Integrated Math 1 · Chapter 1 · Functions

Representing a Growing Pattern

One dot pattern adds four dots every minute. Its twin doubles. Picture, table, graph, and rule are four ways of saying either one, and each makes a different question easy.

bookSHelf · Integrated Math 1 · §1.2 · a self-paced section

Growing Dots and Growing, Growing Dots are the two anchor tasks; everything else hangs off the contrast.

Representing a Growing Pattern · bookSHelf Integrated Math 1§1.2

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

Objectives

Describe a growing pattern of figures in words and extend it one step §1.2.2
Build a table of values from a picture, a story, or a graph Table 1.2.4
Decide from a table whether a pattern adds or multiplies each step Def 1.2.1, 1.2.4
Write a recursive and an explicit rule, and pick the easier one Def 1.2.2, 1.2.3
Sketch and read a graph of a situation — flat, rising, falling §1.2.6
Move in any direction among picture, table, graph, and rule Table 1.2.9

Six SLOs, one line each. The third is the section's spine.

Representing a Growing Pattern · bookSHelf Integrated Math 1§1.2

§1.2.1 — Four ways to show the same pattern

Four languages, one sentence

Representation	What it looks like	What it is good at
Picture / diagram	a sequence of figures, one per step	showing why the rule works — you can point at the new pieces
Table	inputs in one column, outputs in the next	spotting how the output changes from one step to the next
Graph	points or a curve on a coordinate plane	showing the overall shape of the growth at a glance
Rule (equation)	symbols such as $B = 8N + 16$	jumping to a far-away step without listing everything before it

Table 1.2.1: Each representation is a full description of the pattern; they differ only in what they make easy.

Not every pattern is friendly enough to have all four. A price list taped to a shop window is a table and nothing else — and that is where this section starts.

A price list is a small machine: put a number in, get a number out.

Representing a Growing Pattern · bookSHelf Integrated Math 1§1.2

§1.2.1 — Worked example

Four cuts, one expression

Example 1.2.1 — an $N \times N$ centre inside a border two tiles thick; simplify all four cuts

Student	How they cut it	Expression
Marisol	floor minus centre hole	$(N+4)^2 - N^2$
Jamal	four pinwheel strips	$4 \cdot 2(N+2)$
Priya	four sides + four corners	$4(2N) + 4(4)$
Wen	two across, two up	$2 \cdot 2(N+4) + 2 \cdot 2N$

Table 1.2.2: Four students, four ways of seeing one border.

Multiply each one out:

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

4 \cdot 2(N+2) = 8(N+2) = 8N + 16

4(2N) + 4(4) = 8N + 16

2 \cdot 2(N+4) + 2 \cdot 2N = 4N + 16 + 4N = 8N + 16

Answer: all four give $8N + 16$ . They were four correct descriptions of one border the whole time.

Marisol's is the one worth slowing on: the

N^2

terms cancel, which is why a quadratic-looking expression describes something that grows steadily.

Representing a Growing Pattern · bookSHelf Integrated Math 1§1.2

§1.2.1 — From one floor to several

Reading along the row, not down a column

One floor cannot show growth — several, side by side, can

Centre size, $N$	1	2	3	4	5	6
Border tiles, $B$	24	32	40	48	56	64

Table 1.2.3: The $N = 5$ column is §1.1's floor, and it says 56 — what all four students counted.

Each entry is 8 more than the one before it. Every time the centre grows by one tile on a side, the border needs exactly 8 more — and the rule says so out loud, because the $8$ in $8N + 16$ is the number multiplying $N$ .

Four expressions became one by simplifying. One expression became a visible pattern here by tabulating. Each representation makes a different fact obvious, and none makes every fact obvious.

A new tool has to pass a test you already know the answer to: before the table says anything about

N = 6

, it has to agree with §1.1 about

N = 5

.

Representing a Growing Pattern · bookSHelf Integrated Math 1§1.2

Insight — one sentence, four languages

A translation never changes what was said

A picture, a table, a graph, and a rule are four translations of the same sentence. A translation can be clumsy in one language and graceful in another, but it never changes the meaning.

If two of your four disagree, one of the translations is wrong — and now you know to go find it.

This is the checking habit the whole section rests on.

