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Play Together, Learn Together
Connect 4 Dice Games
Play Together, Learn Together: Connect Four Dice Games
Bob Albrecht & George Firedrake  [email protected]  Begin 2016-09-01
This FREE eBook is published under a Creative Commons Attribution-NonCommercial-ShareAlike license.
You may give this eBook to anyone. You may snip stuff from this eBook, edit it, and paste it into your
work. You may use it in other ways described at http://creativecommons.org/licenses/by-nc-sa/4.0/.
Play is the work of the child. – Friedrich Froebel, Jean Piaget, and others
When tools become toys, then work becomes play. – Bernie DeKoven
Dice & dice games are among our favorite tools/toys for learning and teaching math. – Bob & George

This eBook has Connect Four Dice Games featuring
addition, subtraction, multiplication, division,
evaluating algebraic expressions, and probability
alakazams.

Browse the Table of Contents on page 2.
Bob’s favorite 7-year-old math kid is in 2nd Grade. Her teacher says that they are moving toward a
games-based math curriculum. She brought home some homeplay instead of homework from school.
We played together, learned together!
We will call her ‘K’ in this eBook. A game K played at school is called Connect Four. This eBook is inspired
by Connect Four as played at K’s school. K and Bob contrived Connect Four variations and played them.
They created play sheets that – we think – might help players learn a bit about mathemagical alakazams
such as dice probabilities.

This eBook is intended for teachers, tutors, parents, and others who help
learners learn math. Download our free math & science eBooks at:

http://i-a-e.org/downloads/cat_view/86-free-ebooks-by-bob-albrecht.html
Bob & George? Bob is an 87-year-old human (as of February 2017). George is a Dragon. Read about Bob
& George at Information Age Education (IAE): http://iae-pedia.org/Robert_Albrecht.
Information Age Education (http://iae-pedia.org/Main_Page) publishes a large number
of free books, a free blog, the free IAE-Pedia, and the free IAE Newsletter.
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Play Together, Learn Together
Connect 4 Dice Games
Table of Contents (TOC)
Connect Four Dice Games – Addition
Roll 2D6 and Pick 1D6 Connect Four Dice Game
Connect Four Subtraction Dice Games
Connect Four Multiplication Dice Games
Connect Four Division Dice Games
Connect Four Multiple Operations Dice Games
Connect Four Algebraic Expression Dice Games
Your Turn: Create Connect Four Digit Dice Games
DragonFun image by Marcie Hawthorne http://marciehawthorne.com/
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Play Together, Learn Together
Connect 4 Dice Games
Connect Four Dice Games – Addition | TOC
K brought home a Connect Four play sheet and the rules for playing the game. We played a bunch of
games with her. Fun! Because the play sheet might be copyrighted, we will not show it here. You can
find the play sheet used at K’s school on the Internet. On 2016-10-08, it was in the center of the page at

Pinterest https://www.pinterest.com/pin/175358979217225322/
Grrr. To read more about the Connect Four dice game at this site, you must sign up. We did not sign up.
The Connect Four play sheet used at K’s school has six rows with seven numbers in each row, 42
numbers in all. The numbers are 2 to 12, with two or more of each number appearing in the play sheet.
To play, you roll 2D6 (two 6-faced dice), add ‘em, and then select the sum in the play sheet. Aha! The
numbers in the play sheet are 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12, the possible sums of 2D6. To win, you
must get four in a row horizontally, vertically, or diagonally.
The Connect Four play sheet has six rows with seven numbers in each row – 42 numbers. We counted
the frequency (number of occurrences) of each number in the play sheet and made the frequency
distribution (https://en.wikipedia.org/wiki/Frequency_distribution) shown in Table 01.
Table 01 Connect Four play sheet at K’s school: Numbers and their frequencies.
Number
2
3
4
5
6
7
8
9
10
11
12
Total
Frequency
4
4
3
3
4
4
4
5
4
4
3
42
Well, tra la, tra la, these numbers and their frequencies seem to be somewhat arbitrary, so we decided
to create play sheets that we think might nudge a Connect Four Dice Game player in the direction of a
mathemagical alakazam.

Alas, we suspect that we will never know if the stuff we write encourages a
player to discover a mathemagical alakazam. Hallelujah! Writing it is a joyful
experience, a happy activity. If someone likes it and uses it, yeah! But just
writing it is so much fun that it helps make a happy life for Bob & George.
3
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Play Together, Learn Together
Connect 4 Dice Games
To play the Connect Four dice game, you roll 2D6 and add the numbers on the two dice. 2D6?
Dice notation:
 1D6: One hexahedral (cubic) die with faces labeled 1 to 6 in
pips (dots) or numerals.
 2D6: Two hexahedral (cubic) dice with faces labeled 1 to 6 in
pips (dots) or numerals.
1D6
2D6
Roll 2D6. There are 36 possible outcomes. Here they are in Table 02. We call the two dice die1 and die2.
An outcome is shown as an ordered pair: die1, die2.
Table 02 Roll 2D6 (die1 and die2) possible outcomes (die1, die2)
die1 ↓ die2 
1
2
3
4
5
6
1
1, 1
1, 2
1, 3
1, 4
1, 5
1, 6
2
2, 1
2, 2
2, 3
2, 4
2, 5
2, 6
3
3, 1
3, 2
3, 3
3, 4
3, 5
3, 6
4
4, 1
4, 2
4, 3
4, 4
4, 5
4, 6
5
5, 1
5, 2
5, 3
5, 4
5, 5
5, 6
6
6, 1
6, 2
6, 3
6, 4
6, 5
6, 6
Your Turn When K and Bob play, they use a green die and a red die. Green die = 1, red die = 2 is a
different outcome than green die = 2, red die = 1. What say your students complete Table 03 before
they see our Table 02? When they complete Table 03, it will show the 36 possible outcomes of rolling a
green die and a red die.
Table 03 Roll 2D6 (green die, red die) possible outcomes (green die, red die)
green ↓ red 
1
2
1
1, 1
1, 2
2
2, 1
3
3
4
5
6
4
4
5
6
Play Together, Learn Together
Connect 4 Dice Games
Roll 2D6 and add the outcomes on the two dice. Table 04 lists the possible outcomes.

We suggest: Your students create this table before they see our Table 04
Table 04 Roll 2D6 (die1, die2) and add the dice. Possible outcomes (die1 + die2)
die1 ↓ die2 
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
7
8
9
10
11
12
Did your students create their Table 04 without peeking at our Table 04? If yes, yeah and bravo!
If they didn’t create a table showing the 36 possible outcomes of rolling 2D6 and adding the outcomes
of the two dice, we encourage them to complete Table 05 directly below. Yep, it’s a green die, red die
rollathon. If you prefer different colors, use your nimble fingers to edit our Table 05.
Table 05 Roll 2D6 (green, red) and add the dice. Possible outcomes (green + red)
green  red 
1
2
1
2
3
2
3
3
3
4
5
6
5
4
5
6
Play Together, Learn Together
Connect 4 Dice Games
If your students roll a green die and a red die, add ‘em, and post the sums in their proper places in Table
05, perhaps they will discover that some outcomes are more likely to occur than other outcomes. It
would be good, we think, if your students find all the ways to get each dice sum 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, and 12. They can count the number of ways to get each sum in their Table 05.
There is only one way to roll 2D6, add ‘em, and get 2: 1 on the green die and 1 on the red die (1 + 1 = 2)
There are two ways to get 3.
 One way: 1 on the green die and 2 on the red die: 1 + 2 = 3
 Another way: 2 on the green die and 1 on the red die: 2 + 1 = 3
There are three ways to get 4.
 One way: 1 on the green die and 3 on the red die: 1 + 3 = 4
 Another way: 2 on the green die and 2 on the red die: 2 + 2 = 4
 Another way: 3 on the green die and 1 on the red die: 3 + 1 = 4
There are four ways to get 5.
 One way: 1 on the green die and 4 on the red die: 1 + 4 = 5
 Another way: 2 on the green die and 3 on the red die: 2 + 3 = 5
 Another way: 3 on the green die and 2 on the red die: 3 + 2 = 5
 Another way: 4 on the green die and 1 on the red die: 4 + 1= 5
There are five ways to get 6,
 One way: 1 on the green die and 5 on the red die: 1 + 5 = 6
 Another way: 2 on the green die and 4 on the red die: 2 + 4 = 6
 Another way: 3 on the green die and 3 on the red die: 3 + 3 = 6
 Another way: 4 on the green die and 2 on the red die: 4 + 2= 6
 Another way: 5 on the green die and 1 on the red die: 5 + 1= 6
There are six ways to get 7,
 One way: 1 on the green die and 6 on the red die: 1 + 6 = 7
 Another way: 2 on the green die and 5 on the red die: 2 + 5 = 7
 Another way: 3 on the green die and 4 on the red die: 3 + 4 = 7
 Another way: 4 on the green die and 3 on the red die: 4 + 3 = 7
 Another way: 5 on the green die and 2 on the red die: 5 + 2= 7
 Another way: 6 on the green die and 1 on the red die: 6 + 1= 7
Your Turn Inspire your students to show the number of ways to roll 2D6, add ‘em, and get the sums 8,
9, 10, 11, and 12.
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Play Together, Learn Together
Connect 4 Dice Games
Roll 2D6 and add the dice. Table 06 is a frequency distribution of the possible sums.
Table 06 Roll 2D6 and add the two dice. Possible dice sums.
Number
Frequency
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
5
4
3
2
1
Roll 2D6 and add the outcomes on the two dice.
Ways to get each dice sum 2 to 12.
1+1
1 + 2, 2 + 1
1 + 3, 2 + 2, 3 + 1
1 + 4, 2 + 3, 3 + 2, 4 + 1
1 + 5, 2 + 4, 3 + 3, 4 + 2, 5 + 1
1 + 6, 5 + 2, 4 + 3, 3 + 4, 2 + 5, 6 + 1
2 + 6, 3 + 5, 4 + 4, 5 + 3, 6 + 2
3 + 6, 4 + 5, 5 + 4, 6 + 3
4 + 6, 5 + 5, 6 + 4
5 + 6, 6 + 5
6+6
Roll 2D6 and add the dice. Table 07 is a frequency distribution and histogram of the possible sums.
Table 07 Roll 2D6 and add the two dice, frequency distribution and histogram
Number
Frequency
2
3
4
5
6
7
8
9
10
11
12
1
2
3
4
5
6
5
4
3
2
1
Histogram: Number of ways to get each dice sum 2 to 12.
Each  represents one way to get the number in column 1.











