Basic Understandings. Recipes for Functions Guess My Rule!


 Hilary Cole
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1 Activity: TEKS: Recipes for Functions Guess My Rule! (a). (3) Function concepts. A function is a fundamental mathematical concept; it expresses a special kind of relationship between two quantities. Students use functions to determine one quantity from another, to represent and model problem situations, and to analyze and interpret relationships. (5) Tools for algebraic thinking. Techniques for working with functions and equations are essential in understanding underlying relationships. Students use a variety of representations (concrete, pictorial, numeric, symbolic, graphic, and verbal), tools, and technology (including, but not limited to, calculators with graphing capabilities, data collection devices, and computers) to model mathematical situations to solve meaningful problems. (A.2) Foundations for functions. The student uses the properties and attributes of functions. The student is expected to: (B) identify mathematical domains and ranges and determine reasonable domain and range values for given situations, both continuous and discrete; (A.4) Foundations for functions. The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills required to simplify algebraic expressions and solve equations and inequalities in problem situations. The student is expected to: (C) connect equation notation with function notation, such as y = x + 1 and f(x) = x + 1. (A.5) Linear functions. The student understands that linear functions can be represented in different ways and translates among their various representations. The student is expected to: (C) use, translate, and make connections among algebraic, tabular, graphic, or verbal descriptions of linear functions. Overview: The purpose of this activity is for students to work on algebraic representations of functions and connections to function tables. Students will create their own linear functions and challenge others to discover the underlying rule. Recipes for Functions Guess My Rule Page 1
2 Materials: Grouping: Time: Transparency 1/Handout 1 Guess My Rule! Input/Output Table Transparency 2/Handout 2 Guess My Rule! Find the Rule Part 1: no groups, teacher works with entire class Parts 24: 3 or 4 students 1 to 2 class periods Lesson: Procedures 1. Part 1: Mapping Notation Show the rule f : N N Describe that this means the rule f says to take the input value, multiply it by four, add one, and that will be the output value. Ask students for more input values and calculate the output values, recording them in a table. Then give some output values and ask students to figure out what the input values must have been. 2. Show the rule f : a a 4 + 1; ask students how this is different from the rule above. Show several more examples, such as f : z z 4 + 1, f : G G 4 + 1, and f : t t 4 + 1, to help students see that the variable name is arbitrary. 3. Ask students to plug in input values for f : N N 2 3. Then ask for output values, and find the corresponding input values. Record all pairs in a table. 4. Do not give students the rule, but provide output values for any input values they give. If they choose, allow them to give you output values, and give the corresponding input values. Write the pairs in a table. Then ask students to guess your rule. Notes Input is another name for domain. Output is another name for range. It is important that students get comfortable with the ideas of input and output using friendly terms, and then move to the algebraic terminology of domain and range. Beginning with the terms input and output helps students understand the basic concepts without intimidating them. It is important that students understand the name of the variable is just that a name. There is nothing significant in the use of any particular letter. Teachers might even use symbols, like smiley faces or stars, to help illustrate that this is just a place holder to direct where to put the input value. For input values include positive, negative, large, and small values, fractions and decimals, and variables and variable expressions. Use several different rules that involve fractions or decimals, not just integers, for the slope and intercept of the linear function. Help students come up with some techniques to find the rule (such Recipes for Functions Guess My Rule Page 2
3 Procedures Notes as finding the difference between the outputs of successive integers). 5. Part 2: Creating a Rule Divide students into small groups (three or four). Have each group create a rule that they do not share with other groups (be sure that it is linear at this point). 6. Each group should fill in a table with input and output values. 7..Each group should also plot their input/output pairs on a rectangular coordinate system. Have students include at least 6 input values. They should include positive, negative, and rational input/output values in their tables. This is a good time to hold a discussion about discrete vs. continuous data. Ask, Should we connect these ordered pairs? Why or why not? Discussion should identify situations in which it is appropriate to connect and situations in which it would not be appropriate. 8. Part 3: Figuring Out the Rule This is played like 20 questions. In turn, each group will come to the front of the classroom to present their input and output, and the rest of the class tries to guess their rule. 9. If the class cannot successfully guess the group s rule, one member of the group will select a member of the class. The class member identifies whether he/she is giving input or output, and gives a value. The group then gives the corresponding output or input value. This should continue until the class guesses the rule or gives up. 10. Points are awarded as follows: The group gets one point for every input or The class should have a limited number of guesses before additional information is given. The teacher should carefully monitor that each group gives correct input/output values for those the class gives them. Be careful not to let guessing get out of hand. To prevent shouting Recipes for Functions Guess My Rule Page 3
4 Procedures output value they are given. The group gets one point for every incorrect guess given by a member of the class. The group loses a point if they give an erroneous input or output value. (This should be monitored by the teacher.) The group that correctly guesses the rule receives one point. Notes out randomly, guessers should be selected by the group presenting their rule by raising their hands. (Groups who guess out of turn can have points deducted if it becomes problematic.) Also watch that the students selecting the guessers are not selecting only their friends. When the rule is guessed correctly, scoring ends, and that group returns to their seats and another group comes to the front. The group with the most points after all groups have presented their rules is declared the winner. 11. Part 4: Making Connections After all rules have been guessed, have each group put the rule and the graph on the board. 12. Show the relationships of each graph to the rule that generated it. 13. Plot the input and output from an unknown rule on the board. Have students guess the rule without seeing the input/output table. Show how the slope and yintercept relate to the rule of a linear function. In the discussion, include the ideas of slope and yintercept, positive and negative slope, parallel lines (rules that have the same slope but different intercepts), etc. Students should be able to identify the slope and yintercept from a graph and connect the graph to the table and to the rule. Homework: Extensions: Assign appropriate homework from the text to graph linear functions from their rules and find the rules from the graphs. Homework should also include problems that ask students to generate tables. Activity can also be used for any other family of functions to teach general characteristics of that family: i.e., quadratic functions, exponential and logarithmic functions, square root functions, etc. Recipes for Functions Guess My Rule Page 4
5 Modifications: Alternate scoring method: Awarding one point to the student who correctly guesses the rule makes the competition more individual. Each student earns one point for each point the group earns, but also earns points for correctly guessing the rules of other groups. This method of scoring gives an individual winner, rather than a group winner. Recipes for Functions Guess My Rule Page 5
6 Name Guess My Rule! 1. In your group, choose a rule and write it down on the back of this worksheet. Do not let other students outside your group see your rule. It should be in the form f(n)=n x Fill in the table below, using your secret rule. Input Output 3. Plot your table values on a rectangular coordinate system using the x coordinate for the input value and the ycoordinate for the output value. Connect the points with a line. Recipes for Functions Guess My Rule Page 6
7 Name Guess My Rule! Use the space below to figure out the rule for each group. Then write the rule on the blank provided. Recipes for Functions Guess My Rule Page 7
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