Length Word Problems | Grade 2 Math

Grade 2 Math Hub › Unit 5: Measurement & Number Lines › Lesson 25

Grade 2 Math — Unit 5: Measurement & Number Lines

Length Word Problems

Solve length word problems with addition and subtraction, compare ribbon and rope lengths, find missing parts, and show an unknown number in an equation. Each answer includes its measurement unit.

Practice with 25 worked examples, 30 independent problems, a 15-question test, and complete answer keys. Review Lesson 24: Estimate, Compare & Measure Lengths when you need a measurement refresher.

Grade 2 MathCCSS 2.MD.B.5Addition & SubtractionWord ProblemsFree Practice & Test

🎯 What You Will Learn

Understand the question

Find the whole length, a missing part, how much longer, or how much is left.

Represent an unknown

Use a box (□) in an equation, then choose an addition or subtraction strategy.

Make helpful drawings

Draw a strip, ruler, or bar model showing the measured parts without gaps or unintended overlaps.

Explain and check

Keep the same units in each calculation, then check that the answer makes sense in the story.

Common Core alignment: This lesson focuses on CCSS 2.MD.B.5: solve word problems with lengths stated in the same unit using addition and subtraction within 100, drawings, and equations with a symbol for the unknown. It reviews measurement and comparison skills from 2.MD.A.1–2.MD.A.4; full number-line instruction is reserved for Lesson 26 (2.MD.B.6).

🧠 Key Vocabulary

Length

The distance measured along an object from one end to the other. For a curved string, follow the string when measuring; the straight distance between its ends can be shorter.

Difference

How much longer or shorter one measured length is than another.

Whole and part

The entire length and its shorter pieces.

Unknown (□)

A number that is not yet known. Find it using the other numbers in the story.

End to end

Pieces touch at their ends without a gap or overlap.

Estimate

A sensible approximation, not a claim of an exact measured length.

Big idea: A length story is about a distance and its parts or changes. Always keep the unit with the answer. A drawing and an equation help you see whether to add, subtract, or find a missing number.

📖 Read and Model a Length Story

A story might say that a 9 cm pencil is next to an eraser that is 4 cm shorter. The question is not asking for a total of two objects: it asks for the eraser’s length. Represent the story with □ + 4 = 9, then find □ = 5. The eraser is 5 cm long.

Compare this with a different story: a 9 cm pencil and a 4 cm eraser are put end to end. Now the unknown is the whole distance: 9 + 4 = □. The total is 13 cm.

Simple plan: (1) Read the question. (2) Circle the numbers and matching units. (3) Draw the parts or write an equation with □. (4) Compute. (5) Check by reversing the operation and writing the unit.

🧩 The Four Common Length Story Types

Join pieces

Add 7 cm and 5 cm placed end to end: 7 + 5 = □.

Take some away

Subtract 18 cm minus 6 cm cut off: 18 − 6 = □.

Compare two lengths

Find the difference 16 cm compared with 11 cm: 16 − 11 = □.

Find a missing piece

Use the whole 8 cm plus an unknown equals 20 cm: 8 + □ = 20.

📏 Handle Rulers, Gaps, and Units Carefully

A ruler starting at zero is easiest to read. If a ribbon starts at 3 cm and ends at 11 cm, its length is 11 cm − 3 cm = 8 cm. Count the equal unit intervals, not the number of numbered marks.

When pieces touch end to end, you add their lengths. If pieces overlap, subtract the overlapped length once from the sum. If there is a gap and the question asks for the distance across the whole span, add the gap. If it asks how much material is present, a gap is not material.

Important: In addition and subtraction word problems aligned to 2.MD.B.5, all length amounts within a calculation must be expressed in the same unit. Do not add inches to centimeters as if they were identical units. Do not use a schematic image on a screen as a physical ruler.

🌟 25 Worked Examples

Work from straightforward one-step stories to multi-step situations with missing parts, comparisons, overlaps, and error analysis. Read each question, think, and then open its solution. Every model is schematic, not a real-size ruler.

1Find the total length

An 8 cm pencil and a 4 cm eraser are put end to end, without gaps or overlaps. What total length do they make?

Two pieces of lengths 8 cm and 4 cm meet end to end. Find the total.

↔ On a small screen, swipe sideways; with a keyboard, focus the model and press the arrow keys.

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Step 1. We need the whole length made by the two parts. We add their measurements.

Step 2. Write an equation with an unknown: 8 + 4 = □.

Step 3. Calculate: 8 cm + 4 cm = 12 cm. Check: the total must be longer than either part.

Answer: 12 cm.

2Shorter than a known length

A playground boundary rope is 16 ft long. Another rope is 5 ft shorter. How long is the shorter rope?