Hey! That histogram in Table 07 looks like a gaggle of confused geese flying east instead of north or
south. Each  represents one goose. Each  represents one occurrence of the number in
column one.
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Play Together, Learn Together
Connect 4 Dice Games
Why not make a 6-row by 6-column (36-number) play sheet with one 2, two 3s, three 4s, four 5s, five 6s,
six 7s, five 8s, four 9s, three 10s, two 11s, and one 12? We did it. Addition Play Sheet 01 contains each
number from 2 in the upper-left corner to 12 in the lower-right corner the number of times shown in the
frequency distribution (Table 06 or Table 07).
Connect Four!
Addition Play Sheet 01
2 3 3 4 4 4
5 5 5 5 6 6
6 6 6 7 7 7
7 7 7 8 8 8
8 8 9 9 9 9
10 10 10 11 11 12
How to play:
1. Roll two dice.
2. Add the numbers on the dice.
3. Select the dice sum in the play sheet.
4. First player to get 4 in a row horizontally, vertically, or diagonally wins!
.

Ahoy early elementary school teacher, might your
players/explorers/investigators play Connect Four on Addition Play Sheet 01
and discover a bit about the probability of rolling each 2D6 sum from 2 to 12?
8

Play Together, Learn Together
Connect 4 Dice Games
We scrambled the numbers in Addition Play Sheet 01 and got Addition Play Sheet 02.
Connect Four!
Addition Play Sheet 02
6
3
7
8
5
6
9 8 10 9
7 4 5 7
10 6 2 4
5 12 8 11
7 11 10 7
9 4 8 9
6
5
8
7
3
6
How to play:
1. Roll two dice.
2. Add the numbers on the dice.
3. Select the dice sum in the play sheet.
4. First player to get 4 in a row horizontally, vertically, or diagonally wins!

Ahoy Teacher: You can make play sheets that present patterns you want your
students to discover. Example: We hope students who play Connect Four using
Addition Play Sheet 01 or Addition Play Sheet 02 with numbers 2, 3, 3, 4, 4, 4,
5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 11, 11, 12
might discover that 6, 7, and 8 are more likely to occur than 2, 3, 11, or 12.
Aha! A teaching moment: As they play, your students can record each sum and
count the number of ways each number 2 to 12 occurs. Some of them might
begin to acquire a conscious or subconscious awareness about probabilities
related to rolling 2D6. Serendipity!
9

Play Together, Learn Together
Connect 4 Dice Games
Addition Play Sheet 03 below has




six 7s on the main diagonal (upper-left to bottom-right),
five 6s on the diagonal above the main diagonal,
five 8s on the diagonal below the main diagonal, and
the rest of the numbers are in diagonal alignments.
Connect Four!
Addition Play Sheet 03
7 6 5
8 7 6
9 8 7
10 9 8
11 10 9
12 11 10
4
5
6
7
8
9
3
4
5
6
7
8
2
3
4
5
6
7
How to play:
1. Roll two dice.
2. Add the numbers on the dice.
3. Select the dice sum in the play sheet.
4. First player to get 4 in a row horizontally, vertically, or diagonally wins!
A Connect Four play sheet is a 2-dimensional array (see https://en.wikipedia.org/wiki/Array) of numbers
arranged in rows and columns. If you are a spreadsheet user, you know about rows and columns. If you
are a matrix aficionado (see https://en.wikipedia.org/wiki/Matrix_(mathematics ), you know about rows
and columns. On the next page, we suggest notation for talking about 2-dimensional arrays, matrices,
and other 2-D alakazams.
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Play Together, Learn Together
Connect 4 Dice Games
Notation. A Connect Four play sheet is an array consisting of horizontal rows and vertical columns. The
intersection of a row and a column is a cell. Table 08 is a 6-row by 6-column array with each of the 36
cells identified by row and column notation. R1C1 is row 1, column 1; R1C2 is row 1, column 2; R2C1 is
row 2, column 1; R2C2 is row 2, column 2, and so it goes to R6C6 – row 6, column 6.
Table 08 Array Notation: Row (R) and Column (C)
R1C1
R1C2
R1C3
R1C4
R1C5
R1C6
R2C1
R2C2
R2C3
R2C4
R2C5
R2C6
R3C1
R3C2
R3C3
R3C4
R3C5
R3C6
R4C1
R4C2
R4C3
R4C4
R4C5
R4C6
R5C1
R5C2
R5C3
R5C4
R5C5
R5C6
R6C1
R6C2
R6C3
R6C4
R6C5
R6C6
A diagonal is a line of two or more cells downhill from upper-left to lower-right.
Table 09 Diagonals in array notation
Main diagonal: R1C1, R2C2, R3C3, R4C4, R5C5, R6C6
Main diagonal: R1C1 to R6C6
Diagonals above the main diagonal:
R1C2, R2C3, R3C4, R4C5, R5C6
R1C3, R2C4, R3C5, R4C6
R1C4, R2C5, R3C6
R1C5, R2C6
Diagonals above the main diagonal:
Diagonal R1C2 to R5C6
Diagonal R1C3 to R4C6
Diagonal R1C4 to R3C6
Diagonal R1C5 to R2C6
Diagonals below the main diagonal:
R2C1, R3C2, R4C3, R5C4, R6C5
R3C1, R4C2, R5C3, R6C4
R4C1, R5C2, R6C3
R5C1, R6C2
Diagonals below the main diagonal:
Diagonal R2C1 to R6C5
Diagonal R3C1 to R6C4
Diagonal R4C1 to R6C3
Diagonal R5C1 to R6C2
We feel a persistent itch to impose a pun that relates to Harry Potter.

Pun: Diagon Allies work/play together in Diagon Alleys, waving magic wands
called games, hoping to help children enjoy and love math.
Ahoy Teacher, join the Diagon Alliance. Use games to make math fun.
11

