Developing Mathematical Thinking Institute (DMTI)

Developing Mathematical Thinking Institute (DMTI)
Jonathan Brendefur, PhD. © DMTI (2019) | Resource Materials |

© DMTI (2019) | Resource Materials | www.DMTinstitute.com
“The Developing Mathematical Thinking Institute (DMTI) is dedicated to enhancing students’ learning of mathematics by supporting educators in the implementation of research-based instructional strategies through high-quality professional development.” For more information contact Dr. Jonathan Brendefur © DMTI (2019) | Resource Materials |

About this Supplemental Module
This module can be used by teachers at many different grade levels to support students’ understanding of whole number place value and to think meaningfully about patterns evident in the base 10 place value system. This module does not replace or supplant lessons provided in the core Unit Modules in grades K-2, but can serve as a supplement to these lessons or as a series of intervention lessons for older students still struggling with these concepts. © DMTI (2019) | Resource Materials |

Place Value Activities for K-2
Supplemental Module © DMTI (2019) | Resource Materials |

Module Sequence Note to Teachers: This supplemental module may be used with a variety of grade levels and in many different instructional settings. It may be unnecessary to proceed from lesson to lesson depending on students’ prior knowledge or the intended purpose of using the module. Part 1: Composing and Decomposing Two-Digit Numbers Problem Solving Games Part 2: Composing and Decomposing Three-Digit Numbers © DMTI (2019) | Resource Materials |

Part 1 Composing and decomposing two-digit numbers © DMTI (2019) | Resource Materials |

Problem Solving Composing and decomposing two-digit numbers © DMTI (2019) | Resource Materials |

Part 1 - Packs of Gum Packs of Chewy Gum hold 10 pieces of gum. How many packs are full if you have 11 pieces of gum? How many pieces of gum are not in a pack? Use models like this to show the answers to the following questions: How many packs are full and how many pieces are not in a pack for the following number of gum pieces? 1 pack is full. 1 pack piece 11 pieces in all 1 piece is not in the pack. 13 – 1 pack, 3 pieces 15 – 1 pack, 5 pieces 18 – 1 pack, 8 pieces 21 – 2 packs, 1 piece 25 – 2 packs, 5 pieces 29 – 2 packs, 9 pieces 31 – 3 packs, 1 piece 13 pieces 15 pieces 18 pieces 21 pieces 25 pieces 29 pieces 31 pieces © DMTI (2019) | Resource Materials |

Part 1 - Packs of Gum Now let’s look at other ways the different pieces of gum could be put in packs. If you had 23 pieces of gum, how many pieces would be left out of a pack if you only had 1 pack to fill? Now try to find the different ways you could package the following number of gum pieces, but with only number of packs given. 1 pack and 13 pieces not in a pack. 24 – 1 pack, 14 pieces 34 – 2 packs, 14 pieces 34 – 1 pack, 24 pieces 24 pieces with 1 pack to fill 34 pieces with 2 packs to fill 34 pieces with 1 packs to fill © DMTI (2019) | Resource Materials |

Part 1 - Collecting Stickers
Martel puts 10 stickers on each page. How many pages are full if he has 13 stickers? How many stickers are left over? Use models like this to show the answers to the following questions: How many pages are full and how many stickers are left over for the following numbers of stickers? 1 page is full. 1 page stickers 13 stickers in all 3 stickers are left over. 13 – 1 pack, 3 pieces 15 – 1 pack, 5 pieces 18 – 1 pack, 8 pieces 21 – 2 packs, 1 piece 25 – 2 packs, 5 pieces 29 – 2 packs, 9 pieces 31 – 3 packs, 1 piece 17 stickers 19 stickers 23 stickers 29 stickers 34 stickers 36 stickers 43 stickers © DMTI (2019) | Resource Materials |

Part 1 - Collecting Stickers
Now let’s look at other ways the stickers could be placed on pages. If you had 31 stickers, how stickers would not be on a page if you only had 2 pages to fill? Now try to find the different ways you could fill pages with 10 stickers and how many stickers that would leave not on a page for the following number of stickers and pages. 2 pages are full and 11 stickers are not on a page. 35 – 2 pages, 15 stickers 35 – 1 page, 25 stickers 47 – 3 pages, 17 stickers 47 – 2 pages, 27 stickers 47 – 1 page, 37 stickers 35 stickers with 2 pages to fill 47 stickers with 2 pages to fill 35 stickers with 1 page to fill 47 stickers with 1 page to fill 47 stickers with 3 pages to fill © DMTI (2019) | Resource Materials |