The first rope is 16 feet; the second is 5 feet shorter.

↔ On a small screen, swipe sideways; with a keyboard, focus the model and press the arrow keys.

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Step 1. The first rope is 16 feet, and the second has 5 fewer feet.

Step 2. Write the missing length as 16 − 5 = □.

Step 3. Subtract: 16 ft − 5 ft = 11 ft. Check: 11 feet is shorter than 16 feet.

Answer: 11 ft.

3How much longer?

A blue strip is 18 cm long, and a red strip is 11 cm long. How much longer is the blue strip?

Two strips share one starting point; compare 18 cm and 11 cm.

↔ On a small screen, swipe sideways; with a keyboard, focus the model and press the arrow keys.

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Step 1. Both lengths use centimeters, so a direct comparison is fair.

Step 2. Represent the difference with 11 + □ = 18.

Step 3. Subtract: 18 cm − 11 cm = 7 cm. Check by adding 11 + 7 = 18.

Answer: 7 cm longer.

4A cut leaves a shorter strip

A strip of paper is 20 cm long. Maya cuts off 7 cm. What length remains?

A 20-centimeter strip consists of a 7-centimeter cut piece and a remaining piece.

↔ On a small screen, swipe sideways; with a keyboard, focus the model and press the arrow keys.

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Step 1. The original strip is the whole, and the cut piece is removed.

Step 2. Write 20 − 7 = □.

Step 3. Subtract: 20 cm − 7 cm = 13 cm. Check: 13 + 7 = 20.

Answer: 13 cm remains.

5Find the original length

After 6 cm is cut from a ribbon, 9 cm remains. How long was the ribbon before cutting?

A cut-off piece of 6 cm and a remaining piece of 9 cm made the original ribbon.

↔ On a small screen, swipe sideways; with a keyboard, focus the model and press the arrow keys.

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Step 1. This time we know the removed part and the part left. The unknown is the original whole.

Step 2. Write □ − 6 = 9, or think 6 + 9 = □.

Step 3. Add: 6 cm + 9 cm = 15 cm. Check: 15 − 6 = 9.

Answer: 15 cm originally.

6Follow a two-step comparison

A red craft strip is 11 cm long. A blue strip is 4 cm longer than the red strip. A green strip is 3 cm shorter than the blue strip. How long is the green strip?

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Step 1. First find the blue strip: 11 cm + 4 cm = 15 cm.

Step 2. Next find the green strip: 15 cm − 3 cm = 12 cm.

Step 3. Check the story: the green strip must be 3 cm shorter than the blue strip.

Answer: 12 cm.

7Find the ending ruler mark

An 11 cm ribbon begins at the 3-cm mark on a ruler. At which mark does it end?

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Step 1. A length is the distance between the two marks, not the number at the last mark.

Step 2. We know 3 + 11 = □.

Step 3. Add: 3 + 11 = 14. Check: 14 cm − 3 cm = 11 cm.

Answer: At the 14-cm mark.

8Read a ruler starting after zero

A ribbon runs from the 2-cm mark to the 10-cm mark. What is its measured length?

A schematic centimeter ruler shows a ribbon beginning at 2 and ending at 10.

↔ On a small screen, swipe sideways; with a keyboard, focus the model and press the arrow keys.

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Step 1. The number 10 is the ending reading, not the entire length.

Step 2. Write 10 − 2 = □.

Step 3. There are 8 whole-centimeter spaces between the marks, so 10 cm − 2 cm = 8 cm.

Answer: 8 cm.

9Two strips overlap

One craft strip is 7 cm long, and another is 9 cm long. They are placed in one straight line with a 2 cm overlap. What is the total length from the first start to the last end?

Schematic: two measured strips overlap by the indicated amount. Count the overlapped part only once.

↔ On a small screen, swipe sideways; with a keyboard, focus the model and press the arrow keys.

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Step 1. If the strips did not overlap, their total would be 7 cm + 9 cm = 16 cm.

Step 2. The 2 cm overlap was counted twice; subtract it once.

Step 3. Compute 16 cm − 2 cm = 14 cm. Check: the total is shorter than 16 cm.

Answer: 14 cm.

10Find a shorter unknown length

A blue ribbon is 6 cm longer than a red ribbon. The blue ribbon is 17 cm long. How long is the red ribbon?

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Step 1. The blue ribbon is the longer one. We need the shorter length.

Step 2. Write □ + 6 = 17.

Step 3. Subtract: 17 cm − 6 cm = 11 cm. Check: 11 + 6 = 17.

Answer: 11 cm.

11Check how far off an estimate was

Liam guessed a strip was 20 cm long. A ruler showed 18 cm. How much too high was the estimate?

