Function Junction, What’s your function? • Function Junction, what's your function? Hooking up a numbers and making 'em run right. Algebraic equations, verbal descriptions, tables, graphs, and concrete, active representations. • Milk and honey, bread and butter, peas and rice. • y = 4x, y = 2x + 1, y = x2 Hey that's nice! Function Junction: Beginning Ideas of Function Lynn Stallings, Ph.D. email@example.com Kennesaw State University, Atlanta, Georgia, USA National Council of Teachers of Mathematics 2004 Conference Building with Toothpicks • Look for patterns in the number of toothpicks in the perimeter of each shape. • Shape 1, Shape 2, Shape 3, Shape n? • Can you predict how many toothpicks would be in the perimeter of shape 7? Shape 10? Shape 22? • How did you do it? Function Junction, What’s your Function? • Functions may be ―the most single important concept from kindergarten to graduate school‖ (Harel & Dubinsky, 1992, The Concept of Function, MAA) • ―Systematic experience with patterns can build up to an idea of function.‖ (PSSM, 2000) • In the middle school, students ―can begin to use variables and algebraic expressions as they describe and extend patterns.‖ (PSSM, 2000) • As students ―work with multiple representations of functions—including numeric, graphic, and symbolic—they will develop a more comprehensive understanding of functions.‖ (PSSM, 2000) Starring . . . PSSM Process Standard Representation Also featuring Communications, Connections, Reasoning, and Problem Solving algebraic equation table graph concrete or active verbal context description (real to students) Where do we typically start? What’s a good mnemonic for these? Number Patterns from Cutting String (or tearing paper) Fold a strip of paper in half. Cut once. How many pieces will you have? Cut a second time. How many pieces? Cut a third time. How many pieces? Number of cuts 1 2 3 Etc. Number of pieces How’s that Function in Middle School? Middle school students need developmentally appropriate instruction which includes • Connections between function representations and what is real to them. • Concrete and active experiences with patterns and actions that they can represent visually, orally, in diagrams, in tables, in graphs, and in symbols. • Visual and active experiences building up to the abstraction of symbols. • Experience interpreting various representations and the connections between them. Number Patterns from Triangles Triangle 1 Triangle 2 Triangle 3 • Form these triangles. • What relationship do you see between the triangle number and the area of the triangle formed? from PSSM, 2000 ChitChat vs. Keep-in-Touch • Two cellular phone plans: • Keep-in-Touch - a basic fee of $20.00 a month, plus $0.10 per minute • ChitChat - no monthly basic fee but charges $0.45 a minute • Both companies use technology that allows them to charge for the exact amount of time used; they do not round up the time to the nearest minute as many competitors do. • Compare their charges. ChitChat vs. Keep-in-Touch L1 = number of minutes, L2 = Keep-in-Touch cellular phone rates, L3 = ChitChat rates ChitChat vs. Keep-in-Touch L1 = number of minutes, L2 = Keep-in-Touch cellular phone rates, L3 = ChitChat rates How to graph this data. Press 2nd Y= (which is STAT PLOT). Press ZOOM and select 9 ZoomStat. Press Graph. ChitChat vs. Keep-in-Touch (cont.) What can we tell from a graph of the values in our table? Would it make sense to connect these points? Keep-in-Touch ChitChat ChitChat vs. Keep-in-Touch (cont.) What equations would connect these dots? Keep-in- Touch (57.14, 25.71) ChitChat What does the point of intersection mean? What do the slopes and y-intercepts mean? Guess the relationship . . . If 8 is the input, what will the output be? What’s the relationship? 7 was the input more than once. What was the output? Is that important? From Navigations Pledge Plans Pledge Plans for a 10K Walk-a-Thon • Jeff: $1.50 per kilometer • Rachel: $2.50 per kilometer • Annie: $4.00 donation plus $0.75 per km Km 0 1 2 3 4 5 6 7 8 9 10 Jeff's plan $0 $1.50 $3 $4.50 $6 $7.50 $9 $10.50 $12 $13.50 $15 Rachel's plan $0 $2.50 $5 $7.50 $10 $12.50 $15 $17.50 $20 $22.50 $25 Annie's plan $4 $4.75 $5.50 $6.25 $7 $7.75 $8.50 $9.25 $10 $10.75 $11.50 From Navigations Pledge Plans What equation should Jeff: y = 1.5x we use for each plan? Rachel: y = 2.5x • Jeff: $1.50 per kilometer Annie: y = 4 + .75x • Rachel: $2.50 per kilometer • Annie: $4.00 donation plus Press Y= and input the $0.75 per kilometer three equations. What should those equations represent? • The relationship between the number of kms (let’s call Press WINDOW. it x) walked and the amount of money pledged (let’s call it y). From Navigations Pledge Plans Let’s use verbal Jeff: y = 