In this lesson:
Last lesson: — one cube, one computation.
New question: What if the formula works for any side length?
Can we write the calculation once for all cases?
You've used a placeholder before:
The letter is a flexible name for "some number we haven't fixed yet."
"Subtract from 5"
A.
B.
Commit to A or B before advancing — think about what "from 5" tells you.
"Subtract from 5" → start at 5, move back
Check with numbers: ✓ → replace 3 with : ✓
Write: "8 more than " → · "Subtract from 12" →
Read back:
Multiple correct phrasings exist — capture the meaning.
Write an expression for each:
Write all three, then advance for the answers.
If item 2 gave : "from 12" means you start at 12 — so 12 comes first.
Zoom out: outer operation: multiplication → a product of two factors
Zoom in: inside Factor 2, addition → a sum of two terms (8 and 7)
is a factor and a sum at the same time.
Analyze :
Then try : name the product structure and the sum inside.
Answer all four, then advance.
: addition → sum of two terms
: multiplication → product of two factors
is a sum — addition is outermost.
You can now write expressions and name every part.
Next: run the machine — substitute a value and compute.
The vocabulary pays off now:
The same three-step protocol handles every expression.
Step 1: Write the expression as given Step 2: Replace each variable — always in parentheses Step 3: Compute using order of operations
Why parentheses matter:
Without: at → (twenty-five) ✗
With parentheses: → ✓
A student evaluated at and wrote:
Your task:
Write your diagnosis before advancing.
Error: replaced without parentheses — digits merged into twenty-five
Correct:
Parentheses separate the coefficient from the value — signaling multiplication, not a two-digit number.
Evaluate at :
The vertical two-pass format — substitute first, compute second — keeps steps separate.
Evaluate at , :
Replace both variables before computing anything:
Both replacements happen in Step 2 — then Step 3 does all arithmetic.
Evaluate each using Write–Replace–Compute:
Show the vertical two-pass format. Advance for answers.
at :
Got 64 for item 2? You added first, then squared. Exponent before addition — always.
The same protocol works for real-world formulas — even with fractions.
Last lesson: was the volume of a cube with side 4.
The formula works for any — including .
Let's run the machine for the standard's capstone example.
Rectangle at , :
Interpret: 16 units — total distance around the rectangle.
Same protocol works for any formula.
Apply Write–Replace–Compute on your own:
Work it out fully, then advance to check.
Subtraction: "from 5" → , not
Juxtaposition: at → , not
OOO: at → exponent first → , not
Vocabulary: is a sum — outer operation names it
✓ Variable: promoted placeholder; operation unchanged ✓ Translation: meaning, not word order ✓ Outer operation names expression type ✓ Write–Replace–Compute: parentheses, then OOO
Three traps to avoid: subtraction reversal · parentheses on substitution · OOO after substitution
In 6.EE.A.3, you'll ask a new question:
Can two different-looking expressions produce the same output for every input?
Same machine? You'll use today's vocabulary — terms, coefficients, factors — to find out.
Subtraction: "from 5" → , not Juxtaposition: at → , not OOO: at → exponent first → , not Vocabulary: is a sum — outer operation names it