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Step 1. Both numbers are in centimeters, so the size of the error is their difference.

Step 2. Write 20 − 18 = □.

Step 3. Subtract: 20 cm − 18 cm = 2 cm. Because 20 is greater than 18, the guess was 2 cm too high.

Answer: 2 cm too high.

12Join a strip, then trim it

A ribbon made of 7 cm and 9 cm pieces is joined end to end. Then 3 cm is trimmed off. How long is the ribbon now?

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Step 1. First find the joined length: 7 cm + 9 cm = 16 cm.

Step 2. Then subtract the trimmed piece: 16 cm − 3 cm = 13 cm.

Step 3. The answer is shorter than the untrimmed 16 cm length, which makes sense.

Answer: 13 cm.

13Find the missing part of a whole

A paper strip is 17 cm long. The green part is 9 cm long. How long is the rest?

A 17-centimeter strip is divided into a 9-centimeter part and an unknown part.

↔ On a small screen, swipe sideways; with a keyboard, focus the model and press the arrow keys.

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Step 1. The whole is 17 cm. The known part is 9 cm.

Step 2. Write 9 + □ = 17.

Step 3. Subtract: 17 cm − 9 cm = 8 cm. Check: 9 + 8 = 17.

Answer: 8 cm.

14Compare two offset segments

Segment A runs from 1 cm to 9 cm. Segment B runs from 3 cm to 13 cm on the same centimeter ruler. Which is longer and by how much?

Two schematic segments on one centimeter scale; A runs 1 to 9, B runs 3 to 13.

↔ On a small screen, swipe sideways; with a keyboard, focus the model and press the arrow keys.

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Step 1. Find A: 9 cm − 1 cm = 8 cm.

Step 2. Find B: 13 cm − 3 cm = 10 cm.

Step 3. Compare: 10 cm − 8 cm = 2 cm. Segment B is 2 cm longer.

Answer: Segment B, by 2 cm.

15Two equal pieces and one leftover

A 24 cm paper strip has two 8 cm pieces cut from it. How much paper remains?

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Step 1. Find the length cut off: 8 cm + 8 cm = 16 cm.

Step 2. Subtract the removed length from the whole: 24 cm − 16 cm = 8 cm.

Step 3. Check: 8 + 8 + 8 = 24, so the parts fit the whole.

Answer: 8 cm remains.

16Join pieces and compare

A blue ribbon is 8 cm long and a red ribbon is 6 cm long. They are joined end to end. How much longer is the joined ribbon than a 12 cm yellow ribbon?

The joined ribbon contains an 8 cm piece and a 6 cm piece; compare with a 12 cm ribbon.

↔ On a small screen, swipe sideways; with a keyboard, focus the model and press the arrow keys.

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Step 1. Add the joined lengths: 8 cm + 6 cm = 14 cm.

Step 2. Compare with the yellow ribbon: 14 cm − 12 cm = 2 cm.

Step 3. Check: the joined ribbon is longer by exactly 2 cm.

Answer: 2 cm longer.

17Find how much more is needed

A class has used 18 ft of string to outline a display. They need 31 ft in all. How many more feet are needed?

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Step 1. The needed total is 31 feet, and 18 feet have already been used.

Step 2. Write 18 + □ = 31.

Step 3. Subtract: 31 ft − 18 ft = 13 ft. Check: 18 + 13 = 31.

Answer: 13 ft more.

18Find the mistake on a ruler

A ribbon begins at the 4-cm mark and ends at the 18-cm mark. A student says it is 18 cm long. What mistake did the student make?

A ribbon starts at 4 cm and ends at 18 cm on a schematic centimeter ruler.

↔ On a small screen, swipe sideways; with a keyboard, focus the model and press the arrow keys.

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Step 1. The student read only the ending mark, not the distance between the two ends.

Step 2. Compute 18 − 4 = □.

Step 3. The correct length is 18 cm − 4 cm = 14 cm. Check: 4 + 14 = 18.

Answer: The correct length is 14 cm.

19A missing middle piece

Three pieces make an 18 cm ribbon. The first piece is 5 cm and the last is 7 cm. How long is the middle piece?

A ribbon totals 18 centimeters, with a 5 cm first part, an unknown middle, and a 7 cm last part.

↔ On a small screen, swipe sideways; with a keyboard, focus the model and press the arrow keys.

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Step 1. Add the known outside pieces: 5 cm + 7 cm = 12 cm.

Step 2. Write 12 + □ = 18 for the remaining part.

Step 3. Subtract: 18 cm − 12 cm = 6 cm. Check: 5 + 6 + 7 = 18.

Answer: 6 cm.