Play Together, Learn Together
Connect 4 Dice Games
Hark! What is that we hear? Aha! We hear other diagonals calling for attention. They say they are
contrary diagonals and go uphill from lower-left to upper-right. Contrary diagonals claim to be in great
physical condition from all that hill climbing.
Here again is our ‘notation array’. The cells in the main contrary diagonal are highlighted in gray.
Table 10 Array notation: Row (R), column (C)
Cells in the main contrary diagonal are highlighted in gray.
R1C1
R2C1
R3C1
R4C1
R5C1
R6C1
R1C2
R2C2
R3C2
R4C2
R5C2
R6C2
R1C3
R2C3
R3C3
R4C3
R5C3
R6C3
R1C4
R2C4
R3C4
R4C4
R5C4
R6C4
R1C5
R2C5
R3C5
R4C5
R5C5
R6C5
R1C6
R2C6
R3C6
R4C6
R5C6
R6C6
Ahoy Contrarians, table 11 lists contrary diagonals in Table 10 array notation. In Table 11, ‘MCD’ means
‘Main contrary diagonal’.
Table 11 Contrary diagonals shown in array notation
Main contrary diagonal (MCD): R6C1 to R1C6
Main contrary diagonal (MCD): R6C1, R5C2, R4C3,
R3C4, R2C5, R1C6
Contrary diagonals above the MCD:
Contrary diagonal R5C1 to R1C5
Contrary diagonal R4C1 to R1C4
Contrary diagonal R3C1 to R1C3
Contrary diagonal R2C1 to R1C2
Contrary diagonals above the MCD:
R5C1, R4C2, R3C3, R2C4, R1C5
R4C1, R3C2, R2C3, R1C4
R3C1, R2C2, R1C3
R2C1, R1C2
Contrary diagonals below the MCD:
Contrary diagonal R6C2 to R2C6
Contrary diagonal R6C3 to R3C6
Contrary diagonal R6C4 to R4C6
Contrary diagonal R6C5 to R5C6
Contrary diagonals below the MCD:
R6C2, R5C3, R4C4, R3C5, R2C6
R6C3, R5C4, R4C5, R3C6
R6C4, R5C5, R4C6
R6C5, R5C6
Imagine play sheets that feature the numbers 2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8,
8, 8, 9, 9, 9, 9, 10, 10, 10. 11, 11, and 12 in contrary diagonals. Oops. Can’t fit one into this page – scroll
on down to the next page.
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Play Together, Learn Together
Connect 4 Dice Games
Especially for you, contrary diagonal fans, here is a play sheet with the numbers 2, 3, 3, 4, 4, 4, 5, 5, 5, 5,
6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10. 11, 11, and 12 in contrary diagonals.
Connect Four!
Addition Play Sheet 04
2
3
4
5
6
7
3
4
5
6
7
8
4
5
6
7
8
9
5 6 7
6 7 8
7 8 9
8 9 10
9 10 11
10 11 12
How to play:
1. Roll two dice.
2. Add the numbers on the dice.
3. Select the dice sum in the play sheet.
4. First player to get 4 in a row horizontally, vertically, or diagonally wins!
Ahoy, Contrarians. Fill in the blanks below. In Addition Play Sheet 04, there are:

six 7s in the main contrary diagonal R6C1 to R1C6.

five 6s in contrary diagonal _____ to _____ and five 8s in contrary diagonal _____ to _____.

four 5s in contrary diagonal _____ to _____ and four 9s in contrary diagonal _____ to _____.
Carry on. Continue the above listing for three 4s, three 10s, two 3s, two 11s, one 2, and one 12.
13
Play Together, Learn Together
Connect 4 Dice Games
Your Turn Connect Four addition game. Put the numbers 2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7,
7, 7, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 9, 10, 10, 10, 11, 11, 12 where you want them in Blank Play Sheet 01.
Connect Four!
Blank Play Sheet 01
How to play:
1. Roll two dice.
2. Add the numbers on the dice.
3. Select the dice sum in the play sheet.
4. First player to get 4 in a row horizontally, vertically, or diagonally wins!
Why not put our numbers or your numbers into play sheets in interesting patterns that might nudge a
student towards a discovery?
How does the placement of numbers in the play sheet influence game play? [We don’t know.]
How does the placement of numbers in the play sheet influence where the winning four in a row might
occur? [We don’t know.]
For a given play sheet, is there a strategy that will enhance your chance of winning a multi-person
game? [We don’t know.] Lucky you - your students can invent and test strategies!
14
Play Together, Learn Together
Connect 4 Dice Games
Roll 2D6 and Pick 1D6 Connect Four Dice Game | TOC
We like the Connect Four addition dice games for early elementary-school math learners. Play Connect
Four, roll dice, add the numbers on the dice, and select the sum of the dice in the play sheet.


Play game, have fun, reinforce addition skill.
Design a strategy for winning. We like designing strategies for different play sheets that contain
the numbers 2, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 7, 8, 8, 8, 8, 8, 9. 9. 9. 9, 10, 10, 10,
11, 11, and 12. What say? Intuitive probability theory at 1st grade, 2nd grade? Yeah! It’s a game.
The addition game doesn’t require deep thinking, so we thought of a variation that requires a little more
cogitation. To play this Bob & George version of Connect Four, roll 2D6, pick 1D6 (one of the dice), and
select that number in the play sheet. Possible selections are 1, 2, 3, 4, 5, and 6. The selections are
equally probable, so the play sheet below contains six of each number 1 to 6.
Connect Four!
Roll 2D6, Pick 1D6 Play Sheet 01
1
2
3
4
5
6
2
3
4
5
6
1
3
4
5
6
1
2
4
5
6
1
2
3
5
6
1
2
3
4
6
1
2
3
4
5
How to play:
1. Roll two dice.
2. Pick one of the dice.
3. Select the number on the die you picked in the play sheet.
4. First player to get 4 in a row horizontally, vertically, or diagonally wins!
15
Play Together, Learn Together
Connect 4 Dice Games
Roll 2D6, Pick 1D6 Play Sheet 01 (above):






The play sheet has six rows and six columns – 36 cells.
Each number 1 to 6 appears six times.
Each row contains all six numbers: 1, 2, 3, 4, 5, and 6.
No number appears more than once in any row.
Each column contains all six numbers: 1, 2, 3, 4, 5, and 6.
No number appears more than once in any column.
Your Turn Challenge your students to create a different play sheet that has the above characteristics,
repeated below:






The play sheet has six rows and six columns – 36 cells.
Each number 1 to 6 appears six times.
Each row contains all six numbers: 1, 2, 3, 4, 5, and 6.
No number appears more than once in any row.
Each column contains all six numbers: 1, 2, 3, 4, 5, and 6.
No number appears more than once in any column.
Your Turn Challenge your students to create another play sheet that has the same characteristics as
those above, but is different from both of the above play sheets. Required characteristics:






The play sheet has six rows and six columns – 36 cells.
Each number 1 to 6 appears six times.
Each row contains all six numbers: 1, 2, 3, 4, 5, and 6.
No number appears more than once in any row.
Each column contains all six numbers: 1, 2, 3, 4, 5, and 6.
No number appears more than once in any column.
Ahoy Teacher: Many, very many play sheets are possible. Might it be good to
organize your class into, say, teams of three students? During a period of time
you choose, each team creates one or more (maybe many more!) play sheets
that have the characteristics described above. Then compare all of the play
sheets created by your students:




Are all of the play sheets different from one another?
Are some play sheets identical to other play sheets?
Is there a ‘most popular’ play sheet?
Conjecture: One or more play sheets will have the same number in each cell
down the main diagonal or up the contrary main diagonal.
16

Play Together, Learn Together
Connect 4 Dice Games
Here are blank 6 by 6 arrays. Copy this page and give each team of investigators (also kwon as students)
a page or two or more.
Ahoy Student, use the 6 row by 6 column arrays below to create Roll 2D6, Pick 1D6 play sheets that have
the following characteristics, also called attributes:






The play sheet has six rows and six columns – 36 cells.
Each number 1 to 6 appears six times.
Each row contains all six numbers: 1, 2, 3, 4, 5, and 6.
No number appears more than once in any row.
Each column contains all six numbers: 1, 2, 3, 4, 5, and 6.
No number appears more than once in any column.
Good words to know: row, column, cell, characteristic, attribute.
17
Play Together, Learn Together
Connect 4 Dice Games
Connect Four Subtraction Dice Games | TOC
After playing Connect Four addition games for a while, Bob & K contrived Connect Four subtraction
games. In the first subtraction game, you roll 2D6, subtract the lesser die from the greater die, and
select that number in the play sheet. Oops – what if the dice roll is a double – same number on both
dice? The difference is 0, so the play sheet includes 0. Here is Subtraction Game Play Sheet 01.
Connect Four!
Subtraction Game Play Sheet 01
0
1
2
3
4
5
1
0
1
2
3
4
2
1
0
1
2
3
3
2
1
0
1
2
4
3
2
1
0
1
5
4
3
2
1
0
How to play:
1. Roll two dice.
2a. If the dice are unequal, subtract the lesser die from the greater die.
2b. If the dice are equal, the difference is zero (0)
3. Select the difference of the dice in the play sheet.
4. First player to get 4 in a row horizontally, vertically, or diagonally wins!
Can 1st-graders play this game? We don’t know. If you are a 1st-grade teacher, you know! We think that
1st-graders can play this subtraction game with the help of a great coach – you. Is our optimistic outlook
correct?
18
Play Together, Learn Together
Connect 4 Dice Games
Subtraction Game Play Sheet 01 contains six 0s, ten 1s, eight 2s, six 3s, four 4s, and two 5s. Why did we
choose these numbers? Roll 2D6 and subtract the numbers on the dice:


