Play Together, Learn Together: Connect Four Dice Games

Transcription

1 Play Together, Learn Together: Connect Four Dice Games Bob Albrecht & George Firedrake Begin 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 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. Bob s favorite 7-year-old math kid is in nd 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: Bob & George? Bob is an 87-year-old human (as of February 017). George is a Dragon. Read about Bob & George at Information Age Education (IAE): Information Age Education ( publishes a large number of free books, a free blog, the free IAE-Pedia, and the free IAE Newsletter. 1

2 Table of Contents (TOC) Connect Four Dice Games Addition Roll D6 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

3 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 , it was in the center of the page at Pinterest 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, 4 numbers in all. The numbers are to 1, with two or more of each number appearing in the play sheet. To play, you roll D6 (two 6-faced dice), add em, and then select the sum in the play sheet. Aha! The numbers in the play sheet are, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 1, the possible sums of D6. 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 4 numbers. We counted the frequency (number of occurrences) of each number in the play sheet and made the frequency distribution ( shown in Table 01. Table 01 Connect Four play sheet at K s school: Numbers and their frequencies. Number Total Frequency 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

4 To play the Connect Four dice game, you roll D6 and add the numbers on the two dice. D6? Dice notation: 1D6: One hexahedral (cubic) die with faces labeled 1 to 6 in pips (dots) or numerals. D6: Two hexahedral (cubic) dice with faces labeled 1 to 6 in pips (dots) or numerals. 1D6 D6 Roll D6. There are 36 possible outcomes. Here they are in Table 0. We call the two dice die1 and die. An outcome is shown as an ordered pair: die1, die. Table 0 Roll D6 (die1 and die) possible outcomes (die1, die) die1 die , 1 1, 1, 3 1, 4 1, 5 1, 6, 1,, 3, 4, 5, 6 3 3, 1 3, 3, 3 3, 4 3, 5 3, 6 4 4, 1 4, 4, 3 4, 4 4, 5 4, 6 5 5, 1 5, 5, 3 5, 4 5, 5 5, 6 6 6, 1 6, 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 = is a different outcome than green die =, red die = 1. What say your students complete Table 03 before they see our Table 0? When they complete Table 03, it will show the 36 possible outcomes of rolling a green die and a red die. Table 03 Roll D6 (green die, red die) possible outcomes (green die, red die) green red , 1 1,,

5 Roll D6 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 D6 (die1, die) and add the dice. Possible outcomes (die1 + die) die1 die 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 D6 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 D6 (green, red) and add the dice. Possible outcomes (green + red) green red

6 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, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 1. They can count the number of ways to get each sum in their Table 05. There is only one way to roll D6, add em, and get : 1 on the green die and 1 on the red die (1 + 1 = ) There are two ways to get 3. One way: 1 on the green die and on the red die: 1 + = 3 Another way: on the green die and 1 on the red die: + 1 = 3 There are three ways to get 4. One way: 1 on the green die and 3 on the red die: = 4 Another way: on the green die and on the red die: + = 4 Another way: 3 on the green die and 1 on the red die: = 4 There are four ways to get 5. One way: 1 on the green die and 4 on the red die: = 5 Another way: on the green die and 3 on the red die: + 3 = 5 Another way: 3 on the green die and on the red die: 3 + = 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: = 6 Another way: on the green die and 4 on the red die: + 4 = 6 Another way: 3 on the green die and 3 on the red die: = 6 Another way: 4 on the green die and on the red die: 4 + = 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: = 7 Another way: on the green die and 5 on the red die: + 5 = 7 Another way: 3 on the green die and 4 on the red die: = 7 Another way: 4 on the green die and 3 on the red die: = 7 Another way: 5 on the green die and on the red die: 5 + = 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 D6, add em, and get the sums 8, 9, 10, 11, and 1. 6

7 Roll D6 and add the dice. Table 06 is a frequency distribution of the possible sums. Table 06 Roll D6 and add the two dice. Possible dice sums. Number Frequency Roll D6 and add the outcomes on the two dice. Ways to get each dice sum to , , +, , + 3, 3 +, , + 4, 3 + 3, 4 +, , 5 +, 4 + 3, 3 + 4, + 5, , 3 + 5, 4 + 4, 5 + 3, , 4 + 5, 5 + 4, , 5 + 5, , Roll D6 and add the dice. Table 07 is a frequency distribution and histogram of the possible sums. Number Table 07 Roll D6 and add the two dice, frequency distribution and histogram Frequency Histogram: Number of ways to get each dice sum to 1. 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. 7