20Estimate what remains after cutting

A paper strip starts at the 0-cm mark and ends just past the 13-cm mark, clearly closer to 13 cm than to 14 cm. Ana cuts off 5 cm from its right end. Estimate the remaining length to the nearest whole centimeter and explain why it is not an exact measurement.

A schematic strip begins at 0 and ends just past 13 centimeters, well before 14.

↔ On a small screen, swipe sideways; with a keyboard, focus the model and press the arrow keys.

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Step 1. The original strip ends just past 13 cm, much closer to 13 cm than to 14 cm. Its reasonable whole-centimeter estimate is about 13 cm.

Step 2. Cut off 5 cm. Using the whole-centimeter estimate, write 13 − 5 = □ and find □ = 8. So about 8 cm remains.

Step 3. Since the original was a little more than 13 cm, the amount left is also a little more than 8 cm. It is still closer to 8 cm than 9 cm. Therefore, report about 8 cm, not exactly 8 cm.

Answer: About 8 cm remains.

21Compare two directions of change

A green ribbon is 15 cm long. A blue ribbon is 2 cm longer than the green ribbon. A red ribbon is 4 cm shorter than the green ribbon. How much longer is blue than red?

Three ribbons: blue is 2 cm longer than a 15 cm green ribbon and red is 4 cm shorter.

↔ On a small screen, swipe sideways; with a keyboard, focus the model and press the arrow keys.

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Step 1. Find blue: 15 cm + 2 cm = 17 cm.

Step 2. Find red: 15 cm − 4 cm = 11 cm.

Step 3. Compare the two: 17 cm − 11 cm = 6 cm.

Answer: Blue is 6 cm longer.

22A shorter object and a combined length

A ribbon is 15 cm long. A bow has a straight ribbon piece that is 4 cm shorter. If the two ribbon pieces are placed end to end, what is their combined length?

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Step 1. First find the bow’s ribbon piece: 15 cm − 4 cm = 11 cm.

Step 2. Now add the two pieces: 15 cm + 11 cm = 26 cm.

Step 3. Check: the total must exceed the length of the longer piece.

Answer: 26 cm total.

23Include a gap when measuring a span

A 25 cm display strip has a 9 cm label starting at its left edge. After a 1 cm blank gap, an 8 cm label follows, with no overlap. How much of the strip remains uncovered to the right of the second label?

A 9 cm label, a 1 cm gap, and an 8 cm label appear in that order, starting at the left edge of a 25 cm display strip.

↔ On a small screen, swipe sideways; with a keyboard, focus the model and press the arrow keys.

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Step 1. The labels cover 9 cm + 8 cm = 17 cm.

Step 2. The gap adds 1 cm to the already-accounted-for span: 17 cm + 1 cm = 18 cm.

Step 3. Subtract from the whole strip: 25 cm − 18 cm = 7 cm.

Answer: 7 cm.

24Three-step craft problem

A rope is 22 cm long. Mina cuts off 8 cm for one craft and 5 cm for another. She then finds a new rope that is 3 cm longer than what remains. How long is the new rope?

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Step 1. Add what was cut off: 8 cm + 5 cm = 13 cm.

Step 2. Find what remains: 22 cm − 13 cm = 9 cm.

Step 3. Find the new rope: 9 cm + 3 cm = 12 cm. Check that the new rope is longer than the 9 cm remainder.

Answer: 12 cm.

25Challenge: uncover the center segment

A 31 cm banner has, from left to right, a 4 cm end piece, a 9 cm piece, an unknown middle piece, an 11 cm piece, and a final 4 cm end piece. How long is the middle piece?

A 31 cm banner is divided into five end-to-end pieces: 4 cm, 9 cm, an unknown middle, 11 cm, and 4 cm.

↔ On a small screen, swipe sideways; with a keyboard, focus the model and press the arrow keys.

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Step 1. Add the known pieces: 4 + 9 + 11 + 4 = 28. Their combined length is 28 cm.

Step 2. Write the unknown-part equation 28 + □ = 31.

Step 3. Subtract: 31 cm − 28 cm = 3 cm. Check by adding all five pieces to make 31 cm.

Answer: 3 cm.

✍️ Independent Practice — 30 Problems

Directions: Solve all 30 problems independently. Use a drawing or an equation with □ when helpful. Give units with all numeric length answers. The last five problems require several steps or careful explanations.