Representing a Growing Pattern · bookSHelf Integrated Math 1§1.2

§1.2.1 — Your turn

Try It Now 1.2.1

Try It Now 1.2.1 — (a) find $B$ when $N = 7$ , twice: by continuing the table, and from the rule; (b) a border took 104 tiles — how big was the centre square?

(a) From the table. It ends at $N = 6$ with 64, and each step adds 8. From the rule.

64 + 8 = 72 \qquad\qquad B = 8(7) + 16 = 56 + 16 = 72

(b) Working backwards. We want $8N + 16 = 104$ .

8N = 104 - 16 = 88 \qquad\qquad N = \frac{88}{8} = 11

Answer: (a) 72 tiles, two representations agreeing. (b) The centre square was 11 by 11.

Part (b) is a rule question, not a table question — continuing the table to

N = 11

works, but it is five more rows of arithmetic for the same answer.

Representing a Growing Pattern · bookSHelf Integrated Math 1§1.2

§1.2.2 — A pattern that grows by adding

Growing Dots: every minute, each arm gains one dot

Minutes, $t$	0	1	2	3	4	5
Dots, $D$	1	5	9	13	17	21

Table 1.2.4: Counting the dots turns the picture into a table — 13 dots at 3 minutes, no drawing required.

5 - 1 = 4 \qquad 9 - 5 = 4 \qquad 13 - 9 = 4 \qquad 17 - 13 = 4

The same number every time. The outputs tell you where the pattern is; the gaps tell you what it is doing.

One dot, then four arms; the arms each grow by one dot a minute.

Representing a Growing Pattern · bookSHelf Integrated Math 1§1.2

§1.2.2 — Definition

Constant Difference

Definition 1.2.1 — Constant Difference

A pattern has a constant difference when you get from each output to the next by adding the same number. That fixed number is the difference.

Here the difference is $4$ , and the picture says where the 4 comes from: four arms, one new dot on each, every minute. That is the picture doing a job the table cannot do.

Definition 1.2.1: four new dots arrive every minute, and that is where the 4 comes from.

You can point at the four new dots. That is the whole argument.

Representing a Growing Pattern · bookSHelf Integrated Math 1§1.2

Context Pause — the gap between the numbers is the real information

Read the gaps, not the entries

The outputs tell you where the pattern is. The gaps tell you what it is doing, and doing is what a rule has to capture.

Almost every question in this section is answered by looking at the gaps — subtract them, or divide them, but always look between the numbers rather than at them.

Sets up the two-test procedure in §1.2.4.

Representing a Growing Pattern · bookSHelf Integrated Math 1§1.2

Insight — climbing stairs versus taking the elevator

Both reach floor 100; only one is worth doing by hand

A recursive rule is the staircase. Each step is easy, and you have to take every one of them. An explicit rule is the elevator: press the floor number and arrive.

Neither rule is the better rule in general. The question decides — which is exactly the choice §1.2.4 turns into a habit.

Introduce the two rule types before either is defined.

Representing a Growing Pattern · bookSHelf Integrated Math 1§1.2

§1.2.2 — Definition

Recursive Rule

Definition 1.2.2 — Recursive Rule

A recursive rule gives the first value of a pattern and then tells you how to build each new value from the value before it.

D(0) = 1, \qquad D(t) = D(t-1) + 4

"Start with 1 dot. Every minute, the count is whatever it was last minute, plus 4." Some books write this as now = previous + 4.

Definition 1.2.2: one hop gets you the next step, and the hundredth costs a hundred hops.

D(t-1)

means the value one step back — nothing more.

Representing a Growing Pattern · bookSHelf Integrated Math 1§1.2

§1.2.2 — Definition

Explicit Rule

Definition 1.2.3 — Explicit Rule

An explicit rule computes the value at any step directly from the step number, without needing the value of the step before it.

D(t) = 4t + 1

The $4$ is the number of arms, the $t$ is how long they have been growing, and the $+1$ is the center dot. Checked at $t = 0,1,2,3$ it returns 1, 5, 9, 13 — four agreements with Table 1.2.4.

Definition 1.2.3: four arms of $t$ dots plus one center dot is exactly what $4t + 1$ says.

Build the rule from the picture rather than by guessing, then check it against the table.

