time.
\[ 3 + 2 \cdot 4^2 \]
Add 3 and 2 first, because they come first on the page
Why: This feels natural — it is how English is read — but it ignores the ladder entirely.
\[ 5 \cdot 4^2 = 5 \cdot 16 = 80 \]
The answer 80 is wrong, and nothing in the arithmetic itself looks suspicious. That is what makes this trap durable.
Evaluate the same expression by asking which rung is highest, not which symbol is leftmost.
\[ 3 + 2 \cdot 4^2 \]
Powers, then multiplication, then addition
Why: Position on the page never decides; rank on the ladder decides. Left to right is only the tie-breaker within a single rung.
\[ 3 + 2 \cdot 16 = 3 + 32 = 35 \]
Reading order and evaluation order are two different things, and only one of them is mathematics.
Section
Section 1.4
Concept
An equation says two expressions name the same number. It might be true, and it might be false.
Figure (svg): A balance scale with the chips three x and minus five on the left pan and the number seven on the right pan, hanging level
solution — A value for the variable that makes the equation a true statement. Checking a solution is always the same move: substitute it, then compare the two sides.
Picture it
Picture the equation as a scale. A number that balances it is a solution; a number that tips it is not.
Figure (svg): Two balance scales: the first is level with x equal to four, the second is tipped because x equal to two makes the left side too light
So testing a candidate is never guesswork: substitute, compute both sides, and see whether they agree.
Worked example
Is four a solution of the equation below?
\[ 3x - 5 = 7 \]
Substitute 4 for x on the left side only
Why: Test one side at a time so you can see the two numbers before comparing them. Changing both at once hides where a mismatch came from.
\[ 3(4) - 5 = 12 - 5 = 7 \]
Compare with the right side
Why: The right side is already the plain number 7, so there is nothing to compute there.
\[ 7 = 7 \quad \checkmark \]
Figure (svg): A two column check showing the left side computing down to seven and the right side already being seven, with a tick between them
Verify: test a nearby number to be sure the check discriminates
Why: Trying 2 gives 3 times 2 minus 5, which is 1, and 1 is not 7 — so the check really does reject wrong values rather than accepting everything.
Prediction
Commit first. No working on the slide, just a decision.
\[ 2x + 3 = 11 \qquad \text{is } x = 5 \text{ a solution?} \]
Predict first
Is five a solution of two x plus three equals eleven?
Correct: No. Substituting 5 gives 13, not 11, so the scale tips.
\[ 2(5) + 3 = 13 \neq 11 \]
Why: Two times five is ten, and ten plus three is thirteen. Thirteen is not eleven, so five fails the check. The value that does work is four, and the fastest way to see that is to test rather than to stare.
Concept
An inequality compares two expressions with greater-than or less-than instead of equals. Usually a whole stretch of numbers works.
Figure (svg): A number line with an open circle at three and the whole line to the right of three shaded, showing x is greater than three
The open circle matters: it says three itself is not included, because three is not greater than three.
Sorting
For the inequality shown, sort each candidate number.
\[ x > 3 \]
Sort into buckets
Which of these numbers make the statement true?
Pattern
This single procedure works for every equation and every inequality in the whole course.
Notice what is not on the list: solving. Checking and solving are different jobs, and checking is the one you can always do.
Section
Section 1.5
Concept
Word problems are built from a small vocabulary. Learn these four and most sentences decode themselves.
Figure (svg): Four word cards, sum, difference, product and quotient, each pointing to the matching arithmetic symbol
The operation words are the easy half. The hard half is order, and that is the next slide.
Translation
Pair each English phrase with the expression that says the same thing.
Match the pairs
Why: The two middle phrases are the whole point. Five less than a number starts at the number and goes down five, giving n minus 5. A number less than five starts at five and goes down by the number, giving 5 minus n. English puts the words in the opposite order to the symbols in one case and not the other.
Worked example
Translate: five less than twice a number.
Name the unknown
Why: Give the number a letter before doing anything else, so the sentence has something to attach to. Let n be the number.
Build the inside part first: twice a number
Why: Work from the innermost phrase outward, exactly like unwrapping brackets. Twice a number is 2n.
\[ 2n \]
Now apply five less than that
Why: Five less than a quantity means start from the quantity and subtract five — the quantity comes first in the symbols even though it comes second in the sentence.
\[ 2n - 5 \]
Figure (svg): The phrase five less than twice a number broken into two labelled chunks with arrows showing twice a number built first and the subtraction applied after
Verify: test the expression on a concrete number
Why: If the number is 10, then twice it is 20, and five less than 20 is 15. Substituting 10 into 2n minus 5 gives 20 minus 5, which is also 15, so the expression matches the English.
Trap
Translate: seven less than a number.
Write the symbols in the order the words appear
Why: Seven comes first in the sentence, so it goes first in the expression. This is exactly how the other phrases behaved, so it feels consistent.
\[ 7 - n \]
Now test it with n equal to 10: seven less than ten should be three, but this expression gives negative three.
\[ 7 - 10 = -3 \quad \text{but the phrase means } 3 \]
Translate the same phrase by asking what the sentence is about.
Identify the starting quantity, then subtract from it
Why: Less than always means take away from the thing named after it. The sentence is about the number, so the number goes first and the seven comes off it.
\[ n - 7 \]
Test it: with n equal to 10 this gives three, which is what seven less than ten actually means.
\[ 10 - 7 = 3 \quad \checkmark \]
Check
Solve it on paper before you click. Watch the order.
Check your understanding
Which expression means four less than three times a number n?
Answer: A
Why: Three times a number is 3n, and four less than that quantity means subtract four from it, giving 3n minus 4. Test it with n equal to 5: three times five is fifteen, and four less than fifteen is eleven, which is what 3n minus 4 gives.
Section
Section 1.6
Concept
A verbal model is the relationship written in words. Getting it right is most of the work; the algebra afterwards is short.
Figure (svg): A four stage flow chart: write a verbal model, assign labels, write an algebraic model, then solve and answer the question
algebraic model — The verbal model with letters and numbers substituted for the words. It is an equation you can actually solve.
Step zero
A car travels at 55 miles per hour for 3 hours. Do not compute anything yet.
Discussion prompt
Write the verbal model in plain English — what times what gives what? Name the quantity you are asked for.
Hint: What is the question actually asking you to find — a distance, a speed, or a time?
Answer:
Verbal model: distance equals rate times time.
Labels: rate is 55 miles per hour, time is 3 hours, distance is the unknown.
Only now is it worth writing symbols. Deciding the relationship first is what stops you multiplying the wrong pair of numbers in a longer problem.
Worked example
A car travels at 55 miles per hour for 3 hours. How far does it go?
Step 1, verbal model: distance equals rate times time
Why: State the relationship in words first, so it can be checked against common sense before any symbols hide it.
Step 2, assign labels: rate is 55, time is 3, distance is d
Why: Labels pin each word to a number and a unit, which is where unit mistakes get caught.
Step 3, algebraic model
Why: Substitute the labels straight into the verbal model, changing nothing else.
\[ d = 55 \cdot 3 \]
Figure (svg): A bar model split into three equal hours of fifty five miles each, totalling one hundred and sixty five miles
Step 4, solve and answer the question asked
Why: Compute, then write the answer with its unit — a bare number is not an answer to a distance question.
\[ d = 165 \text{ miles} \]
Verify: check the size of the answer against the situation
Why: Roughly 55 miles each hour for 3 hours should be a bit over 150 miles, and 165 sits right there. A wrong operation such as dividing would have given about 18, which is obviously too small for three hours of driving.
Real world
A pizza place charges a flat 4 dollars for delivery plus 12 dollars per pizza. You want to describe the total cost of any order.
Discussion prompt
Write the verbal model, assign labels, and then write the algebraic model. What does each letter mean?
Hint: Which number changes when you order more, and which one stays put?
Answer:
Verbal model: total cost equals delivery fee plus cost per pizza times number of pizzas.
\[ C = 4 + 12p \]
Labels: C is the total cost in dollars and p is the number of pizzas. The 4 does not depend on p, and that is exactly what makes it a flat fee.
Missing information
A problem reads: a rectangle has a length of 12 centimetres. Find its area.
Discussion prompt
This cannot be answered as written. What single piece of information must you ask for, and why does the question collapse without it?
