SMI 2011
Race to 20
Race to 20
Procedure:
•Place 20 objects in a pile in front of a group of two students.
•Students take turns taking either one or two objects from the pile.
•As students take the objects, they count up orally AND represent their
game on the game board.
•The winner is the person who removes the 20th object from the pile.
Example:
•Student A takes one object and says “one”
•Student B takes two objects and says “two, three”
•Student A takes two objects and says “four, five”
•Student B takes one object and says “six”
Representation Sheet
Objective:
•To develop a strategy which will guarantee you will win each time the
game is played:
no matter how many objects are used
no matter how many objects can be removed each time
1
2
3
4
5
6
A
B
B
A
A
B
7
8
9
10
11
12
13
14
15
16
17
18
19
20
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
1
2
3
4
5
6
7
8
9
10
11
12
13
14
1
2
3
4
5
6
7
8
9
10
11
1
2
3
4
5
1
2
1
2
3
4
5
18
6
19
7
20
8
Round 1
Round 2
Winner
Round 3
Winner
Round 4
Winner
Round 5
Winner
Round 6
Winner
Round 7
Winner
Winner
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
A
A
R
A
A
R
R
A
R
A
A
R
R
A
R
R
A
R
A
A
Look at rounds 2, 3, 4, 5, 6, and 7.
• Notice how sometimes the winner says one number and other times the winner says two numbers.
–
Now, notice how the winner always says the third number in the round.
How does the winner guarantee that they can always say the winning number?
• The strategy is to take the maximum number that can be chosen by the opponent (in this case two
numbers) and then add one to this number. That is the control number for the game. Let’s use the
example above to better understand this statement:
–
In round 2 R said only one number (3) so A said two numbers (4 and 5). In round 3 R said two numbers (6
and 7) so A said one number (8). By taking the maximum number that can be said by an opponent and
adding one to it, the winner can guarantee that they can fill up each round.
• You may be asking yourself why 2 or 4 aren’t control numbers. In this example if my opponent
chooses to say 2 numbers, I cannot say 0 numbers so two is not a control number. I cannot
guarantee that I can say every second number. If my opponent chooses to say 1 number, I
cannot say 3 numbers, so four is not a control number. I cannot guarantee that I can say every
fourth number.
So we now know why you need to say numbers 5, 8, 11, 14, 17, and 20. This helps us to know that the
winner must also say the number 2. The winner must control every round (including the first round) to
win the game. The only way the winner can guarantee they control the first round is to go first. If there
were three numbers in the first round they would, of course, want to go second.
Using Lessons Learned from Race to 20 to play Race to 21
Procedure:
• Place 21 objects in a pile in front of a group of two students.
• Students take turns taking either one or two objects from the pile.
• As students take the objects, they count up orally AND represent their game on
the game board.
• The winner is the person who removes the 21st object from the pile
Objective:
• Students will recognize that the goal number impacts whether you want to be the
person who goes first or second (see strategy pages for an explanation of how to
teach this to students).
– Students will probably play 5-6 games before they come to this conclusion.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
1
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
7
8
9
1
2
3
4
5
6
1
2
3
16
16
17
17
18
18
19
20
21
A Winning Strategy for “Race to 20” type games
So, how can I generalize this strategy to other games with a different goal number, or when I am allowed to say more
than two numbers? Follow these two steps:
Step One:
Determine what the largest possible number that can be chosen is and add one to this number
– This answer allows you to determine the control numbers.
• In the game Race to 20 the most you can say was two numbers. So (the largest number) + 1 = 3.
Step Two:
To determine if you want to go first or second, take the goal number and divide it by the answer from above.
•
In the game Race to 20 the goal number is 20. Three was our control number. So 20 (goal number) ÷ 3
(control number) = 6 with a remainder of 2
This tells us we will have 6 full rounds, and one round with only two numbers in it.
– Anytime you have a game that will not have all the rounds full, you want to go first. Another
way to say that, is any time you have a remainder when you divide the goal number by the
control number, you want to go first. If it divides evenly, you want to go second.
What if I was playing a game to 21 and I could say either one number or two numbers? Would I want to go first or
second? What would the control numbers be?
What if I was playing a game to 24 and I could say one number, two numbers or three numbers? Would I want to go
first or second? What would the control numbers be?