Representing a Growing Pattern · bookSHelf Integrated Math 1§1.2

§1.2.2 — Worked example

Dots at 100 minutes, and at any minute

Example 1.2.2 — use $D(t) = 4t + 1$ to find the dots at 100 minutes, and write the dots at $t$ minutes

Step 1 — substitute the step number, then do the arithmetic.

D(100) = 4(100) + 1 = 400 + 1 = 401

Step 2 — read it back into the picture. Four arms, each 100 dots long, is 400 dots, plus the one center dot that never moves.

Answer: 401 dots at 100 minutes; $D(t) = 4t + 1$ dots at $t$ minutes — the rule is the answer for every $t$ at once.

Nothing more has to be computed for the general step; that is the point of an explicit rule.

Representing a Growing Pattern · bookSHelf Integrated Math 1§1.2

§1.2.2 — Your turn

Try It Now 1.2.2

Try It Now 1.2.2 — for Growing Dots: (a) dots at 7 minutes; (b) dots at 250 minutes; (c) which rule for each, and why

(a) The table stops at 21 dots, two recursive steps away — and the explicit rule agrees.

21 + 4 = 25 \qquad 25 + 4 = 29 \qquad D(7) = 4(7) + 1 = 29

(b) Recursion would mean 245 more additions, so use the explicit rule.

D(250) = 4(250) + 1 = 1001

Answer: (a) 29; (b) 1001; (c) recursive when a known value is nearby, explicit when the step is far away.

Part (c) is the real question; (a) and (b) only set it up.

Representing a Growing Pattern · bookSHelf Integrated Math 1§1.2

§1.2.3 — A pattern that grows by multiplying

Same start, opposite engine

Growing Dots	0	1	2	3	4	5
Dots, $D$	1	5	9	13	17	21
Differences		+4	+4	+4	+4	+4

Four new dots arrive each minute, whatever the figure already holds.

Growing, Growing Dots	0	1	2	3	4	5
Dots, $N$	1	2	4	8	16	32
Ratios		×2	×2	×2	×2	×2

Every dot is replaced by two, so growth depends on how big the figure already is.

Table 1.2.5: Both index from $t = 0$ : constant difference $4$ against constant ratio $2$ .

Run the difference test on the right-hand table and you get $1, 2, 4, 8$ — not constant. Failing a test is information: it tells you to go try the other one. The rules are $D(t) = 4t + 1$ and $N(t) = 2^{t}$ .

This slide is the section. Everything before it builds one column; everything after compares them.

Representing a Growing Pattern · bookSHelf Integrated Math 1§1.2

§1.2.3 — Definition

Constant Ratio

Definition 1.2.4 — Constant Ratio

A pattern has a constant ratio when you get from each output to the next by multiplying by the same number. That fixed number is the ratio.

N(0) = 1, \qquad N(t) = 2 \cdot N(t-1), \qquad N(t) = 2^{t}

Exponent notation is only shorthand for repeated multiplication here — $2^{t}$ is "doubled $t$ times." The rules for working with exponents come in §1.5.

Definition 1.2.4: every dot becomes two dots, so the whole count is multiplied by 2.

The recursive rule swaps the + for a ×; the explicit rule needs an exponent.

Representing a Growing Pattern · bookSHelf Integrated Math 1§1.2

§1.2.3 — Worked example

Reading the doubling pattern

Example 1.2.3 — (a) dots at 8 minutes for Growing, Growing Dots; (b) compare with Growing Dots at 8 minutes

N(8) = 2^{8} = 256 \qquad D(8) = 4(8) + 1 = 33

Compare. The doubling picture started out behind — 2 dots against 5 at one minute — and at eight minutes it is ahead by a factor of about 8.

Answer: (a) 256 dots; (b) 33 dots for Growing Dots, so the doubling pattern is now far ahead.

Losing early and winning permanently is the shape of every multiplying pattern.

Representing a Growing Pattern · bookSHelf Integrated Math 1§1.2

Insight — a rumor beats a paycheck

The paycheck adds; the rumor multiplies

Give someone $4 a day and after three weeks they have $84. Have one person tell two people, who each tell two more, and after three weeks the rumor has passed a million.

That is the whole gap between the two columns of Table 1.2.5, and it does not depend on the size of the amount being added.

Repeated multiplying eventually overwhelms repeated adding, always.

