Breaking up double brackets

What this sequence teaches

To break up double brackets, split the first bracket. Each of its terms, with its sign, gets its own copy of the second bracket, and the second bracket is copied exactly. Nothing gets multiplied out: the response to (3x + 2)(4x − 5) is 3x(4x − 5) + 2(4x − 5), not 12x² − 7x − 10. This is the first line of expanding double brackets, pulled out so students can practise it on its own and you can see it on every whiteboard.

The teaching sequence

Example 1 — (3x + 2)(4x − 5). The general case. Each term in the first bracket gets a copy of (4x − 5).

Example 2 — (4x − 5)(3x + 2). The same two brackets, swapped. Students watch a second full break-up, with their attention on the one thing that matters: which bracket is split and which is copied. The answer is 4x(3x + 2) − 5(3x + 2), built from different pieces even though the brackets are the same. The −5 keeps its minus sign, and the copy after it is still exactly (3x + 2).

Two examples are enough. A negative first term and a constant written first are left to the testing sequence, where students have to produce the answer rather than watch it. The testing sequence also swaps a pair of brackets, so students produce the contrast they have just watched.

Running the sequence

• Keep it uninterrupted — about a minute. The testing sequence is where students take over.
• Reveal follows the rule in order, one piece at a time, and takes about seven seconds. The first term pulses in the question, drops to its place in the answer, and the page pauses. The second term does the same, with its sign. A longer pause then shows the bracket broken up, with a faint line marking the gap after each term — a good moment to ask “what goes in each gap?” Then the second bracket pulses and a copy flies into the first gap; it pulses again and a copy flies into the second. One copy per term.
• Each piece is blue while it moves, and so is the place it came from. It turns black once it has landed, so the finished answer looks like the one on students’ boards.
• Moving to Example 2 is deliberately slow: Example 1’s answer dims and is labelled “Previous answer”, the page pauses, then the two brackets slide past each other. Ask: “Same two brackets — will the answer be the same?” Reveal then clears the previous answer and breaks up the new question from scratch.
• Replay on Example 2 plays the swap again from Example 1 — before Reveal, just the swap; after it, the swap and then the reveal. On Example 1, Replay runs the reveal again.

The rule and the script

The rule, exactly as Show rule displays it: For the following examples: to break up double brackets, split the first bracket. Each term keeps its sign and gets its own copy of the second bracket. The second bracket stays exactly the same.

• Example 1 — “I split the first bracket. 3x gets a copy of (4x − 5), and so does +2. The second bracket stays exactly the same.”
• Example 2 — “Same two brackets, swapped. I split the first bracket, the one written first. 4x gets a copy of (3x + 2), and so does −5. The −5 keeps its minus sign.”

When Why? is open, the page steps through the terms one at a time: the term turns blue, is underlined and pulses in both the question and the answer, and its copy of the second bracket is underlined alongside the second bracket itself. Colour is never the only cue, so it works for colour-blind students and on a washed-out projector. Whether students see the rule and the reasons is your call.

Why these conventions?

Why always the first bracket? Either bracket would work, but a choice every time is a chance to go wrong, and a fixed rule makes the whiteboards easy to read. Example 2 swaps the two brackets to show that position decides, not content, and the testing sequence checks it.

Why write the sign where it was? The response is then the first bracket’s terms, in order, with a copy of the second bracket after each one. Nothing moves except the copies arriving, so “everything else stays the same” is visible on the board. What −2(4x − 5) comes to is the next atom’s job.

Where this atom sits

Before it: reading 3x(4x − 5) as 3x times the whole bracket.

After it: expanding each single bracket, then simplifying the products and collecting like terms. Each of those is a separate step and is never asked for here.

