Sprouts Teachers’ Guide

OVERVIEW & STANDARDS

Materials:

Handout (or something similar)

Paper and writing utensils

Students will explore the game of Sprouts, and can investigate the following major questions:

1.Does the game always end? What can the end look like?

2.Who wins?

Possible mathematical approaches to question 1 & performance indicators:

  • Arithmetic:
  • Observe how the game can end, looking for patterns
  • Observe patterns of linear differences in tables of values and extrapolate the pattern to figure out the game always ends.
  • Selected performance indicators that may be met:

7.A.7 Draw the graphic representation of a pattern from an equationor from a table of data

  • Algebra
  • Model the game using linear functions, possibly based on tables.
  • Determine the maximum number of turns by finding intersections of linear functions.
  • Selected performance indicators that may be met:

7.A.10 Write an equation to represent a function from a table of values

8.A.19 Interpret multiple representations using equation, table of values,and graph

A.A.5 Write algebraic equations or inequalities that represent asituation

A.A.10 Solve systems of two linear equations in two variablesalgebraically (See A.G.7)

A.A.23 Solve literal equations for a given variable

A.A.32 Explain slope as a rate of change between dependent andindependent variables

A.A.38 Determine if two lines are parallel, given their equations in anyform

8.G.18 Solve systems of equations graphically (only linear, integralsolutions, y = mx + b format, no vertical/horizontal lines)

7.PS.6 Represent problem situations verbally, numerically,algebraically, and graphically

A.PS.8 Determine information required to solve a problem, choosemethods for obtaining the information, and define parametersfor acceptable solutions

A.PS.9 Interpret solutions within the given constraints of a problem

A.PS.10 Evaluate the relative efficiency of different representations andsolution methods of a problem

Possible mathematical approaches to question 2 & performance indicators:

  • Experimental Probability
  • Play many games, keeping track of whether the first or second player wins. Determine the experimental probability of winning. (This addresses the question of who wins when playing at random, but does not deal with strategy!)
  • Selected performance indicators that may be met:

7.S.8 Interpret data to provide the basis for predictions and to establishexperimental probabilities

7.S.9 Determine the validity of sampling methods to predict outcomes

A.S.19 Determine the number of elements in a sample space and thenumber of favorable events

7.CM.1 Provide a correct, complete, coherent, and clear rationale forthought process used in problem solving

  • Game Trees (only practical to at most 3 vertices at the start)
  • Consider the different moves at each stage and create a game tree that maps out the different possibilities.
  • To simplify the game tree, use graph isomorphisms to identify some graphs.
  • Analyze the tree to see who can win.
  • Selected performance indicators that may be met:

7.CM.4 Share organized mathematical ideas through the manipulation ofobjects, numerical tables, drawings, pictures, charts, graphs,tables, diagrams, models and symbols in written and verbalform

  • If both, selected performance indicators that may be met

7.S.12 Compare actual results to predicted results

7.CN.3 Connect and apply a variety of strategies to solve problems

NYS Performance indicators that might be met by either question:

7.PS.2 Construct appropriate extensions to problem situations

7.PS.4 Observe patterns and formulate generalizations/ A.PS.3 Observe and explain patterns to formulate generalizations andconjectures

7.PS.7 Understand that there is no one right way to solvemathematical problems but that different methods have

advantages and disadvantages

7.PS.8 Understand how to break a complex problem into simplerparts or use a similar problem type to solve a problem

7.PS.13 Set expectations and limits for possible solutions

A.PS.1 Use a variety of problem solving strategies to understand newmathematical content

A.PS.7 Work in collaboration with others to propose, critique,evaluate, and value alternative approaches to problem solving

7.RP.4 Provide supportive arguments for conjectures

7.RP.5/A.RP.4 Develop, verify, and explain an argument, usingappropriate mathematical ideas and language

A.RP.5 Construct logical arguments that verify claims orcounterexamples that refute them

7.CM.1 Provide a correct, complete, coherent, and clear rationale forthought process used in problem solving

7.CM.4 Share organized mathematical ideas through the manipulation ofobjects, numerical tables, drawings, pictures, charts, graphs,tables, diagrams, models and symbols in written and verbalform

A.CM.5 Communicate logical arguments clearly, showing why a resultmakes sense and why the reasoning is valid

