Integrated Math 1 · Unit 0 · Review of the Essentials

Use the Language of Algebra

Place value, variables, translation, exponents and like terms — the vocabulary every later chapter assumes you already speak.

bookSHelf · Integrated Math 1 · §0.1 · a self-paced section

Titlepage in LaTeX grammar: italic venue kicker, double rule over the serif title, one-sentence lede, hairline rule, small-print byline.

Use the Language of Algebra · bookSHelf Integrated Math 1§0.1

Outline — by the end of this section you will be able to

Objectives

Read, write, and round whole numbers using place value §0.1.1
Distinguish a variable from a constant, an expression from an equation Def 0.1.1–0.1.4
Translate an English phrase into algebra, and read one back §0.1.3
Write repeated multiplication using exponent notation Def 0.1.5
Simplify by the order of operations, and evaluate for a given value §0.1.4
Identify like terms and combine them Def 0.1.6

The six section SLOs verbatim, revealed one at a time so you can pace the introduction.

Use the Language of Algebra · bookSHelf Integrated Math 1§0.1

§0.1.1 — Whole numbers, place value, and rounding

A digit's value depends on where it sits

The whole numbers are $0, 1, 2, 3, 4, \dots$ — the counting numbers together with zero. They go on forever, so there is no largest one.

Our system is positional. The number 4,072 is not "four, zero, seven, two":

4{,}072 = 4(1000) + 0(100) + 7(10) + 2(1)

The zero is doing real work. Take it away and 4,072 collapses into 472. A zero that holds a place open is a placeholder, and it is the reason the whole positional system works.

Each place is worth ten times the place to its right. Commas group digits in threes — "four thousand, seventy-two."

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§0.1.1 — The same digit, four different numbers

What the hundreds digit is worth

Number	Digit in the hundreds place	What that digit is worth
4,072	0	0
4,572	5	500
41,072	0	0
900,318	3	300

Table 0.1.1: The hundreds digit of four numbers, and the value it carries in each.

Read it: the digit alone tells you nothing. 4,572 and 900,318 both hold a non-zero hundreds digit, but 5 there is worth 500 and 3 there is worth 300.

Booktabs discipline: 2px top and bottom rules, one hairline under the header, no vertical rules.

Use the Language of Algebra · bookSHelf Integrated Math 1§0.1

§0.1.1 — Place value, drawn

Every digit of 4,072

Reading right to left the places are ones, tens, hundreds, thousands, ten-thousands — each worth ten times the place to its right.

Figure 0.1.1: Each digit of 4,072 sits in a place that fixes what it is worth.

Graphic right, prose left — the locked layout convention. Double-click the figure for fullscreen.

Use the Language of Algebra · bookSHelf Integrated Math 1§0.1

§0.1.1 — Writing a number out

Read it in comma-separated groups

68,204,915 is "sixty-eight million, two hundred four thousand, nine hundred fifteen." Name each group as you reach it.

Two conventions cause most of the mistakes

No "and" inside a whole number. It is "two hundred four," not "two hundred and four" — the word "and" is saved for the decimal point, which matters in §0.4.

Hyphenate two-word numbers from 21 to 99. "Sixty-eight," "ninety-five." The multiples of ten in that range are single words and take no hyphen: "thirty," "ninety."

Both conventions reveal on click so each can be said before the next appears.

Use the Language of Algebra · bookSHelf Integrated Math 1§0.1

§0.1.1 — Rounding

Trading accuracy for convenience

Procedure — rounding a whole number

1. Find the digit in the place you are rounding to.

2. Look at the digit immediately to its right.

3. If that digit is 5 or more, add 1 to the rounding digit. Otherwise leave it alone.

4. Replace every digit to the right with zeros.

Round 6,473 to the nearest hundred: the hundreds digit is 4, the digit right of it is 7, so the 4 becomes 5 — 6,500.

Round the same 6,473 to the nearest thousand: the rounding digit is 6, the digit right of it is 4, so the 6 stays — 6,000.

† When the rounding digit is 9 and has to increase, it rolls over: 3,962 to the nearest hundred turns the 9 into 10, which carries, giving 4,000.

Same number, two places, two different answers — the setup for the Context Pause that follows.

