# Race to 100

Achievement Objectives
NA3-8: Connect members of sequential patterns with their ordinal position and use tables, graphs, and diagrams to find relationships between successive elements of number and spatial patterns.
Student Activity

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?

Specific Learning Outcomes
Describe in words number patterns
Devise and use problem solving strategies to explore situations mathematically (guess and check, be systematic, make a drawing, use equipment).
Description of Mathematics

This problem involves students in finding number patterns and solving 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.

Required Resource Materials
Activity

### Problem

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?

### Teaching sequence

1. Introduce the 2 characters.
2. Pose the problem and have students demonstrate and explain the movement of each character.
3. Discuss possible strategies and ways in which the students will record their solutions.
4. As the students 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?
5. Share solutions
6. If the students 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.

#### Extension to the problem

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, 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.

#### Solution to the extensions:

Using the equation 2# + 51 = 100 for Freda,  guess and check can be used 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.

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.

If 2# + 51 = 4# + 1, then 2# = 50, so # = 25. They land together at the end of the 25th jump but that is the first time that they are together. Freda is ahead up to that point.

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