Calculus+and+Making+Connections

A game of chance where the dice may roll, but you still can control your odds! As the dice is rolled, each player needs to figure out how to use the total amount on the dice to better their odds of winning. Mental math skills and the breaking up of numbers and using substitution for other variables are key components to this game. As students becoming better mathematicians, the next step is to give them a fun assessment that challenges their mental math skills and to help them see the connection of math concepts and how one idea can be used in a different situation. Crossing is a hands-on way to help students become aware and make connections about math concepts being a whole, rather than pieces (i.e. mental math, substitution, formations of algorithms). **Student Learning Objective**: · 6.6.C Analyze and compare mathematical strategies for solving problems, and select and use one or more strategies to solve a problem. · 6.6.F Apply a previously used problem-solving strategy in a new context. · 6.6.G Extract and organize mathematical information from symbols, diagrams, and graphs to make inferences, draw conclusions, and justify reasoning. · 6.6.H Make and test conjectures based on data (or information) collected from explorations and experiments.
 * Title** : Crossing

**Materials:** · Game Board (one for each set of two students) · A pair of die for each Game Board · Game pieces (chips)

**Game Directions:** · Each player gets a section of the game board and places his or her chips at the “start” position. · Each player rills the die and moves one chip that many space or a combination of chips that many spaces. For example, if a child rolls a four, he/she could move four spaces with one chip or one space with two chips and one chip two spaces, etc. The player must tell the other player what he/she is going to do. · Each player concludes his/her turn by picking up the dice. Once a dice has been picked up, that player‘s turn is over. · If the player’s chip lands on a black spot, that chip must return to the start position. · A player must roll an exact value to get his/her piece(s) to the “10” block to finish. · All the value of a roll must be used or the player loses their turn. · The first player to get all three of his/her chips to the “10” blocks is the winner.

** Modifications: **

There are a few ways this game can be adapted to make it easier or harder for students depending on their skill levels. To make the game easier, you could create boards with less black dots, or using two markers instead of three. This game could become more difficult in more ways than it can be to make it easier. One could make the board bigger, have more markers, and add black dots. A teacher could also make the students role four dice, two that are one color and the other two a different color, and one color could be negative and the other color could be positive. This game could be adapted to be played with almost any group of students with any skill level.

**Game Questions:** One thing about the game is to encourage thinking, not a singular style of thinking. Open questions keep this game interesting and can help the student find new or creative approaches to solving problems. · “How did you split the five you just rolled?” · “Do you think this game is simple?” · “Have you thought of a strategy to play?” · “Can you tell who is ahead right now?” · "How did you know that you could avoid those black spaces when you rolled an (x amount)?" · "Now that the game is over, do you think you can make less rolls to end the game?" These kinds of questions don’t lead the children to understand that their thinking is “wrong,” what it does is help the student become aware to more forms of thinking.

Assessing the students on what they learned can be derived from a couple different methods. One way to measure what the students learned is to have a tournament with double elimination. During the tournament the teacher would walk around the classroom and evaluate the students on their play by asking questions about their strategy to win the game. As the teacher does this the teacher can tell what layer of understanding they have obtained. Examples: · The student is moving pieces at random. ..   · The student moves pieces because they think they move faster one at a time over all at once or vice versa. · The student devises a way to move the pieces fastest without hitting black spots. · The student can leave themselves the greatest about of  opportunity to use the final rolls quickest. · The student can do all of these and verbalize it.
 * Assessment and Extensions: **

As you can tell, there are many layers to the game involved with moving the pieces the least amount of turns and you want the student aware of the “implied” goal of the game. A formal way of testing the students, as well as entertaining for them, is to have them teacher other students of  the same grade or students in a lower grade. This allows the teacher to see them demonstrate the skills involved, assist them in exampling sportsmanship, and this activity reinforces their understanding.

**Teacher Reflections: by Joey Le Beau and Ben Hoover** The game Crossing was an easy game to grasp and teach to students. The strategy behind it though, much akin to backgammon, was the place that students could make the most of this game. We were surprised as a group about how fast the students understood the game and able to play it, even without verbal confirmation of strategies being used. They also surprised us by how many times they wanted to keep playing the game. It was apparent that the students really enjoyed this game by the number of times they wanted to play and the thought they invested into trying to understand strategies, or in the math world, algorithms. This lesson was successful because the students wanted to keep playing, they were able to verbalize their strategies (how they kept score, split the dice, and movement of each piece), and the ideas they wanted to try in the next game. While Ben and I wouldn’t change this game, we do think that another lane can be added or the board extended to make it a bit more challenging for the students. Also, the next step in this unit could be to bring in other games such as Mancala and Backgammon to see if the students can apply what they learned in one game and adapt that knowledge to a new game/skill.

See Also

Game Board: