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Race to 100

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?

Achievement Objectives:

Achievement Objective: 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.
AO elaboration and other teaching resources

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: 
Number lines (1-100) (or use metre rulers)
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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RaceTo100.pdf113.59 KB
TePekeMawhitiwhiti.pdf201.83 KB