Represent length sums and differences on a number line

Learning goal: Solve length stories within 100 when all amounts use the same whole-number unit, and represent whole numbers, sums and differences as lengths on a number line. A number line has equal spaces for equal numerical differences. The location of a point is determined by its distance from zero in the chosen unit.

Suppose one ribbon is 37 centimeters long and another is 25 centimeters long. Placed end to end without overlap, their combined length is 37 + 25 = 62 centimeters. The question concerns total length, not the number of ribbon pieces. A model must retain the centimeter unit throughout.

On the illustration's number line, start at 37. Decompose 25 into 20 and 5. A jump of twenty reaches 57 and a jump of five reaches 62. The jump labels are lengths of movement, while 37, 57 and 62 label locations. Calling the first jump 57 confuses its endpoint with its size.

Zero matters even when calculation begins at another point. The number 37 represents a length of thirty-seven unit intervals from zero. Starting a jump at 37 does not relabel that location zero. Equal ticks continue across the line so positions can be interpreted consistently.

Subtraction can show a removed length. If a 62-centimeter ribbon has 25 centimeters cut off, start at 62 and move back twenty to 42, then back five to 37. The difference is 37 centimeters. The backward direction represents removing length; it does not mean the tick spacing should reverse or become uneven.

A missing addend is another length relationship. A strip is 37 centimeters long and must reach a total of 62 centimeters. Write 37 + □ = 62. The required additional length is 25 centimeters. Counting from 37 to 62 or subtracting 62 − 37 finds the same gap.

A comparison asks how much longer one object is than another. If two rods are 62 and 37 centimeters, their difference is 25 centimeters. Align the rods' beginnings or compare their endpoints on a common zero-based model. Their total length, 99 centimeters, answers a different question.

The line drawn on a screen is a representation, not an actual centimeter ruler. Its tick intervals have consistent visual size but may appear larger or smaller on different devices. Use its numerical labels to reason about the story; use a physical measuring tool to measure an actual ribbon. Detailed ribbon decorations are unnecessary when a simple labeled line shows the mathematics.

Worked example 1: Add in useful jumps. From 37, +20 reaches 57 and +5 reaches 62. The jumps total 25, matching the second ribbon. Check by calculating 37 + 20 + 5.

Worked example 2: Subtract in reverse direction. From 62, −20 reaches 42 and −5 reaches 37. These jumps remove the full 25. A jump of twenty does not necessarily land on a multiple of ten; its size is the difference between start and endpoint.

Worked example 3: Explain a comparison. For rods of 62 and 37 centimeters, the longer contains a matching 37-centimeter portion and 25 extra centimeters. Write 37 + 25 = 62 or 62 − 37 = 25.

Guided practice: Build a number line with equal unit spacing and label zero. Represent a chosen whole number as a length from zero, then show an addition and subtraction using labeled jumps. Connect each model with a same-unit length story.

Independent tasks: Solve total-length, removed-length and difference problems, including an unknown symbol in the equation. Construct a number-line model with consistent spacing and explain the difference between a jump size and its endpoint. Do not infer construction skill only from reading a supplied diagram.

Feedback: If a jump is counted by marks instead of intervals, trace the gaps crossed. If units differ, first restate the task with comparable measurements rather than silently adding unlike quantities. Require the final unit and an answer to the actual question.