This problem solving activity has an algebra focus.
Two ladybirds, Freda and Fred, are playing a game on a numberline.
Fred can jump three numbers at a time and Freda can only jump two.
Fred starts at 1 and Freda starts at 30.
If they both jump together, who gets to 100 first and how long do they have to wait for the other one?
This problem involves students in finding number patterns and using them to solve algebraic problems. Have the students first make an estimate and prediction. Students may approach the problem in a range of ways including drawing jumps along a number line, making a table, seeing a relationship and using guess and check, using division.
Note that in extension 1, Fred and Freda don’t land exactly on the number 100.
Two ladybirds, Freda and Fred, are playing a game on a number line. Fred can jump three numbers at a time and Freda can only jump two. Fred starts at 1 and Freda starts at 30. If they both jump together, who gets to 100 first and how long do they have to wait for the other one?
This can be done by using equipment, by drawing, by algebra (see Toothpick Squares problem), or by using a table such as this.
0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
Freda | 30 | 32 | 34 | 36 | 38 | 40 | 42 | 44 | 46 | 48 | 50 |
Fred | 1 | 4 | 7 | 10 | 13 | 16 | 19 | 22 | 25 | 28 | 31 |
11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 | 21 | |
Freda | 52 | 54 | 56 | 58 | 60 | 62 | 64 | 66 | 68 | 70 | 72 |
Fred | 34 | 37 | 40 | 43 | 46 | 49 | 52 | 55 | 58 | 61 | 64 |
22 | 23 | 24 | 25 | 26 | 27 | 28 | 29 | 30 | 31 | 32 | |
Freda | 74 | 76 | 78 | 80 | 82 | 84 | 86 | 88 | 90 | 92 | 94 |
Fred | 67 | 70 | 73 | 76 | 79 | 82 | 85 | 88 | 91 | 94 | 97 |
33 | 34 | 35 | |||||||||
Freda | 96 | 98 | 100 | ||||||||
Fred | 100 |
The table shows that Fred gets to the 100th square and has to wait two jumps for Freda to catch up.
The table is a valid (if tedious) way to solve the problem. 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. The table shows that Fred gets to the 100th square in 33 jumps, two ahead of Freda.
Fred's equation is 4# + 1= 100. By using guess and check, a table, or some other means, it can be seen that # must be more than 24 (4 x 24 + 1 = 97) and less than 25 (4 x 25 + 1 = 101).
They both landed on the 101st square on their 25th jump.
Printed from https://meaningfulmaths.nt.edu.au/mmws/nz/resource/race-100 at 8:56pm on the 26th February 2024