1. A pencil and its cap measure 17 cm together. The cap adds 5 cm to the pencil’s length when fitted end to end. How long is the pencil without its cap?
2. A toy caterpillar is 9 cm long. A toy snake is 14 cm long. If a 2 cm tail is added to the snake, how much longer is the snake than the caterpillar?
3. A toy robot moves 8 cm to the right along a straight track and then moves 3 cm back toward its starting point. (a) How many centimeters did it travel altogether? (b) How many centimeters from its starting point does it stop? Explain why the two answers are different.
4. A toy car is 9 cm long. A toy truck is 3 cm longer than the car. Both are placed on a centimeter ruler with their back ends at the 4-cm mark, facing the direction of larger numbers. At which centimeter mark is the front end of the truck? Write an equation with □.
5. Maya joins two 13 cm pieces and one 9 cm piece end to end, without gaps or overlaps. She wants a banner exactly 32 cm long. How many centimeters must she trim from the end? Write an equation with □.
6. Two paper strips are 9 cm and 14 cm long. Sam joins them end to end without gaps or overlaps. First the 9-cm strip is on the left; then he reverses their order. Does the total length change? Calculate both totals and explain.
7. A 29 cm strip is cut into three pieces with no waste. The first piece is 8 cm. The second piece is 6 cm longer than the first. How long is the third piece?
8. A 13 cm ribbon begins somewhere on a centimeter ruler and ends at mark 21. What is the starting mark?
9. Which story matches 8 + □ = 23? (A) A 23 cm ribbon is made 8 cm longer. (B) An 8 cm strip is joined to an unknown strip to make 23 cm. Explain.
10. A rope had an unknown original length. After 6 ft was cut off, a 4-ft piece was joined end to end without overlap. The resulting rope was 18 ft long. How long was the original rope? Write an equation with □.
11. Three pieces of string measuring 6 cm, 9 cm, and 4 cm are laid end to end with no gaps. What is their total length?
12. A straight piece of string is 30 cm long. A 7 cm piece is cut off. The remaining string is bent into a curve without stretching or cutting it again. How long is the curved piece? A student says that bending the string must make it shorter because its ends are closer together. Explain the mistake.
13. The class estimated a poster width of 29 cm, but the measured width was 34 cm. The class then cuts 2 cm from the width of the poster. How much wider than the original estimate is the poster now? Is the estimate still too low, or is it too high?
14. A plant is 19 cm tall. It grows 7 cm one week and another 4 cm the next week. Its class project goal is a height of 32 cm. How many centimeters short of the goal is the plant after the two weeks? Write an equation using □ for the missing growth still needed.
15. A 24 cm ribbon is 5 cm longer than another ribbon. Then 3 cm is cut from the 24 cm ribbon. Which ribbon is longer now, and by how much?
16. A 46 cm straight border strip is colored blue along 17 cm. A red section is 9 cm shorter than the blue section. The colored sections do not overlap. How many centimeters are uncolored?
17. A rope is 18 cm long. 6 cm is cut off. A 3 cm piece is then joined end to end to what remains. How long is the resulting rope?
18. A 100 cm rope and a 64 cm rope are compared. How much longer is the first rope?
19. A classmate says, “My ribbon is 12 cm. The other ribbon is 7 cm longer, so it is 5 cm.” Explain the mistake and give the correct length.
20. Two ribbons are 23 cm and 16 cm long. A 4-cm piece is cut from each ribbon. How much longer is the first ribbon now? Explain whether removing the same length from both ribbons changes the difference between their lengths.
21. A 47 cm ribbon is shortened, and 29 cm remains. How much was removed?
22. A 12 cm display strip is partly covered by a blue 4 cm label and a red 3 cm label. Both labels lie entirely within the strip, but they overlap by 1 cm. How many centimeters of the display strip remain uncovered? Explain why the overlap is counted only once.
23. Two craft ribbons measure 19 cm and 13 cm. They are placed along one straight line with a 2 cm overlap. Then 5 cm is trimmed from the right end of the combined ribbon. What is its final length? Explain why the overlap must not be counted twice.
24. A paintbrush is 26 cm long and a pencil is 7 cm shorter. A friend reports the pencil is 8 cm longer than an 11 cm craft stick. Could both reports be correct?
25. A ribbon is measured twice on the same centimeter ruler without being stretched or cut. First it extends from the 1-cm mark to the 17-cm mark. After being moved, it extends from the 4-cm mark to the 20-cm mark. Are the measurements consistent? What is the ribbon’s length? Explain.
26. A paper chain is made by joining a 9 cm strip to another strip of unknown length, end to end without overlap. Then 4 cm is cut from the chain’s right end. The chain is now 21 cm long. How long was the unknown strip before the cut? Write an equation with □.
27. A straight rope lies beside a centimeter ruler with one end at the 4-cm mark and the other at the 31-cm mark. A 9 cm piece is cut off. Another piece of unknown length is then added end to end, without a gap or overlap, making a rope 26 cm long. How long is the added piece? Write an equation with □ and check your answer.
28. A sign is 32 cm wide. The class leaves an 8 cm blank border on the left and a 6 cm blank border on the right. They want to place a 19 cm title strip between the borders. Will the title fit without crossing either border? If not, by how much is the available space too short?
29. A 60 cm ribbon is used for two crafts. First, 22 cm is cut off. The second piece cut off is 18 cm shorter than the first piece. How much of the original ribbon remains after both cuts? Explain each step.
30. A 48 cm rope has two pieces of 8 cm each cut away. The remaining rope is compared with a 27 cm rope. How much longer is the remaining rope?

