Place Value & Decimals

5.A.1Read & write decimals01/04

Read and write decimals to thousandths.

Step01of 04

Decimal places extend to the right of the decimal point: TENTHS, HUNDREDTHS, THOUSANDTHS, ten-thousandths, etc. Each place is 1/10 the previous.

First place after .	→tenths (1/10)
Second place	→hundredths (1/100)
Third place	→thousandths (1/1000)
Fourth place	→ten-thousandths (1/10,000)

5.A.1 — Read and write decimals to thousandths.

Step 01 of 04

Decimal places extend to the right of the decimal point: TENTHS, HUNDREDTHS, THOUSANDTHS, ten-thousandths, etc. Each place is 1/10 the previous.

First place after .	→tenths (1/10)
Second place	→hundredths (1/100)
Third place	→thousandths (1/1000)
Fourth place	→ten-thousandths (1/10,000)

Step 02 of 04

Read a decimal aloud. Say the whole-number part, "and," then the decimal part as if it were a whole number with the place-value name of the LAST digit.

7.342 → "seven AND three hundred forty-two THOUSANDTHS." (The last digit is in the thousandths place, so the whole decimal portion is named thousandths.)

Step 03 of 04

Write from words. "Twelve and seven hundred five thousandths."

Whole part: 12. Decimal part: "705 thousandths" → fill three places with 705. Result: 12.705.

Twelve and forty-three thousandths. "043 thousandths" — pad with leading zero. Result: 12.043.

Step 04 of 04

Three forms.

Standard form	→4.273
Word form	→four and two hundred seventy-three thousandths
Expanded form	→ $4 + 0.2 + 0.07 + 0.003$

Key insight

Decimal places: tenths, hundredths, thousandths. Read the whole part, then "and," then the decimal part with the place name of the LAST digit. Pad with leading zeros if needed when writing.

5.A.2 — Compare decimals.

Step 01 of 04

Pad to the same length. First, make both decimals have the same number of digits after the point by adding trailing zeros.

0.7 = 0.700, 0.65 = 0.650, 0.654 = 0.654

Step 02 of 04

Compare digit by digit, left to right.

Compare 0.700 and 0.654. Tenths: 7 vs. 6 — 7 wins. So 0.7 > 0.654.

Step 03 of 04

Common trap. 0.7 is NOT smaller just because it has fewer digits. The leading digit value is what matters.

0.7 vs. 0.07	→0.7 is BIGGER (7 tenths > 0 tenths)
0.5 vs. 0.50	→EQUAL
0.4 vs. 0.39	→0.4 is BIGGER (4 tenths > 3 tenths)

Step 04 of 04

Order three or more. Pad them all to the same length, then sort.

Order 0.5, 0.42, 0.6, 0.55. Pad: 0.50, 0.42, 0.60, 0.55. Sort: 0.42 < 0.50 < 0.55 < 0.60.

Key insight

Pad with trailing zeros so all decimals have the same number of digits after the point. Then compare digit by digit, left to right. Trailing zeros after a decimal don't change the value.

5.A.3 — Round decimals.

Step 01 of 04

Same recipe as rounding whole numbers: look at the digit IMMEDIATELY AFTER the rounding place. If 5 or more, round UP. If less than 5, round DOWN.

Step 02 of 04

Round 4.367 to the nearest tenth.

Tenths digit is 3. Next digit (hundredths) is 6. 6 ≥ 5, round UP: tenths digit becomes 4. Drop everything to the right.

4.367 \to 4.4

Step 03 of 04

Round 4.367 to the nearest hundredth.

Hundredths digit is 6. Next (thousandths) is 7 ≥ 5, round UP. Hundredths becomes 7.

4.367 \to 4.37

Step 04 of 04

Cascading round-up. If rounding up creates a 10, the carry cascades up the places.

2.999 to nearest hundredth \to 3.00

The hundredths 9 + 1 = 10, carry to tenths (9 + 1 = 10), carry to ones (2 + 1 = 3). Note: keep the trailing zeros — they show the precision (rounded to hundredths).

Key insight

Look at digit after the rounding place. ≥5 round up, <5 round down. Drop everything to the right. Keep trailing zeros to show the precision when rounding up cascades.

5.A.4 — Decimals on the number line.

Step 01 of 04

Each whole-number gap on the number line splits into 10 equal pieces for tenths, 100 for hundredths, etc.

Step 02 of 04

Locate 0.7 on a number line from 0 to 1. Divide the segment into 10 equal pieces. The 7th tick from 0 is 0.7.

Step 03 of 04

Locate 0.43 on a tenths-numbered line. 0.43 is between 0.4 and 0.5. Mentally split that gap into 10 equal pieces; 0.43 is at the 3rd tick.

Step 04 of 04

Order decimals visually. Plotting them on the same number line makes order obvious — leftmost is smallest.

Use the number line to round too: a value less than halfway to the next tick rounds down; more than halfway rounds up.

Key insight

Each whole-number gap splits into 10 pieces (tenths), 100 (hundredths), etc. Plotting decimals on a number line makes ordering and rounding visual.

5.A.5 — Place-value relationships — 10× larger or smaller.

Step 01 of 04

In our number system, EACH place is 10× the place to its RIGHT and 1/10 the place to its LEFT. This works on BOTH sides of the decimal.

Hundreds → Tens	→× 1/10
Tens → Ones	→× 1/10
Ones → Tenths	→× 1/10
Tenths → Hundredths	→× 1/10
Each step left	→× 10
Each step right	→÷ 10

Step 02 of 04

Same digit, different position. The 5 in 5.0 is 10× the 5 in 0.5. The 5 in 0.05 is 1/10 the 5 in 0.5.

5.0/0.5 = 10

0.5/5.0 = 0.1 = 1/10

Step 03 of 04

Multiplying by 10, 100, 1000 = shift left.

4.7 × 10	→47 (shift one place left)
4.7 × 100	→470 (two places)
0.063 × 1000	→63 (three places)

Step 04 of 04

Dividing by 10, 100, 1000 = shift right.

82 ÷ 10	→8.2
82 ÷ 100	→0.82
82 ÷ 1000	→0.082

Key insight

Each place is 10× its right neighbor. Multiplying / dividing by powers of 10 just shifts the decimal point (or imagine the digits sliding left/right). No actual computation needed.

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