Part 1– Going to the Zoo Some kids are going on a field trip to the zoo. Each van holds 10 kids. How many vans are full if there are 23 kids? How many kids are left that won’t fill a van? Use models like this to show the answers to the following questions: How many vans are full and how many kids are not filling the vans for the following numbers of kids? 2 vans are full. 4 kids are left over. 2 vans kids 24 kids in all 27 – 2 vans, 7 kids 34 – 3 vans, 4 kids 36 – 3 vans, 6 kids 41 – 4 vans, 1 kid 45 – 4 vans, 5 kids 48 – 4 vans, 8 kids 53 – 5 vans, 3 kids 27 kids 34 kids 36 kids 41 kids 45 kids 48 kids 53 kids © DMTI (2019) | Resource Materials |

Part 1– Going to the Zoo Now let’s look at other ways the kids could ride in vans. If there were 42 kids but only 3 vans, how many kids would not be in a van? Now try to find the different ways you could fill vans with 10 kids and how many kids would not be in a van. 3 vans are full and 12 kids are not in a van. 45 kids with 3 vans to fill 45 – 3 vans, 15 kids 45 – 2 vans, 25 kids 57 – 4 vans, 17 kids 57 – 2 vans, 37 kids 62 – 5 vans, 12 kids 62 – 4 vans, 22 kids 62 – 3 vans, 32 kids 62 kids with 5 vans to fill 45 kids with 2 vans to fill 62 kids with 4 vans to fill 57 kids with 4 vans to fill 62 kids with 3 vans to fill 57 kids with 2 vans to fill © DMTI (2019) | Resource Materials |

Part 1– Candies in a Pack Each pack of candy holds 10 pieces. How many packs are full if there are 35 pieces of candy? How many pieces of candy are left that won’t fill another pack? Use these models to show the answers to the following questions: How many packs are full and how many pieces are left over for the following number of candies? 3 packs are full. 5 pieces are left over. 3 packs pieces 35 pieces of candy in all 47 – 4 packs, 7 candies 51 – 5 packs, 1 candy 64 – 6 packs, 4 candies 71 – 7 packs, 1 candy 82 – 8 packs, 2 candies 88 – 8 packs, 8 candies 91 – 9 packs, 1 candy 47 candies 51 candies 64 candies 71 candies 82 candies 88 candies 91 candies © DMTI (2019) | Resource Materials |

Part 1– Candies in a Pack Now let’s look at other ways candies can put in packs of 10. If there were 51 candies but only 4 packs, how many candies would not be in a pack? Now try to find the different ways you could fill packs with 10 candies and how many candies would not be in a pack. 4 packs are full and 11 candies are not in a pack. 54 candies with 4 packs to fill 54 – 4 packs, 14 candies 54 – 3 packs, 24 candies 54 – 2 packs, 34 candies 54 – 1 pack, 44 candies 82 – 7 packs, 12 candies 82 – 5 packs, 32 candies 82 – 3 packs, 52 candies 102 – 9 packs, 12 candies 82 candies with 7 packs to fill 54 candies with 3 packs to fill 82 candies with 5 packs to fill 54 candies with 2 packs to fill 82 candies with 3 packs to fill 54 candies with 1 pack to fill 102 candies with 9 packs to fill © DMTI (2019) | Resource Materials |

Games Composing and decomposing two-digit numbers © DMTI (2019) | Resource Materials |

Part 1 - Place Value Dice Game with 2-Digit Numbers
Directions for 2 players: 1. Player 1 rolls two six-sided dice. 2. Player 1 decides which number on the dice will be the tens’ digit and which will be the ones’ digit. 3. Player 1 draws a model (either the bar model or the place value model shown below of 42) of the largest number the digits can compose. The game can also be played with the goal of drawing a model of the smallest number. 4. Player 2 repeats this process. 5. The winner of the round is the player with the largest (or smallest) number composed. The winner of the game is the first player to win 5 to 10 rounds depending on the time allotted to game play. Place value model of 42 Bar model of 42 © DMTI (2019) | Resource Materials |

