Measure a length when the ruler does not start at zero

model. A paper strip starts at the 1 1/4-inch mark and ends at the 3-inch mark. How long is it?

1. Locate both ends of the strip. The left end sits at 1 1/4 inches, so the first 1 1/4 inches of the ruler are outside the strip.
2. Count equal quarter-inch intervals from 1 1/4 to 3. Count the spaces traversed, not both endpoint marks as extra pieces.
3. Group four quarter-inch intervals into one inch. The seven intervals give one inch and three quarters of another inch.
4. Check by moving the strip's left end to zero on a drawing without stretching it. Its right end would be at 1 3/4, showing that changing the starting position does not change its length.

Worked example: 1 3/4 inches. Seven quarter-inch intervals connect the start to the end; the end reading alone is not the strip length.

p1. A strip begins at 1/2 inch and ends at 2 1/4 inches. Enter its length in inches.

Answer: 1 3/4

Teacher response: Seven quarter-inch intervals separate the two marks.

p2. A strip begins at 2 inches and ends at 4 1/2 inches. Enter its length in inches.

Answer: 2 1/2

Teacher response: The distance includes two full inches and one half inch.

p3. A strip begins at 3/4 inch and ends at 1 1/2 inches. Enter its length in inches.

Answer: 3/4

Teacher response: Three quarter-inch intervals separate the marks.

e1. A strip begins at 1 3/4 inches and ends at 4 inches. Enter its length and explain your interval count.

Answer: 2 1/4

Look for:

• Records a distance rather than a coordinate.
• Counts equal intervals and keeps inch units.

Teacher response: Nine quarter-inch intervals make two inches and one quarter.

transfer. An object measured from 2 inches to 5 inches is moved so its left end is at 1 inch. Where should its right end be?

Answer: At 4 inches. The object is 3 inches long; moving both ends one inch left preserves that distance.

teach-1. 3 minutes: Present two identical strips starting at different ruler marks. Ask whether their lengths should differ.

teach-2. 4 minutes: Count the highlighted intervals and record start, finish and distance as separate quantities.

teach-3. 5 minutes: Use the first three measurement cards. Have partners check each other's interval counts.

teach-4. 3 minutes: Complete the independent card and explain why the endpoint reading is not the answer.

misconception. The end mark reads 3, so the strip is 3 inches long.

Teacher response: Ask whether the strip includes the part of the ruler from zero to its starting mark. Trace only the intervals covered by the strip.

support. Mark each covered quarter-inch space with a small arc before counting; start the count after the left endpoint.

extend. Draw two different start-and-end pairs for a strip of length 1 1/2 inches and explain the agreement.