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Mathematical Architects · Algebra I · Foundations

Get Ready: Module 1 — Searching for Patterns

Everyone starts somewhere. Before we explore patterns and sequences, let's build the floor before the building — the handful of skills that make Module 1 click. Go at your own pace; there's no rush and no wrong place to begin.

Foundations Module 01 · Prerequisites Self-Paced
Feeling shaky on the basics is completely normal — it just means you're about to fill in a gap and get stronger. Work through one card at a time. Try each example, then tackle a "Try it" with the answer hidden so you can check yourself honestly. When the cards feel comfortable, take the Readiness Check at the bottom and head into the module.
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Skills to build first

Five prerequisite skills, each with a plain-language "why," a worked example, and practice you can check yourself.

1

The coordinate plane & plotting ordered pairs

What it is: A grid with a horizontal number line (the \(x\)-axis) and a vertical number line (the \(y\)-axis). Any point is named by an ordered pair \((x, y)\) — the first number tells you how far right or left, the second how far up or down.

Why you need it for this module: In Module 1 you'll watch a pattern show up as dots on a graph. To see the pattern, you have to be able to place those dots correctly — one for each term of the sequence.

Worked example — plot \((3, 2)\)
  1. Start at the origin \((0,0)\), the center where the two axes cross.
  2. The first number is \(x = 3\). Move 3 units right along the \(x\)-axis (positive = right).
  3. The second number is \(y = 2\). From there, move 2 units up (positive = up).
  4. Mark the point. You landed at \((3, 2)\).
Remember the order: \((x, y)\) is always (right/left, then up/down). "Run, then jump."

Try it 1 Where do you end up if you plot \((-2, 4)\)?

Show answer
Move 2 units left (because \(x=-2\) is negative), then 4 units up (because \(y=4\) is positive). The point sits in the upper-left region of the grid.

Try it 2 A point is plotted 5 right and 1 down from the origin. Write its ordered pair.

Show answer
Right is positive \(x\), down is negative \(y\), so the ordered pair is \((5, -1)\).
Practice more (free) Khan Academy ↗ IXL ↗
2

Reading tables and graphs

What it is: A table lines up two related quantities in rows or columns; a graph shows the same pairs as points. Reading them means matching an input to its output — "when this is …, that is …."

Why you need it for this module: A sequence is shown three ways at once — as a table of terms, as plotted points, and as a rule. If you can read a table and a graph, you can spot the pattern jumping between them.

Worked example — read this table

Term number \(n\): 1, 2, 3, 4  →  value: 5, 8, 11, 14.

  1. Pair them up: when \(n=1\), value \(=5\); when \(n=2\), value \(=8\); and so on.
  2. Look down the value row. From 5 to 8 is \(+3\); from 8 to 11 is \(+3\); from 11 to 14 is \(+3\).
  3. Name the pattern: the value goes up by a steady \(3\) each step.
  4. As points: these become \((1,5),(2,8),(3,11),(4,14)\) — and they'd line up straight on a graph.
Tip: to read a graph, just reverse this — pick a point, read across to the value, and back down to the input.

Try it 1 In the table above, what is the value when \(n = 3\)?

Show answer
Read the column for \(n=3\): the value is 11.

Try it 2 The points \((1,2),(2,4),(3,6)\) are on a graph. By how much does the value change each step?

Show answer
From 2 to 4 is \(+2\), and 4 to 6 is \(+2\). The value increases by 2 each step.
Practice more (free) Khan Academy ↗ IXL ↗
3

Evaluating expressions with order of operations

What it is: When an expression mixes operations, everyone follows the same order so we all get the same answer: Parentheses, Exponents, Multiply/Divide (left to right), then Add/Subtract (left to right) — PEMDAS.

Why you need it for this module: A pattern's rule looks like \(a_n = 3 + 2(n-1)\). To find a specific term you plug in a number for \(n\) and evaluate — and only the right order gives the right term.

Worked example — evaluate \(3 + 2(n-1)\) when \(n = 5\)
  1. Substitute: replace \(n\) with 5 → \(3 + 2(5 - 1)\).
  2. Parentheses first: \(5 - 1 = 4\), giving \(3 + 2(4)\).
  3. Multiply before adding: \(2 \times 4 = 8\), giving \(3 + 8\).
  4. Add: \(3 + 8 = 11\).
Answer: \(11\). The parentheses and the multiplication had to happen before the addition.

Try it 1 Evaluate \(4 + 3 \times 2\).

Show answer
Multiply first: \(3 \times 2 = 6\). Then add: \(4 + 6 = \mathbf{10}\) (not 14 — multiplication comes before addition).

Try it 2 Evaluate \(2(3 + 4) - 5\).

Show answer
Parentheses: \(3+4=7\). Multiply: \(2 \times 7 = 14\). Subtract: \(14 - 5 = \mathbf{9}\).
Practice more (free) Khan Academy ↗ IXL ↗
4

Integer operations (\(+\ -\ \times\ \div\) with negatives)

What it is: Working with positive and negative whole numbers. The two rules that catch people: adding/subtracting moves you along the number line, and same signs multiply/divide to a positive, different signs to a negative.