Use the Language of Algebra · bookSHelf Integrated Math 1§0.1

Context Pause — always round from the original

Chained rounding compounds its error

Round 6,473 to the nearest hundred and you get 6,500. Round that result to the nearest thousand and you get 7,000 — but rounding 6,473 directly gives 6,000.

Each rounding step adds a little error, and chaining them lets the error compound into a wrong answer. Go back to the original number every time.

The kicker names the callout kind exactly once; the box header is not repeated inside.

Use the Language of Algebra · bookSHelf Integrated Math 1§0.1

§0.1.1 — Worked example

Rounding to two different places

Example 0.1.1 — round 27,548 to the nearest thousand, then to the nearest ten thousand

Step 1 — Nearest thousand. The thousands digit is 7. The digit to its right is 5, which is 5 or more, so the 7 becomes 8.

27{,}548 \rightarrow 28{,}000

Step 2 — Nearest ten thousand. Go back to the original 27,548. The ten-thousands digit is 2, the digit right of it is 7, so the 2 becomes 3.

27{,}548 \rightarrow 30{,}000

Answer: 28,000 and 30,000 — started from 27,548 both times.

Commit-first: the prompt shows, one click reveals the whole solution below a hairline rule.

Use the Language of Algebra · bookSHelf Integrated Math 1§0.1

§0.1.1 — Your turn

Try It Now 0.1.1

Try It Now 0.1.1 — round 8,461 to the nearest hundred, and 5,097 to the nearest hundred

8,461. The hundreds digit is 4; the digit to its right is 6, which is 5 or more, so the 4 becomes 5.

8{,}461 \rightarrow 8{,}500

5,097. The hundreds digit is 0; the digit to its right is 9, so the 0 becomes 1.

5{,}097 \rightarrow 5{,}100

Answer: 8,500 and 5,100.

Let the room commit to an answer before clicking.

Use the Language of Algebra · bookSHelf Integrated Math 1§0.1

§0.1.2 — Variables, expressions, and equations

One plan, two kinds of number

A phone plan charges a flat $25 every month plus $10 for each gigabyte of data you use.

The $25 never changes

A number whose value always stays the same.

The gigabytes change

And so does your bill — month to month.

That difference has names, and the next four definitions are those names.

The whole subsection hangs off this one concrete plan — keep coming back to it.

Use the Language of Algebra · bookSHelf Integrated Math 1§0.1

§0.1.2 — Definition

Variable

Definition 0.1.1 — Variable

A variable is a letter that represents a number whose value may change.

Definition 0.1.1: one line covers every month, because the letter is the only thing that changes.

Any letter works, though $x$ , $y$ , $z$ , $a$ , $b$ and $n$ are usual. Picking one that hints at its meaning — $g$ for gigabytes, $t$ for time, $C$ for cost — makes your own work readable a week later.

The definition travels with its figure — the statement alone is a bare assertion.

Use the Language of Algebra · bookSHelf Integrated Math 1§0.1

§0.1.2 — Definition

Constant

Definition 0.1.2 — Constant

A constant is a number whose value always stays the same.

If we let $g$ stand for gigabytes used, the monthly bill is

25 + 10g

Definition 0.1.2: the numbers that are settled before you start, in both senses the section uses.

That line is not the bill for one month — it is the rule that produces the bill for any month, as soon as you know $g$ .

The power of the notation is that one line covers every case at once.

Use the Language of Algebra · bookSHelf Integrated Math 1§0.1

§0.1.2 — Operation symbols

Algebra adjusts two arithmetic symbols

Operation	Arithmetic writes	Algebra prefers	Why the change
Addition	$a + b$	$a + b$	no change needed
Subtraction	$a - b$	$a - b$	no change needed
Multiplication	$3 \times 4$	$3 \cdot 4$ , $3(4)$ , or $3b$	$\times$ looks too much like the variable $x$
Division	$12 \div 3$	$\dfrac{12}{3}$	the fraction bar also groups, which $\div$ does not

Table 0.1.2: The notation used from here on, and the reason for each change.

Two habits follow. Writing two things side by side means multiply — $5n$ is "5 times $n$ ". And the fraction bar groups: $\dfrac{a+b}{2}$ adds first, where $\div$ would need $(a+b) \div 2$ .

Juxtaposition only means multiply when a variable is involved — 54 is fifty-four, not five times four.

Use the Language of Algebra · bookSHelf Integrated Math 1§0.1

§0.1.2 — Definition

Expression

Definition 0.1.3 — Expression

An expression is a combination of numbers, variables, and operation symbols that names a quantity. It contains no equals sign.

Definition 0.1.3: numbers, variables and operation symbols naming a quantity, with no equals sign anywhere.

25 + 10g

names a quantity. It asserts nothing.

Use the Language of Algebra · bookSHelf Integrated Math 1§0.1

Insight — a phrase versus a sentence

You simplify a phrase; you solve a sentence

Expression = phrase

"The total cost." It names something but does not claim anything.

Equation = full sentence

"The total cost is 85 dollars." It makes a claim that can be true or false.

The grammar analogy is the one students remember — lean on it before the formal comparison table.

Use the Language of Algebra · bookSHelf Integrated Math 1§0.1

§0.1.2 — Definition

Equation

Definition 0.1.4 — Equation

An equation is two expressions joined by an equals sign, asserting that they have the same value.

Definition 0.1.4: two expressions joined by an equals sign, making a claim that one value makes true.

25 + 10g = 85

asks "how much data would give me an $85 bill?"

Use the Language of Algebra · bookSHelf Integrated Math 1§0.1

§0.1.2 — Keeping the two straight

Expression against equation

Expression	Equation
A phrase. Names a quantity, like $25 + 10g$ .	A sentence. Makes a claim, like $25 + 10g = 85$ .
No equals sign. Nothing is asserted.	Has an equals sign. Something is asserted.
You simplify it. Rewrite it more cleanly.	You solve it. Find the values making it true.
Result is a quantity.	Result is a value for the variable.

Table 0.1.3: The four differences that matter for the rest of the course.

Asking someone to "solve" $25 + 10g$ does not mean anything — there is no claim to make true. Asking them to "simplify" $25 + 10g = 85$ is confused the same way.

The mismatched-verb test is the quickest diagnostic when a student is stuck.

Use the Language of Algebra · bookSHelf Integrated Math 1§0.1

§0.1.2 — Not every relationship is an equality

Symbols for comparing, and for grouping

Symbol	Read as
$a = b$	$a$ equals $b$
$a \neq b$	$a$ is not equal to $b$
$a < b$	$a$ is less than $b$
$a > b$	$a$ is greater than $b$
$a \le b$	$a$ is less than or equal to $b$
$a \ge b$	$a$ is greater than or equal to $b$

Table 0.1.4: The six comparison symbols.

The strict symbols $<$ and $>$ leave equality out; $\le$ and $\ge$ include it. That distinction does real work in Chapter 9.

Parentheses $(\ )$ , brackets $[\ ]$ and braces $\{\ \}$ all group and all mean the same thing. The shapes differ only so nested groupings stay readable, as in $2[3 + 4(5 - 1)]$ . The fraction bar and the radical sign group as well.

Grouping symbols come back immediately in the order of operations.

Use the Language of Algebra · bookSHelf Integrated Math 1§0.1

§0.1.2 — Your turn

Try It Now 0.1.2

Try It Now 0.1.2 — a gym charges a $40 sign-up fee plus $15 per month

Write an expression for the total cost after $m$ months. Then write an equation stating the total cost is $145. Which one finds how many months you have been a member?

Step 1 — The expression. The $40 is paid once, so it is a constant; the $15 repeats every month.

40 + 15m

Step 2 — The equation. Set that expression equal to the stated total.

40 + 15m = 145

Step 3. The question gives a total and asks for months, so you need the claim that can be true or false — the equation, solved for $m$ .

Same shape as the phone plan — deliberately, so the transfer is visible.

Use the Language of Algebra · bookSHelf Integrated Math 1§0.1

§0.1.3 — Translating words into algebra

The phrases map onto operations reliably

Operation	Phrases that signal it
Addition	the sum of; increased by; more than; total of; plus
Subtraction	the difference of; decreased by; less than; minus; subtracted from
Multiplication	the product of; times; twice; of; multiplied by
Division	the quotient of; divided by; the ratio of; per
Equals	is; gives; yields; results in; will be

Table 0.1.5: The signal phrases for each operation.

A lot of what makes word problems hard is translation, not mathematics. Learning this map removes most of the difficulty.