8 Why not make a 6-row by 6-column (36-number) play sheet with one, two 3s, three 4s, four 5s, five 6s, six 7s, five 8s, four 9s, three 10s, two 11s, and one 1? We did it. Addition Play Sheet 01 contains each number from in the upper-left corner to 1 in the lower-right corner the number of times shown in the frequency distribution (Table 06 or Table 07). Connect Four! Addition Play Sheet How to play: 1. Roll two dice.. 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 D6 sum from to 1? 8

9 We scrambled the numbers in Addition Play Sheet 01 and got Addition Play Sheet 0. Connect Four! Addition Play Sheet How to play: 1. Roll two dice.. 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 0 with numbers, 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, 1 might discover that 6, 7, and 8 are more likely to occur than, 3, 11, or 1. Aha! A teaching moment: As they play, your students can record each sum and count the number of ways each number to 1 occurs. Some of them might begin to acquire a conscious or subconscious awareness about probabilities related to rolling D6. Serendipity! 9

10 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 How to play: 1. Roll two dice.. 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 -dimensional array (see 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 ), you know about rows and columns. On the next page, we suggest notation for talking about -dimensional arrays, matrices, and other -D alakazams. 10

11 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; R1C is row 1, column ; RC1 is row, column 1; RC is row, column, and so it goes to R6C6 row 6, column 6. Table 08 Array Notation: Row (R) and Column (C) R1C1 R1C R1C3 R1C4 R1C5 R1C6 RC1 RC RC3 RC4 RC5 RC6 R3C1 R3C R3C3 R3C4 R3C5 R3C6 R4C1 R4C R4C3 R4C4 R4C5 R4C6 R5C1 R5C R5C3 R5C4 R5C5 R5C6 R6C1 R6C 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, RC, R3C3, R4C4, R5C5, R6C6 Main diagonal: R1C1 to R6C6 Diagonals above the main diagonal: R1C, RC3, R3C4, R4C5, R5C6 R1C3, RC4, R3C5, R4C6 R1C4, RC5, R3C6 R1C5, RC6 Diagonals below the main diagonal: RC1, R3C, R4C3, R5C4, R6C5 R3C1, R4C, R5C3, R6C4 R4C1, R5C, R6C3 R5C1, R6C Diagonals above the main diagonal: Diagonal R1C to R5C6 Diagonal R1C3 to R4C6 Diagonal R1C4 to R3C6 Diagonal R1C5 to RC6 Diagonals below the main diagonal: Diagonal RC1 to R6C5 Diagonal R3C1 to R6C4 Diagonal R4C1 to R6C3 Diagonal R5C1 to R6C 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

12 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 R1C R1C3 R1C4 R1C5 R1C6 RC1 RC RC3 RC4 RC5 RC6 R3C1 R3C R3C3 R3C4 R3C5 R3C6 R4C1 R4C R4C3 R4C4 R4C5 R4C6 R5C1 R5C R5C3 R5C4 R5C5 R5C6 R6C1 R6C R6C3 R6C4 R6C5 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 Contrary diagonals above the MCD: Contrary diagonal R5C1 to R1C5 Contrary diagonal R4C1 to R1C4 Contrary diagonal R3C1 to R1C3 Contrary diagonal RC1 to R1C Contrary diagonals below the MCD: Contrary diagonal R6C to RC6 Contrary diagonal R6C3 to R3C6 Contrary diagonal R6C4 to R4C6 Contrary diagonal R6C5 to R5C6 Main contrary diagonal (MCD): R6C1, R5C, R4C3, R3C4, RC5, R1C6 Contrary diagonals above the MCD: R5C1, R4C, R3C3, RC4, R1C5 R4C1, R3C, RC3, R1C4 R3C1, RC, R1C3 RC1, R1C Contrary diagonals below the MCD: R6C, R5C3, R4C4, R3C5, RC6 R6C3, R5C4, R4C5, R3C6 R6C4, R5C5, R4C6 R6C5, R5C6 Imagine play sheets that feature the numbers, 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, , 11, and 1 in contrary diagonals. Oops. Can t fit one into this page scroll on down to the next page. 1

13 Especially for you, contrary diagonal fans, here is a play sheet with the numbers, 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, , 11, and 1 in contrary diagonals. Connect Four! Addition Play Sheet How to play: 1. Roll two dice.. 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, and one 1. 13