Why you need it for this module: Patterns don't only go up. A sequence can step down by a negative amount, like \(-3\) each time. Handling negatives confidently lets you follow decreasing patterns without getting tripped up.

Worked example — the four operations with \(-6\) and \(2\)
  1. Add: \(-6 + 2\). Start at \(-6\), move 2 right (toward zero) → \(-4\).
  2. Subtract: \(-6 - 2\). Move 2 further left (more negative) → \(-8\).
  3. Multiply: \(-6 \times 2\). Different signs → negative → \(-12\).
  4. Divide: \(-6 \div 2\). Different signs → negative → \(-3\).
Signs: same → positive, different → negative (for \(\times\) and \(\div\)). Subtracting a number is the same as adding its opposite.

Try it 1 Compute \(5 + (-8)\).

Show answer
Start at 5, move 8 left → \(\mathbf{-3}\). (Adding a negative is the same as subtracting: \(5 - 8 = -3\).)

Try it 2 Compute \(-4 \times -3\).

Show answer
Same signs (both negative) → the product is positive: \(-4 \times -3 = \mathbf{12}\).
Practice more (free) Khan Academy ↗ IXL ↗
5

Recognizing ratios & patterns

What it is: A pattern is a repeating rule that gets you from one number to the next. Sometimes you add the same amount each time (a constant difference); sometimes you multiply by the same amount each time (a constant ratio). A ratio compares two quantities by division.

Why you need it for this module: This is the whole heart of "Searching for Patterns." You'll decide whether a list is built by adding the same number (arithmetic) or multiplying by the same number (geometric) — that's the difference and ratio idea in action.

Worked example — find the rule for \(4, 12, 36, 108\)
  1. Check for a constant difference (try adding): \(12 - 4 = 8\), but \(36 - 12 = 24\). Not the same — so it's not "add the same amount."
  2. Check for a constant ratio (try dividing): \(12 \div 4 = 3\), and \(36 \div 12 = 3\), and \(108 \div 36 = 3\).
  3. Same ratio every time: each term is \(3\) times the one before.
  4. Predict the next term: \(108 \times 3 = 324\).
Rule: multiply by \(3\) each step (a constant ratio of \(3\)). Next term: \(324\).

Try it 1 What is the rule for \(7, 10, 13, 16\), and what comes next?

Show answer
Differences: \(10-7=3\), \(13-10=3\), \(16-13=3\). It's add 3 each step (constant difference). Next term: \(16 + 3 = \mathbf{19}\).

Try it 2 Is \(2, 6, 18, 54\) built by adding or by multiplying? Find the next term.

Show answer
Dividing gives \(6\div2=3\), \(18\div6=3\), \(54\div18=3\) — a constant ratio, so it's multiplying by 3. Next term: \(54 \times 3 = \mathbf{162}\).
Practice more (free) Khan Academy ↗ IXL ↗
✅

Quick Readiness Check

Six short questions spanning all five skills. Try each one first, then reveal the answer to see how you did.

  1. Plot \((4, -3)\): which direction do you move for each number?

    Show answer
    4 units right (positive \(x\)), then 3 units down (negative \(y\)).

  2. A table reads \(n: 1,2,3\) and value \(: 6, 10, 14\). By how much does the value change each step?

    Show answer
    \(10-6=4\) and \(14-10=4\): it increases by 4 each step.

  3. Evaluate \(5 + 3(4 - 2)\).

    Show answer
    Parentheses: \(4-2=2\). Multiply: \(3\times2=6\). Add: \(5+6=\mathbf{11}\).

  4. Compute \(-7 + 3\) and \(-7 - 3\).

    Show answer
    \(-7 + 3 = \mathbf{-4}\) (move right toward zero); \(-7 - 3 = \mathbf{-10}\) (move further left).

  5. Compute \(-8 \div 4\) and \(-2 \times -5\).

    Show answer
    \(-8 \div 4 = \mathbf{-2}\) (different signs → negative); \(-2 \times -5 = \mathbf{10}\) (same signs → positive).

  6. Is \(5, 10, 20, 40\) built by adding or by multiplying? What's next?

    Show answer
    Each term is \(2\times\) the one before (a constant ratio), so it's multiplying by 2. Next term: \(40\times2=\mathbf{80}\).

If these feel comfortable, you're ready for the module.
If a couple felt hard, that's totally fine — scroll back up, re-read that one card, and try its "Practice more" links. Then come back.

When you're ready →

Step into Module 1

You built the floor. Now let's build on it. Open the module, play with the Visual Lab, or check where Module 1 sits in the year.

Module 1 Overview → Open the Visual Lab Course Syllabus

Foundations content is original review material aligned to the prerequisites for the TEA Bluebonnet Learning — Secondary Mathematics open curriculum (Edition 1, adapted from Carnegie Learning), licensed CC BY-NC 4.0. Classroom use is non-commercial.