A.CM.8 Reflect on strategies of others in relation to one’s own strategy

A.CM.13 Draw conclusions about mathematical ideas through decoding,comprehension, and interpretation of mathematical visuals,symbols, and technical writing

7.CN.1/A.CN.1 Understand and make connections among multiple representationsof the same mathematical idea

7.CN.3 Connect and apply a variety of strategies to solve problems

7.CN.4/A.CN.3 Model situations mathematically, using representations to drawconclusions and formulate new situations

A.R.2 Recognize, compare, and use an array of representational forms

7.R.5 Use standard and non-standard representations with accuracy anddetail

PEDAGOGICAL COMMENTS:

If you want to give students experience with process performance indicators, it is important that students have time to play the game and explore some of their own ideas regarding the questions before being given too much structure to answer the questions. This will force them to think of their own strategies and enrich later work, even if you give them specific structure for a solution method. Students will have their own ideas which will make different connections to other ideas possible and will also add some surprise elements. Also, they will get more ownership of the problem as something they are curious about rather than just as an assignment.

DIFFERENTIATING LEARNING:

Different approaches to the questions can allow different learners to participate. Two handouts are provided that model different approaches, one (titled How to Play Sprouts) which is intended to emphasize student exploration and individual methods to answer questions about the game and one (titled Graph Theory Unit) which is intended to be teacher-led and give more background on Graph Theory. The firstmight be used over a longer period of time in parallel to other course material while the second is the focus of one or two class periods. The first handout is intended to allow students with a variety of background and ability to make progress, but some students will need more support in working with the ideas than others. The second allows students to follow their interests further than what is done is class, but brings everyone along at the same rate. Beyond using the different mathematical approaches given above as appropriate for the level of your students, here are some ideas to challenge and support all your students:

Question 1: Does the game have to end?

  • More support:
  • Ask students to cross out vertices once they have degree 3 to help know when a vertex can no longer be used.
  • Provide the table for students to fill in to help them see the pattern of decreasing available degree and ask them to look for and explain patterns. The table could look something like that below.

Turn #Resulting number of verticesAvailable degree

(The available degree is found by adding up the number of edges needed at each vertex to reach a degree of 3.)

  • Ask students to consider what happens to degrees as one turn is taken. This helps explain the pattern in the table above.
  • Suggest specific functions for students to write and ask what the intersection of these functions mean. For example, ask students to write functions that model the total degree and the degree used.
  • More challenge:
  • Ask students to extend the game so that the pattern of decreasing available degree doesn’t occur – this might push them toward Brussel Sprouts or other extensions.
  • Ask students to model the game in at least two different ways.
  • Work toward the extensions listed below.
  • Give students only the two main questions and have them discuss what strategies they might use to investigate the problems.

Question 2: Who wins?

  • More support:
  • Provide more structure to the experimental games, focusing only on 2-vertex Sprouts and telling students to keep track of which player wins and organizing the material for the whole class.
  • Start the game tree together and give different groups portions to finish.
  • Accept graphs that are isomorphic as separate entries in the game tree (it doesn’t change the result, but makes it more difficult to analyze.)
  • Start with the one-vertex starting case to analyze the game tree and who wins. From a few games, students will probably discover this winning strategy.
  • Look at sub-trees of the two-vertex game tree first. In any case, you need to work from the bottom leaves of the tree to figure out who has a winning strategy in the end.
  • Work through the game tree as a whole class.
  • More challenge:
  • Have students consider whether the options in the game tree are equally likely (they aren’t because of the isomorphisms and player choice) and whether it is possible to adjust to use the game tree to find the probability of the first player winning.
  • Have students find the expected number of moves in a game, both experimentally and from the game tree. Compare the different results and attempt to explain them.
  • Have students create a game tree starting with 3 vertices (this can also be split up between different groups) and decide who has a winning strategy.

OTHER EXTENSIONS:

Have students explore other ways the game might change:

  • What if loops are not allowed? (This leads to a different game tree & does not allow vertices to be trapped.)
  • What if we do not play Brussel Sprouts with crosses as our vertices so that we still allow vertices to have degree 4, but don’t say how the four edges must connect? (This allows an infinite game!)
  • What if we allow vertices to have degree 5? Or 6?
  • Allow students to find and explore their own variation.