LESSON PLAN

Operations with Decimals

Apothem Team

Grade 7 · Computational Fluency

LESSON AT A GLANCE

Warm-up

5 min

Explore

15 min

Formalize

10 min

Practice

12 min

Exit ticket

3 min

Warm-up

Put four computations on the board and ask which can be done ENTIRELY in the head: $0.25 \times 8$ , $3.6 \div 0.4$ , $7.2 - 3.99$ , $0.1 \times 0.1$ . Pairs claim and defend — $0.25 \times 8$ is "a quarter of 8" = 2; $3.6 \div 0.4$ is "how many 0.4s in 3.6" = 9; $7.2 - 3.99$ begs compensation ( $-4$ , then $+0.01$ ): 3.21; $0.1 \times 0.1 = 0.01$ still fools half the room.

The unit consolidates all four decimal operations with fraction-thinking as the mental engine and place-value as the written one.

Explore

Strategy-pairing workshop: each computation type gets a written METHOD and a mental TRANSLATION side by side. Multiplication: $0.6 \times 0.35$ by whole-number product + place count, AND as $\frac{6}{10} \times \frac{35}{100} = \frac{210}{1000}$ . Division: $4.8 \div 0.6$ by point-shifting, AND as "48 tenths shared into groups of 6 tenths → 8 groups."

The estimation gauntlet closes the lab: ten expressions, answers chosen from three magnitudes ( $0.42 \times 0.5$ : is it ≈ 0.02, 0.2, or 2?) — no computing allowed, only reasoning about size.

Formalize

Formalize the operations as place-value bookkeeping over whole-number arithmetic, with the fraction forms as the WHY:

$0.6 \times 0.35 = \frac{6}{10}\times\frac{35}{100} = \frac{210}{1000} = 0.21 \qquad 4.8 \div 0.6 = \frac{48}{6} = 8$

The four operations' alignment rules diverge and must be kept distinct: add/subtract stack the POINTS; multiply counts PLACES after whole-number work; divide SHIFTS both numbers until the divisor is whole. One sentence each, attached to its reason.

Practice

Practice: mixed 12-item set spanning all four operations with three mandatory estimates-first; two multi-step money problems; one error autopsy (points aligned for multiplication).

Exit ticket: compute $2.4 \times 0.15$ and $2.4 \div 0.15$ , then explain in one sentence why one answer is tiny and the other huge. ( $0.36$ and $16$ — multiplying by a small number shrinks; dividing by one grows.)

Exit ticket

Practice: mixed 12-item set spanning all four operations with three mandatory estimates-first; two multi-step money problems; one error autopsy (points aligned for multiplication).

TIP The "multiply-makes-bigger" ghost returns at every grade: exorcise it weekly with one pair like

2.4 \times 0.15

2.4 \div 0.15

until the size-instinct flips reliably.

WORKED EXAMPLES

Example 1 — The fabric bill: multiplication with an estimate bodyguard

The purchase: 3.4 m of fabric at \$12.75 per metre.

Step 1: Estimate: $3.4 \approx 3.5$ and $12.75 \approx 13$ → about $45.5$ . Expect low-to-mid forties.

Step 2: Whole-number multiply: $34 \times 1275 = 43{,}350$ .

Step 3: Place the point: one decimal place ( $3.4$ ) + two ( $12.75$ ) = three → $43.350$ → $43.35.

Step 4: Reconcile: \$43.35 vs ≈\$45.5 ✓ (we rounded both factors up, so the true answer sits below the estimate — even the DIRECTION of the gap checks out).

Step 5: The receipts habit: money answers get two decimals and a size-check against the estimate, every time, forever.

Example 2 — The gas-tank division: how many fills?

The question: a jerrycan holds 18.9 L; a lawnmower tank takes 0.7 L per fill. How many full tanks does the can provide?

Step 1: Estimate: "how many 0.7s in about 19?" — 0.7 × 20 = 14, 0.7 × 27 = 18.9… expect around 27.

Step 2: Shift both one place: $189 \div 7 = 27$ .

Step 3: Answer: exactly 27 fills.

Step 4: Why the shift is legal, one more time with feeling: $\frac{18.9}{0.7} = \frac{189}{7}$ — both scaled by 10, the quotient unmoved. Division compares SIZES, and both sizes grew tenfold together.

Step 5: The interpretation reflex: had it come out 26.4, the story wants 26 full fills with a splash left — the remainder conversation from whole-number division, alive and well in decimals.

Example 3 — Order of operations with decimals: the deposit puzzle

Evaluate: $12 - 3 \times (4.2 - 1.7) + 0.5^2$ .

Step 1: Parentheses: $4.2 - 1.7 = 2.5$ .

Step 2: Exponent: $0.5^2 = 0.25$ (half of a half — the value that still surprises; a half squared is a QUARTER).

Step 3: Multiplication: $3 \times 2.5 = 7.5$ .

Step 4: Add/subtract left to right: $12 - 7.5 = 4.5$ ; $4.5 + 0.25 = 4.75$ .

Step 5: The two landmines dodged, named: $0.5^2 \ne 1$ (squaring a number below 1 shrinks it), and the subtraction happened before the addition only because it stood to the LEFT — tier rules, decimal edition. Old grammar, new vocabulary.

MATERIALS

•Estimation gauntlet cards

•Place-value charts

•Money manipulatives

•Practice set (PDF)

WATCH FOR

!Points aligned for multiplication (imported from addition). Different operations, different bookkeeping — the three-rules table.

!Division by a decimal dodged by dividing the wrong direction (

0.15 \div 2.4

). Estimate first: "how many 0.15s fit in 2.4? Lots" → answer must exceed 1.

!Trailing zeros dropped mid-computation changing place counts:

2.50 \times 0.4 = 1.000 = 1

✓ but students who wrote 2.5 × 0.4 = 0.1 counted places on vanished zeros.