Decimals to Thousandths and Decimal Operations
Warm-up
Athletic timing: the 100 m record is 9.58 seconds. A local runner finishes in 10.004 seconds. How much slower? The answer requires decimal subtraction. Estimate first: about 0.4 seconds. Compute: 10.004 - 9.580 = 0.424 seconds. The thousandths digit in 9.580 is a trailing zero: it does not change the value.
Explore
Precision measurement: students measure 5 classroom objects using kitchen scales (to the nearest gram = 3 decimal places in kilograms) and graduated cylinders (to the nearest mL = decimal in litres). Record in decimal notation. Order from smallest to largest. Add two measurements and subtract another.
Formalize
Decimal comparison misconception attack: compare 0.7 and 0.70 (equal: trailing zero). Compare 0.07 and 0.7 (0.7 > 0.07: different tenths digits). Compare 0.375 and 0.4 (0.4 = 0.400 > 0.375: hundredths decide). Each comparison requires finding where the digits first differ.
Decimals to Thousandths and Decimal Operations
Fraction connection: 0.625 = 625/1000. GCF of 625 and 1000: both divisible by 125. 625/125=5, 1000/125=8. So 0.625 = 5/8. Check: 5/8 as a decimal: 5 divided by 8 = 0.625. The connection between decimal and fraction representations is a two-way street.
Practice
Students compare and order 6 sets of decimals to thousandths, convert 4 decimals to simplified fractions, and solve 6 decimal addition/subtraction problems. Exit ticket: add 3.5 + 1.275.
Exit ticket
Students compare and order 6 sets of decimals to thousandths, convert 4 decimals to simplified fractions, and solve 6 decimal addition/subtraction problems. Exit ticket: add 3.5 + 1.275.
Step 1: Ground the places physically: on a metre stick, tenths are decimetres (10 cm chunks), hundredths are centimetres, thousandths are millimetres. 0.407 m = 4 dm + 0 cm + 7 mm.
Step 2: Say it right: "four hundred seven thousandths" — the whole tail reads as one number of the SMALLEST named place. (Also legal: 4 tenths + 7 thousandths.)
Step 3: Place it: between 0.4 and 0.5, very close to 0.4 — just 7 mm past the 40 cm mark.
Step 4: The comparison that exposes the classic bug: order 0.407, 0.42, 0.4. Longer-is-bigger thinkers rank 0.407 first. Align by place (pad with zeros): 0.407 / 0.420 / 0.400 → 0.4 < 0.407 < 0.42.
Step 5: The density surprise: "name a number between 0.41 and 0.42." (0.415. Then between 0.415 and 0.416? 0.4155…) There is ALWAYS another decimal between two decimals — the line has no gaps. Let that sit; it's the strangest true thing they've met this year.
Step 1: Estimate both: 3.75 + 2.8 ≈ 3¾ + 2¾ ≈ 6½. And 6.3 − 2.45 ≈ 6⅓ − 2½ ≈ 3.8-ish.
Step 2: Add with points stacked and a helper zero: 3.75 + 2.80: hundredths 5+0=5; tenths 7+8=15 → 5 carry 1; ones 3+2+1 = 6. Sum 6.55 ✓ (matches ≈6½).
Step 3: Subtract with the helper zero doing real work: 6.30 − 2.45: hundredths 0−5 → borrow: 10−5=5; tenths (now 2)−4 → borrow: 12−4=8; ones (now 5)−2=3. Result 3.85 ✓.
Step 4: Show the disaster the helper zero prevents: right-aligning 6.3 − 2.45 as if whole numbers pairs the 3 tenths with the 5 hundredths — nonsense worth seeing once, labelled, and buried.
Step 5: The money check that makes it all obvious: 2.45 — nobody pairs dimes with pennies at a cash register. Decimals ARE money with more places; when in doubt, put a dollar sign on it.
The data: four runners' 100 m legs — 14.62 s, 13.9 s, 14.08 s, 13.87 s.
Step 1: Order the legs fastest to slowest (align places mentally: 13.87, 13.90, 14.08, 14.62). Fastest runner: 13.87. Watch the 13.9 vs 13.87 comparison — the shorter numeral is the SLOWER time here… no wait: 13.87 < 13.90, and in racing less time = faster. So 13.87 wins. Two inversions at once (place value AND smaller-is-better) — say both aloud.
Step 2: Total the relay time: 14.62 + 13.90 + 14.08 + 13.87. Group friendly pairs: (14.62 + 13.87) = 28.49? — check: 14.62+13.88 would be 28.50, so 28.49 ✓; (13.90 + 14.08) = 27.98. Total: 28.49 + 27.98 = 56.47 s.
Step 3: The comparison question: the rival team ran 56.5 s flat. Who won? 56.47 < 56.50 — we did, by 0.03 s. Three hundredths of a second: the width of a lean at the tape. Decimals exist because whole seconds are too blunt to separate finishers — precision has a PURPOSE, and this is it.