Part 1 - Digit Cards with 2 Digit Numbers
Directions for 2 players: 1. Cut out the set of Digit Cards (see template) and lay them face down in a shuffled deck. 2. Player 1 draws two digit cards. 3. Player 1 draws a model (either the bar model or the place value model shown below of 42) of the largest number the digits can compose. The game can also be played with the goal of drawing a model of the smallest number. 4. Player 2 repeats this process. 5. The winner of the round is the player with the largest (or smallest) number composed. The winner of the game is the first player to win 5 to 10 rounds depending on the time allotted to game play. 4 2 Place value model of 42 If one of the digits is 0, there are two options: Make a rule that the tens’ digit cannot be 0 Make a rule that if the digit in the tens’ place is 0, this is another way to write a single digit number (e.g. 08 = 8) Bar model of 42 © DMTI (2019) | Resource Materials |

Digit Cards - TEMPLATE 1 2 3 4 5 6 7 8 9 © DMTI (2019) | Resource Materials |

Part 1 - Make the Next 10 36 Directions for 2 players: 1. Cut out the three pages of Number Cards (see template) and lay them face down in a shuffled deck. 2. Player 1 draws a Number Card from the deck and lays it face up. 3. Player 1 rolls two dice. If the sum of the two dice is enough to compose the next unit of ten past the number on the Number Card, Player 1 keeps the card. Player 2 then repeats the process. However, if the sum of Player 1’s dice is exactly enough to compose the next ten (e.g = 40) then Player 1 not only keeps the card with 36 on it, Player 1 gets to draw again. 4. If Player 1 rolls a sum that is not enough to compose the next ten, Player 2 rolls using the same number on the Number Card as the starting number. 5. The winner of the game is the player with the most cards when the deck is empty. When the deck gets down to the last card, both players alternate rolling the dice until one player rolls enough to compose the next ten or to go past the next ten. © DMTI (2019) | Resource Materials |

Number Cards- TEMPLATE Page 1 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 © DMTI (2019) | Resource Materials |

Part 1 - I Have ____. Who Has____?
Directions: 1. Play with 3-6 players. Print out the three pages of Game Cards (see template) and cut out the cards. 2. Deal the Game Cards out equally to each player. Any extra cards can be held by the dealer or set aside to be removed from the game. 3. Player A reads the, “I have….who has….?” statement on the card. 4. The player with the answer to the, “…who has…?” statement indicates he/she has the correct number. This is Player B. 5. Player B receives the card from Player A. Player B then reads the, “I have…who has…” statement on the card that won him/her the card from Player B. The card is placed face down in front of Player B and is now out of the game and gives Player B a card to win the game with. 6. This process repeats until the cards are all face down. The player with the most cards wins. Example: Player A: I have 27. Who has 3 units of ten and 4 units of one?” Player B: I have 34. Who has 1 unit of ten and 6 units of one?” Player C: I have 16…..etc. © DMTI (2019) | Resource Materials |

Game Cards- TEMPLATE Page 1 I have 13. Who has 1 unit of ten and 4 units of one? I have 14. Who has 1 unit of ten and 7 units of one? I have 17. Who has 2 units of ten and 3 units of one? I have 23. Who has 2 units of ten and 9 units of one? I have 29. Who has 3 units of ten and 2 units of one? I have 32. Who has 4 units of ten and 6 units of one? I have 46. Who has 5 units of ten and 2 units of one? I have 52. Who has 5 units of ten and 9 units of one? I have 59. Who has 6 units of ten and 8 units of one? I have 68. Who has 7 units of ten and 4 units of one? I have 74. Who has 8 units of ten and 5 units of one? I have 85. Who has 9 units of ten and 1 unit of one? © DMTI (2019) | Resource Materials |

Game Cards- TEMPLATE Page 2 I have 91. Who has 2 units of ten and 14 units of one? I have 34. Who has 2 units of ten and 17 units of one? I have 37. Who has 3 units of ten and 15 units of one? I have 45. Who has 3 units of ten and 19 units of one? I have 49. Who has 4 units of ten and 13 units of one? I have 53. Who has 5 units of ten and 16 units of one? I have 66. Who has 6 units of ten and 12 units of one? I have 72. Who has 7 units of ten and 19 units of one? I have 89. Who has 7 units of ten and 18 units of one? I have 88. Who has 7 units of ten and 14 units of one? I have 84. Who has 8 units of ten and 15 units of one? I have 95. Who has 9 units of ten and 11 units of one? © DMTI (2019) | Resource Materials |