Representing a Growing Pattern · bookSHelf Integrated Math 1§1.2

§1.2.3 — Your turn

Try It Now 1.2.3

Try It Now 1.2.3 — a pattern starts at 3 and doubles every step: 3, 6, 12, 24, … (a) write both rules; (b) find the 7th value, counting the starting 3 as the 1st

(a) The ratio is 2. The $n$ th value has been doubled $n - 1$ times, because the 1st has not been doubled at all.

f(1) = 3, \quad f(n) = 2 \cdot f(n-1) \qquad f(n) = 3 \cdot 2^{\,n-1}

(b) Check by listing: 3, 6, 12, 24, 48, 96, 192.

f(7) = 3 \cdot 2^{6} = 3 \cdot 64 = 192

The

n - 1

exponent is where students lose a doubling; the listing check catches it.

Representing a Growing Pattern · bookSHelf Integrated Math 1§1.2

§1.2.4 — Telling the two apart from a table

Subtract first, then divide

The two-test procedure

1. Subtract each output from the next. All the same? It adds — record the difference.

2. If not, divide each output by the one before. All the same? It multiplies — record the ratio.

3. Neither constant? The pattern is neither — still a pattern, just not one of these two.

Run both tests on every neighboring pair. Two matching differences prove nothing: $1, 3, 5, 11, 21$ starts out looking like it adds 2, and then stops.

1st	2nd	3rd	4th	5th	Test	Verdict
2	4	8	16	32	ratios all $2$	multiplying
66	50	34	18	2	differences all $-16$	adding (negative)
160	80	40	20	10	ratios all $\tfrac{1}{2}$	multiplying (shrinking)
$-9$	$-2$	5	12	19	differences all $+7$	adding

Table 1.2.6: Shrinking happens both ways — a negative difference, or a ratio between 0 and 1.

"Growth" is the traditional word even when the values fall; nothing about the test changes.

Representing a Growing Pattern · bookSHelf Integrated Math 1§1.2

Insight — naming comes later

The test is the skill, not the label

These two kinds of growth do have names, and Chapter 4 gives them. A name is what you call the answer after the test has given it to you; it is not a shortcut to the answer.

Someone who remembers only "the fast one grows by multiplying" will say that about every fast-growing table, and will be wrong about half the time. For now, run the test and report what it found.

A pattern that passes neither test is neither — a real verdict, not a failure to classify. Run the test; report what it found.

Representing a Growing Pattern · bookSHelf Integrated Math 1§1.2

§1.2.4 — Worked example

A classification worked start to finish

Example 1.2.4 — $x = 0,1,2,3,4$ pairs with $y = 5, 8, 11, 14, \; ?$ ; describe the next term, write both rules, and classify

Step 1 — difference test. Constant at 3, so it grows by adding, and the missing value is $14 + 3 = 17$ .

8 - 5 = 3 \qquad 11 - 8 = 3 \qquad 14 - 11 = 3

Steps 2–5 — both rules. Here $x$ starts at 0, so by step $n$ you have taken $n$ steps.

f(0) = 5, \quad f(n) = f(n-1) + 3 \qquad f(n) = 5 + 3n

Answer: add 3; $f(n) = 5 + 3n$ ; grows by adding. Check at $n = 3$ : $5 + 9 = 14$ .

Where the index starts changes the explicit rule; the recursive rule never notices.

Representing a Growing Pattern · bookSHelf Integrated Math 1§1.2

§1.2.4 — Worked example · Table A

Table A — the differences fail, the ratios do not

Example 1.2.5 — $x = 1,2,3,4$ pairs with $y = 5, 10, 20, 40$ ; next term, both rules, classification

Step 1 — differences are $5, 10, 20$ . Not constant, so it does not grow by adding. Ratio test:

\dfrac{10}{5} = 2 \qquad \dfrac{20}{10} = 2 \qquad \dfrac{40}{20} = 2

Step 2 — the rules. Start at 5 and double $n - 1$ times; the 5th term is $40 \times 2 = 80$ .

f(1) = 5, \quad f(n) = 2 \cdot f(n-1) \qquad f(n) = 5 \cdot 2^{\,n-1}

Answer: multiply by 2, next term 80; grows by multiplying, constant ratio 2. Check at $n = 4$ : $5 \cdot 8 = 40$ .