Worth having students keep this table visible for the rest of the unit.

Use the Language of Algebra · bookSHelf Integrated Math 1§0.1

Context Pause — "less than" reads backwards

Word order is the trap

"5 less than $n$ " is $n - 5$ , not $5 - n$ . The phrase names the amount being removed first and the starting quantity second, so the algebra comes out in the opposite order from the reading. "Subtracted from" flips it the same way.

Addition and multiplication can be written either way round without changing the answer. Subtraction and division cannot — and those are exactly the two English reverses.

This single slide prevents more errors than anything else in §0.1.3.

Use the Language of Algebra · bookSHelf Integrated Math 1§0.1

§0.1.3 — Four phrases that look alike

How they actually translate

Read each one carefully

"5 more than $n$ " is $n + 5$ .

"5 less than $n$ " is $n - 5$ .

" $n$ subtracted from 5" is $5 - n$ .

"the difference of $n$ and 5" is $n - 5$ .

When unsure, substitute a number. If $n = 12$ , then "5 less than $n$ " ought to be 7.

Testing $n - 5$ gives 7 — right. Testing $5 - n$ gives $-7$ — not.

The substitute-and-check habit is the transferable skill here, not the four memorized forms.

Use the Language of Algebra · bookSHelf Integrated Math 1§0.1

§0.1.3 — Both directions

English into algebra, and back

English	Algebra
the sum of $x$ and 9	$x + 9$
4 less than twice $y$	$2y - 4$
the product of 6 and $m$ , decreased by 1	$6m - 1$
the quotient of $p$ and 3	$\dfrac{p}{3}$
8 more than the product of 5 and $k$	$5k + 8$
twice the sum of $a$ and 7	$2(a + 7)$

Table 0.1.6: Six translations; the last one is why parentheses matter.

Why that last row. "Twice the sum of $a$ and 7" doubles the whole sum — $2(a+7)$ . Without parentheses, $2a + 7$ doubles only the $a$ .

Reading back. Name the last operation first. $3(m - 2)$ is "three times the difference of $m$ and 2"; $3m - 2$ is "two less than three times $m$ ."

Any time an operation applies to a whole phrase rather than a single quantity, that phrase needs grouping symbols.

Use the Language of Algebra · bookSHelf Integrated Math 1§0.1

§0.1.3 — Worked example

Turning a situation into an equation

Example 0.1.2 — a theater sells adult tickets for $12 and student tickets for $8; one night it takes in $960

Step 1 — Name the changing quantities. Let $a$ be adult tickets sold and $s$ student tickets.

Step 2 — Build each piece of revenue. Adult revenue is $12a$ ; student revenue is $8s$ .

Step 3 — Combine, then set equal to the stated fact.

12a + 8s = 960

That is the whole modeling process in miniature: quantities become variables, relationships become expressions, and a stated fact becomes an equation.

Chapters 2 and 6 do exactly this at a larger scale.

Use the Language of Algebra · bookSHelf Integrated Math 1§0.1

§0.1.3 — Your turn

Try It Now 0.1.3

Try It Now 0.1.3 — write an algebraic expression for each phrase

(a) 7 less than $w$ ; (b) three times the sum of $k$ and 2; (c) the quotient of 30 and $n$ , increased by 4.

(a) "Less than" reverses the reading order, so the 7 is removed from $w$ : $w - 7$ .

(b) The tripling applies to the entire sum, so it needs parentheses: $3(k + 2)$ .

(c) Take the quotient first, then add 4: $\dfrac{30}{n} + 4$ .

(a) is the reversal, (b) is the grouping, (c) is the order — one of each trap.

Use the Language of Algebra · bookSHelf Integrated Math 1§0.1

§0.1.4 — Exponent notation and the order of operations

Repeated multiplication gets a shorthand

Rather than writing $7 \cdot 7 \cdot 7 \cdot 7$ , we write $7^4$ — read "seven to the fourth power."

Two powers get special names from where they come from geometrically: $a^2$ is " $a$ squared," from the area of a square, and $a^3$ is " $a$ cubed," from the volume of a cube.

Chapter 1 comes back to exponents properly in §1.3.