If the dice are unequal, subtract the lesser die from the greater die. The result is 1, 2, 3, 4, or 5.
If the dice are equal, subtract either die from the other die. The result is 0.
Your Turn Your students can complete Table 12 showing the 36 outcomes that can occur when you roll
a green die and a red die, and then subtract them to get a non-negative (zero or positive) difference.
Table 12 Roll 2D6 (green, red) and subtract: green – red or red – green.
Subtract the lesser (or equal) die from the greater (or equal) die.
green ↓/ red 
1
2
1
0
1
2
1
3
4
5
6
3
4
5
6
Here is our Table 12. It displays the 36 possible outcomes rolling 2D6 (green die, red die) and subtracting
one die from the other die so that the difference is non-negative (zero or positive).
Table 12 Roll 2D6 (green, red) and subtract them (green – red or red – green)
Subtract the lesser (or equal) die from the greater (or equal) die
green ↓/ red 
1
2
3
4
5
6
1
0
1
2
3
4
5
2
1
0
1
2
3
4
3
2
1
0
1
2
3
4
3
2
1
0
1
2
5
4
3
2
1
0
1
6
5
4
3
2
1
0

Ahoy students, notice that Table 12 is symmetric about the main diagonal.
19

Play Together, Learn Together
Connect 4 Dice Games
Here again is our Table 12.
Table 12 Roll 2D6 (green, red) and subtract them (green – red or red – green)
Subtract the lesser (or equal) die from the greater (or equal) die
green ↓/ red 
1
2
3
4
5
6
1
0
1
2
3
4
5
2
1
0
1
2
3
4
3
2
1
0
1
2
3
4
3
2
1
0
1
2
5
4
3
2
1
0
1
6
5
4
3
2
1
0
Your Turn Complete Table 13. It is a frequency distribution and histogram of the number of ways to get
each number 1 to 5. Count the number of ways to get each number in Table 12.
Table 13 Frequency distribution and histogram of the numbers in Table 12
Frequency
Histogram https://en.wikipedia.org/wiki/Histogram
0
1
2
3
4
5
.
Here is our Table 13.
Table 13 Frequency distribution and histogram of the numbers in Table 12
Number
Frequency
0
1
2
3
4
5
6
10
8
6
4
2
Histogram https://en.wikipedia.org/wiki/Histogram






20
Play Together, Learn Together
Connect 4 Dice Games
Subtraction Game Play sheet 02 features 0s on the contrary main diagonal.
Connect Four!
Subtraction Game Play Sheet 02
5
4
3
2
1
0
4
3
2
1
0
1
3
2
1
0
1
2
2
1
0
1
2
3
1
0
1
2
3
4
0
1
2
3
4
5
How to play:
1. Roll two dice.
2a. If the dice are unequal, subtract the lesser die from the greater die.
2b. If the dice are equal, the difference is zero (0)
3. Select the difference of the dice in the play sheet.
4. First player to get 4 in a row horizontally, vertically, or diagonally wins!

Ahoy students, notice that Subtraction Play Sheet 02 is symmetric about the
main contrary diagonal.
21

Play Together, Learn Together
Connect 4 Dice Games
At what grade level do math learners learn that a lesser number minus a greater number produces a
negative number? Example: 2 – 5 = -3. We don’t know and couldn’t figure it out by browsing the
Common Core Math Standards. 2nd grade? 3rd grade? Oh well, we will go for it – a Connect Four
subtraction game with positive, zero, and negative differences of subtracting 2D6. Voila! Tada! Roll 2D6
(die1, die2) and subtract die2 from die1 (die1 – die2). Table 14 enumerates the 36 possible outcomes.
Table 14 Roll 2D6 (die1, die2) and subtract die2 from die1 (die1 – die2)
die1 ↓/ die2 
1
2
3
4
5
6
1
0
1
2
3
4
5
2
1
0
-1
-2
-3
-4
3
2
1
0
-1
-2
-3
4
3
2
1
0
-1
-2
5
4
3
2
1
0
-1
6
5
4
3
2
1
0
Table 15 is a frequency distribution https://en.wikipedia.org/wiki/Frequency_distribution and histogram
https://en.wikipedia.org/wiki/Histogram of the numbers in Table 14.
Table 15 Frequency distribution and histogram of the numbers in Table 14 (die1 – die2)
die1 – die2
Frequency
5
4
3
2
1
0
1
2
3
4
5
1
2
3
4
5
6
5
4
3
2
1
Histogram https://en.wikipedia.org/wiki/Histogram











Add the frequencies in Table 15. We got 36. Count the little black squares (} in the histogram. We
got 36. Yeah! 36 is the right number of numbers for a 6-row by 6-column play sheet that
contains one -5, two -4s, three -3s, four -2s, five -1s, six 0s, five 1s, four 2s, three 3s, two 2s, and
one 1. Coming soon down yonder: a subtraction game play sheet that contains each number -5
to 5 as frequently as in the frequency distribution (Table 15). Oh happy day!
22
Play Together, Learn Together
Connect 4 Dice Games
Your Students’ Turn Hide our Table 14 and challenge your students to bring into existence Table 16
showing the possible outcomes rolling a green die and a red die, and then subtracting the red die from
the green die. The difference can be positive, negative, or zero.
Table 16 Roll 2D6 (green die, red die) and subtract: green die – red die
green ↓/ red 
1
2
3
4
5
6
1
2
3
4
5
6
Your Students’ Turn After constructing Table 16, your students can complete Table 17: frequency
distribution and histogram of the numbers in their Table 16.
Table 17 Frequency distribution and histogram of the numbers in Table 16 (green die – red die)
green – red
Frequency
Histogram https://en.wikipedia.org/wiki/Histogram
5
4
3
2
1
0
1
2
3
4
5
23
Play Together, Learn Together
Connect 4 Dice Games
Subtraction Game Play Sheet 03 has one -5, two -4s, three -3s, four -2s, five -1s, six 0s, five 1s, four 2s,
three 3s, two 4s, and one 5. The numbers appear in the order and frequency they enjoy as denizens of
Tables 14 & 15 up yonder.
Connect Four!
Subtraction Game Play Sheet 03
-5
-2
-1
0
1
3
-4
-2
-1
0
1
3
-4
-2
-1
0
2
3
-3
-2
0
1
2
4
-3
-1
0
1
2
4
-3
-1
0
1
2
5
How to play:
1. Roll 2D6 (two 6-faced dice).
2. Subtract the dice numbers. The difference can be positive, negative, or zero.
3. Select the difference in the play sheet.
4. First player to get 4 in a row horizontally, vertically, or diagonally wins!
Ahoy Teacher, the numbers in Subtraction Game Play Sheet 03 appear in the
order -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 with the frequencies shown in the frequency
distribution. We hope some of your students wonder why some numbers are

more likely to occur and other numbers less likely to occur when they roll 2D6
and subtract the dice numbers. Perhaps you gently nudge them toward
understanding this mathemagical manifestation of probabilistic propensity.
24

Play Together, Learn Together
Connect 4 Dice Games
Especially for contrary diagonal aficionados, we recommend Subtraction Game Play Sheet 04.
Connect Four!
Subtraction Game Play Sheet 04
5
4
3
2
1
0
4
3
2
1
0
-1
3
2
1
0
-1
-2
2
1
0
-1
-2
-3
1
0
-1
-2
-3
-4
0
-1
-2
-3
-4
-5
How to play:
1. Roll 2D6 (two 6-faced dice).
2. Subtract the dice numbers. The difference can be positive, negative, or zero.
3. Select the difference in the play sheet.
4. First player to get 4 in a row horizontally, vertically, or diagonally wins!
Your explorers/adventurers/investigators can peruse Subtraction Game Play Sheet 04 and discover:








The main contrary diagonal contains six 0s.
Positive numbers reside above the main contrary diagonal.
Negative numbers dwell below the main contrary diagonal.
Five 1s occupy contrary diagonal R5C1 to R1C5 above the main contrary diagonal.
Five -1s populate contrary diagonal R6C2 to R2C6 below the main contrary diagonal.
Four 2s inhabit contrary diagonal R4C1 to R1C4 above the main contrary diagonal.
Four -2s nest in contrary diagonal R6C3 to R3C6 below the main contrary diagonal.
Oops – running out of room on this page. You and your students can continue the above list and
also find patterns that we missed. Is there a sum of rows pattern? Is there a sum of columns
pattern? Is there a sum of diagonals pattern? Is there …? Carry on!
25
Play Together, Learn Together
Connect 4 Dice Games
As an equal opportunity play sheet provider, we are happy to show Subtraction Game Play Sheet 05 for
fans of downhill diagonals.
Connect Four!
Subtraction Game Play Sheet 05
0
1
2
3
4
5
-1
0
1
2
3
4
-2
-1
0
1
2
3
-3
-2
-1
0
1
2
-4
-3
-2
-1
0
1
-5
-4
-3
-2
-1
0
How to play:
1. Roll 2D6 (two 6-faced dice).
2. Subtract the dice numbers. The difference can be positive, negative, or zero.
3. Select the difference in the play sheet.
4. First player to get 4 in a row horizontally, vertically, or diagonally wins!
Your students can cast their eyes upon Subtraction Game Play Sheet 05 and discover:








The main diagonal contains six 0s.
Negative numbers reside above the main diagonal;
Positive numbers dwell below the main diagonal.
Five -1s occupy diagonal R1C2 to R5C6 above the main diagonal.
Five 1s populate diagonal R2C1 to R6C5 below the main diagonal.
Four -2s inhabit diagonal R1C3 to R4C6 above the main diagonal.
Four 2s nest in diagonal R3C1 to R6C4 below the main diagonal.
Patterns abound! Your students can find many more patterns.
26
Play Together, Learn Together
Connect 4 Dice Games
In Subtraction Game Play Sheets 01, 02, 03, 04, and 05, the negative numbers are scrunched together,
and the positive numbers are scrunched together. Let’s intertwingle the positive and negative numbers.
Subtraction Play Sheet 06 shows one way.
Connect Four!
Subtraction Game Play Sheet 06
5
-4
3
-2
1
0
-4
3
-2
1
0
-1
3
-2
1
0
-1
2
-2
1
0
-1
2
-3
1
0
-1
2
-3
4
0
-1
2
-3
4
-5
How to play:
1. Roll 2D6 (two 6-faced dice).
2. Subtract the dice numbers. The difference can be positive, negative, or zero.
3. Select the difference in the play sheet.
4. First player to get 4 in a row horizontally, vertically, or diagonally wins!
Hey! Every positive number is next to a negative number horizontally, vertically, or diagonally. Every
negative number is next to a positive number horizontally, vertically, or diagonally. Yeah!
0 is next to 1, -1, 2, and -2, but not next to 3, -3, 4, -4, 5, or -5.
Zowie! Imagine the patterns that your students can create.
27
Play Together, Learn Together
Connect 4 Dice Games
Your Turn Here are 6 by 6 blank arrays in which your students can scribble play sheet numbers.
Ahoy Teacher, This eBook is published under a Creative Commons License
https://creativecommons.org/.
 You may copy our stuff, paste it into your documents, edit it, enhance it, gaze at
your changes admiringly, print it, and hand out your enhanced document to
students or send the file to their computer gadgets.
28

Play Together, Learn Together
Connect 4 Dice Games
Connect Four Multiplication Dice Games | TOC
After playing addition and subtraction games, we began thinking about multiplication games and made
Table 18. It displays the 36 possible outcomes when you roll 2D6 and multiply the numbers on the two
dice. Table 19 is a frequency distribution and histogram of the possible outcomes exposed in Table 18.
Table 18 Roll 2D6 (die1, die2) and multiply the dice (die1  die2)
die1 ↓/ die2 
1
2
3
4
5
6
1
1
2
3
4
5
6
2
2
4
6
8
10
12
3
3
6
9
12
15
18
4
4
8
12
16
20
24
5
5
10
15
20
25
30
6
6
12
18
24
30
36
Table 18 is symmetric about the main diagonal. Example: R2C1 = R1C2 (2  1 = 1  2). Sure, of course,
that is because of the commutative property of multiplication: die1  die2 = die2  die1.
Table 19 Frequency distribution and histogram of the numbers in Table 18 (die1  die2)
die1  die2
1
2
3
4
5
6
8
9
10
12
15
16
18
20
24
25
30
36
Frequency
1
2
2
3
2
4
2
1
2
4
2
1
2
2
2
1
2
1
Histogram https://en.wikipedia.org/wiki/Histogram


















29
Play Together, Learn Together
Connect 4 Dice Games
Connect Four Multiplication Play Sheet 01 contains the 36 possible products displayed in Tables 18 and
19. Each possible product appears as frequently as shown in the frequency distribution (Table 19)..
Connect Four!
Multiplication Game Play Sheet 01
1 2 2 3 3 4
4 4 5 5 6 6
6 6 8 8 9 10
10 12 12 12 12 15
15 16 18 18 20 20
24 24 25 30 30 36
How to play:
1. Roll 2D6 (two 6-faced dice).
2. Multiply the numbers on the dice.
3. Select the product in the play sheet.
4. First player to get 4 in a row horizontally, vertically, or diagonally wins!
About Multiplication Play Sheet 01:





There are one 1, two 2s, two 3s, three 4s, two 5s, four 6s, two 8s, one 9, two 10s, four 12s, two
15s, one 16, two 18s, two 20s, two 24s, one 25, two 30s, and one 36.
There are nine odd numbers and 27 even numbers. Your students: Why more even than odd?
Square numbers: one 1, three 4s, one 9, one 16, one 25, and one 36. Your students: Elucidate.
Prime numbers: two 2s, two 3s, and two 5s. Your students: Why no 7? Why no 11? Why no 13?
Why no 17? Why no 19? Why no 23? Why no 29? Why no 31? Why no 37?
Triangular numbers: one 1, two 3s, four 6s, two 10s, and two 15s.
Why, oh why? Your students can scan, browse, and peruse Tables 18 and 19 and Multiplication Play
Sheet 01. What might they discover? What might they find that escaped the eyes of Bob & George?
30
Play Together, Learn Together
Connect 4 Dice Games
Creating Multiplication Game Play Sheet 01 was easy. We started at R1C! (row 1, column 1) and inserted
the numbers as frequently as they occur in the frequency distribution: one 1, two 2s, two 3s, et cetera,
et cetera. To create Multiplication Game Play Sheet 02 below, we swapped rows 1 & 2, swapped rows 3
& 4, and swapped rows 5 & 6. Yeah! Swapping rows in a 2D array is a handy math alakazam.
Connect Four!
Multiplication Game Play Sheet 02
4 4 5 5 6 6
1 2 2 3 3 4
10 12 12 12 12 15
6 6 8 8 9 10
24 24 25 30 30 36
15 26 18 18 20 30
How to play:
1. Roll 2D6 (two 6-faced dice).
2. Multiply the numbers on the dice.
3. Select the product in the play sheet.
4. First player to get 4 in a row horizontally, vertically, or diagonally wins!
Your students can create new multiplication play sheets by swapping rows in any multiplication play
sheet.
31
Play Together, Learn Together
Connect 4 Dice Games
Your students can create new multiplication play sheets by swapping rows in an existing multiplication
play sheet. As you probably know or suspect, they can also create new play sheets by swapping columns
in an existing play sheet. Swapping columns of a 2D array is an important math operation.
We began with Multiplication Play Sheet 01, swapped columns 1 & 2, swapped columns 3 & 4, and
swapped columns 4 & 5. Voila! Here is Multiplication Play Sheet 03:
Connect Four!
Multiplication Game Play Sheet 03
2 1 3 2
4 4 5 5
6 6 8 8
12 10 12 12
16 15 18 18
24 24 30 25
4 3
6 6
10 9
15 12
20 20
36 30
How to play:
1. Roll 2D6 (two 6-faced dice).
2. Multiply the numbers on the dice.
3. Select the product in the play sheet.
4. First player to get 4 in a row horizontally, vertically, or diagonally wins!
Your students can create new play sheets by swapping rows and then swapping columns. Or by
swapping columns and then swapping rows. Start with, say, Multiplication Play Sheet 01 and:



Swap rows and then swap columns.
Swap columns and then swap rows
Et cetera, et cetera, et cetera.
32
Play Together, Learn Together
Connect 4 Dice Games
In order to aid, abet, and facilitate your students’ row-swapping and column-swapping activities, we
here provide some small 6 by 6 2D arrays. Up, up, and scribble away!
33
Play Together, Learn Together
Connect 4 Dice Games
Connect Four Division Dice Games | TOC
We were puzzled for a while about how to do division games. Then one night in the wee hours we
awoke with an idea, eased out of bed in old Bob’s slow way, and recorded the idea.