14 Your Turn Connect Four addition game. Put the numbers, 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, 1 where you want them in Blank Play Sheet 01. Connect Four! Blank Play Sheet 01 How to play: 1. Roll two dice.. 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

15 Roll D6 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, 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, , 10, 10, 10, 11, 11, and 1. What say? Intuitive probability theory at 1st grade, nd 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 D6, pick 1D6 (one of the dice), and select that number in the play sheet. Possible selections are 1,, 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 D6, Pick 1D6 Play Sheet How to play: 1. Roll two dice.. 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

16 Roll D6, 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,, 3, 4, 5, and 6. No number appears more than once in any row. Each column contains all six numbers: 1,, 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,, 3, 4, 5, and 6. No number appears more than once in any row. Each column contains all six numbers: 1,, 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,, 3, 4, 5, and 6. No number appears more than once in any row. Each column contains all six numbers: 1,, 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

17 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 D6, 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,, 3, 4, 5, and 6. No number appears more than once in any row. Each column contains all six numbers: 1,, 3, 4, 5, and 6. No number appears more than once in any column. Good words to know: row, column, cell, characteristic, attribute. 17

18 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 D6, 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 How to play: 1. Roll two dice. a. If the dice are unequal, subtract the lesser die from the greater die. b. 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

19 Subtraction Game Play Sheet 01 contains six 0s, ten 1s, eight s, six 3s, four 4s, and two 5s. Why did we choose these numbers? Roll D6 and subtract the numbers on the dice: If the dice are unequal, subtract the lesser die from the greater die. The result is 1,, 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 1 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 1 Roll D6 (green, red) and subtract: green red or red green. Subtract the lesser (or equal) die from the greater (or equal) die. green / red Here is our Table 1. It displays the 36 possible outcomes rolling D6 (green die, red die) and subtracting one die from the other die so that the difference is non-negative (zero or positive). Table 1 Roll D6 (green, red) and subtract them (green red or red green) Subtract the lesser (or equal) die from the greater (or equal) die green / red Ahoy students, notice that Table 1 is symmetric about the main diagonal. 19

20 Here again is our Table 1. Table 1 Roll D6 (green, red) and subtract them (green red or red green) Subtract the lesser (or equal) die from the greater (or equal) die green / red 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 1. Table 13 Frequency distribution and histogram of the numbers in Table 1 Frequency Histogram Here is our Table 13. Table 13 Frequency distribution and histogram of the numbers in Table 1 Number Frequency Histogram

21 Subtraction Game Play sheet 0 features 0s on the contrary main diagonal. Connect Four! Subtraction Game Play Sheet How to play: 1. Roll two dice. a. If the dice are unequal, subtract the lesser die from the greater die. b. 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 0 is symmetric about the main contrary diagonal. 1

22 At what grade level do math learners learn that a lesser number minus a greater number produces a negative number? Example: 5 = -3. We don t know and couldn t figure it out by browsing the Common Core Math Standards. nd grade? 3rd grade? Oh well, we will go for it a Connect Four subtraction game with positive, zero, and negative differences of subtracting D6. Voila! Tada! Roll D6 (die1, die) and subtract die from die1 (die1 die). Table 14 enumerates the 36 possible outcomes. Table 14 Roll D6 (die1, die) and subtract die from die1 (die1 die) die1 / die Table 15 is a frequency distribution and histogram of the numbers in Table 14. Table 15 Frequency distribution and histogram of the numbers in Table 14 (die1 die) die1 die Frequency 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 -s, five -1s, six 0s, five 1s, four s, three 3s, two s, 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!

23 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 D6 (green die, red die) and subtract: green die red die green / red 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

24 Subtraction Game Play Sheet 03 has one -5, two -4s, three -3s, four -s, five -1s, six 0s, five 1s, four s, 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 How to play: 1. Roll D6 (two 6-faced dice).. 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, -, -1, 0, 1,, 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 D6 and subtract the dice numbers. Perhaps you gently nudge them toward understanding this mathemagical manifestation of probabilistic propensity. 4

25 Especially for contrary diagonal aficionados, we recommend Subtraction Game Play Sheet 04. Connect Four! Subtraction Game Play Sheet How to play: 1. Roll D6 (two 6-faced dice).. 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 R6C to RC6 below the main contrary diagonal. Four s inhabit contrary diagonal R4C1 to R1C4 above the main contrary diagonal. Four -s 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! 5