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§0.1.4 — Definition

Exponential notation

Definition 0.1.5 — Exponential notation

For a counting number $n \ge 1$ , the expression $a^n$ means the product of $n$ factors of $a$ :

a^n = \underbrace{a \cdot a \cdot a \cdots a}_{n \text{ factors}}

$a$ is the base, the repeated factor; $n$ is the exponent, counting how many times the base appears.

Definition 0.1.5: the exponent counts the factors, and it grabs only what it touches.

The figure's second half is the setup for the Context Pause on the next slide.

Use the Language of Algebra · bookSHelf Integrated Math 1§0.1

Context Pause — an exponent grabs only what it touches

The parentheses are the whole difference

In $3x^2$ the exponent applies to $x$ alone, so you square $x$ and then multiply by 3. In $(3x)^2$ it applies to the whole product, giving $9x^2$ .

These are two different quantities, and nothing but a pair of parentheses separates them.

Pair this immediately with the minus-sign table that follows.

Use the Language of Algebra · bookSHelf Integrated Math 1§0.1

§0.1.4 — Worth memorizing

What the exponent applies to

Expression	Value	What the exponent applies to
$-4^2$	$-16$	just the 4; the negation happens afterward
$(-4)^2$	$16$	the whole quantity $-4$
$3x^2$	$3 \cdot x \cdot x$	just the $x$
$(3x)^2$	$9x^2$	the whole product $3x$

Table 0.1.7: Four expressions that differ only in what the exponent reaches.

The first two rows are the pair students most often get wrong on a test.

Use the Language of Algebra · bookSHelf Integrated Math 1§0.1

§0.1.4 — One fixed convention

The order of operations

Procedure — the order of operations

1. Grouping symbols — parentheses, brackets, braces, fraction bars, radicals. Innermost outward.

2. Exponents.

3. Multiplication and division — left to right, as they appear.

4. Addition and subtraction — left to right, as they appear.

Steps 3 and 4 are where people misremember. Multiplication does not outrank division, and addition does not outrank subtraction.

20 \div 5 \cdot 2 = 4 \cdot 2 = 8

10 - 4 + 3 = 6 + 3 = 9

† Multiplying before dividing in the first line gives 2; adding before subtracting in the second gives 3. Both are wrong.

2 + 3 \cdot 4

could mean 20 or 14 — the convention exists so every expression has exactly one value.

Use the Language of Algebra · bookSHelf Integrated Math 1§0.1

§0.1.4 — Worked example

Working from the inside out

Example 0.1.3 — simplify $\;5 + 2\left[3^2 - (8 - 6)\right]$

Step 1 — Innermost grouping. The parentheses sit inside the brackets.

5 + 2[3^2 - 2]

Step 2 — Exponent inside the brackets.

5 + 2[9 - 2] = 5 + 2[7]

Steps 3–4 — Multiplication, then addition.

5 + 14 = 19

Answer 19. Every line names the rule that produced it — the same habit the figure that follows draws.

Use the Language of Algebra · bookSHelf Integrated Math 1§0.1

§0.1.4 — The same simplification, drawn

One step at a time

Naming the rule beside each line is what turns a correct answer into a reproducible method.

Figure 0.1.2: One expression simplified a step at a time, each line naming the rule that produced it.

Double-click the figure to blow it up if the back row cannot read the steps.

Use the Language of Algebra · bookSHelf Integrated Math 1§0.1

§0.1.4 — Evaluating an expression

Substitute, then simplify

To evaluate an expression, substitute a number for each variable and simplify what is left. Always wrap the substituted value in parentheses — this one habit prevents most sign errors.

Evaluate $n^2 - 5n$ when $n = -3$ :

(-3)^2 - 5(-3) = 9 - (-15) = 9 + 15 = 24

Without those parentheses, $-3^2$ reads as $-9$ and the answer comes out 6 instead of 24. Signed-number arithmetic is §0.2, and this is where careful notation starts paying for itself.

The parentheses habit is cheap to teach here and expensive to retrofit later.

Use the Language of Algebra · bookSHelf Integrated Math 1§0.1

§0.1.4 — Terms and coefficients

The pieces an expression breaks into

A term is a single number, a single variable, or a product of them — the pieces an expression separates into at its $+$ and $-$ signs. In $7x^2 + 3x - 4$ the terms are $7x^2$ , $3x$ and $-4$ . The number multiplying a variable is its coefficient.