Game Cards- TEMPLATE Page 3 I have 101. Who has 2 units of ten and 24 units of one? I have 44. Who has 2 units of ten and 27 units of one? I have 47. Who has 3 units of ten and 25 units of one? I have 55. Who has 3 units of ten and 28 units of one? I have 58. Who has 4 units of ten and 35 units of one? I have 75. Who has 3 units of ten and 46 units of one? I have 76. Who has 6 units of ten and 22 units of one? I have 82. Who has 2 units of ten and 59 units of one? I have 79. Who has 5 units of ten and 23 units of one? I have 73. Who has 6 units of ten and 34 units of one? I have 94. Who has 8 units of ten and 25 units of one? I have 105. Who has 1 unit of ten and 3 units of one? © DMTI (2019) | Resource Materials |

Part 2 Composing and decomposing three-digit numbers © DMTI (2019) | Resource Materials |

Problem Solving Composing and decomposing Three-digit numbers © DMTI (2019) | Resource Materials |

Part 2 – Packaging Oranges
Each box of oranges holds 100 oranges. Each bag holds 10 oranges. There are also single oranges. How can you package 123 oranges using boxes, bags and single oranges? What if you could not use a box and could only use bags and single oranges? Now try to find at least two different ways you could package the following number of oranges using boxes, bags and singles. 1 box of 100, 2 bags of 10 and 3 single oranges. 12 bags of 10 oranges and 3 single oranges. Answers will vary. Focus students’ attention on how they can limit the use of any type of packaging and then find a way to use a different packaging method from smaller units. For example, if packaging 234 oranges, you could limit the option to only 1 box of 100 therefore needing 10 more bags of 10 to replace the box. 87 378 134 405 234 567 © DMTI (2019) | Resource Materials |

Part 2 – Flower seeds A box of flower seeds holds 100 seeds. A packet of seeds holds 10 seeds. There are also single seeds. How can you package 245 seeds using boxes, packets and single seeds? What if you could only use 1 box of 100 to package 245 seeds? Now try to find at least two different ways you could package the following number of seeds using boxes, packets and singles. 2 boxes of 100, 4 packets of 10 and 5 single seeds. 1 box of 100, 14 packets of 10 seeds and 5 single seeds. 146 578 306 502 417 631 © DMTI (2019) | Resource Materials |

Part 2 – The Library A bookcase holds 100 books. A shelf holds 10 books. There are also single books. How could you place 316 books on the cases, shelves or as single books? What if you could not use any shelves but only cases and single books? Now try to find at least two different ways you could place the following number of books on cases, shelves or as single books. 3 cases of 100 books, 1 shelf of 10 books and 6 single books. 3 cases of 100 books and 16 single books. The picture is of a children’s library in Japan. 251 601 508 713 851 444 © DMTI (2019) | Resource Materials |

Part 2 - Place Value Dice Game with 3-Digit Numbers
Directions for 2 players: 1. Player 1 rolls three six-sided dice. 2. Player 1 decides which number on the dice will be the hundreds’ digit, the tens’ digit and the ones’ digit. 3. Player 1 draws a model (see the place value model shown below of 432) of the largest number the digits can compose. The game can also be played with the goal of drawing a model of the smallest number. 4. Player 2 repeats this process. 5. The winner of the round is the player with the largest (or smallest) number composed. The winner of the game is the first player to win 5 to 10 rounds depending on the time allotted to game play. Place value model of 432 © DMTI (2019) | Resource Materials |

Part 2 - Digit Cards with 3-Digit Numbers
Directions for 2 players: 1. Cut out the set of Digit Cards (see template) and lay them face down in a shuffled deck. 2. Player 1 draws three digit cards. 3. Player 1 draws a model (see the place value shown below of 432) of the largest number the digits can compose. The game can also be played with the goal of drawing a model of the smallest number. 4. Player 2 repeats this process. 5. The winner of the round is the player with the largest (or smallest) number composed. The winner of the game is the first player to win 5 to 10 rounds depending on the time allotted to game play. 4 3 2 If one of the digits is 0, there are two options: Make a rule that the hundreds’ digit cannot be 0 Make a rule that if the digit in the hundreds’ place is 0, this is another way to write a two digit number (e.g. 098 = 98) Place value model of 432 © DMTI (2019) | Resource Materials |