The idea: Roll 2D6, add the dice, and then divide 24 by the sum of the dice.
The possible 2D6 sums are 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 12. Sums 2, 3, 4, 6, 8, and 12 are divisors of
24. Sums 5, 7, 9, 10, and 11 are not divisors of 24. Let’s divide 24 by each 2D6 sum that is a divisor of 24:
24 / 2D6 sum 
24/12 = 2
24/8 = 3
24/6 = 4
24/4 = 6
24/3 = 8
24/2 = 12
Aha! We will construct a 6-row by 6-column play sheet with 36 numbers. What numbers might we
include? Answer: the six quotients obtained by dividing 24 by the 2D6 sums 2, 3, 4, 6, 8, and 12.
Connect Four!
Division Game Play Sheet 01
2
3
4
6
8
12
3
4
6
8
12
2
4
6
8
12
2
3
6
8
12
2
3
4
8
12
2
3
4
6
12
2
3
4
6
8
How to play:
1. Roll 2D6 (two 6-faced dice).
2. Add the numbers on the two dice.
3. Calculate the quotient of 24 divided by the result of Step 2.
4. Select the quotient of Step 3 in the play sheet.
5. First player to get 4 in a row horizontally, vertically, or diagonally wins!
34
Play Together, Learn Together
Connect 4 Dice Games
That was easy. We entered the numbers 2, 3, 4, 6, 8, and 12 in each of the six rows, but in different
order in each row. Ahoy Contrarians, notice that 12 appears in every cell in the main contrary diagonal.
Division Game Play Sheet 02 sports the same numbers, but in a different order. The main diagonal has
six 12s. The other divisors of 24 (2, 3, 4, 6, and 8) each occupy a downhill diagonal.
Connect Four!
Division Game Play Sheet 02
12
2
3
4
6
8
8
12
2
3
4
6
6
8
12
2
3
4
4
6
8
12
2
3
3
4
6
8
12
2
2
3
4
6
8
12
How to play:
1. Roll 28D6 (two 6-faced dice).
2. Add the numbers on the two dice.
3. Calculate the quotient of 24 divided by the result of Step 2.
4. Select the quotient of Step 3 in the play sheet.
5. First player to get 4 in a row horizontally, vertically, or diagonally wins!

Ahoy Teacher, you can make play sheets that reinforce mathemagical alakazams
that you want your students to learn.

We wonder: Roll 2D6 and add the dice. How many ways to get a divisor of 24 (1, 2, 3, 4, 6, 8, or 12)?




The least 2D6 sum is 2, so there is no way to get 1.
How many ways to get 2?
How many ways to get 3?
Et cetera, et cetera. We think this is a good task for your students.
35
Play Together, Learn Together
Connect 4 Dice Games
Roll 2D6, a green die and a red die. Table 20 shows the possible outcomes as ordered pairs (green, red).
Table 20 Roll 2D6, a green die and a red die. Possible outcomes: green, red
green ↓/ red 
1
2
3
4
5
6
1
1, 1
1, 2
1, 3
1, 4
1, 5
1, 6
2
2, 1
2, 2
2, 3
2, 4
2, 5
2, 6
3
3, 1
3, 2
3, 3
3, 4
3, 5
3, 6
4
4, 1
4, 2
4, 3
4, 4
4, 5
4, 6
5
5, 1
5, 2
5, 3
5, 4
5, 5
5, 6
6
6, 1
6, 2
6, 3
6, 4
6, 5
6, 6
Your students Roll 2D6 and add the dice. Complete Table 21 showing the 36 outcomes. Highlight
outcomes that are divisors of 24.
Table 21 Roll 2D6 (green die, red die) and add the dice. Outcomes: green die + red die
Highlight outcomes that are divisors of 24
1
2
3
4
5
6
green  / red 
1
2
3
4
5
6
Table 21 Roll 2D6 and add the dice. Outcomes that are divisors of 24 are on a gray background
green ↓/ red 
1
2
3
4
5
6
1
2
3
4
5
6
7
2
3
4
5
6
7
8
3
4
5
6
7
8
9
4
5
6
7
8
9
10
5
6
7
8
9
10
11
6
7
8
9
10
11
12
36
Play Together, Learn Together
Connect 4 Dice Games
Roll 2D6 and add the dice. 17 of the outcomes are divisors of 24. The probability of getting a divisor of
24 is 17/36.
Ahoy contrarians, did you notice: in Table 21, every contrary diagonal has the same number in each
contrary cell? Amazing? Righteous? Other?
Ahoy teacher, did your students notice any number pattern in the main diagonal in Table 21?







Number patterns in diagonals and contrary diagonals.
Number patterns in rows and columns.
Add the numbers in the rows and get six row sums. Pattern?
Add the numbers in the columns and get six column sums. Pattern?
Go  across row 1, then  column 6, then  across row 6, then  column 1. Pattern?
Traverse other rectangular trajectories , , , . Patterns?
Best: Patterns that your students discover!
How many ways to roll 2D6, add the dice, and get each divisor of 24? Table 22 shows the ways.
Table 22 Roll 2D6 and add the dice, outcomes that are divisors of 24
Outcome
2
3
4
6
8
12
Frequency
1
2
3
5
5
1
Are we there yet? Well, not quite. We are itching to display the 2D6 sums that are divisors of 24 in a
frequency table and histogram. But first, what say your students construct a frequency distribution and
histogram by completing Table 23?
Table 23 Roll 2D6 and add the dice: divisors of 24 frequency distribution and histogram
2D6 sum
Frequency
2
___
3
___
4
___
6
___
8
___
12
___
Histogram
37
Play Together, Learn Together
Connect 4 Dice Games
Here is our Table 23
Table 23 Roll 2D6 and add the dice: divisors of 24 frequency distribution and histogram
2D6 sum
2
3
4
6
8
12
Frequency
1
2
3
5
5
1
Histogram






We want to construct a 6-row by 6-column division play sheet that uses the divisors of 24 according to
their frequencies in Table 23.





We added the frequencies in Table 23. 1 + 2 + 3 + 5 + 5 + 1 = 17.
2  17 = 34. Our play sheet will have two 2s, four 3s, six 4s, ten 6s, ten 8s, and two 12s.
Oops, we need two more numbers. We decided on another 6 and another 8.
Our play sheet will have two 2s, four 3s, six 4s, eleven 6s, eleven 8s, and two 12s.
2 + 4 + 6 + 11 + 11 + 2 = 36. Yeah!
Mosey on down to Connect Four Division Game Play Sheet 03.
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Play Together, Learn Together
Connect 4 Dice Games
Connect Four Division Game Play Sheet 03 has two 2s, four 3s, six 4s, eleven 6s, eleven 8s, and two 12s.
Connect Four!
Division Game Play Sheet 03
2
4
6
6
8
8
2
4
6
6
8
8
3
4
6
6
8
8
3
4
6
6
8
8
3 3
4 4
6 6
6 8
8 8
12 12
How to play:
1. Roll 28D6 (two 6-faced dice).
2. Add the numbers on the two dice.
3. Calculate the quotient of 24 divided by the result of Step 2.
4. Select the quotient of Step 3 in the play sheet.
5. First player to get 4 in a row horizontally, vertically, or diagonally wins!
Your Turn What say your students make more division play sheets by scrambling the numbers in our
Division Play Sheet 03? Or scrap our play sheet and begin from scratch.





One way: Pick two numbers in Division Play Sheet 03 and swap them.
Another way: Swap rows of Division Play Sheet 03.
Another way: Swap columns of Division Play Sheet 03.
Another way: Swap rows and then swap columns of Division Play Sheet 0.
The best ways: Your way or your students’ ways.
39
Play Together, Learn Together
Connect 4 Dice Games
Here there be blank 6-row by 6-column arrays that your students can scribble in to create play sheets.
40
Play Together, Learn Together
Connect 4 Dice Games
Connect Four Multiple Operations Dice Games | TOC
Up, up, and away! Connect Four Dice Games for players at a higher math maturity level. Multiple
Operations Game Play Sheet 01 lurks below. Before you venture onward, browse the how-to-play rules:
HOW TO PLAY:
1. Roll 2D6 and then do ONE of the following operations (a or b or c).
a. Add the dice. Possible outcomes: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
b. Subtract the dice and get a non-negative outcome: 0, 1, 2, 3, 4, 5
c. Multiply the dice. Possible outcomes: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 25, 30, 36
2. Select the number you cleverly calculated in Step 1 in the play sheet.
3. First player to get four in a row horizontally, vertically, or diagonally wins!
Possible numbers to imbed in Multiple-Operations Game Play Sheet 01:

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 16, 18, 20, 24, 25, 30, and 36
There are 21 possible numbers, so we will make a 6-row by 7-column play sheet, put the 21 numbers in
rows 1, 2, and 3, and again in rows 4, 5, and 6. Neato!
Connect Four!
Multiple-Operations Game Play Sheet 01
0
7
16
0
7
16
1
8
18
1
8
18
2
9
20
2
9
20
3
10
24
3
10
24
41
4
11
25
4
11
25
5
12
30
5
12
30
6
15
36
6
15
36
Play Together, Learn Together
Connect 4 Dice Games
That was easy. We stared in R1C1 (row 1, column 1) and entered all of the possible numbers listed
above the play sheet in rows 1, 2, and 3. Then we started over and entered the possible numbers again
until we ran out of room after R6C6 (row 6, column 6). Your students might like a different play sheet. If
so, they can put their numbers in the Multiple Operations Game Play Sheet below.
Possible numbers to populate your Multiple-Operations Game Play Sheet (or ignore our numbers):