Tables A, B, E and F are lettered as the MVP handout letters them — the gap is deliberate.

Representing a Growing Pattern · bookSHelf Integrated Math 1§1.2

§1.2.4 — Worked example · Table B

Table B — falling values, still the adding kind

Example 1.2.5 — $x = 1$ through $6$ pairs with $y = -8, -17, -26, -35, -44, -53$ ; next term, both rules, classification

Step 1 — difference test. Constant at $-9$ , so it grows by adding. Falling is exactly what a negative difference looks like.

-17 - (-8) = -9 \qquad -26 - (-17) = -9 \qquad -35 - (-26) = -9

Step 2 — the rules, then distribute.

f(1) = -8, \quad f(n) = f(n-1) - 9 \qquad f(n) = -8 - 9(n-1) = -9n + 1

Answer: subtract 9; $f(n) = -9n + 1$ ; grows by adding. Check at $n = 4$ : $-36 + 1 = -35$ .

Distributing the

-9

is where sign errors live; the check at

n = 4

is not optional.

Representing a Growing Pattern · bookSHelf Integrated Math 1§1.2

§1.2.4 — Worked example · Table E

Table E — the reason step 3 exists

Example 1.2.6 — $x = 0$ through $5$ pairs with $y = 3, 4, 7, 12, 19, \; ?$ ; next term, an explicit rule, classification

Step 1 — both tests fail. Differences $1, 3, 5, 7$ ; ratios $\tfrac{4}{3}, \tfrac{7}{4}, \tfrac{12}{7}$ nowhere near constant. The verdict is neither.

Step 2 — the differences climb by 2, so the next difference is 9 and the next term is $19 + 9 = 28$ . The outputs sit 3 above the perfect squares:

f(n) = n^{2} + 3

Answer: next term 28; $f(n) = n^{2} + 3$ ; neither kind.

Calling a table the multiplying kind because it grows quickly is a guess, not a verdict.

Plenty of perfectly orderly patterns are neither. That is not a failure of the table.

Representing a Growing Pattern · bookSHelf Integrated Math 1§1.2

§1.2.4 — Worked example · Table F

Table F — alarming signs, ordinary test

Example 1.2.7 — $x = 1$ through $6$ pairs with $y = -1,\; 0.2,\; -0.04,\; 0.008,\; -0.0016,\; 0.00032$

Step 1 — the values flip sign, which looks alarming. The test does not care.

\dfrac{0.2}{-1} = -0.2 \qquad \dfrac{-0.04}{0.2} = -0.2 \qquad \dfrac{0.008}{-0.04} = -0.2

Step 2 — the rules.

f(1) = -1, \quad f(n) = -0.2 \cdot f(n-1) \qquad f(n) = -1 \cdot (-0.2)^{\,n-1}

Answer: grows by multiplying, constant ratio $-0.2$ . The negative sign flips the values; the size below 1 shrinks them toward zero.

Both facts fall straight out of the ratio; neither requires a new idea.

Representing a Growing Pattern · bookSHelf Integrated Math 1§1.2

§1.2.4 — Your turn

Try It Now 1.2.4

Try It Now 1.2.4 — classify and write an explicit rule: (a) $66, 50, 34, 18, 2$ ; (b) $160, 80, 40, 20, 10$ ; (c) which rule finds the 30th term of (a)?

(a) Differences all $-16$ , so it grows by adding, counting the first term as $n = 1$ :

f(n) = 66 - 16(n-1) = 82 - 16n

(b) Differences vary; ratios are all $\tfrac{1}{2}$ , so it grows by multiplying:

f(n) = 160 \cdot \left(\dfrac{1}{2}\right)^{\,n-1}

(c) The explicit rule — 25 subtractions otherwise: $f(30) = 82 - 480 = -398$ .

Both answers check:

82 - 80 = 2

at

n = 5

, and

160 \cdot \tfrac{1}{8} = 20

at

n = 4

.

Representing a Growing Pattern · bookSHelf Integrated Math 1§1.2

Context Pause — "different from the book" is not the same as "wrong"

Do not erase it. Simplify both and compare.

This is the single most useful thing in the subsection. When your expression does not match the answer key, the first move is not to reach for the eraser.

Most of the time the two collapse to the same thing, and you have found a second way of seeing the figure rather than a mistake.

