Download DragonBox Algebra 12+

 Where is the math?
DragonBox Algebra 12+
1 DragonBox – Where is the math? How does the game work? The goal of the game is to get the Dragon Box alone on one side of the gameboard. If there is more than one Dragon box, you have to get rid of all but one box. The box card (alias the unknown “x”): There are two special cards. The green vortex (alias zero): The one (alias one): All other cards have no “special” properties and have two sides: The algebraic rules are gradually introduced with the cards that represent monsters or dice. Over time, these cards are replaced by letters and numbers, and the equal sign and plus sign are introduced. By then, the player is comfortable with the algebraic logic and can solve equations in the game. 2 DragonBox – Where is the math? Presentation of the tutorials and didactical approach Here are presentations of the main levels and introductions of algebraic rules. Level 1-­‐1: we present the goal of the game, and the player is presented Rule 1. To get rid of the green vortices, just click on them. The vortices represent the number zero. Mathematically speaking, it would have been better to drag and drop the zero on other cards, but we decided to perform the operation by clicking on them from an entertaining point of view. Rule 1: Additive Identity Property. 𝑎 + 0 = 𝑎 𝑥 + 0 = 𝑥 Level 1-­‐3: the player is presented Rule 2. The player has to drag and drop cards on their opposite cards (night background). Opposite cards are differentiated by colors. When later on, letters and numbers are introduced, the minus sign will be the differentiator, but the properties of opposites are by then understood. We didn´t introduce numbers on purpose, because it can scare people. From our experience, it is harder to distinguish letters and numbers than characters, so we introduce all rules with characters. Rule 2: Additive Inverse Property If we add a number by the opposite of itself, we will end up with 0. 𝑎 + (−𝑎) = 0 𝑥 + (−𝑥) = 0 Level 1-­‐5: the player is presented her first equation, in two dimensions. Even if there are no equal or plus signs, it is a real mathematics equation in two dimensions. It allows us to communicate commutative properties without explaining this property. Players experience it naturally. Level 1-­‐9: Rule 3 is introduced. The player has to add the same card to both sides from the deck. We are well aware that when you solve an equation, you have access to any number. Our deck of cards helps us introduce the property of equality in a progressive way and at the same time use the constraint of cards to create an interesting game play. 3 DragonBox – Where is the math? Rule 3: Properties of Equality (I) If 𝑎 = 𝑏, 𝑡ℎ𝑒𝑛 𝑎 + 𝑐 = 𝑏 + 𝑐 Add c to each side Level 1-­‐16: we introduce to the student, that cards have a day side as well as a night side. This creates the necessary link in the mind of the student that each card has an opposite. This way, the concept of opposite is naturally introduced. Level 1-­‐19: the DragonBox is replaced by the letter x, but thanks to the small light particles around the box and the x, young players identify the x as the box without any problem. It usually takes a “where is the box?” thought and a few seconds before the player kcontinues playing. When transferring to pencil and paper, it will not be any trouble introducing other unknowns to players. The game creates the concept of the unknown as the box. Thus, referring to any unknown as the box and the work is done. Level 2-­‐1: Rule 4 is introduced. Students can now drag one card up onto it's twin. Rule 4: Multiplicative Inverse Property If we multiply a number by its reciprocal, we will end up with 1. 1
𝑎 ∙ = 1 𝑎
1
𝑥 ∙ = 1 𝑥