26 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 How to play: 1. Roll D6 (two 6-faced dice).. 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 R1C to R5C6 above the main diagonal. Five 1s populate diagonal RC1 to R6C5 below the main diagonal. Four -s inhabit diagonal R1C3 to R4C6 above the main diagonal. Four s nest in diagonal R3C1 to R6C4 below the main diagonal. Patterns abound! Your students can find many more patterns. 6

27 In Subtraction Game Play Sheets 01, 0, 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 How to play: 1. Roll D6 (two 6-faced dice).. 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,, and -, but not next to 3, -3, 4, -4, 5, or -5. Zowie! Imagine the patterns that your students can create. 7

28 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 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. 8

29 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 D6 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 D6 (die1, die) and multiply the dice (die1 die) die1 / die Table 18 is symmetric about the main diagonal. Example: RC1 = R1C ( 1 = 1 ). Sure, of course, that is because of the commutative property of multiplication: die1 die = die die1. Table 19 Frequency distribution and histogram of the numbers in Table 18 (die1 die) die1 die Frequency Histogram

30 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 How to play: 1. Roll D6 (two 6-faced dice).. 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 s, two 3s, three 4s, two 5s, four 6s, two 8s, one 9, two 10s, four 1s, two 15s, one 16, two 18s, two 0s, two 4s, one 5, two 30s, and one 36. There are nine odd numbers and 7 even numbers. Your students: Why more even than odd? Square numbers: one 1, three 4s, one 9, one 16, one 5, and one 36. Your students: Elucidate. Prime numbers: two s, two 3s, and two 5s. Your students: Why no 7? Why no 11? Why no 13? Why no 17? Why no 19? Why no 3? Why no 9? 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

31 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 s, two 3s, et cetera, et cetera. To create Multiplication Game Play Sheet 0 below, we swapped rows 1 &, swapped rows 3 & 4, and swapped rows 5 & 6. Yeah! Swapping rows in a D array is a handy math alakazam. Connect Four! Multiplication Game Play Sheet How to play: 1. Roll D6 (two 6-faced dice).. 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

32 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 D array is an important math operation. We began with Multiplication Play Sheet 01, swapped columns 1 &, swapped columns 3 & 4, and swapped columns 4 & 5. Voila! Here is Multiplication Play Sheet 03: Connect Four! Multiplication Game Play Sheet How to play: 1. Roll D6 (two 6-faced dice).. 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. 3

33 In order to aid, abet, and facilitate your students row-swapping and column-swapping activities, we here provide some small 6 by 6 D arrays. Up, up, and scribble away! 33

34 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 D6, add the dice, and then divide 4 by the sum of the dice. The possible D6 sums are, 3, 4, 5, 6, 7, 8, 9, 10, 11, and 1. Sums, 3, 4, 6, 8, and 1 are divisors of 4. Sums 5, 7, 9, 10, and 11 are not divisors of 4. Let s divide 4 by each D6 sum that is a divisor of 4: 4 / D6 sum 4/1 = 4/8 = 3 4/6 = 4 4/4 = 6 4/3 = 8 4/ = 1 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 4 by the D6 sums, 3, 4, 6, 8, and 1. Connect Four! Division Game Play Sheet How to play: 1. Roll D6 (two 6-faced dice).. Add the numbers on the two dice. 3. Calculate the quotient of 4 divided by the result of Step. 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

35 That was easy. We entered the numbers, 3, 4, 6, 8, and 1 in each of the six rows, but in different order in each row. Ahoy Contrarians, notice that 1 appears in every cell in the main contrary diagonal. Division Game Play Sheet 0 sports the same numbers, but in a different order. The main diagonal has six 1s. The other divisors of 4 (, 3, 4, 6, and 8) each occupy a downhill diagonal. Connect Four! Division Game Play Sheet How to play: 1. Roll 8D6 (two 6-faced dice).. Add the numbers on the two dice. 3. Calculate the quotient of 4 divided by the result of Step. 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 D6 and add the dice. How many ways to get a divisor of 4 (1,, 3, 4, 6, 8, or 1)? The least D6 sum is, so there is no way to get 1. How many ways to get? How many ways to get 3? Et cetera, et cetera. We think this is a good task for your students. 35

