Lesson type: Discovering — meeting a quadratic through its first face, the number face.
Stepping Stone: generate the terms of a growing pattern, and notice that the gap between terms is not constant — the pattern speeds up.
You will need: just your attention, and the optional ✏️ Scratchpad. No calculator needed today.
NEO Cornerstones: Connection · Movement · Reflection · Creativity · Rest · Nutrition
45-minute shape: Connect 12 · Move 12 · Reflect 12 · Create 5 · Rest/Nourish 4
Choose only the support that helps now. Every route returns you to this lesson.
Some patterns grow by the same amount each time — 3, 5, 7, 9 goes up by 2 over and over. Those are the linear patterns you met with straight lines. But some patterns grow by more and more each time. Watch a square of dots grow: 1 dot, then 4, then 9, then 16. The jumps are 3, then 5, then 7 — each jump is bigger than the last. The pattern is speeding up. A pattern that speeds up in this steady way is called a quadratic.
Today you only need to notice one thing: the gap between the terms keeps growing. Next lesson we will find the surprising number hiding underneath it.
What to do: pick a pattern, then press Add the next stage and watch the dots — and the orange gaps between the terms — grow.
What you are trying to notice: the number of dots grows, but so does the jump from one stage to the next. In a straight-line pattern the jump stays the same; here it gets bigger every time.
Carry this forward: a linear pattern grows by the same jump each time; a quadratic pattern's jumps keep growing.
What to do: a sequence appears with its last term hidden. Use the growing gaps to work out the next term and type it in. Press Show the gaps if you would like to see the jumps first.
What you are trying to notice: the gaps (we call them the first differences — the jumps between terms) are not equal, but they grow in a tidy, predictable way. That lets you predict what comes next.
Carry this forward: the jumps between terms are the first differences. In a quadratic they grow — but they grow steadily, which is the clue we chase next lesson.
What to do: each card shows a sequence. Decide whether it is steady (a straight-line, linear pattern — equal jumps) or growing (a quadratic pattern — jumps that get bigger). Press Show the gaps if it helps, then choose.
What you are trying to notice: you can tell the two kinds apart just from the jumps. Equal jumps → steady/linear. Jumps that change by the same amount → growing/quadratic. A pattern can go down and still be steady, as long as the jump never changes.
Carry this forward: equal jumps mean linear; jumps that grow by a fixed step mean quadratic.
What to do: make a growing pattern of your very own. Start with a small picture — a growing square, a growing staircase, a growing triangle of dots — and count the dots at each stage.
What you are trying to notice: almost any picture that grows in two directions at once (wider and taller) produces a quadratic pattern — its jumps grow. Patterns that grow in only one direction stay steady.
On the Scratchpad, sketch three or four stages of your growing pattern. Underneath, write the number of dots at each stage, and then the gaps between them. Do your gaps grow?
You have met a new kind of pattern — one that speeds up. There is nothing to work out here. Take a breath before moving on.
Notice one thing: was it the picture of the growing dots, or the numbers and their jumps, that made the "speeding up" clearest for you? There is no right answer — just what your own attention noticed.
You can jot it on the Scratchpad, say it aloud, or simply hold it in mind. You can pause here and return whenever you are ready.
Mode: intellectual. Growing patterns of dots are honest pictures, not the nourishment itself — so we name honestly what the learning feeds.
What it feeds: today feeds one of the most powerful habits in all of mathematics — looking for what stays steady while everything else changes. The terms grew, the jumps grew, yet something underneath was perfectly regular (you will uncover exactly what next lesson). Learning to ask "amid all this change, what is constant?" is how mathematicians find order in things that look messy — in patterns, in motion, in data. It turns "this is just getting bigger and bigger" into "ah, I can see the rule."
Carry one idea forward: when something seems to be changing faster and faster, look at the changes themselves — they often hold the pattern.
Your task: show that you can spot a growing (quadratic) pattern and tell it apart from a steady (linear) one.
Submission choices: an annotated Scratchpad image, a written explanation, a labelled poster, or a spoken/video walkthrough.
Filename pattern: NEO_MATH_QUAD_GROWING_PATTERNS_FirstName_Date
| Criterion | Approaching Proficiency | Developing Proficiency | Proficiency | Refining Proficiency |
|---|---|---|---|---|
| Understanding | Sees that some patterns grow. | Knows a quadratic pattern's jumps get bigger. | Distinguishes steady jumps from growing jumps. | Explains "growing jumps" as the sign of a quadratic. |
| Application | Reads terms from a pattern. | Finds the gaps between terms. | Predicts the next term from the growing gaps. | Invents a growing pattern and justifies why it grows. |
| Communication | Records a sequence. | Records a sequence and its gaps. | Explains how the gaps behave. | Explains clearly how linear and quadratic patterns differ. |
| Reflection | Names one thing noticed. | Notices the jumps change. | Notices the jumps change steadily. | Wonders what stays constant underneath — ready for second differences. |
Use support in this order: 🤔 Socratic prompt → 💡 hint → ✅ check → 👀 worked solution. The worked solution unlocks separately for each question, once you have used the prompt, used the hint and checked two answers.
Use the Scratchpad to predict, sketch, test and revise.
You can use this board, paper or another way of recording your thinking. The Scratchpad is optional.
Angles are in degrees. Examples: 9 × sin(37) = 5.42; sin⁻¹(0.5) = 30