Fractions, Decimals, and Their Operations
Warm-up
Show $3.47 in coins and bills. What does the 4 represent? (4 dimes = 4 tenths of a dollar.) The 7? (7 pennies = 7 hundredths.) Write: 3.47 = 3 + 4/10 + 7/100. Now write it as a fraction: 347/100. These are all the same number.
Explore
Decimal comparison pairs: given pairs of fractions and decimals (e.g., 0.7 and 3/4; 0.45 and 1/2), students place both on a 0-to-1 number line and determine which is greater. The number line makes the comparison visual and avoids mechanical conversion.
Formalize
Decimal addition: 2.34 + 1.57. Ones: 2+1=3. Tenths: 0.3+0.5=0.8. Hundredths: 0.04+0.07=0.11. Regroup: 0.11 = 0.1 + 0.01. So: 3 + 0.8 + 0.1 + 0.01 = 3.91. Lining up decimal points ensures tenths add to tenths and hundredths to hundredths: place value alignment.
Fractions, Decimals, and Their Operations
Fraction benchmarks: is 5/8 greater or less than 1/2? 4/8 = 1/2, so 5/8 > 1/2. Is 7/12 closer to 1/2 or 1? 6/12 = 1/2, so 7/12 is just above 1/2. Compare: 0.6 and 7/12. 0.6 = 6/10 = 0.600. 7/12 = 0.583. So 0.6 > 7/12.
Practice
Students convert 6 decimals to fractions and vice versa, place 8 fractions on a 0-1 number line, and solve 4 decimal addition and subtraction problems. Exit ticket: is 0.8 greater or less than 3/4?
Exit ticket
Students convert 6 decimals to fractions and vice versa, place 8 fractions on a 0-1 number line, and solve 4 decimal addition and subtraction problems. Exit ticket: is 0.8 greater or less than 3/4?
Step 1: Shade 3 columns of a 10-strip: three tenths, written 3/10 as a fraction.
Step 2: Introduce decimal notation as place value continuing RIGHTWARD: one place right of the ones, past the decimal point, lives the tenths. Three tenths → 0.3. Same amount, second outfit.
Step 3: Place both on a 0–1 number line cut into ten hops: 3 hops in. One point, two names.
Step 4: Extend to hundredths on the 10×10 grid: shade 30 squares of 100 — that's 30/100 = 0.30. And LOOK: 30 squares is exactly 3 full columns — the same shading as 0.3. So 0.3 = 0.30, proven by grid, not by rule.
Watch for: reading 0.3 as "point three" only. Require "three tenths" aloud — the place-value name carries the meaning; "point three" is just spelling.
Step 1: The trap, stated openly: 35 beats 5, so students vote 0.35 > 0.5. Log the vote.
Step 2: Build both on 10×10 grids: 0.5 = 5 tenths = 5 full columns = 50 little squares. 0.35 = 35 little squares. 50 > 35 — the vote flips.
Step 3: Explain the illusion: the digits after the point aren't a whole number "35" — they're 3 tenths and 5 hundredths. Comparing decimals digit-by-digit from the LEFT (like whole numbers, tenths first): 5 tenths vs 3 tenths — decided immediately.
Step 4: The equalizing trick for skeptics: write 0.5 as 0.50. Now 50 hundredths vs 35 hundredths — no illusion survives.
Step 5: Order these to cement: 0.4, 0.35, 0.09, 0.41. (0.09 < 0.35 < 0.4 < 0.41.) The one that fools people is 0.09 — "nine" sounds big until it's nine HUNDREDTHS.
Step 1: Estimate: 2.35 ≈ 2⅓, 1.8 ≈ 2 — expect around 4.
Step 2: The error to stage and dissect: right-aligning the digits like whole numbers — 2.35 + 1.8 → aligned at the right edge, the 8 sits under the 5 — gives 2.35 + .18 territory, landing near 2.53. The estimate (≈4) screams.
Step 3: State the real rule and its reason: line up the DECIMAL POINTS, because addition must join tenths WITH tenths, hundredths with hundredths — same-place with same-place, exactly like ones-with-ones in whole numbers.
Step 4: Recompute aligned: 2.35 + 1.80 (write the helper zero): hundredths 5+0=5; tenths 3+8=11 → write 1 carry 1; ones 2+1+1 = 4. Answer 4.15 ✓ matches the estimate.
The habit: any decimal sum starts with points stacked in a column and helper zeros filling the ragged edge. Then it's just place-value addition wearing a dot.