Problem
Two ladybirds, Freda and Fred, were playing a game on a numberline. Fred can jump three numbers at a time and Freda can only jump two. Fred started at 1 and Freda started off at 30. If they both jumped together, who got to the 100 first and how long did they have to wait for the other one?

What is this problem about?
This is about number patterns and also about solving simple algebraic problems. Of course, it can be done going laboriously from jump to jump. If your children can only do it this way that’s fine, however, you should point out to them that there are other ways to do it.

It’s worth noting that in problems like this, it is always worth asking the class to estimate what will happen before they try to work it out. It helps them to get a feel for the problem and it also helps their intuition.

The extra twist in Extension 1, is that both Fred and Freda don’t land exactly on the number 100. Extension 2 is hard unless done by a constructive approach like making a table or by guess and check. It can be done by solving two equations but this is beyond this Level. The extension problems here could be development band work.

Achievement Objectives
Algebra (Level 3)
- describe in words, rules for continuing number and spatial sequence patterns
- state the general rule for a set of practical problems

Mathematical Processes
- devise and use problem solving strategies to explore situations mathematically (guess and check, be systematic, make a drawing, use equipment).

Resources
Number lines (1-100) (or use metre rulers)
Blackline master of the problem (English)
Blackline master of the problem (Maaori)

Specific learning outcomes
The children will be able to:
- describe in words number patterns
- work systematically to solve a problem involving number patterns

Teaching sequence

1. Introduce the 2 characters in the problem.
   Say that Fred jumps using this pattern: snap, clap, snap, clap. Discuss ways to record this (in twos)
   Say that Freda jumps using this pattern: snap, clap, clap, snap, clap, clap. Discuss ways to record this on the numberline (in 3's)
2. Pose the problem for the children to work on in pairs.
3. As the children work ask questions that focus on the thinking that they are using.
   What are you doing? Why are you solving it this way?
   Who do you think will get there first? Why do you think that?
   What can you tell me about the numbers in Freda's pattern?
   What can you tell me about the numbers in Fred's pattern?
4. Share solutions
5. If the children have all acted or drawn the problem ask them to look back and think about other ways that they could have used to solve the problem eg, use division.

1. Let Freda start on 51 and jump two numbers at a time. Let Fred start on 1  and jump four numbers at a time. Who is first to 100?

2. In Extension 1, on what number does the overtaking take place?

 Solution
This can be done by using equipment, by drawing or by algebra (see Toothpick Squares problem). We will do it here using a table.

So from the table you can see that Fred gets to the 100^th square and had to wait two jumps for Freda to catch up.

The other thing that you and the children can see from the table is that this is a tedious way to do this problem. If you note that Freda is jumping on the squares numbered 2# + 30, then she gets to the 100th square when 2# + 30 = 100. This is when # = 35 (check this with the table).

On the other hand, Fred is using the pattern 3# + 1. So he gets to 100 when 3# + 1 = 100. In other words when 3# = 99 or when # = 33. As we saw in the table, Fred gets to the 100th square in 33 jumps, two ahead of Freda.

Extensions: Using the equation 2# + 51 = 100 for Freda, we can use guess and check to see that # must be bigger than 24 (2 x 24 + 51 = 99) and less than 25 (2 x 25 + 51 = 101). So Freda will need 25 steps to get to the 100th square.

For Fred we get the equation 4# + 1= 100. In this case using guess and check or a table or some other means we can see that # must be more than 24 (4 x 24 + 1 = 97) and less than 25 (4 x 25 + 1 = 101).

So what happened? Well they both landed on the 101st square on their 25th jump. As it happens I think that a photo finish would have shown Freda just ahead. But what do you think?