Jaeger AFA

Forms of Functions

MCC Content Expectations

Strand A: Algebra and Functions

Standard A2: Functions

Topic A2.1: Definitions, Representations, and Attributes of Functions

Expectations:

A2.1.1 Determine whether a relationship (given in contextual, symbolic, tabular, or graphical form) is a function and identify its domain and range.

A2.1.2 Read, interpret, and use function notation and evaluate a function at a value in its domain.

A2.1.3 Represent functions in symbols, graphs, tables, diagrams, or words and translate among representations.

A2.1.5 - Recognize that functions may be defined recursively. Compute values of and graph simple recursively defined functions.

Outcomes for the Mini-unit

In this beginning of the course unit, students will learn what a function is. Once they understand what a function is they will be introduced to the different representations and move from one representation to another while reviewing needed math skills and learning appropriate vocabulary.

Materials and Resources

  • Computers for students with internet access
  • Smart board
  • Active votes system for formative assessment
  • Graphing calculators
  • Graph papers
  • McDougal LittleAlgebra 1
  • index cards

Procedures

Day 1:Defining Functions Recursively

The Hook: Hand out the field trip permission slips to the students. Explain to the students that in order to go on the canoe trip we need to figure out how much the trip will cost. There are several costs. The first is the cost of the bus – a constant. The second cost is based on how many students will be in canoes.

Content Expectations:

Strand: Algebra and Functions

Standard: A2 Functions

Topic: A2.1 Definitions, representations, and attributes of functions

Content Expectation: A2.1.5 - Recognize that functions may be defined recursively. Compute values of and graph simple recursively defined functions.

Lesson Outcome:

Students will be able to use recursive notation when describing a situation that can be represented by a linear function.

Materials and Resources: smart board, computer with internet access hooked up to smart board, AFA Recursion on-line activity, McDougla Littel Algebra 1.

AFA activity:

Background help:

Procedures:

  1. Introduce the first sequence to be studied. Project this table for the students using a smart board or document camera – referred to as smart board from this point

THE SENARIO

Team Renaissance will be traveling to the Happy Mohawk Canoe Livery to take a canoe trip. The bus will cost $600 to get us there. Each person traveling in a canoe will cost $13. How much will it cost the team to take a class of 30 students?

Number of People Riding in Canoes / Total Price
1 / $613
2 / $626
3 / $639
4 / $652
  • Allow students to work in pairs to determine that the sequence increases by $13 each or that each person going increases the price by $13. Define recursion and recursive notation.
  1. Explain that this relationship is a function because for each input, the number of people, there can only be one output.
  2. Introduce NEXT-NOW notation for the function above.
  • Ask students where the table is NOW (4 tickets). Write NOW = $652. Discuss how we get to the NEXT term.
  • Write NEXT = NOW + $13
  • Discuss where the tabled started from. Write START = $600
  1. Introduce using subscribes to show the same sequence. Ask students to compare the following to the NEXT-NOW statements made above

Un = Un-1 + 13

U1 = 652

  • Have students recognize Un is NEXT
  • Have students recognize Un-1 is NOW
  • Have students recognize that U1 is START
  1. Guided practice – use the smart board and the AFA Recursion module. Click forward to reach the second practice. Practice several times with students, and then allow them to try some on their own.
  1. Homework. Algebra text p. 310 10-15, change the directions to “define each sequence using NEXT-NOW notation and subscript notation”

Sequence from p. 310

10. -12, -5, 2, 9…

11. 51, 72, 93,114…

12. 0.25, - 0.75, -1.25…

13. ¼, 3/8, ½, 5/8…

14. 0, -5, -10, -15…

15. 1, 1 1/3, 1 2/3, 2…

Formative assessment 1: On a ticket to leave write the following sequence in next-now notation. 3,7,11,15

Formative Assessment 2: The next class period’s warm-up will be:

Write a next-now and a sequential representation for each of the following. Find the next three values

x / 1 / 2 / 3 / 4 / 5 / x / 1 / 2 / 3 / 4 / 5
y / 19 / 17 / 15 / 13 / 11 / y / 4 / 6 / 8 / 10 / 12

Summative assessment: Exam will contain the following question

Write a next-now and a sequential representation for each of the following. Find the next three values

x / 1 / 2 / 3 / 4 / 5
y / 100 / 150 / 200 / 250 / 300

Additional Notes:

This lesson is just an introduction to the notation. The next lessons will follow with graphing of these functions as well as writing the equations that represent situations.

An extension activity “Counting Trains” can be found at

Remediation or a precursor lesson “Curses and Re-Curses It’s Happening Again” can be found at

Day 2: Writing Functions as Expressions

Content Expectations:

Strand: Algebra and Functions

Standard: A2 Functions

Topic: A2.1 Definitions, representations, and attributes of functions

Content Expectation:A2.1.1 Determine whether a relationship (given in contextual, symbolic, tabular, or graphical form) is a function and identify its domain and range.

A2.1.3 Represent functions in symbols, graphs, tables, diagrams, or words and translate among representations

.

Lesson Outcome: Students will be learning to write functions symbolically and translate verbal descriptions of functions into equations. Students will learn the meaning and understand function, domain, range, variable, expression, equation, algebraic expression.

PROCEDURE

  1. Begin the day with a warm-up

Write a next-now and a sequential representation for each of the following. Find the next three values

x / 1 / 2 / 3 / 4 / 5 / x / 1 / 2 / 3 / 4 / 5
y / 19 / 17 / 15 / 13 / 11 / y / 4 / 6 / 8 / 10 / 12
  1. Have students enter their answers using the active-votes software. Use their answers to guide how you go over the warm-up.
  2. Pass out homework answers – students should work in pairs to evaluate their work. Ask each group to share one success or one difficulty.
  3. Project the following scenario on the board.