Roll 2D6 and add the dice. Possible outcomes: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12
Roll 2D6 and subtract the dice to get a non-negative outcome: 0, 1, 2, 3, 4, 5
Roll 2D6 and multiply the dice. Possible outcomes: 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24,
25, 30, 36
Connect Four!
Multiple-operations Game Play Sheet ____
Ahoy Teacher, you can make multiple-operation play sheets that reinforce
mathemagical alakazams that you want your students to learn. Example: you can
 add algebraic subtraction with possible outcomes -5, -4, -3, -2, -1, 0, 1, 2, 3, 4,
and 5. You can add the operation of of division (see division games up yonder).
Wave your magic wand!
42

Play Together, Learn Together
Connect 4 Dice Games
After making Multiple-Operations Game Play Sheet 01, we gazed upon it, took a nap, and had a
persistent dream that it might be good to construct a frequency distribution of the numbers in the play
sheet. When we awakened, we thrashed about a bit, and then contrived Table 24. It is a frequency
distribution showing the number of ways to obtain each possible outcome if you roll 2D6 and add the
dice, subtract the dice, or multiply the dice.
Table 24 Roll 2D6: multiple operations frequency distribution
Frequency
(number of ways)
Possible
number
0
1
2
3
4
5
6
7
8
9
10
11
12
15
16
18
20
24
25
30
36
Total
Add
dice
1
2
3
4
5
6
5
4
3
2
1
36
Subtract Multiply
dice
dice
6
10
1
8
2
6
2
4
3
2
2
4
Scribble space. Your students can scan this table and jot
down numerous notable notations.
2
1
2
36
4
2
1
2
2
2
1
2
1
36
Ahoy Teacher, why oh why did we not add the rows in Table 24 to obtain the
total number of ways for each number in the possible numbers column? Please

think upon this question and, if a student asks why, explain. Hint: some ways
overlap other ways.
43

Play Together, Learn Together
Connect 4 Dice Games
Connect Four Algebraic Expression Dice Games | TOC
We thought we were almost done writing this eBook, and then Ideas for algebraic expression games
bubbled up and demanded to be included in this eBook. OK.
Connect Four Algebraic Expression Game 01.


Roll 3D6 (three 6-faced dice). Assign the outcomes as values of the variables a, b, and c.
Calculate a number by evaluating the algebraic expression a  b + c.
Possible values of a  b + c: 1  1 + 1 = 2 to 6  6 + 6 = 42.
Roll 3D6, place the dice as values of a, b, and c in the 1-row table below, and write the value of a  b + c
in the rightmost column. We suggest: put this table in a plastic envelope and use an erasable marker
pen to write the value of the algebraic expression in the last column over there on the right.
a

b
+
c
=
value
Example: Roll 3D6 and enjoy outcomes 2, 3, and 5.
Place the die with outcome 2 as the value of a, the die with outcome 3 as the value of b, and the die
with outcome 5 as the value of c in the table, and write the value of the algebraic expression a  b + c in
the column way over there on the right .

+
=
11
Or place the die with outcome 3 as the value of a, the die with outcome 2 as the value of b, and the die
with outcome 5 as the value of c in the table, and write the value of the algebraic expression a  b + c in
the column way over there on the right .

+
Aha! 2  3 + 5 and 3  2 + 5 are both equal to 11.
44
=
11
Play Together, Learn Together
Connect 4 Dice Games
An ancient Euclidean axiom: Things which are equal to the same thing are equal to one another.

2  3 + 5 and 3  2 + 5 are both equal to 11, so they are equal to each other.
Why is 2  3 + 5 = 3  2 + 5? Well, as you know, because of


the commutative property of multiplication (2  3 = 3  2)
and the additive property of equality (2  3 = 3  2, so 2  3 + 5 = 3 2 + 5).
When we play with the resident math kid (Bob’s 7-year-old granddaughter), we don’t mention
commutative property of multiplication or additive property of equality. We just enjoy watching her.
play and learn. Games are mathemagical wands for learning and enjoying math.
There are four more ways to place 2, 3, and 5 as the values of a, b, and c on the a  b + c table.

+
=
13

+
=
13

+
=
17

+
=
17
Your students might notice (perhaps with a nudge from you):



2  3 + 5 = 11 and 3  2 + 5 = 11
2  5 + 3 = 13 and 5  2 + 3 = 13
3  5 + 2 = 17 and 5  3 + 2 = 17
Serendipity! We chose 2, 3, and 5 for our 3D6 example because 2, 3, and 5 are the first three prime
numbers. We hope that one or more of your students notice that. Until we crunched the a  b + c
numbers, we did not know that the values of a  b + c would be prime numbers.
45
Play Together, Learn Together
Connect 4 Dice Games
Here is a recap. Roll 1D6:






Outcomes 2, 3, and 5 are the 1st three prime numbers.
Outcomes 4 and 6 are composite numbers.
Outcome 1 is neither prime nor composite. From an old folk song: One is one and all alone, and
evermore shall be so. [Green Grow the Rushes, O]
2  3 + 5 = 11 and 3  2 + 5 = 11. 11 is a prime number.
2  5 + 3 = 13 and 5  2 + 3 = 13. 13 is a prime number.
3  5 + 2 = 17 and 5  3 + 2 = 17. 17 is a prime number.
Conjecture: Much (most?) of elementary-school math can be learned by playing
games, if we invent the games. We think that kids who learn math by playing
games are likely to like/love math.
Kids who play the Connect Four games in this eBook use selection (roll 2D6 and
pick 1D6), addition, subtraction, multiplication, division, multiple arithmetic
operations and algebraic expressions to play a game. Some kids will develop

strategies for playing the game – they will frequently win. Serendipity! All this
with only one game. Imagine: you and we and many others creating games that
introduce, expand, elaborate, reinforce, enhance, et cetera, et cetera, the
Common Core Math Standards.

Aha! Ho ho! Alakazam! We can surround the Common Core Math Standards with
games and so they become the Common Core Joyful Math Standards.
Connect Four Algebraic Expression Play Sheet 01 is down yonder.



The least possible value of a  b + c is 1  1 + 1 = 2.
The greatest possible value of a  b + c is 6  6 + 6 = 42.
The play sheet contains the numbers 2 through 42 in a 6-row by 7-column array.
The possible values of the variables a, b, and c are 1, 2, 3, 4, 5, and 6. Can all
numbers 2 to 42 be calculated by evaluating a  b + c using only the numbers 1
to 6 as values of a, b, and c?
 Challenge your students to show that every number from 2 to 42 can be
calculated by evaluating a  b + c using only 1, 2, 3, 4, 5, and 6 as values of a, b,
and c.
Examples: 10 = 2  3 + 4, 23 = 4  5 + 3, and 37 = 6  6 + 1.
46

Play Together, Learn Together
Connect 4 Dice Games
HOW TO PLAY:
1.
2.
3.
4.
5.
Roll 3D6 (three 6-faced dice).
Place the dice as the values of a, b, and c in the algebraic expression a  b + c.
Calculate the value of a  b + c.
Select the value of a  b + c in the play sheet.
First player to get 4 in a row horizontally, vertically, or diagonally wins!
a

b
+
c
=
value
Connect Four!
Algebraic Expressions Game Play Sheet 01
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
47
Play Together, Learn Together
Connect 4 Dice Games
Connect Four Algebraic Expression Game 02.


Roll 3D6 (three 6-faced dice) and assign the outcomes as values of the variables a, b, and c.
Calculate a number by evaluating the algebraic expression a  (b + c)
Possible values of a  (b + c): 1  (1 + 1) = 1 to 6  (6 + 6) = 72.
Roll 3D6, place the dice as values of a, b, and c in the table below, and write the value of a  (b + c) in
the rightmost column. We suggest: put this table in a plastic envelope and use an erasable marker pen
to write the value of the algebraic expression in the last column over there on the right.
a

(
b
+
c
)
=
value
Example: Roll 3D6 and enjoy outcomes 2, 3, and 5.
Place the die with outcome 2 as the value of a, the die with outcome 3 as the value of b, and the die
with outcome 5 as the value of c in the table, and write the value of the algebraic expression a  (b + c)
in the column way over there on the right .

(
+
)
=
16
Or place the die with outcome 2 as the value of a, the die with outcome 5 as the value of b, and the die
with outcome 3 as the value of c in the table, and write the value of the algebraic expression a  (b + c)
in the column way over there on the right .

(
+
)
=
Tada! 2  (3 + 5) and 2  (5 + 3) are both equal to 16.
Things which are equal to the same thing are equal to one another.

