Measuring and solving length problems
Measuring with a real ruler, solving length word problems, and using a number line to add and subtract
About three lessons of 40 to 45 minutes
A number line is a ruler you can jump along
In Grade 1, you measured with paperclips and blocks laid end to end. Now you measure with a real ruler, marked in standard units like centimetres, so anyone anywhere can measure the same object and get the same answer.
A ruler and a number line are really the same idea: both show whole numbers spaced evenly along a line. That means you can use a number line to add and subtract lengths, hopping forward to add and backward to subtract, exactly the way you already hop to add and subtract other numbers.
- A ruler starting at 0, ending where the object endsalways start measuring from the 0 mark, not the 1
- A 45cm ribbon, and a second ribbon 28cm longer45 + 28 = 73cm, the length word problem
- 47 + 25 on a number line: hop 20, then hop 547, then 67, then 72
- Two pieces of string, 34cm and 19cm, joined end to end34 + 19 = 53cm total
What students will be able to do
Students will measure the length of an object using standard tools such as a ruler or metre stick, solve addition and subtraction word problems involving lengths given in the same units, and represent whole numbers on a number line, using it to add and subtract within 100.
- I can measure an object's length using a ruler, starting from the 0 mark.
- I can solve a word problem that adds or subtracts two lengths given in the same unit.
- I can use a number line, hopping by tens and ones, to add or subtract within 100.
Standards this unit teaches
- 2.MD.A.1Common Core (US)Measure with standard tools
Measure the length of an object using rulers, yardsticks, metre sticks, and measuring tapes.
- 2.MD.A.5Common Core (US)Length word problems
Solve addition and subtraction word problems involving lengths given in the same units.
- 2.MD.B.6Common Core (US)Number lines for length
Represent whole numbers as lengths on a number line and use it to add and subtract within 100.
- AC9M2M01Australian Curriculum v9 (ACARA)Measure length with units (Year 2)
Measure the length of shapes and objects with informal units placed end to end, keeping the units the same size.
Prior knowledge
This unit builds on skills students should already have met. Revisit any that are shaky first.
Words to teach and display
- Standard unit
- a fixed, agreed-upon unit like a centimetre, used so any two measurements can be fairly compared
- Ruler
- a standard tool marked with evenly spaced units, used to measure length
- Same units
- both lengths in a word problem measured with the same unit, such as both in cm, so they can be added or subtracted directly
Teach it: concrete, pictorial, abstract
The lesson moves from things students can hold, to pictures and diagrams, to the written maths. The diagrams below are drawn from data, so they are accurate and print cleanly. Teach straight from them.
1. Measuring with a ruler
ConcreteLine an object up against a ruler, starting exactly at the 0 mark, not at the very end of the ruler and not at the number 1. Read the mark where the object ends: that number is the object's length in that unit. A ruler, a metre stick, and a measuring tape all work the same way, just at different lengths.
Starting at 0 matters because the ruler's numbers count the whole units passed since the start. Starting at 1 by mistake would add one extra, incorrect unit to every measurement.
An eraser is lined up starting exactly at the 0cm mark on a ruler, and its other end reaches the 7cm mark. How long is the eraser?
- The eraser starts at 0.
- The eraser ends at the 7cm mark.
- The length is the reading at the end, since the start was 0.
Answer: The eraser is 7cm long.
- Why should you always start measuring at the 0 mark on a ruler?
- If an object starts at 0 and ends at the 12cm mark, how long is it?
2. Solving length word problems
PictorialLength word problems add or subtract two lengths given in the same unit, just like any other addition or subtraction word problem, once you have decided which operation the story calls for. 'Longer than' or 'joined together' usually means adding; 'shorter by' or 'how much was cut off' usually means subtracting.
Always check both lengths are in the same unit before adding or subtracting them. 45cm and 28cm can be added directly; 45cm and 2m cannot, until one is converted to match the other's unit.
A rope is 82cm long. It is cut down to 55cm. How much rope was cut off?
- 'Cut down to' means the rope got shorter, so this is subtraction.
- Subtract the new length from the original length: 82 - 55.
- 82 - 55 = 27.
Answer: 27cm of rope was cut off.
- A ribbon is 45cm. A second ribbon is 28cm longer. How long is the second ribbon, and is this addition or subtraction?
- Why must both lengths be in the same unit before adding or subtracting them?
3. Using a number line to add and subtract within 100
AbstractA number line marked with whole numbers is really a length model: the distance from 0 to any number is that number's length. To add on a number line, start at the first number and hop forward, first by tens, then by ones. To subtract, start at the first number and hop backward the same way.
Hopping by tens first, then ones, keeps the jumps few and easy to count. For 47 + 25, hop forward 20 (two big hops of ten) landing on 67, then hop forward 5 more landing on 72.
Use a number line to work out 84 - 37.
- Start at 84.
- Hop back 30 (three hops of ten): 84 - 30 = 54.
- Hop back 7 more: 54 - 7 = 47.
Answer: 84 - 37 = 47.
- To add 36 + 47 on a number line, which hop should you make first: tens or ones?