Hint: Write the algebraic model first, then look at which letters have no number attached.
Answer:
You must be told the width. Area of a rectangle is length times width, and a length on its own is compatible with infinitely many rectangles — a 12 by 1 strip and a 12 by 12 square share that length and have completely different areas.
\[ A = \ell w \quad \text{needs both } \ell \text{ and } w \]
Noticing a missing quantity is a real skill: the algebraic model tells you exactly which labels you still owe.
Section
Section 1.7
Concept
A table of values lists inputs beside their outputs. Plotting those pairs turns arithmetic into a shape you can read at a glance.
Figure (svg): A small table of hours and dollars beside a coordinate plane with the same four points plotted and rising steadily
Both representations hold the same information. The graph is faster to read, and the table is easier to be exact with.
Worked example
A job pays 9 dollars per hour. Make a table for one through four hours, then plot it.
Write the rule
Why: Pay equals rate times hours, so the model is nine times h.
\[ p = 9h \]
Substitute each input in turn
Why: Do them one at a time and record the pair, rather than computing in your head and plotting from memory.
| hours h | pay p | point |
|---|---|---|
| 1 | 9 | (1, 9) |
| 2 | 18 | (2, 18) |
| 3 | 27 | (3, 27) |
| 4 | 36 | (4, 36) |
Plot each pair, reading across then up
Why: The first number in a pair is the horizontal move and the second is the vertical move. Getting that order backwards is the classic plotting error.
Figure (svg): A coordinate plane with the four pay points plotted, lying on a straight rising path
Verify: read one point back off the graph and test it in the rule
Why: The point at 3 hours sits level with 27 dollars, and nine times three is 27, so the plotted point agrees with the rule that generated it.
Tweak it
The pay rate is the only thing that changes below. Watch what it does to the steepness.
Parameter explorer
Drag the hourly rate. What happens to the line, and what stays fixed?
\[ p = {r}h \]
Estimation
Using the same 9 dollars per hour job: you work a full week of 38 hours.
Predict first
Roughly how much do you earn — to the nearest hundred dollars?
Correct: About 350 dollars — the exact figure is 342.
\[ 9 \cdot 38 = 342 \]
Why: Round 9 up to 10 and 38 down to 35 and you get about 350, which is close enough to catch a wrong answer. The exact value is nine times thirty-eight, which is 342. Estimating first is what tells you instantly that 3400 came from a misplaced decimal rather than from the situation.
Comparison
Fill the blanks. Both representations are useful, but not for the same questions.
Comparison matrix
| question you are asking | table | graph |
|---|---|---|
| exactly how much for 7 hours | better | you have to read between marks |
| is the growth steady or speeding up | hard to see from a column | better |
| when does pay pass 100 dollars | scan down the rows | read across at 100 |
The habit worth building: make the table, then plot it, then answer from whichever one fits the question.
Section
Section 1.8
Concept
A function is a rule that gives exactly one output for each input. No input is allowed two different answers.
Figure (svg): A function machine labelled multiply by three then add one, with the input x entering on the left and the output y leaving on the right
domain and range — The domain is the collection of inputs the rule accepts; the range is the collection of outputs it produces.
Picture it
Only one thing can disqualify a rule, and this picture is it.
Figure (svg): Two mapping diagrams side by side: the left one is a function with one arrow from each input, the right one is not because one input has two arrows
Notice what is allowed: two different inputs may share an output. It is only the reverse that is forbidden.
Worked example
For the rule below, find the output at each input in the given domain.
\[ y = 3x + 1 \quad \text{domain: } 0,\, 1,\, 2,\, 3 \]
Substitute each input in turn
Why: Take the inputs in order and record each pair, so nothing is skipped and no output is invented.
| input x | compute | output y |
|---|---|---|
| 0 | 3(0) + 1 | 1 |
| 1 | 3(1) + 1 | 4 |
| 2 | 3(2) + 1 | 7 |
| 3 | 3(3) + 1 | 10 |
Read off the range
Why: The range is simply the collection of outputs actually produced, listed once each.
\[ \text{range: } 1,\, 4,\, 7,\, 10 \]
Figure (svg): A mapping diagram from the inputs zero to three onto the outputs one, four, seven and ten, one arrow from each input
Verify: substitute the largest input again
Why: Three times three is nine, and nine plus one is ten, which matches the last row of the table and the last arrow in the diagram.