🏆 Lesson Test — 15 Questions

Directions: Work without opening the solutions or practice answer key. Show your reasoning for unknown lengths, comparisons, and error checks. All numbers in a calculation must refer to the same unit.

1. A 25 cm ribbon is cut into pieces measuring 10 cm and 15 cm, without waste. A student says, “The second piece is exactly 4 cm longer than the first.” Is that statement correct? Check both the total and the difference.
2. A border ribbon is 31 cm long. A 9-cm green section is at its left end; the rest is red. The red section is 5 cm longer than a separate yellow ribbon. How long is the yellow ribbon? Write an equation with □ and explain both steps.
3. A red ribbon is 16 cm long. A blue ribbon is 5 cm shorter than red. Then a 7 cm piece is joined end to end to blue. Which ribbon is longer now, and by how much?
4. For a class display, Noah puts a piece of ribbon beside a centimeter ruler. Its ends are at the 3-cm mark and the 15-cm mark. Noah says, “It is 15 cm long.” He then joins an 8 cm ribbon to it end to end without a gap or overlap. The display needs 26 cm of ribbon. What was Noah’s measurement mistake, and how many more centimeters of ribbon are needed?
5. A 40 cm ribbon has 12 cm cut off. Another piece is cut off, leaving 17 cm. How long was the second cut?
6. A blue ribbon is 18 cm long and a red ribbon is 25 cm long. A 6-cm piece is added end to end to the blue ribbon without overlap. Then 4 cm is cut from the red ribbon. Which ribbon is longer now, and by how much?
7. A school banner must be 50 cm long. Four identical strips, each 8 cm long, have already been joined end to end without gaps or overlaps. How much additional strip length is needed? Write an equation with □.
8. A 29 cm paper strip has marks at 9 cm and 21 cm, both measured from its left end. Maya cuts away the section between those two marks. What is the total length of the two pieces left? Write an equation with □.
9. Two ribbons are 26 cm and 9 cm long. They lie in one straight line with a 4 cm gap between them. What is the distance from the first start to the last end, and how many centimeters of ribbon material are there?
10. A class needs 37 cm of ribbon for a display. It already has a 12 cm piece. After adding one more piece end to end, the class has 5 cm more ribbon than it needs. How long is the added piece?
11. One board is 18 inches long and another is 12 inches long. How many inches must be added end to end to the shorter board so that it becomes 3 inches longer than the longer board?
12. A student knows that 8 cm was cut from a ribbon and 19 cm remained. The student says the original length was 11 cm, because 19 − 8 = 11. Explain the mistake, write a correct equation, and find the original length.
13. The left end of a ribbon stays at the 13-cm mark on a ruler. The ribbon extends 8 cm to the right. Then 3 cm is cut off its right end without moving the left end. At which centimeter mark is the new right end? Write an equation with □.
14. A 28 cm ribbon is divided into a 9 cm piece and an unknown piece. A student claims the unknown piece is 37 cm, because 28 + 9 = 37. Explain the mistake and find the unknown piece.
15. Maya needs a continuous 45 cm ribbon across an art display. She lays a 12 cm ribbon and an 18 cm ribbon in one straight line so that they overlap by 3 cm. She will add a third ribbon end to end, without a further gap or overlap, to reach exactly 45 cm. How long must the third ribbon be? Write an equation using □.

🔐 Complete Answer Key

Keep both sections closed until you have finished the corresponding work. Each answer includes a brief explanation or verification.

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Practice Answers

1. 12 cm. Let □ be the pencil alone. □ + 5 = 17, so 17 − 5 = 12.

2. 7 cm. The longer snake is 14 cm + 2 cm = 16 cm; 16 cm − 9 cm = 7 cm.

3. (a) 11 cm traveled; (b) 5 cm from the starting point. The robot travels 8 cm to the right plus 3 cm back, for 8 cm + 3 cm = 11 cm in all. Moving back toward the start changes its final position: 8 cm − 3 cm = 5 cm to the right of the start. The distance traveled counts both parts of the trip, while the final distance from the start does not.

4. At the 16-cm mark. First find the truck's length: 9 cm + 3 cm = 12 cm. The back of the truck is at mark 4, so write 4 + 12 = □. Then □ = 16. Check: 16 cm − 4 cm = 12 cm, matching the truck's length.

