Substituting into an expression

What this sequence teaches

To substitute, replace the letter with its value, in brackets. Everything else stays exactly as it was. Nothing gets worked out: the response is 10(3) − 7y + 11, not 30 − 7y + 11. Because the page never asks for a value, students’ arithmetic can’t hide whether they understand substitution — and you see the substitution itself on every whiteboard.

The teaching sequence

Example 1 — x = 3 into 10x − 7y + 11. The base case: a letter with a number in front, and a whole-number value.

Example 2 — x = 38, same expression. Only the value changes, and only the inside of the brackets changes in the response. What goes in the brackets is whatever the letter equals.

Example 3 — y = 19a, same expression. A different letter, a value with a letter in it, and a minus sign in front — the same process. The x is left untouched: “everything else stays the same” includes the other letters.

The expression is identical in all three examples, so the only thing that ever changes is the substitution line. That’s deliberate: the substitution line decides what goes in the brackets, and where.

Running the sequence

• Keep it fast and uninterrupted — about thirty seconds for all three. Don’t stop for students to try one between examples; the testing sequence is where they take over.
• Reveal plays the substitution: a copy of the expression appears, the letter fades, the value drops in from the substitution line, and everything after it shifts along to make room for the brackets.
• Replay runs the reveal again on any example.

The rule and the script

The rule, exactly as Show rule displays it: For the following examples: to substitute, replace the letter with its value, in brackets. Everything else stays the same.

• Example 1 — “I replaced x with 3, in brackets. Everything else stays the same.”
• Example 2 — “I replaced x with 38, in brackets. Everything else stays the same.”
• Example 3 — “I replaced y with 19a, in brackets. Everything else stays the same — x stays as x, and the minus sign stays where it was.”

When Why? is open, the replaced letter in the expression and the bracketed value in the answer both turn blue and are underlined, so students see where the value went. Whether students see the rule and the reasons is your call: with some classes the words help, and with others the three examples say it better.

Why brackets every time?

10(3) could be written 10 × 3, and a lone positive number doesn’t strictly need brackets. But a rule with exceptions means students have to decide each time whether brackets are needed — and that decision is exactly what goes wrong later with negatives and powers: 10(−4b)² is not 10 − 4b². Brackets every time means there’s no decision to get wrong.

Where this atom sits

Before it: reading 10x as 10 times x, so that 10(3) reads as 10 times 3.

After it: the testing and expansion sequences below. Working out the value of a substituted expression is a separate job, which belongs to order of operations, and is never asked for here.

About the testing sequence

Six items in a designed order. Each one changes exactly one thing from the item before, and the first changes one thing from Example 3. Show the item, students write on mini-whiteboards, then reveal. Go through them in order and reveal each answer before moving on: the next item’s change only makes sense against the previous answer. There is no scoring — the whiteboards are the data. There is no Show rule button here.

What each item is diagnosing

• y = 2.5 into 10x − 7y + 11. Evaluating: a board showing −17.5 has worked something out.
• y = 2.5 into 10x − y + 11. A letter on its own: 10x − 2.5 + 11 means the brackets were dropped. Watch for a lost minus sign too.
• x = 2.5 into the same expression. Reading the substitution line: 10x − (2.5) + 11 means the student copied the last answer’s pattern.
• x = 2.5 into 11 − y + 10x. Position: 11 − (2.5) + 10x means they replaced the first letter they came to.
• x = 2.5 into 11 − y + 10x². Powers: 10(2.5²) or 10(6.25) means the squared went inside or got worked out.
• x = 4b into 11 − y + 10x². A letter value under a power: 10(4b²) or 40b² means only the letter got squared.

Common confusions

The two most common are writing 10x − 2.5 + 11 when the letter stands alone, and writing a worked-out number. Both come from the same place: substituting feels like “putting the number in”, so it doesn’t feel as if it needs brackets or should stop before calculating.

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 have the same expression and the same value. Why are the answers different?”

About the expansion sequence

The rule was introduced with “For the following examples”. These seven items show how far it reaches. Every one uses the same rule, applied to something new. The order does the teaching: each item adds one new thing, so a class that has just done the previous item can almost always do the next. Don’t skip items or run them out of order. The last item given cold would stop most students; at the end of this sequence it should feel easy.

What each step adds

1. A negative value: (−4b)².
2. The letter appears twice: replace both.
3. A value with two terms: (a + 1).
4. The letter on top of a fraction.
5. Two letters at once.
6. A fraction as the value.
7. Substituting into an expression that already has brackets — the answer becomes the question.

Where classes stall

• The negative value. Students write −4b² and lose the brackets.
• The fraction. Someone may point out that (a + 1) over 2 doesn’t need its brackets, because the fraction bar already holds a + 1 together. They’re right — say so, and say the rule keeps brackets every time so nobody has to decide.
• The last item. The nested ((5) + 1) looks alarming until students notice they’re doing exactly what they did in Example 1.

Discussion prompts

• “What’s new in this one? Did the rule change?”
• “Look at the last answer. Point to where the 5 went.”