Multiplication and Division of Decimals

Warm-up

Hold up a store flyer: "Granola bars, $0.85 each. I'm buying 6." Ask for the total by mental math and collect strategies: some do $6 \times 85$ cents $= 510$ cents $= \$5.10$; some do $6\times0.8 + 6\times0.05$. Both work — and both dodge the decimal until the end or split it sensibly.

Name the theme of the unit: decimal multiplication and division are whole-number operations plus PLACE-VALUE bookkeeping. The digits do what they always did; only the point needs managing.

Explore

Estimation-first workshop: for each of $4.2 \times 0.5$, $0.3 \times 0.4$, $6.4 \div 0.8$, pairs first bracket the answer ("more or less than 4.2? than 1?"), then compute with any method (grids, money, repeated reasoning), then reconcile.

The two shockers to let land: multiplying by a number less than 1 SHRINKS ($0.3 \times 0.4 = 0.12$ — smaller than both factors), and dividing by a number less than 1 GROWS ($6.4 \div 0.8 = 8$). The 10×10 grid shows why: 0.3 of 0.4 is a sliver of a sliver.

Formalize

Formalize both operations through whole-number arithmetic. Multiplication: multiply as whole numbers, then place the point so the answer has as many decimal places as the factors combined — because each factor was scaled by a power of ten:

$3.6 \times 0.24 = \frac{36 \times 24}{10 \times 100} = \frac{864}{1000} = 0.864$

Division: shift BOTH numbers' points until the divisor is whole ($6.4 \div 0.8 \to 64 \div 8$) — legal because both scaled by the same factor, and a quotient is unchanged when both numbers grow tenfold. Then divide as usual. The estimate made first referees the point's final position.

Practice

Practice: six computations mixed with three estimation-only items ("is $0.9 \times 52$ closer to 5, 50, or 500?"), one money problem, and one "place the point" puzzle where the digits 336 are given and students must position points to make $4.8 \times 0.7 = ?$ true.

Exit ticket: compute $0.6 \times 0.35$ AND explain with the grid or the fraction form why the answer is smaller than both factors.

Exit ticket

Practice: six computations mixed with three estimation-only items ("is $0.9 \times 52$ closer to 5, 50, or 500?"), one money problem, and one "place the point" puzzle where the digits 336 are given and students must position points to make $4.8 \times 0.7 = ?$ true.

Exit ticket: compute $0.6 \times 0.35$ AND explain with the grid or the fraction form why the answer is smaller than both factors.