36 Roll D6, a green die and a red die. Table 0 shows the possible outcomes as ordered pairs (green, red). Table 0 Roll D6, a green die and a red die. Possible outcomes: green, red green / red , 1 1, 1, 3 1, 4 1, 5 1, 6, 1,, 3, 4, 5, 6 3 3, 1 3, 3, 3 3, 4 3, 5 3, 6 4 4, 1 4, 4, 3 4, 4 4, 5 4, 6 5 5, 1 5, 5, 3 5, 4 5, 5 5, 6 6 6, 1 6, 6, 3 6, 4 6, 5 6, 6 Your students Roll D6 and add the dice. Complete Table 1 showing the 36 outcomes. Highlight outcomes that are divisors of 4. Table 1 Roll D6 (green die, red die) and add the dice. Outcomes: green die + red die Highlight outcomes that are divisors of 4 green / red Table 1 Roll D6 and add the dice. Outcomes that are divisors of 4 are on a gray background green / red

37 Roll D6 and add the dice. 17 of the outcomes are divisors of 4. The probability of getting a divisor of 4 is 17/36. Ahoy contrarians, did you notice: in Table 1, 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 1? 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 D6, add the dice, and get each divisor of 4? Table shows the ways. Table Roll D6 and add the dice, outcomes that are divisors of 4 Outcome Frequency Are we there yet? Well, not quite. We are itching to display the D6 sums that are divisors of 4 in a frequency table and histogram. But first, what say your students construct a frequency distribution and histogram by completing Table 3? Table 3 Roll D6 and add the dice: divisors of 4 frequency distribution and histogram D6 sum Frequency Histogram

38 Here is our Table 3 Table 3 Roll D6 and add the dice: divisors of 4 frequency distribution and histogram D6 sum Frequency Histogram We want to construct a 6-row by 6-column division play sheet that uses the divisors of 4 according to their frequencies in Table 3. We added the frequencies in Table = = 34. Our play sheet will have two s, four 3s, six 4s, ten 6s, ten 8s, and two 1s. Oops, we need two more numbers. We decided on another 6 and another 8. Our play sheet will have two s, four 3s, six 4s, eleven 6s, eleven 8s, and two 1s = 36. Yeah! Mosey on down to Connect Four Division Game Play Sheet

39 Connect Four Division Game Play Sheet 03 has two s, four 3s, six 4s, eleven 6s, eleven 8s, and two 1s. Connect Four! Division Game Play Sheet How to play: 1. Roll 8D6 (two 6-faced dice).. Add the numbers on the two dice. 3. Calculate the quotient of 4 divided by the result of Step. 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

40 Here there be blank 6-row by 6-column arrays that your students can scribble in to create play sheets. 40

41 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 D6 and then do ONE of the following operations (a or b or c). a. Add the dice. Possible outcomes:, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1 b. Subtract the dice and get a non-negative outcome: 0, 1,, 3, 4, 5 c. Multiply the dice. Possible outcomes: 1,, 3, 4, 5, 6, 8, 9, 10, 1, 15, 16, 18, 0, 4, 5, 30, 36. 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,, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1, 15, 16, 18, 0, 4, 5, 30, and 36 There are 1 possible numbers, so we will make a 6-row by 7-column play sheet, put the 1 numbers in rows 1,, and 3, and again in rows 4, 5, and 6. Neato! Connect Four! Multiple-Operations Game Play Sheet

42 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,, 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 D6 and add the dice. Possible outcomes:, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1 Roll D6 and subtract the dice to get a non-negative outcome: 0, 1,, 3, 4, 5 Roll D6 and multiply the dice. Possible outcomes: 1,, 3, 4, 5, 6, 8, 9, 10, 1, 15, 16, 18, 0, 4, 5, 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, -, -1, 0, 1,, 3, 4, and 5. You can add the operation of of division (see division games up yonder). Wave your magic wand! 4

43 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 4. It is a frequency distribution showing the number of ways to obtain each possible outcome if you roll D6 and add the dice, subtract the dice, or multiply the dice. Table 4 Roll D6: multiple operations frequency distribution Frequency (number of ways) Possible number Add dice Subtract dice Multiply dice Total Scribble space. Your students can scan this table and jot down numerous notable notations. Ahoy Teacher, why oh why did we not add the rows in Table 4 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

44 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: = to = 4. 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, 3, and 5. Place the die with outcome 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 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 Aha! and are both equal to

