# Well, well!

Achievement Objectives
NA1-1: Use a range of counting, grouping, and equal-sharing strategies with whole numbers and fractions.
Student Activity

Freddo wants to get out from the bottom of the well.
Each day, he climbs 3m up the wall of the well and then rests.
But the wall is slippery and he then slips down 1m.
The next day he does the same thing.
In fact he does this every day until he gets out of the well.

The well is 13m deep. How long does it take Freddo to climb out of the well?

Specific Learning Outcomes
Skip count in twos
Investigate and find patterns in addition and subtraction
Devise and use problem solving strategies (act it out, draw a picture)
Description of Mathematics

At its simplest this problem involves the students in a series of single digit additions and subtractions (3 – 1 + 3 – 1 ...). It can also be used to reinforce skip counting in twos with a starting point of 3.

Required Resource Materials

Paper for drawing

Copymaster of problem (English)

A frog (from Copymaster) to "tell the problem"

Copymaster of the problem (Māori)

Activity

### The Problem

Freddo wants to get out from the bottom of the well. Each day, he climbs 3m up the wall of the well and then rests. But the wall is slippery and he then slips down 1m. The next day he does the same thing. In fact he does this every day until he gets out of the well.

The well is 13m deep. How long does it take Freddo to climb out of the well?

### Lesson Sequence

1. Use Freddo to share his problem with the class.
2. Ask the students to suggest ways that they might solve the problem.  You may get them to act it out with Freddo on the board.
3. Discuss with students how they are going to keep track of what they are doing as they solve the problem.
4. As they work on the problem, circulate and encourage the students to identify the leaps and slips that Freddo makes. Check to see if they are looking for patterns in the numbers that they are recording.
5. Have them share and demonstrate their answers. Discuss the range of strategies that students use.
For those who give 7 as the answer get them to think about where Freddo was before he went to sleep the night before.

#### Extension to the problem

Freddo has a brother in a deeper well. Each day he climbs 5m and then slips back 2m each night. How many days does it take him to reach the top of a 20m well?

Other contexts for the problem

A koala climbing a tree
A snail climbing a wall

### Solution

Think about is how high Freddo gets each day and not where he ends up before he goes to sleep. So each day in turn he reaches the 3m mark, the 5m mark, 7m, 9m, 11m, 13m. Freddo climbs out after 6 days! (A picture will show you all this quite quickly.)

#### Solution to the Extension:

Here the sequence of numbers is 5, 8, 11, 14, 17, 20.  So it takes 6 days for Freddo's brother to climb out of the 20m well.

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