Two people can count the same tiles and write rules that do not look alike, and both be right.

Representing a Growing Pattern · bookSHelf Integrated Math 1§1.2

§1.2.5 — Graphs of situations

Shape is the part a table hides

What the graph does	What it means about the situation
rises / falls left to right	the output quantity is increasing / decreasing
stays flat (horizontal)	the output quantity is not changing
rises steeply / gently	it is increasing quickly / slowly
a straight segment	the change is happening at a constant rate
a bending curve	the rate of change is itself changing

Table 1.2.8: Each row is a translation between something the graph does and something the situation does.

Not every relationship arrives as a picture of dots. Often it arrives as a story — and the shape of a graph carries meaning a column of numbers does not.

The vocabulary is short. Everything in §1.2.6 is an application of this table.

Representing a Growing Pattern · bookSHelf Integrated Math 1§1.2

§1.2.5 — Worked example · the same afternoon, graphed twice

Laying the cafeteria border — tiles laid so far

Example 1.2.8 — Marisol and Jamal are laying the 56 border tiles of the N = 5 design, and they lay 8 tiles an hour between them. Three hours in they stop for an hour, then finish the job. Graph the tiles laid so far against hours elapsed.

Setting up the timeline. At 8 tiles an hour, three hours of work lays 24 tiles. The break adds none. The remaining 32 tiles take 4 more hours, finishing at hour 8.

Hours	What is happening	Tiles laid	Shape of the graph
0 to 3	laying at a steady 8 an hour	0 up to 24	a straight segment rising
3 to 4	the break	stays at 24	a horizontal segment
4 to 8	back to a steady 8 an hour	24 up to 56	rising again, same steepness

Answer: climbs to 24, flat through the break, then climbs to 56 by hour 8.

At 8 tiles an hour: 3 hours lays 24, the break adds none, 32 more tiles take 4 more hours — done at hour 8.

Representing a Growing Pattern · bookSHelf Integrated Math 1§1.2

§1.2.5 — Worked example · the same afternoon, graphed twice

Same afternoon, opposite graph

Example 1.2.8, continued — same afternoon, same workers, same break. Now graph the tiles still to lay against hours elapsed.

Hours	Tiles remaining	Shape of the graph
0 to 3	56 down to 32	a straight segment falling
3 to 4	stays at 32	a horizontal segment
4 to 8	32 down to 0	a straight segment falling

The point. One graph climbs from 0 to 56; the other drops from 56 to 0. Nothing about the afternoon changed — what changed is which quantity went on the vertical axis.

Both graphs have a flat piece from hour 3 to hour 4, and both are made of straight pieces because the work happened at a constant rate.

One climbs while the other drops — the axis choice did that, not the workers. A flat graph does not mean nothing is happening; it means the quantity being graphed is not changing.

Representing a Growing Pattern · bookSHelf Integrated Math 1§1.2

§1.2.5 — Your turn

Try It Now 1.2.5

Try It Now 1.2.5 — Priya fills a bucket for 2 minutes, carries it for 1, then pours it out over 30 seconds. (a) Sketch the water in the bucket against time and say what each piece means. (b) One piece is flat — why is that right when Priya is working hardest during it?

(a) Three pieces.

Time	What is happening	Water in the bucket	Shape
0 to 2 min	filling from the tap	rising from empty to full	a straight segment rising
2 to 3 min	carrying it	stays full	a horizontal segment
3 to 3.5 min	pouring it out	full to empty	a steep segment falling

(b) Why flat is right. The graph tracks water in the bucket, and carrying it changes where the water is, not how much.

Answer: rising, flat, then falling steeply — steep on the way out because the same water leaves in a quarter of the time; flat because the graphed quantity is unchanged.

Same shape as Example 1.2.8: rising, flat, falling. The flat piece is flat because the graphed quantity is unchanged.

Representing a Growing Pattern · bookSHelf Integrated Math 1§1.2

§1.2.6 — Moving between the representations

One skill, practiced in eight directions

From	To	How
Picture	Table	count the objects in each figure
Table	Rule	test differences, then ratios
Rule	Table	substitute step numbers and evaluate
Table	Graph	plot each (input, output) pair

From	To	How
Graph	Table	read coordinates off labeled points
Story	Graph	increasing / unchanged → rising / flat
Graph	Story	describe each piece in the situation's language
Picture	Rule	name what each visible part contributes

Table 1.2.9: Eight roads, not eight procedures.