In the game, if you drag a card up onto its twin card, it will disappear into the 1 dice card. Level 2-­‐5: Rule 5 is introduced. Just click on the 1 dice card. The 1 dice card will disappear under the neighbour card. Rule 5: Multiplicative Identity Property If we multiply 1 to any number, we will end up with the same number. 𝑎 ∙ 1 = 𝑎 𝑥 ∙ 1 = 𝑥 4 DragonBox – Where is the math? Level 2-­‐11: Rule 6 is presented. Place a card under another, this is division. Rule 6: Properties of Equality (II) 𝑖𝑓 𝑎𝑐 = 𝑏𝑐 and 𝑐 ≠ 0, then 𝑎 = 𝑏 Divide both sides by c Level 3-­‐1: You can drag a group of cards from one side of the board to the other side of the board. Rule 8: Shortcut 𝑖𝑓 𝑥 + 𝑎 = 𝑐, 𝑡ℎ𝑒𝑛 𝑥 = 𝑐 + (−1) ∙ 𝑎 Level 3-­‐7: Rule 7 is introduced. You can attach a card alongside another card. This is multiplication Rule 7: Properties of Equality (III) 𝑖𝑓 𝑎 = 𝑏, 𝑡ℎ𝑒𝑛 𝑎𝑐 = 𝑏𝑐 Multiply both sides by c Level 4-­‐1 We introduce the addition of numbers. Drag and drop two numbers and they will add together. The addition of numbers cannot exceed 100. Level 4-­‐4 We introduce the factorization of numbers. By clicking on the 6, it will become 2 times 3. Level 4-­‐8 We introduce multiplicaton of numbers. Drag and drop the 2 on the 3, and it becomes 6. 5 DragonBox – Where is the math? Level 5-­‐1: Multiplication of signs. When you double click on a card in a group it will flip to its opposite and also flip an attached card to its opposite. If the clicked on card is negative then it will flip to its positive side and also flip an attached card to its opposite side. If the clicked on card is positive it will flip to its negative side and also flip another attached card to its opposite side. If there is only one negative card, double clicking on it will flip it to its positive side and a -­‐1 card will appear in front of the group. If it is a single positive card, it will flip to its negative side and a -­‐1 card will appear in front of the group. Rule 9: Properties of Negation (−1)𝑎 = −𝑎 (−1)1 = −1 −(−𝑎) = 𝑎 − (−2) = 2 (−𝑎)𝑏 = −(𝑎𝑏) = 𝑎(−𝑏) (−2)𝑥 = −(2𝑥) = 2(−𝑥) (−𝑎)(−𝑏) = 𝑎𝑏 (−2)(−𝑥) = 2𝑥 −(𝑎 + 𝑏) = (−𝑎) + (−𝑏) − (𝑥 + 2) = (−𝑥) + (−2) = −𝑥 − 2 Rule 10: Rules of Signs for fractions 𝑎 −𝑎
𝑎
−𝑎 𝑎
− =
=
𝑎𝑛𝑑 = 𝑏
𝑏
−𝑏
−𝑏 𝑏
Level 6-­‐1: introduction of parenthesis as bubbles. When the bubble is soft, it can burst by double clicking on it. When it is an iced bubble, you can not burst it. Double clicking on it will trigger an animation showing the reason why it can not be burst. Level 6-­‐5. You can get a card inside a bubble when this card is attached to a bubble by dragging it to the top of the bubble. 6 DragonBox – Where is the math? Rule 11: Distributive Property (I) When we are adding and multiplying with a parenthesis, we can distribute the multiplication through the addition. 𝑎(𝑏 + 𝑐) = 𝑎𝑏 + 𝑎𝑐 𝑥(2 + 3𝑦) = 2𝑥 + 3𝑥𝑦 Level 6-­‐7. You can get a card under a bubble into the bubble by dragging it to the bottom of a bubble Rule 12: Distributive Property (II) When we are adding and dividing with a parenthesis, we can distribute the division through the addition. (𝑏 + 𝑐) 𝑏 𝑐
= + 𝑎
𝑎 𝑎
(2 + 3𝑦) 2 3𝑦
= + 𝑥
𝑥
𝑥
level 7-­‐1: you can get a card out of a bubble if this card is common to all groups in a bubble. To do it, drag these cards to the top of the bubble. Rule 13: Factoring property (I) 𝑎𝑏 + 𝑎𝑐 = 𝑎(𝑏 + 𝑐) 2𝑥 + 3𝑥𝑦 = 𝑥(2 + 3𝑦) Level 7-­‐5: you can create parentheses by dragging a bubble to a group of card and dragging other cards to this bubble. Level 7-­‐10: you can get a card out of a bubble if it is common to all groups in the lower part of the groups in a bubble (ie. in the denominator). Drag this card to the bottom of the bubble. 7 DragonBox – Where is the math? Rule 14: Factoring property (II) (𝑏 + 𝑐)
𝑏 𝑐
! + !=
𝑎 𝑎
𝑎
(2 + 3𝑦)
2 3𝑦
! + !=
𝑥 𝑥
𝑥
Level 7-­‐14: you can make a 1 card appear by sliding a card to the side. Also, sliding a 1 card to the bottom will create two empty cards. Rule 15: Generate Equivalent Fractions 𝑎 𝑎𝑐
= 𝑐 ≠ 0 𝑏 𝑏𝑐
multiplying the top and bottom by the same thing keeps the fraction the same value Rule 16: Add/Subtract with Like Denominators 𝑎 𝑐 𝑎∓𝑐
∓ =
𝑏 𝑏
𝑏
if the denominators are the same, add or subtract the top of the fraction Level 8-­‐1: you can add “like” cards or groups of cards (i.e. collecting “like” terms) Level 9-­‐1: you can create your own cards by collecting groups of cards into a special container. These containers will behave the same way as other cards and you will be able to manipulate them as such. (ie. substitution) Rule 19: Create a parameter 𝑥 ∙ (𝑎 + 𝑏) = 𝑐 𝑖𝑠 𝑒𝑞𝑢𝑖𝑣𝑎𝑙𝑒𝑛𝑡 𝑡𝑜 𝑥 ∙ 𝑦 = 𝑐 𝑎𝑛𝑑 𝑦 = (𝑎 + 𝑏) 8 DragonBox – Where is the math?