The Dangerous-ers’ Drop-Line Test

At a glance

• Part 1: Match hook 73 to rope 45, hook 61 to rope 33, and hook 54 to rope 26. Circle the 29-mark rope; it is left behind.
• Part 2: The checks are 73 −45 =28, 61 −33 =28, and 54 −26 =28.
• Part 3: The rope changes from 45 to 33, so it is 12 marks shorter. The hook and rope both decrease by 12, so the landing stays at mark 28.

Key idea

A rope that starts at hook mark 73 and is 45 marks long ends at 73 −45 =28. Every safe match must have the same difference: hook mark minus rope length equals 28.

Addition can check the same match. For example, 28 +45 =73, so a 45-mark rope reaches from mark 73 down to mark 28.

1. Match the hooks and ropes

Start with each hook and find the rope that leaves 28.

• Hook 73: 73 −45 =28, so match 73 to 45.
• Hook 61: 61 −33 =28, so match 61 to 33.
• Hook 54: 54 −26 =28, so match 54 to 26.

Those three matches use the 45-, 33-, and 26-mark ropes. The 29-mark rope is the one left behind.

2. Complete the check strip

The completed equations are:

Each equation starts at a hook mark, subtracts the matched rope length, and ends at the safe ledge mark.

3. Compare two drop lines

The hook changes from 73 to 61. That is 12 marks lower because 73 −61 =12.

The matching rope changes from 45 to 33. That rope is also 12 marks shorter because 45 −33 =12.

Both parts of the subtraction decrease by the same amount:

The second hook starts 12 marks lower, but its rope is also 12 marks shorter. Those equal changes balance, so both lines still land at mark 28.

Check and completeness

All three selected ropes reach the required mark:

• 28 +45 =73;
• 28 +33 =61; and
• 28 +26 =54.

The only unused rope is 29 marks long. It does not reach mark 28 from any hook:

• 73 −29 =44;
• 61 −29 =32; and
• 54 −29 =25.

There are three hooks, all three working matches have been checked, and the fourth rope fails at every hook. Therefore the matching is complete and the 29-mark rope is exactly the rope that must be left behind.

Teaching note

Accept subtraction, addition, a number-line jump, or place-value work. For Part 3, a complete response identifies the rope change as 45 to 33, states that it is 12 marks shorter, and connects that equal 12-mark change to the unchanged landing mark of 28.