Download Race-to-20-Gameboard-and-Strategy-Sheet

Small Group Time Day One
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”
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
Representation Sheet
Objective:
 To develop a strategy which will guarantee you will win each time the game is played:
o no matter how many objects are used
o no matter how many objects can be removed each time
To guide students in this process have them stop after every two games and talk in a group of six about
patterns they notice. Initially everyone should note that getting to “17” guarantees a win.
 Have students play race to 17 for two games and have them again talk in groups to discuss
patterns they notice. Everyone should notice that getting to “14” guarantees a win.
 Have students play race to 14 for two games and have them again talk in groups to discuss
patterns they notice. Everyone should notice that getting to “11” guarantees a win. (Students
may begin to note that the “winning numbers” are decreasing by 3. Even so, continue to
adjust the target number because students will need to grapple with determining who should go
first.)
 Have students play race to 11 for two games and have them again talk in groups to discuss
patterns they notice. Everyone should notice that getting to “8” guarantees a win.


Have students play race to 8 for two games and have them again talk in groups to discuss
patterns they notice. Everyone should notice that getting to “5” guarantees a win.
Have students play race to 2. Everyone should notice that the winner must go first.
Students will play 11 games with their partner but have the opportunity to look at the game
boards of others in their group when identifying patterns to develop the wining strategy.
Race to 20
1
2
3
4
5
6
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
18
19
20
1
2
3
4
5
6
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
18
19
20
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Small Group Time Day Two and Three
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).
o Students will probably play 5-6 games before they come to this conclusion.
SMI Objectives for Small Groups:
 By the end of Day 3 students should be able to record in their journals:
o how they would use a table identify the “winning numbers” for any game (no matter the
goal number or number of objects taken)
o 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?
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”
o 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).
Race to 21
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
Strategy to Win “Race to 20” Type Games
Round 1
Winner
Round 2
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.
o Notice how sometimes the winner says one number and other times the winner says two numbers.
o 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?
o 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:
o 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.
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
o 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.
o 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 anytime 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?