Is this a quadratic expression?

What this sequence teaches

A quadratic expression has an x squared, and nothing higher than x squared. That one feature can fail in two directions: no x² at all, or something higher alongside it. Everything else — how many terms there are, their order, the signs and sizes of the numbers — doesn’t matter, and the examples are chosen to show that.

The second direction is the one that catches students out. “It’s got an x², so it’s quadratic” is the common error, which is why the sequence ends on it.

The teaching sequence

Example 1 — 9x + 4 (not quadratic). Nothing is higher than x², but there’s no x² at all.

Example 1 → Example 2. The + 4 slides across to make room, and a small 2 appears on the x. Nothing else changes.

Example 2 — 9x² + 4 (quadratic). The only change is the square on the x. Notice there’s no x term: a quadratic doesn’t need all three kinds of term.

Example 2 → Example 3. A clean fade: a new example, not a transformation.

Example 3 — −¾x² (quadratic). One term, with a negative fraction in front. Still quadratic.

Example 3 → Example 4. A clean fade.

Example 4 — 8 − 5x + x² (quadratic). All three kinds of term, with x² last. Order doesn’t matter. This example sets up the last one.

Example 4 → Example 5. The + x² slides across, and a small 3 appears on the 5x.

Example 5 — 8 − 5x³ + x² (not quadratic). The x² is still there, in the same place, and it isn’t enough. This is the contrast students should leave with.

Why the sequence is shaped this way

• The two animated changes answer each other. In both, a power appears on an x term and nothing else changes. A square makes the expression quadratic; a cube stops it being one.
• The three quadratics in the middle are as different as possible: two terms with no x term, one term with a negative fraction in front, and three terms with x² last. None looks like the textbook shape ax² + bx + c.
• Example 5 keeps three terms and keeps x² last, so neither “too many terms” nor “x² in the wrong place” can explain why it isn’t quadratic. Only the cube can.
• No number doubles as a power. There’s no + 2 beside an x² and no 3 beside an x³, so a small 2 or 3 can only be read as a power.

What this sequence doesn’t address

Left to the testing card, where the rule decides every item: 2x read as x²; x² on its own; a cube hidden at the end; and x⁴.

Left to the expansion card: other letters, terms that cancel, brackets, and x in a fraction. Equations come in a later atom.

The rule and the script

The rule, exactly as the Show rule button displays it: For the following examples: if it has an x squared, and nothing higher than x squared, then it is a quadratic expression.

The spoken justification for each example:

• The three quadratics — “This is a quadratic expression. How do I know? Because it has an x squared, and nothing higher than x squared.”
• 9x + 4 — “This is not a quadratic expression. How do I know? Because it has nothing higher than x squared — but it doesn’t have an x squared.” Said aloud, “there’s no x squared” is fine.
• 8 − 5x³ + x² — “This is not a quadratic expression. How do I know? Because it has an x squared — but it also has x cubed, which is higher.”

Whether students see the words is your call; nothing appears unless you click. Show rule puts the rule above the stage — you can turn it on before Example 1 — and it stays until you turn it off. Why? appears after each reveal and points at what the reason names: the x² on each quadratic; the bare x in 9x + 4; and on Example 5, first the x² and then the x³. On the two fades (Examples 3 and 4), where no animation has drawn the eye, it is worth asking “What decided it?” before you click.

Vocabulary. The page says “nothing higher than x squared” rather than “the highest power is 2”, so that no example relies on students knowing that x means x¹. Once students are fluent, the shorter version is worth teaching.

Where this atom sits

Before it: recognising an expression (Is this an expression?), and knowing what x² and x³ mean.

This atom: a quadratic expression has an x squared, and nothing higher.

After it: the expansion card below. Beyond that, a later Is this a quadratic equation? atom, whose new feature is collecting everything on one side first.

Running the sequence

• Pause on each example before revealing. The pause is the prediction moment — students commit to an answer before the marker appears.
• The Replay button appears on Examples 2 and 5, where a single change is the lesson.
• The number of terms changes a lot between Examples 2, 3 and 4. Don’t comment on it; let the examples show that it doesn’t matter.

About the testing sequence

Ten items in a new random order each time: four quadratic expressions and six that aren’t. Every item uses x, needs no simplifying, and can be decided by the rule as shown. There is no Show rule button here — students apply the rule without it on screen. Why? works as in the teaching sequence, after each answer.

