Problem
Distance between points on a line
Four points P, Q, R, S lie on one line, but the order is unknown. PQ is 210 cm, QR is 330 cm and RS is 420 cm. Each next point may sit to the left or to the right. Find every distance PS could be.
Reasoning · Grade 2-2 Working with Short Lengths
Your answer
Watch out Assuming the points run in order gives only 210 + 330 + 420 = 960. Counting the steps that double back yields four.
How to solve
Strategy Make a Systematic List — Because the order is unknown, the smart move is to draw a number line and place the points step by step. At each step the next point can go left or right, so we list every choice in an orderly way and read off P-to-S for each. A diagram keeps the directions clear.
1STEP 1
Anchor Q and place P
With Q at 0, P sits at +210 or −210.
Q = 0, P = +210 or -210
Pinning one point down and measuring everything from it is exactly how a ruler works.
2.MD.B.5Draw A Diagram2STEP 2
Place R relative to Q
R independently sits at +330 or −330.
R = +330 or -330
Each new point only cares about its own given gap, so left-or-right is a fresh choice.
2.MD.B.5Make A Systematic List3STEP 3
Place S relative to R, then list P-to-S
Placing S and reading PS gives 120, 300, 540, 960 cm.
|P-S| ∈ {120, 300, 540, 960} cm
Adding and subtracting the segment lengths (within 1000) is all that is needed; the systematic list guarantees none is missed.
2.NBT.B.7Make A Systematic List4STEP 4
Convert back to meters and centimeters
In metres: 1 m 20 cm, 3 m, 5 m 40 cm, 9 m 60 cm.
120=1 m 20 cm, 300=3 m, 540=5 m 40 cm, 960=9 m 60 cm
100 cm make 1 m, so splitting a centimeter count into meters plus leftover centimeters is just place-value with lengths.
2.MD.B.5Make A Systematic ListA length in centimetres is read as metres plus leftover centimetres by pulling out whole hundreds.
Why?
One metre is always exactly 100 centimetres, so the same length can be renamed between the two units without changing.
🧱A bigger unit is a fixed count of smaller unitsA bigger unit always equals the same fixed number of smaller units (1 hour = 60 minutes, 1 km = 1000 m), so a measure can be renamed between them.
Why?
Counting how many whole hundreds fit and what is left over is the same question division with a remainder answers.
🧱Division with remainderAny number splits into a whole count of equal groups plus a leftover, and the leftover is always smaller than the group size.
Answer
1 m 20 cm, 3 m, 5 m 40 cm, 9 m 60 cm
The smallest possible gap is when the segments fold back and partly cancel (330 - 210 = 120 cm), and the largest is when they all stretch the same way (210 + 330 + 420 = 960 cm). Every answer lands between 120 cm and 960 cm, which matches the bounds.
Takeaway
Lining the points up on a number line and trying left-or-right for each step lets you find every answer with just Grade 2 adding and subtracting!
- Anchor Q and place P
- Place R relative to Q
- Place S relative to R, then list P-to-S
- Convert back to meters and centimeters