Counterexample
A classmate says: if two inputs give the same output, the rule cannot be a function.
Discussion prompt
Find a rule that proves this claim false, and explain what the classmate has confused.
Hint: Try a rule that treats a number and its opposite the same way.
Answer:
The rule that squares its input is a counterexample: negative two and positive two both give four, and it is still a perfectly good function.
\[ f(x) = x^2 \quad f(-2) = 4, \;\; f(2) = 4 \]
The classmate has the condition backwards. A function forbids one input with two outputs; it says nothing at all about two inputs sharing an output.
Definition probe
Each rule below either is a function or fails for one specific reason. Sort them.
Sort into buckets
Sort each rule by whether it satisfies the definition.
Two truths and a lie
Three statements about functions. Knock out the one that is false.
Eliminate the wrong options
Which statement about functions is false?
Survives elimination: C
Why: Keep the false statement, which is C. The definition of a function constrains inputs only: each input gets exactly one output. Sharing outputs across different inputs is completely ordinary — the squaring rule sends both negative three and positive three to nine and is still a function.
Discrimination
For each task, say which tool from this chapter you would reach for. Do not compute anything.
Sort into buckets
Sort each task by the tool it calls for.
Fill the middle
Fill in the missing middle line of this evaluation.
Fill in the blanks
5 + 3(2)^2 \;=\; 5 + 3 \cdot 4 \;=\; 5 + 12 \;=\; 17
Why: Powers fire before multiplication, so two squared becomes four first. Then three times four is twelve, and only then does the five join in to give seventeen. Doing the addition early would give eight times four, which is thirty-two, and that is the mistake this exercise is built to catch.
Explain it
A younger student asks why anyone bothers with letters when numbers work fine.
Discussion prompt
Answer in two sentences a sixth-grader would follow. No jargon.
Hint: Try starting with: instead of writing it out a hundred times...
Answer:
Something like: a letter lets you write one rule that works for every number at once, instead of writing the same sum over and over with different numbers in it.
If you can say it plainly, you understand it. If your explanation needs the word variable to make sense, you are still leaning on the vocabulary rather than the idea.
Commit first
Answer, then rate your confidence. Being confidently wrong is worth finding now rather than in a test.
\[ \text{Evaluate } 2 + 6 \div 2 \cdot 3 \]
Predict first
What is the value of two plus six divided by two times three?
Correct: 11
\[ 2 + 6 \div 2 \cdot 3 = 2 + 3 \cdot 3 = 2 + 9 = 11 \]
Why: Division and multiplication share a rung, so they fire left to right: six divided by two is three, then three times three is nine, and finally two plus nine is eleven. Doing the multiplication first because of a half-remembered mnemonic gives six divided by six, which is one, and then three — a very common wrong answer.
Check
One question that touches four sections at once. Paper first.
Check your understanding
If the rule is y = 2x + 5, which statement is true?
Answer: A
Why: Substitute three for x: two times three is six, and six plus five is eleven. The rule is also a perfectly good function, because each input produces exactly one output no matter how many operations are involved.
Exit ticket
Last commitment of the chapter, and the most useful one.
Predict first
Which of these would you be least confident doing right now, with no notes?
Correct: There is no wrong answer here — the honest one is the useful one.
Why: Whatever you picked is the thing to practise first, and it is also the thing to bring to a session. Students who name their weak spot out loud fix it far faster than students who quietly hope it does not come up, because a named gap can be aimed at directly.
Connect it up
One page, drawn by you. This is the fastest revision tool in the chapter.
Draw it
Draw a map linking: variable, power, order of operations, equation, solution, translation, model, table, graph, function. Draw an arrow wherever one idea is used by another, and label each arrow with how it is used.
If your map has no arrows into function, look again: every earlier idea feeds it.
Recap
You started this chapter with arithmetic and finished it with rules that work on every number at once.
| if you remember one thing | it should be |
|---|---|
| about letters | a variable is a name for a number, so you may always swap it back |
| about order | rank on the ladder beats position on the page |
| about words | less than reverses; more than does not |
| about functions | one input, one output — sharing outputs is fine |
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