About the testing sequence

Six items in a designed order. The first is a reset: a new pair of brackets, unlike either teaching example, so students have to use the rule from scratch rather than adapt a teaching answer. After that, some items change exactly one thing from the item before, and a second and third reset start again from nothing. Show the item, students write on mini-whiteboards, then reveal. Go through them in order and reveal each answer before moving on. There is no scoring — the whiteboards are the data.

After a one-change item, the previous answer stays on screen, dimmed and labelled, and Reveal changes only what needs changing (item 2 is the exception: nearly everything changes, so it plays in full). After a reset the previous answer is wiped: nothing carries over, so students have to use the rule, not edit their last board.

The change to the question is slow on purpose, and the changed part pulses once when it lands. If anyone missed it, press Replay: before Reveal it replays the change to the question; after it, the change to the question and then the change to the answer. On a reset, Replay appears after Reveal and replays the reveal.

A full reveal plays one piece at a time, as in the teaching sequence: each term pulses, drops and pauses, then each copy of the second bracket flies into its gap.

What each item is diagnosing

• (5x − 4)(2x − 9). Reset. Can students use the rule on a pair of brackets they haven’t seen? A board that isn’t two copies of (2x − 9) means the rule isn’t there yet.
• (2x − 9)(5x − 4). The same brackets, swapped. Students watched this contrast in the teaching sequence; now they produce it. A board showing 5x(2x − 9) − 4(2x − 9) split the bracket it recognised, not the first one. The previous answer stays on screen, but the reveal starts from scratch: “Same two brackets — why doesn’t the answer look the same?”
• (−2x − 9)(5x − 4). A negative first term. 2x(5x − 4) − … means the leading minus was lost.
• (−7x + 6)(8x − 9). Reset. A board that isn’t two copies of (8x − 9) means the student was editing their last board, not splitting.
• (6 − 7x)(8x − 9). The constant comes first. −7x(8x − 9) + 6(8x − 9) means the student always puts the letter term first; 6(8x − 9) + 7x(8x − 9) means the minus didn’t travel with the 7x.
• (−2 + 5x)(9x + 4). Reset: a negative constant first. Watch for the leading minus and the order.

Common confusions

Two stand out. The first is running ahead and multiplying out — the board shows 12x² or −10 somewhere. It’s the right instinct at the wrong time: this step stops at the split. The second is “fixing” the signs inside a copy after a minus, writing −2(4x + 5). The copy is always exactly the second bracket; what the minus does to it comes later.

Discussion prompts

• “Say the rule, then use it on this one.”
• “What stayed the same between these two answers? What changed, and why?”
• “These two questions have the same brackets. Why are the answers different?”

About the expansion sequence

Eight items that show how far the rule reaches. Every one uses the same rule on something new. The order does the teaching: each run of items adds one thing at a time, and a reset starts the next run. Don’t skip items or run them out of order.

What each step adds

1. A third term in the first bracket: three copies.
2. A third term in the second bracket: every copy grows.
3. Reset: a fraction, two letters and powers. The rule doesn’t change.
4. A surd joins the first bracket: three copies of a three-term bracket.
5. Reset: back to a plain pair of brackets, to set up the next change.
6. The 7 becomes 1. The 1 gets a copy like any term, but isn’t written: 6x(2x − 9) + (2x − 9). The reveal drops the 1 like any other term, hands out the copies, then fades the 1 and closes the gap.
7. +1 becomes −1: 6x(2x − 9) − (2x − 9). The minus stays when the 1 goes.
8. Reset: identical brackets. Either split gives the same answer.

Where classes stall

• The long answers. Three copies of a three-term bracket don’t fit on one line, so the answer wraps between two products — never in the middle of a copy — and the second line starts with its sign.
• The 1. Some students write 1(2x − 9). That’s correct, just not how we write it, in the same way we write x, not 1x.
• −(2x − 9). This is exactly where the sign errors of the next step come from. Here, the copy is still just the second bracket.

Discussion prompts

• “What’s new in this one? Did the rule change?”
• “Where did the 1 go? Is it still there?”