- To subtract on a number line, which direction do you hop?
Common misconceptions and how to address them
MisconceptionWhen reading a ruler, the child lines the object up starting at the '1' mark instead of the '0' mark, giving a measurement that is one unit too long.
Why it happens: The number 1 looks like a natural starting point, especially since counting usually starts at 1, not 0.
How to address it: Physically point to the 0 mark and say 'this is where nothing has been measured yet, so this is where we start,' before lining up any object.
MisconceptionIn a length word problem, the child picks the wrong operation, such as adding when the story describes something getting shorter, because the numbers alone do not signal which operation to use.
Why it happens: Without focusing on the story's action words (longer, shorter, cut off, joined together), it is easy to default to whichever operation was practised most recently.
How to address it: Underline the key action word in the problem first (longer than, cut off, joined) and connect that specific word to addition or subtraction before touching the numbers.
MisconceptionOn a number line, subtracting is done by hopping forward instead of backward, or an addition is hopped backward by mistake.
Why it happens: Once a student is comfortable hopping in one direction, switching direction for the other operation takes a deliberate, separate check.
How to address it: Say the rule before every hop: adding moves right (forward, toward bigger numbers), subtracting moves left (backward, toward smaller numbers). Check the answer makes sense: subtracting should give a smaller number.
MisconceptionThe child adds or subtracts two lengths given in different units without converting first, such as adding 45cm and 2m as if they were both the same size of unit.
Why it happens: The numbers alone (45 and 2) look addable, and the different units (cm and m) are easy to overlook.
How to address it: Before calculating, check both lengths use the exact same unit, written the same way. If they do not match, that is a signal something needs converting before adding or subtracting, not a problem to solve as-is.
Guided practice (with answers)
1. A pencil is lined up from 0cm and ends at the 12cm mark on a ruler. How long is it?
Answer: 12cm.
2. A pen is 9cm longer than a 12cm pencil. How long is the pen?
Answer: 21cm, because 12 + 9 = 21.
3. Use a number line to work out 63 - 28 (hop back 20, then back 8).
Answer: 35. 63 - 20 = 43, then 43 - 8 = 35.
4. A garden path is 58m. A second path is 24m shorter. How long is the second path?
Answer: 34m, because 58 - 24 = 34.
5. Use a number line to work out 36 + 47 (hop forward 40, then forward 7).
Answer: 83. 36 + 40 = 76, then 76 + 7 = 83.
6. Two ribbons, 27cm and 38cm, are joined end to end. What is their total length?
Answer: 65cm, because 27 + 38 = 65.
Independent practice worksheets
Set the matching ChalkBee worksheets for independent work. The answer keys are computed in code, so they are never wrong. Reading a ruler practises standard-unit measuring directly, and the number line worksheet practises hopping to add and subtract.
Differentiation
- Practise starting at 0 with a real ruler and real objects before moving to printed ruler diagrams.
- Underline or circle the key action word (longer, shorter, cut off) in a word problem before choosing an operation.
- Keep number-line hops to tens only at first, before adding a second hop of ones.
- Colour-code forward hops and backward hops differently so the direction rule stays visually obvious.
- Solve a two-step length word problem, such as combining three lengths or finding a total then a difference.
- Measure the same object in two different standard units, such as centimetres and a metre stick reading, and compare.
- Use a number line to add or subtract two numbers that are both close to a multiple of ten, choosing an efficient hop strategy.
- Explain why starting at 0, not 1, matters for a ruler using a labelled diagram.
Assessment: exit ticket
A three-question exit ticket for the last five minutes, sampling ruler reading, a length word problem, and a number-line calculation.
1. An object is lined up from 0cm and ends at the 15cm mark. How long is it?
Answer: 15cm.
2. A rope is 40cm. It is cut down to 17cm. How much was cut off?
Answer: 23cm, because 40 - 17 = 23.
3. Use a number line to work out 29 + 34 (hop forward 30, then forward 4).
Answer: 63. 29 + 30 = 59, then 59 + 4 = 63.
Teacher notes and timings
- Rough timing across three lessons: Lesson 1 measuring with a ruler (section 1), Lesson 2 length word problems (section 2), Lesson 3 the number line for adding and subtracting (section 3) plus the exit ticket.
- These three standards cluster naturally as Grade 2's move from Grade 1's informal-unit measuring to standard tools, plus the number-line model that ties length directly back to whole-number addition and subtraction within 100.
- Language to keep saying: start at 0, which action word tells you the operation, and which direction do you hop. These phrases pre-empt the misconceptions above.
- Curriculum note: US Grade 2 states standard-tool measuring (2.MD.A.1), length word problems (2.MD.A.5) and number-line representation (2.MD.B.6) as three standards. ACARA Year 2 covers length measuring with same-size units under one Measurement descriptor (AC9M2M01), without yet distinguishing standard tools from informal units the way the US standard does.
- Present mode and print both work: use the Print button for a student worksheet, or project the page and measure real classroom objects with a real ruler before moving to the number-line diagrams.