The items are static. The animations in the teaching sequence mark the change between two examples; they don’t carry the meaning of a quadratic, so a still expression is a fair test.

What each item is diagnosing

x³ + 4x² − 1; 7x² − 6x + x³. “Any x² makes it quadratic.” The first is the plain form, cube first. The second is harder: a confident 7x² at the front, and a cube with no number in front of it at the end.

x⁴ + 5x². “Quad means four.”

x³ − 8x. “Any power above x makes it quadratic.” Both conditions fail: there’s no x², and there is something higher.

2x + 5. “2x is the same as x².” The only item where a 2 appears as a number rather than a power — deliberately.

12 − x. A twin of 10 − x², for students who spot a number, a minus and an x without looking for the power.

x² − 7x; 10 − x²; x²; 6x − x² + 1. The four quadratics. Each breaks the textbook shape ax² + bx + c: no number; x² last and negative; x² on its own; x² in the middle.

Common confusions to watch for

• The misreadings of 2x and of “quad” usually come from vocabulary and notation, not from the idea of a quadratic.
• A student who rejects x² on its own often says “there’s nothing to it”. Point them back to the rule: it asks for an x squared and nothing higher, and says nothing about other terms.

Discussion prompts

• “Which part of 7x² − 6x + x³ stops it being quadratic?”
• “What’s the only difference between 10 − x² and 12 − x that matters?”
• “Does x² on its own have an x squared, and nothing higher?”

Reading the summary

At the end, all ten items are shown together. Each cell carries two channels of information: a green or red tint shows whether the student answered correctly; the ✓ or ✗ in the corner shows whether the expression actually is quadratic.

Look for clusters. Misses on the items with a cube or x⁴ alongside an x² point to “any x² makes it quadratic”. Misses on the four quadratics point to “it must look like ax² + bx + c”. They need different reteaching.

About the expansion sequence

Six items just outside what the teaching and testing sequences covered, to see how far students’ understanding travels. It works exactly like the testing sequence: random order, Quadratic or Not quadratic, feedback, Why?, and a summary.

The rule begins “For the following examples”. Every one of those examples used x, had no brackets or fractions with x in them, and had nothing left to simplify. This card steps outside each of those.

Two kinds of item

The rule still works on an unfamiliar surface.

• n² + 4n (quadratic) and 5t² − t³ (not quadratic). The letter doesn’t matter. n is the letter students will meet in quadratic sequences.

Applying the rule blindly gives the wrong answer.

• x² + 5x − x² (not quadratic). It looks quadratic, but the x²s cancel, leaving 5x.
• x³ + x² − x³ (quadratic). It looks cubic, but the x³s cancel, leaving x².
• (x + 4)(x − 5) (quadratic). No x² is written, but x times x is hiding in the brackets.
• x² + 6/x (not quadratic). It passes the rule, but it has x on the bottom of a fraction, and a quadratic expression is made only of x squared, x and numbers.

Who it’s for

The two letter items suit anyone who has done the testing card. The cancelling items need students who can collect like terms. The bracket item assumes students can break up double brackets (Breaking up double brackets). The fraction item is best kept for confident students.

The justifications

Here the Why? reasons name what’s new first, then restate the two conditions:

• x³ + x² − x³ — “This is a quadratic expression. How do I know? Because the x cubeds cancel out, leaving x squared. It has an x squared, and nothing higher than x squared.”
• (x + 4)(x − 5) — “This is a quadratic expression. How do I know? Because breaking up the brackets gives x times x, which is x squared, and nothing higher than x squared.”
• x² + 6/x — “This is not a quadratic expression. How do I know? Because it has an x squared, and nothing higher — but it also has x on the bottom of a fraction. A quadratic expression is made only of x squared, x and numbers.”

As on the other cards, Why? points at what the reason names: the two terms that cancel, the two xs in the brackets, the x in the fraction.

Discussion prompts

• “What do you have to do to x³ + x² − x³ before the rule can decide it?”
• “Where is the x squared in (x + 4)(x − 5)?”

Reading the summary

The summary uses the same two channels as the testing card. Misses on the cancelling items suggest students are applying the rule to how an expression looks rather than to what it is — a good moment to return to “For the following examples”.