Check across representations, not within one. A mistake almost never shows up where it was made — it shows up when a second representation disagrees. And say what each symbol means: in $D(t) = 4t + 1$ the 4 is dots-per-minute and the 1 is the starting dot.

A rule you cannot narrate is a string of characters you are hoping is right.

Representing a Growing Pattern · bookSHelf Integrated Math 1§1.2

Insight — a map, not a checklist

Four cities, and the roads between them

Think of the four representations as four cities and Table 1.2.9 as the roads. You are not memorizing eight procedures.

You are learning that from wherever you are standing, there is a road to wherever the question is easy.

Move a hard question to the representation where it is easy, answer it there, move it back.

Representing a Growing Pattern · bookSHelf Integrated Math 1§1.2

§1.2.6 — Your turn

Try It Now 1.2.6

Try It Now 1.2.6 — a rule is given as $y = 5x - 1$ : (a) build a table for $x = 0$ through $4$ ; (b) describe the graph of those points without drawing it; (c) classify the pattern, and say which of the four representations you used to decide

$x$	0	1	2	3	4
$y$	$-1$	4	9	14	19

(b) The differences are $5, 5, 5, 5$ , so the points climb 5 units for every 1 unit across — a straight line, meeting the vertical axis at $(0, -1)$ , which is what the rule gives at $x = 0$ .

(c) Constant difference of 5, so it grows by adding. The table decided it: the rule alone would only tell you if you already knew that $y = mx + b$ always grows by adding, but the differences are the direct evidence.

Rule to table to graph in one question — the three moves 1.2.6 has just collected.

Representing a Growing Pattern · bookSHelf Integrated Math 1§1.2

§1.2 — Key terminology (1 of 2)

The words for how a pattern grows

constant difference	the fixed number added to each output to get the next one
constant ratio	the fixed number each output is multiplied by to get the next one
recursive rule	a rule giving the first value and how to build each new value from the previous one
explicit rule	a rule computing the value at any step directly from the step number

Four of the seven; the graph vocabulary follows.

Representing a Growing Pattern · bookSHelf Integrated Math 1§1.2

§1.2 — Key terminology (2 of 2)

The words for reading a graph

horizontal intercept	a point where a graph meets the horizontal axis, so the output is 0
vertical intercept	a point where a graph meets the vertical axis, so the input is 0
maximum	the highest point of a graph; the largest output it reaches

Seven words, and every one of them names something you can point at in a table, a picture, or a graph.

Intercepts and maximum return constantly from §1.4 onward.

Representing a Growing Pattern · bookSHelf Integrated Math 1§1.2

§1.2 — Key result

Multiplying repeatedly always overwhelms adding repeatedly

However large the amount being added, and however small the ratio above 1, the multiplying pattern eventually passes the adding pattern and never gives the lead back.^†

Minutes	0	1	4	5	10	20
Growing Dots, $4t + 1$	1	5	17	21	41	81
Growing, Growing Dots, $2^{t}$	1	2	16	32	1,024	1,048,576

Table 1.2.10: The adding pattern leads for the first four minutes and never recovers.

† Both patterns are perfectly regular and neither ever skips or stalls — the gap is not a quirk of these particular numbers.

The paycheck and the rumor, in one table.

1.2

Representing a Growing Pattern · bookSHelf Integrated Math 1§1.2

§1.2 — Conclusions

What to carry forward

The one idea

Subtract, then divide. A constant difference means the pattern adds; a constant ratio means it multiplies; neither means neither. Write both rules, use the recursive one for a nearby step and the explicit one for a far-away step, and check every answer in a second representation.

Where it goes wrong

Testing only the first pair; calling a fast-growing table the multiplying kind without running the ratio test; losing a doubling in the $n - 1$ exponent; and drawing a busy graph for a flat quantity because the situation was busy.

Next: this section said "the number of dots at $t$ minutes" as if that always named one definite number. For the bagel sign it did not — 13 bagels produced four prices. §1.3 makes that honest, and calls the well-behaved case a function. Back to start.

Exactly one output per input is the subject of §1.3.