45 An ancient Euclidean axiom: Things which are equal to the same thing are equal to one another and are both equal to 11, so they are equal to each other. Why is = 3 + 5? Well, as you know, because of the commutative property of multiplication ( 3 = 3 ) and the additive property of equality ( 3 = 3, so = 3 + 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, 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): = 11 and = = 13 and = = 17 and = 17 Serendipity! We chose, 3, and 5 for our 3D6 example because, 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

46 Here is a recap. Roll 1D6: Outcomes, 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] = 11 and = is a prime number = 13 and = is a prime number = 17 and = 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 D6 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 =. The greatest possible value of a b + c is = 4. The play sheet contains the numbers through 4 in a 6-row by 7-column array. The possible values of the variables a, b, and c are 1,, 3, 4, 5, and 6. Can all numbers to 4 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 to 4 can be calculated by evaluating a b + c using only 1,, 3, 4, 5, and 6 as values of a, b, and c. Examples: 10 = 3 + 4, 3 = , and 37 =

47 HOW TO PLAY: 1. Roll 3D6 (three 6-faced dice).. Place the dice as the values of a, b, and c in the algebraic expression a b + c. 3. Calculate the value of a b + c. 4. Select the value of a b + c in the play sheet. 5. First player to get 4 in a row horizontally, vertically, or diagonally wins! a b + c = value Connect Four! Algebraic Expressions Game Play Sheet

48 Connect Four Algebraic Expression Game 0. 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) = 7. 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, 3, and 5. Place the die with outcome 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 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. ( + ) = 16 Tada! (3 + 5) and (5 + 3) are both equal to 16. Things which are equal to the same thing are equal to one another. (3 + 5) and (5 + 3) are both equal to 16, so they are equal to each other. 48

49 Why is (3 + 5) = (5 + 3)? Because of the commutative property of addition: = and the multiplicative property of equality: = 5 + 3, so (3 + 5) = (5 + 3). There are four more ways to place, 3, and 5 as the values of a, b, and c in the a (b + c) table. ( + ) = 1 ( + ) = 1 ( + ) = 5 ( + ) = 5 The possible values of a, b, and c are the numbers obtained by rolling D6: 1,, 3, 4, 5, and 6. The least possible value of a (b + c) is 1 (1 + 1) =. The greatest possible value of a (b + c) is 6 (6 + 6) = 7. Question: Can you calculate all numbers from to 7 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 to 7 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):, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1, 14, 15, 16, 18, 0, 1,, 4, 5, 7, 8, 30, 3, 33, 35, 36, 40, 4, 44, 45, 48, 50, 54, 55, 60, 66, 7 [38 numbers]. We will use these numbers, except and 7, to construct Connect Four Algebraic Expression Game Play Sheet 0 We were not able to calculate the following numbers as values of a (b + c): 13, 17, 19, 3, 6, 9, 31, 34, 37, 38, 39, 41, 43, 46, 47, 49, 51, 5, 53, 56, 57, 58, 59, 61, 6, 63, 64, 65, 67, 68, 69, 70, 71 [33 numbers] 49

50 HOW TO PLAY: 1. Roll 3D6 (three 6-faced dice).. Place the dice as the values of a, b, and c in the algebraic expression a (b + c). 3. Calculate the value of a (b + c). 4. Select the value of a (b + c) in the play sheet. 5. First player to get 4 in a row horizontally, vertically, or diagonally wins! a ( b + c ) = value Connect Four! Algebraic Expressions Game Play Sheet

51 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 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 ( 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,. multiply a and c, 3. add the two products. 51

52 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,, 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,, 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 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: Connect Four Addition Game. Roll DD, add the dice, and select the sum in the play sheet. Possible play sheet numbers: 0, 1,, 3, 4, 5, 6, 7, 8, 9, 10, 11, 1, 13, 14, 15, 16, 17, 18 Connect Four Roll DD and Pick One Die Game. Play sheet numbers: 0, 1,, 3, 4, 5, 6, 7, 8, 9 Connect Four Subtraction Game. Roll DD, subtract the dice to get a positive or zero difference and then select the difference in the play sheet. Play sheet numbers: 0, 1,, 3, 4, 5, 6, 7, 8, 9 Connect Four Subtraction Game. Roll DD 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, -, -1, 0, 1,, 3, 4, 5, 6, 7, 8, 9. Connect Four Multiplication Game. Roll DD, 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 DD, 1) pick one die or ) add the dice, divide the result into 4 (or 36 or 48? or???) and select the quotient in the play sheet. Et cetera, et cetera. Games that YOU and YOUR STUDENTS create. 5