An invisible 1

$y$ means $1y$ , and $-y$ means $-1y$ — so the coefficient of $-y$ is $-1$ .

A constant term

A term that is just a number, like the $-4$ above. It has no variable part, and the term is its own coefficient.

Both cases come up constantly; name them now rather than when they cause an error.

Use the Language of Algebra · bookSHelf Integrated Math 1§0.1

§0.1.4 — Definition

Like terms

Definition 0.1.6 — Like terms

Like terms are terms with the same variables raised to the same powers. Only like terms can be combined, and combining them means adding their coefficients.

7x^2 + 3x - 4 + 2x^2 + 5x = 9x^2 + 8x - 4

Definition 0.1.6: same variable, same power, so their coefficients add; everything else stays apart.

In $6b + 4 - 2b + 9$ the $6b$ and $-2b$ give $4b$ , and 4 and 9 give 13, leaving $4b + 13$ . Watch the sign travelling with each term.

Constant terms are like terms with each other too, since none carries a variable.

Use the Language of Algebra · bookSHelf Integrated Math 1§0.1

Insight — why $9x^2$ and $8x$ will not combine

They measure different kinds of thing

If $x$ is a length, then $x^2$ is an area — and there is no way to add an area to a length and report one number.

No conversion exists, which is exactly why the two terms have to stay apart. "Unlike terms" is not a rule to memorize; it is a unit mismatch.

This is the slide that makes the rule stick — reach for it whenever a student combines unlike terms.

Use the Language of Algebra · bookSHelf Integrated Math 1§0.1

§0.1.4 — Your turn

Try It Now 0.1.4

Try It Now 0.1.4 — simplify $\;4 + 3(2^3 - 5)$ , then evaluate $\;x^2 - 2x$ when $x = -4$

Part 1. Inside the parentheses first, exponent before subtraction:

4 + 3(8 - 5) = 4 + 3(3) = 4 + 9 = 13

Part 2. Substitute with parentheses around the $-4$ , square first:

(-4)^2 - 2(-4) = 16 - (-8) = 16 + 8 = 24

Answer: 13 and 24.

Part 2 is the parentheses habit from slide 34, tested.

Use the Language of Algebra · bookSHelf Integrated Math 1§0.1

Key Terminology — the twelve words this section defines

Key terms

whole numbers — the counting numbers together with zero.

placeholder — a zero that holds a place open so the remaining digits keep their values.

rounding — replacing a number with a nearby, simpler one, at the cost of some accuracy.

variable — a letter representing a number whose value may change.

constant — a number whose value always stays the same.

expression — numbers, variables and operations naming a quantity; no equals sign.

equation — two expressions joined by an equals sign.

base — in $a^n$ , the factor $a$ that is repeated.

exponent — in $a^n$ , the number $n$ counting how many times the base is used.

term — the pieces an expression separates into at its $+$ and $-$ signs.

coefficient — the number multiplying the variable part of a term.

like terms — same variables, same powers, and therefore combinable.

Second column reveals on click so the first can be recalled before it is shown.

Use the Language of Algebra · bookSHelf Integrated Math 1§0.1

The headline result

One expression covers every case at once

$25 + 10g$ is not the bill for one month — it is the rule that produces the bill for any month, as soon as you know $g$ .

Every technique in this section — place value, translation, exponent scope, like terms — exists to keep that one line unambiguous.

† This is why notation discipline is not fussiness. A missing pair of parentheses in $(3x)^2$ , or a reversed "less than", changes which quantity the line names — and every later chapter reads that line literally.

The result lives in a ruled, accent-topped box — never a lone giant numeral.

0.1

Use the Language of Algebra · bookSHelf Integrated Math 1§0.1

§0.1 — Conclusions

What to carry forward

The one idea

Algebra is a language with a fixed grammar. A variable names what changes, an expression names a quantity, an equation makes a claim — and the order of operations guarantees each line has exactly one value.

Where it goes wrong

Chained rounding, "less than" read forwards, an exponent assumed to reach further than it does, and unlike terms combined. Every one is a reading error, not an arithmetic error.

Next: §0.2 Integers — the signed-number arithmetic this section's parentheses habit was preparing you for. Back to start.

Two ruled cards — core idea under a heavy top rule, failure case beside it — over the ghost section numeral.