5. 3 cm. The pieces total 13 cm + 13 cm + 9 cm = 35 cm. The unknown trimmed length satisfies 35 − □ = 32, so □ = 3. Trim 3 cm from the end to make exactly 32 cm.

6. No; both arrangements are 23 cm long. First order: 9 cm + 14 cm = 23 cm. Reversed order: 14 cm + 9 cm = 23 cm. Changing which strip comes first does not change the total length.

7. 7 cm. The second piece measures 8 cm + 6 cm = 14 cm. The two known pieces total 8 cm + 14 cm = 22 cm. The third piece is 29 cm − 22 cm = 7 cm.

8. 8 cm. Start + 13 = 21, so start = 21 − 13 = 8. The ribbon begins at mark 8.

9. B. The equation adds a known 8 cm part to an unknown part to make 23 cm; the missing part is 23 cm − 8 cm = 15 cm.

10. 20 ft. Let □ be the original number of feet. The equation is □ − 6 + 4 = 18. Work backward: 18 ft − 4 ft + 6 ft = 20 ft. Check: 20 ft − 6 ft + 4 ft = 18 ft.

11. 19 cm. 6 cm + 9 cm + 4 cm = 19 cm.

12. 23 cm. Subtract the cut piece: 30 cm − 7 cm = 23 cm. Bending the remaining piece changes its shape and the straight-line distance between its ends, not the length of the string itself. Its length is still 23 cm.

13. 3 cm too low. After trimming, the poster width is 34 cm − 2 cm = 32 cm. Compare the new width with the original 29 cm estimate: 32 cm − 29 cm = 3 cm. The original estimate is still 3 cm too low.

14. 2 cm short. The new height is 19 cm + 7 cm + 4 cm = 30 cm. Let □ be the additional growth needed: 30 + □ = 32, so □ = 2. The plant is 32 cm − 30 cm = 2 cm short of the goal.

15. The shortened first ribbon, by 2 cm. Second ribbon: 24 cm − 5 cm = 19 cm. First after cutting: 24 cm − 3 cm = 21 cm. Difference: 21 cm − 19 cm = 2 cm.

16. 21 cm. The red section is 17 cm − 9 cm = 8 cm. The colored length is 17 cm + 8 cm = 25 cm. The uncolored length is 46 cm − 25 cm = 21 cm.

17. 15 cm. 18 cm − 6 cm = 12 cm remain; 12 cm + 3 cm = 15 cm.

18. 36 cm. 100 cm − 64 cm = 36 cm.

19. 19 cm. “Longer” means add 7 cm, not subtract: 12 cm + 7 cm = 19 cm.

20. 7 cm longer. After the cuts, the first ribbon is 23 cm − 4 cm = 19 cm, and the second is 16 cm − 4 cm = 12 cm. Their new difference is 19 cm − 12 cm = 7 cm. The original difference was also 23 cm − 16 cm = 7 cm: shortening both by an equal length does not change their difference.

21. 18 cm. Use 47 − □ = 29, so □ = 47 − 29 = 18. The removed length is 18 cm.

22. 6 cm. The labels cover 4 cm + 3 cm − 1 cm = 6 cm, because the 1 cm overlap belongs to both labels but is only one part of the strip. The uncovered length is 12 cm − 6 cm = 6 cm.

23. 25 cm. Before trimming, the overlapped span is 19 cm + 13 cm − 2 cm = 30 cm. The overlap is shared by the two strips and must be counted only once. After trimming 5 cm, the remaining length is 30 cm − 5 cm = 25 cm.

24. Yes. 26 cm − 7 cm = 19 cm, and 11 cm + 8 cm = 19 cm. Both descriptions give the same pencil length.

25. Yes; the ribbon is 16 cm long. First measurement: 17 cm − 1 cm = 16 cm. Second measurement: 20 cm − 4 cm = 16 cm. Moving a ribbon without stretching or cutting it does not change its length.

26. 16 cm. Let □ be the unknown length in centimeters. The equation is 9 + □ − 4 = 21. Add back the cut piece: 21 cm + 4 cm = 25 cm was the joined length. Then 25 cm − 9 cm = 16 cm for the unknown strip. Check: 9 cm + 16 cm − 4 cm = 21 cm.

27. 8 cm. The original rope measures 31 cm − 4 cm = 27 cm. After cutting, 27 cm − 9 cm = 18 cm remains. Write (31 − 4) − 9 + □ = 26, so 18 + □ = 26 and □ = 8. Check: 18 cm + 8 cm = 26 cm. The added piece is 8 cm long.

28. No; the available space is 1 cm too short. Available space: 32 cm − 8 cm − 6 cm = 18 cm. The title needs 19 cm, so 19 cm − 18 cm = 1 cm of additional space is needed.