2  (3 + 5) and 2  (5 + 3) are both equal to 16, so they are equal to each other.
48
16
Play Together, Learn Together
Connect 4 Dice Games
Why is 2  (3 + 5) = 2  (5 + 3)? Because of


the commutative property of addition: 3 + 5 = 5 + 3
and the multiplicative property of equality: 3 + 5 = 5 + 3, so 2  (3 + 5) = 2  (5 + 3).
There are four more ways to place 2, 3, and 5 as the values of a, b, and c in the a  (b + c) table.

(
+
)
=
21

(
+
)
=
21

(
+
)
=
25

(
+
)
=
25
The possible values of a, b, and c are the numbers obtained by rolling 2D6: 1, 2, 3, 4, 5, and 6.


The least possible value of a  (b + c) is 1  (1 + 1) = 2.
The greatest possible value of a  (b + c) is 6  (6 + 6) = 72.
Question: Can you calculate all numbers from 2 to 72 by evaluating a  (b + c) using only numbers 1 to 6
as values of a, b, and c?
Answer: No.
We tried to calculate each and every number from 2 to 72 by evaluating a  (b + c) using only numbers
from 1 to 6 as values of a, b, and c. We calculated the following numbers as values of a  (b + c):

2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 16, 18, 20, 21, 22, 24, 25, 27, 28, 30, 32, 33, 35, 36, 40, 42,
44, 45, 48, 50, 54, 55, 60, 66, 72 [38 numbers]. We will use these numbers, except 2 and 72, to
construct Connect Four Algebraic Expression Game Play Sheet 02
We were not able to calculate the following numbers as values of a  (b + c):

13, 17, 19, 23, 26, 29, 31, 34, 37, 38, 39, 41, 43, 46, 47, 49, 51, 52, 53, 56, 57, 58, 59, 61, 62, 63,
64, 65, 67, 68, 69, 70, 71 [33 numbers]
49
Play Together, Learn Together
Connect 4 Dice Games
HOW TO PLAY:
1.
2.
3.
4.
5.
Roll 3D6 (three 6-faced dice).
Place the dice as the values of a, b, and c in the algebraic expression a  (b + c).
Calculate the value of a  (b + c).
Select the value of a  (b + c) in the play sheet.
First player to get 4 in a row horizontally, vertically, or diagonally wins!

a
(
b
+
c
)
=
value
Connect Four!
Algebraic Expressions Game Play Sheet 02
3
4
5
6
7
8
9
10
11
12
14
15
16
18
20
21
22
24
25
27
28
30
32
33
35
36
40
42
44
45
48
50
54
55
60
66
50
Play Together, Learn Together
Connect 4 Dice Games
Your turn: Create Connect Four algebraic expression games. Better and best: Inspire your students to
create Connect Four algebraic expression games. We suggest:
Connect Four Algebraic Expression Game 03: a  b  c. Roll 3D6, assign the outcomes as values of a, b,
and c, evaluate the algebraic expression a  b  c, and select that value in the play sheet.


If you allow only non-negative values of a  b  c, what are the least and greatest possible
values? Are all numbers between the least possible value and the greatest possible value
possible? What does the play sheet look like? What numbers reside in the play sheet? If the play
sheet is a rectangular array, how many rows and how many columns does it have?
If you allow negative values of a  b  c, such as 1  2  5 = 3, what are the least and greatest
possible values? Are all numbers between the least possible value and greatest possible value
possible? What does the play sheet look like? What numbers hang out in the play sheet? If the
play sheet is a rectangular array, how many rows and how many columns does it have?
.
Connect Four Algebraic Expression Game 04: c + a  b. Roll 3D6, assign the outcomes as values of a, b,
and c, evaluate the algebraic expression c + a  b, and select that value in the play sheet.
This game is equivalent to the a  b + c game, thanks to the commutative property of addition. The two
games reinforce order of operations.


a  b + c: multiply a and b, and then add the product and c.
c + a  b: multiply a and b, and then add the product and c.
Connect Four Algebraic Expression Game 05: a  (b  c). Roll 3D6, assign the outcomes as values of a, b,
and c, evaluate the algebraic expression a  (b  c), and select that value in the play sheet.


If you allow only non-negative values of a  (b  c), what are the least and greatest possible
values? Are all numbers between the least possible value and the greatest possible value
possible? What does the play sheet look like? What numbers are at home in the play sheet? If
the play sheet is a rectangular array, how many rows and how many columns does it have?
If you allow negative values of a  (b  c) such as 1  (2  3) = 1, what are the least and greatest
possible values? Are all numbers between the least possible value and greatest possible value
possible? What does the play sheet look like? What numbers hang out in the play sheet? If the
play sheet is a rectangular array, how many rows and how many columns does it have?
Connect Four Algebraic Expression Game 06: a  b + a  c. Roll 3D6, assign the outcomes as values of a,
b, and c, evaluate the algebraic expression a  b + a  c, and select that value in the play sheet.
This game is equivalent to the a  (b + c) game, thanks to the distributive property of multiplication over
addition: a  (b + c) = a x b + a  c. The two games reinforce order of operations.


a  (b + c): add b and c, and then multiply the sum and a.
a x b + a  c: 1. multiply a and b, 2. multiply a and c, 3. add the two products.
51
Play Together, Learn Together
Connect 4 Dice Games
Your Turn: Create Connect Four Digit Dice Games | TOC
First-grade students are learning how to add 1-digit numbers. Does that Include 0? We don’t know, but
we include 0 in our list of 1-digit numbers.

1-digit numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9
We think that digit dice (DDs) are the right stuff to help 1st-graders learn how to add 1-digit numbers.
A digit die (DD) is a die with 10 faces numbered 0 through 9 (decimal digits – yeah!)
Roll 1DD (1 digit die): Possible outcomes are 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9. The
outcomes of a fair DD are equally probable. The probability of occurrence of each
outcome is 1/10.
Are DDs (digit dice) among your math manipulatives? If not and you
wish to buy a bunch of digit dice (DDs), one source is Amazon. Amazon
calls digit dice D10s because they have 10 faces. We call them DDs
because the 10 faces are numbered 0 to 9 instead of 1 to 10.

Go to Amazon https://www.amazon.com/ and search for
D10 polyhedral dice.
We wonder if Amazon sends a penny with every DD dice order.
Digit dice at Amazon
It will be bodacious if you and your explorers/investigators/researchers create Connect Four dice games
with DDs (digit dice) such as:
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Connect Four Addition Game. Roll 2DD, add the dice, and select the sum in the play sheet.
Possible play sheet numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18
Connect Four Roll 2DD and Pick One Die Game. Play sheet numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Connect Four Subtraction Game. Roll 2DD, subtract the dice to get a positive or zero difference
and then select the difference in the play sheet. Play sheet numbers: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9
Connect Four Subtraction Game. Roll 2DD and subtract the dice algebraically – the difference
can be positive, zero, or negative. Select the difference in the play sheet. Possible play sheet
numbers: -9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
Connect Four Multiplication Game. Roll 2DD, multiply the outcomes, and select the product in
the play sheet. Possible products: good task for your students, we think.
Connect Four Division Games. Roll 2DD, 1) pick one die or 2) add the dice, divide the result into
24 (or 36 or 48? or ???) and select the quotient in the play sheet.
Et cetera, et cetera. Games that YOU and YOUR STUDENTS create.
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Play Together, Learn Together
Connect 4 Dice Games
This FREE eBook is published under a Creative Commons AttributionNonCommercial-ShareAlike license.
http://creativecommons.org/licenses/by-nc-sa/4.0/
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You and your students may copy stuff from this eBook, modify it, and make it
your/their work. For example, you may copy tables and play sheets up yonder
and tweak them into tables and play sheets for Connect Four Digit Dice Games.
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You and your students can write dynamite articles (nitroglycerine articles?)
about Connect Four Digit Dice Games for a local or state math teachers’
magazine such as the periodical of the National Council of Teachers (NCTM)
affiliate in your state. We will love it if you mention us and our download site.
http://i-a-e.org/downloads/free-ebooks-by-bob-albrecht.html
Conjecture: Much (most?) of elementary-school math can be learned by playing
games, if we invent the games. We think that kids who learn math by playing
games are likely to like/love math.
Kids who play the Connect Four games in this eBook use selection (roll 2D6 and
pick 1D6), addition, subtraction, multiplication, division, multiple arithmetic
operations and algebraic expressions to play a game. Some kids will develop
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strategies for playing the game – they will frequently win. Serendipity! All this
with only one game. Imagine: you and we and many others creating games that
introduce, expand, elaborate, reinforce, enhance, et cetera, et cetera, the
Common Core Math Standards.
Aha! Ho ho! Alakazam! We can surround the Common Core Math Standards with
games and so they become the Common Core Joyful Math Standards.
The End
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