What if I was playing a game to 26 and I could say one number, two numbers or three numbers? Would I want to go
first or second? What would the control numbers be?
Extensions:
Depending on the grade level and ability of your students
you may want to:
• Explain how to use division to determine whether you
want to go first or second
• Help students draw a table to find the “winning
numbers”
– Note number of columns and rows depends on the goal
number and the number of objects that can be taken each
turn (see strategy sheets for further explanation)
• Change the number of objects that can be taken each
time
• Make the person who takes the last object the “loser”
(Marilyn Burns About Teaching Mathematics page
131).
SMI Objectives for Small Groups:
• By the end of Day 3 students should be able to
record in their journals:
– how they would use a table to identify the
“winning numbers” for any game (no matter the
goal number or number of objects taken)
– how they would use division to determine winning
numbers and if they want to go first or second
• A prompt to assess if students have achieved this
objective would be to give them the number 38 and
say they can take one, two, three, or four objects each
turn. What are the winning numbers? Do you want
to go first or second?
SMI Small Group Time
Tuesday – Race to 20
Wednesday –
Race to 21
Extensions
Thursday – Articulate Strategy for Race to “n”
Small Group Time
Determine who will teach on:
Tuesday
Wednesday
Thursday
Race to 12
(players can say one or two numbers)
1
2
3
Each Round
A
B
Each Round
A
B
B
Each Round
A
A
B
Each Round
A
A
B
4
B
Race to 12
(players can say one or two numbers)
1
2
3
Each Round
A
B
B
Each Round
A
A
B
1
2
3
4
5
6
7
8
9
10
11
12
A
B
B
A
A
B
A
A
B
A
B
B
1
2
3
4
5
6
7
8
9
10
11
12
A
A
B
A
B
B
A
B
B
A
A
B
Do you notice how Player B controls the numbers 3, 6, 9, and 12?
How can Player B ensure that they always control these numbers?
Race to 12
(players can say one or two numbers)
1
2
3
1
2
3
Each Round
A
B
B
Each Round
A
A
B
4
5
6
7
8
9
Round 1
1
2
3
Round 2
4
5
6
Round 3
7
8
9
Round 4
10
11
12
10
11
12
12 ÷ 3 = 4
4 full rounds
Race to 13
(players can say one or two numbers)
1
2
3
4
1
2
3
Each Round
A
B
B
Each Round
A
A
B
5
6
7
8
9
10
Round 1
1
2
3
Round 2
4
5
6
Round 3
7
8
9
Round 4
10
11
12
Round 5
13
11
12
13
13 ÷ 3 = 4 with a
remainder of 1
4 full rounds
1 “unfull” round
with one number
Race to 13
1
2
3
4
5
6
7
8
9
10
11
12
13
Will Player B win using the strategy if the “unfull” round is at the end of the game?
1
2
3
4
5
Round 1
6
7
1
Round 2
2
3
4
Round 3
5
6
7
Round 4
8
9
10
Round 5
11
12
13
8
9
10
11
12
13
How should Player B’s
strategy for Race to 13 be
different than Race to 12?
Why?
Race to 16
(players can say one or two numbers)
1
2
3
4
5
6
Round 1
7
8
9
10
11
12
13
14
15
16
1
Round 2
2
3
4
Round 3
5
6
7
Round 4
8
9
10
Round 5
11
12
13
Round 6
14
15
16
16 ÷ 3 = 5 with a remainder
of 1
5 full rounds
1 “unfull” round with one
number
Race to 16
(players can say one, two, or three numbers)
1
2
3
4
5
6
7
8
9
Round 1
1
2
3
4
Round 2
5
6
7
8
Round 3
9
10
11
12
Round 4
13
14
15
16
10
11
12
13
14
15
16
16 ÷ 4 = 4
4 full rounds
How should Player B’s strategy change now that the players can say one,
two, or three numbers? Why?
Race to 38
(players can say one, two, or three numbers)
Round 1
1
2
Round 2
3
4
5
6
Round 3
7
8
9
10
Round 4
11
12
13
14
Round 5
15
16
17
18
Round 6
19
20
21
22
Round 7
23
24
25
26
Round 8
27
28
29
30
Round 9
31
32
33
34
Round 10
35
36
37
38
38 ÷ 4 = 9
with a remainder of 2
9 full rounds
1 “unfull” round with two
numbers