29. 34 cm remains. The second piece is 22 cm − 18 cm = 4 cm long. Together, the two cuts remove 22 cm + 4 cm = 26 cm. Subtract from the original: 60 cm − 26 cm = 34 cm.

30. 5 cm. Remains: 48 cm − 8 cm − 8 cm = 32 cm. Difference: 32 cm − 27 cm = 5 cm.

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Test Answers

1. No. The total is correct: 10 cm + 15 cm = 25 cm. But the second piece is 15 cm − 10 cm = 5 cm longer, not 4 cm.

2. 17 cm. The red section is 31 cm − 9 cm = 22 cm. Because red is 5 cm longer than yellow, yellow measures 22 cm − 5 cm = 17 cm. One equation is 31 − 9 − 5 = □, so □ = 17.

3. Blue by 2 cm. Blue was 16 cm − 5 cm = 11 cm. After joining, blue is 11 cm + 7 cm = 18 cm, which is 18 cm − 16 cm = 2 cm longer than red.

4. 6 cm more. Noah read the ending mark (15 cm) instead of the distance between the ribbon’s endpoints. The first ribbon is 15 cm − 3 cm = 12 cm long. Joining the second piece makes 12 cm + 8 cm = 20 cm. To reach 26 cm, the display needs 26 cm − 20 cm = 6 cm more.

5. 11 cm. After the first cut, 40 cm − 12 cm = 28 cm; second cut 28 cm − 17 cm = 11 cm.

6. Blue is 3 cm longer. The new blue length is 18 cm + 6 cm = 24 cm. The new red length is 25 cm − 4 cm = 21 cm. Compare: 24 cm − 21 cm = 3 cm. The initially shorter ribbon has become longer.

7. 18 cm. The four strips total 8 cm + 8 cm + 8 cm + 8 cm = 32 cm. Equation: 32 + □ = 50, so 50 cm − 32 cm = 18 cm is still needed.

8. 17 cm. The section removed measures 21 cm − 9 cm = 12 cm. Equation: 29 − 12 = □, so 29 cm − 12 cm = 17 cm remains in the two outside pieces together.

9. 39 cm total span; 35 cm ribbon material. The span includes the gap: 26 cm + 4 cm + 9 cm = 39 cm. The material does not: 26 cm + 9 cm = 35 cm.

10. 30 cm. The total after joining is 37 cm + 5 cm = 42 cm. Subtract the original piece: 42 cm − 12 cm = 30 cm. The added piece is 30 cm long.

11. 9 inches. The target is 18 in. + 3 in. = 21 in. The shorter board is 12 inches, so it needs 21 in. − 12 in. = 9 in. added.

12. 27 cm. The original whole includes both the cut-off part and the part left. A correct equation is □ − 8 = 19 or □ = 19 + 8, so □ = 27. The original length is 27 cm.

13. At the 18-cm mark. Before cutting, the right end was at mark 13 + 8 = 21. Cutting 3 cm from that right end moves it left: 21 − 3 = 18. An equation for the new mark is 13 + 8 − 3 = □, so □ = 18. The ribbon is now 5 cm long.

14. 19 cm. The 28 cm is the whole, not another part to add. Subtract the known 9 cm piece: 28 cm − 9 cm = 19 cm. Check: 9 cm + 19 cm = 28 cm.

15. 18 cm. The first two ribbons span 12 cm + 18 cm − 3 cm = 27 cm because the overlapped 3 cm must be counted once. Write 27 + □ = 45. Then 45 cm − 27 cm = 18 cm. Check: 27 cm + 18 cm = 45 cm of continuous span.

💡 Common Mistakes to Watch For

Adding instead of finding the difference

“How much longer?” asks for the difference, not the total of two lengths.

Using the last ruler number

When a ribbon starts after zero, subtract the start reading from the end reading.

Forgetting a gap or an overlap

Check what distance the story is asking about. Gaps are empty space; overlapping parts are not counted twice.

Forgetting units

Use cm with centimeters, in. with inches, and so on. Make sure every length you add or subtract uses the same unit.

🏡 Family Practice Idea

Kitchen-table length stories: Measure two safe household objects with a ruler in centimeters. Ask the child to make a story using the measurements: “How much longer?” “What is the total if placed end to end?” “How long is the missing piece?” Have them draw a strip model and write one equation with □. Change which quantity is unknown to make a new problem.

🔗 Related Lessons

Grade 2 Lesson 23 — Measure LengthReview measuring tools, units, and how to read rulers.Grade 2 Lesson 24 — Estimate, Compare & Measure LengthsReview how to estimate a length and compare two measured distances.

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