The Adele is coming to Grand Rapids and you are trying to buy tickets for yourself and 4 friends from Tickets R Us. How much will it cost?

Number of Tickets / Price
1 / $45
2 / $75
3 / $105
4 / $135
  1. Write the function recursively. Rewrite as an algebraic equation.
  2. Have students make a vocabulary cards for the following words: function, domain, range, variable, expression, equation, algebraic expression. Define and give examples.See Appendix A
  3. Classwork: students add pictures and 2 new examples and non-examples
  4. Share examples with table partners
  5. Pairs practice: Students will practice turning verbal expressions into algebraic expressions. P. 18 15-21
  6. Give students answers
  7. Homework: p. 19-20 31-37

Formative Assessment 1: Ticket to Leave

Complete this sentence: When writing verbal expressions as algebraic expressions the most difficult thing is to…

Summative Assessment: Questions for exam

Your basic monthly cell phone service charges you $30 a month, 3 cents for every text and 5 cents a minute. Write a symbolic representation of the function that describes your cell phone bill.

Day 3: Translating Functions from Words and Expressions into Tables

Content Expectations:

Strand: Algebra and Functions

Standard: A2 Functions

Topic: A2.1 Definitions, representations, and attributes of functions

Content Expectation:

A2.1.3 Represent functions in symbols, graphs, tables, diagrams, or words and translate among representations

.

Lesson Outcome: Students will be learning to make tables from word and symbolic representations into tables. Students will deepen their understanding of functions, variables, expression, equation, and algebraic expression.

PROCEDURE

  1. Warm-up Question:

Write an equation representing the following:

The cost for a family to attend a matinee is the price of the admission times the number of people attending the matinee.

  1. Ask for students to share answers.
  2. Define “evaluate” for the students. Have them write the definition on a new vocabulary card. Then practice “evaluating” the equation if the ticket price was $4.50 for different numbers of movie goers – show how this can be represented in a table.
  3. Correct homework from last night – have students make tables for numbers 31, 32, 35 on page 19.
  4. Have students get a computer, a graphing calculator and go to
  5. Have students work through the tutorial and then use their calculator to make a table for each of the equations from number 4. Students should compare their hand-made tables to the calculator tables.

Formative Assessment:

Answer this question in your ticket to leave: Did your tables that you made using the calculator and by hand match? What were some reasons if they didn’t?

Summative Assessment: Question for the exam

Write an equation for the following situation and then make a table showing the number of beads needed to make 1,2,3,4, and 5 necklaces for the following situation.

Joan is making necklaces to raise money for endangered animals. Each necklace needs 5 blue beads, 10 red beads, and 20 green beads.

Day 4: Translating Functions from Tables into Graphs

Content Expectations:

Strand: Algebra and Functions

Standard: A2 Functions

Topic: A2.1 Definitions, representations, and attributes of functions

Content Expectation:

A2.1.3 Represent functions in symbols, graphs, tables, diagrams, or words and translate among representations

.

Lesson Outcome: Students will be learning to make graphs from the tables. Students will continue to translate functions from equations into tables.

  1. Warm-up: Students should get both computer and a graphing calculator. Once they have the materials they need, they should make a table for the equation y=3x+2.
  2. Demonstrate how to turn the table into a graph using graph paper.
  3. Students should take their tables from yesterday and graph them by hand.
  4. Students should log onto the computer and go to and work through the graphing tutorial.
  5. Then they should learn how to graph multiple equations using the tutorial
  6. They now need to graph the equations from yesterday and compare then to their hand graphs.

Formative Assessment

Ticket to leave – Use the calculator to graph the equation y= -2x+5. Where does the line cross the y axis, where does the line cross the x-axis?

  1. Homework p. 46 1-15

Summative Assessment: Question for Exam

The admission for the class to go to Michigan’s Adventure is $24 per person. The cost of the busses for the entire 8th grade will be $450. Write an equation or rule that represents the function. Then make a table that show how much a trip for 50 students, 100 students, 150 students, and 200 students would be. Finally make a graph of this.

Day 5: Translating Graphs into Words/Describing graphs

Content Expectations:

Strand: Algebra and Functions

Standard: A2 Functions

Topic: A2.1 Definitions, representations, and attributes of functions

Content Expectation:

A2.1.3 Represent functions in symbols, graphs, tables, diagrams, or words and translate among representations

Lesson Outcome: Students will be able to make a story the correctly describes the graph.

  1. Warm-up: Graph the equation that describes this scenario

The county fair charges $5 admission and $1.00 per ride.

  1. Students need to get computers and go to
  2. Have students work through the Describing Graphs module.
  3. On the smart board project the Two Brothers applet
  4. Choose a graph. Students should work in pairs to make story describing what they brothers are doing. Stories can then be shared with the class.

Formative Assessment

Ticket to leave: Set graph in the Two Brothers applet to show one constant speed and one not – ask students to describe what is happening before they leave on their “ticket”

Summative Assessment: Question for Exam

Describe what is happening in this graph


Appendix A

Vocabulary card instructions:

  1. On the front of the card write the word large and in the center
  1. On the back side of the card write
  2. The word at the top in the center – small this time
  3. The definition
  4. Draw a picture or symbol representing the word
  5. Give an example
  6. Give non-examples