Part 2 - I Have ____. Who Has____?
Directions: 1. Play with 3-6 players. Print out the three pages of Game Cards (see template) and cut out the cards. 2. Deal the Game Cards out equally to each player. Any extra cards can be held by the dealer or set aside to be removed from the game. 3. Player A reads the, “I have….who has….?” statement on the card. 4. The player with the answer to the, “…who has…?” statement indicates he/she has the correct number. This is Player B. 5. Player B receives the card from Player A. Player B then reads the, “I have…who has…” statement on the card that won him/her the card from Player B. The card is placed face down in front of Player B and is now out of the game and gives Player B a card to win the game with. 6. This process repeats until the cards are all face down. The player with the most cards wins. Example: Player A: I have 27. Who has 3 units of ten and 4 units of one?” Player B: I have 34. Who has 1 unit of ten and 6 units of one?” Player C: I have 16…..etc. © DMTI (2019) | Resource Materials |

Game Cards- TEMPLATE Page 1 I have 123. Who has 2 units of hundred and 4 units of one? I have 204. Who has 3 units of hundred, 2 units of ten and 7 units of one? I have 327. Who has 3 units of hundred and 3 units of ten? I have 330. Who has 4 units of hundred, 3 units of ten and 9 units of one? I have 439. Who has 40 units of ten and 2 units of one? I have 402. Who has 50 units of ten and 6 units of one? I have 506. Who has 61 units of ten and 2 units of one? I have 612. Who has 4 units of hundred, 8 units of ten and 9 units of one? I have 489. Who has 73 units of ten and 8 units of one? I have 738. Who has 81 units of ten and 4 units of one? I have 814. Who has 8 units of ten and 15 units of one? I have 95. Who has 10 units of ten and 1 unit of one? © DMTI (2019) | Resource Materials |

Game Cards- TEMPLATE Page 2 I have 101. Who has 3 units of ten and 14 units of one? I have 44. Who has 3 units of ten and 17 units of one? I have 47. Who has 4 units of ten and 15 units of one? I have 55. Who has 4 units of ten and 19 units of one? I have 59. Who has 5 units of ten and 13 units of one? I have 63. Who has 6 units of ten and 16 units of one? I have 76. Who has 7 units of ten and 12 units of one? I have 82. Who has 8 units of ten and 19 units of one? I have 99. Who has 8 units of ten and 18 units of one? I have 98. Who has 8 units of ten and 14 units of one? I have 94. Who has 9 units of ten and 15 units of one? I have 105. Who has 10 units of ten and 11 units of one? © DMTI (2019) | Resource Materials |

Game Cards- TEMPLATE Page 3 I have 111. Who has 3 units of ten and 24 units of one? I have 54. Who has 3 units of ten and 27 units of one? I have 57. Who has 4 units of ten and 25 units of one? I have 65. Who has 4 units of ten and 28 units of one? I have 68. Who has 5 units of ten and 35 units of one? I have 85. Who has 4 units of ten and 46 units of one? I have 86. Who has 7 units of ten and 22 units of one? I have 92. Who has 3 units of ten and 59 units of one? I have 89. Who has 9 units of ten and 23 units of one? I have 113. Who has 8 units of ten and 34 units of one? I have 114. Who has 9 units of ten and 25 units of one? I have 115. Who has 1 unit of hundred, 2 units of ten and 3 units of one? © DMTI (2019) | Resource Materials |

Part 2 – Place Value Concentration
Directions for 2 players: 1. Cut out the Place Value Concentration cards, shuffle the cards and place them face down on the Place Value Concentration Game Board. 2. Player 1 flips over any card he/she chooses. Player 1 then flips over any other card. The goal is to match the numbers with place value models. See the example below. If the second card flipped over does not match the first card, the cards are placed face down and it becomes Player 2’s turn. 3. Player 2 repeats the same process. 4. The winner of the game is the player with the most pairs of cards when the cards have all been used. 118 © DMTI (2019) | Resource Materials |

Brendefur and Strother (2019)
For more information contact Dr. Brendefur at © DMTI (2019) | RESOURCE MATERIALS | © DMTI (2019) | Resource Materials |

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