1.5x description to link the Rachel: y = 2.5x graph, the algebraic equations, and the Annie: y = 4 + .75x table Rachel Jeff Annie What do the slopes mean? (5 1/3, 8) (2.3, 5.7) What about y- intercepts? What do the points of intersection mean? From Navigations Walking Strides • We need a designated walker and a timer. • Our walker is going to walk 20 feet three different ways: with baby steps, regular steps, and exaggerated steps while our timer times it. • We’ll record the data for 20 feet and then use it to figure out the rest. Actions graphs From Navigations Actions graphs Walking Strides • How would we fill in the rest of the chart? Baby Regular BIG • Let’s let x = time and Steps Steps Steps y = distance walked. 0 ft. • Can we graph these? 20 ft. What equations could we use? 40 ft. 60 ft. • What do the slopes mean? 80 ft. • What do y-intercepts 100 ft. mean? What would you do to change the y-intercept? Actions graphs Match the Graph Has anyone used the CBR before? How does it work? Would you like to try some sample data first? Given a graph, can we interpret it by moving in a way that will replicate it. How would we replicate this one? Actions graphs Match the Graph • Press PRGM on your calculator. If you don’t see RANGER, follow the directions on the next two pages labeled “Getting Started with CBR” to load the program. • The next page is “Notes for Teachers.” • For right now, you can go on to the page titled “Activity 1: Match the Graph.” later you’ll want to go back to “Notes for Teachers.” Actions graphs Match the Graph • What happened? • What did slope mean here? • What about y-intercept? • What happened when – you were on a section of the graph with a positive slope? – you were on a section of the graph with a negative slope? – you were on a horizontal section of the graph? – you were on a vertical section of the graph? Symbols graphs Transformation Creations • ―Rain‖ is a way to practice your connections between the algebraic equations and graphic representations. • What does –4 < x < 4 and –3 < y < 3 tell you about the graph? • How do we input that into the calculator? Symbols graphs Transformation Creations • How can you make ―Rain‖ on your calculators? •How did you know? From Navigations Words graphs From Graphs to Stories Let’s explain these graphs. 1. What happened when John and his father raced? 2. Describe the lemonade stand’s profit graph. 3. Describe the raising of the flag graph. 4. Describe the race between the two swimmers. From Navigations Words graphs From Stories to Graphs Sketch a graph for each of these three stories. Note that number two will vary the most. Josephine’s walk Mowing the lawn 40 100 Uncut Grass (%) Distance (feet) 20 50 10 20 30 40 50 60 Time Time (seconds) Words graphs More Graphs to Stories The next pages include graphs to match to stories for more practice. You can also make up your own. For example, • the speed of students in the halls during class changes, • the amount of studying done versus the number of days before the test, • hyperactivity versus the number of days until school is out, • What else? PSSM standard on Algebra says . . . In grades 6–8 all students should Understand patterns, relations, and functions • represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules; • relate and compare different forms of representation for a relationship; • identify functions as linear or nonlinear and contrast their properties from tables, graphs, or equations. PSSM standard on Algebra says . . . In grades 6–8 all students should Represent and analyze mathematical situations and structures using algebraic symbols • explore relationships between symbolic expressions and graphs of lines, paying particular attention to the meaning of intercept and slope; • use symbolic algebra to represent situations and to solve problems, especially those that involve linear relationships; PSSM standard on Algebra says . . . In grades 6–8 all students should Use mathematical models to represent and understand quantitative relationships • model and solve contextualized problems using various representations, such as graphs, tables, and equations. Analyze change in various contexts • use graphs to analyze the nature of changes in quantities in linear relationships. Function Junction • Function Junction, what's your function? Hooking up numbers and expressing patterns. • Function Junction, how's that function? I got five great representations, • That get most of my job done. • Function Junction, what's your representation? I got graphs, symbols, tables, contexts, and verbal descriptions! • They'll get you pretty far. • Graphs: That's • Symbols: That's where you say y=3x + 2 • Don’t forget to make a table, talk about it, and act it out! • These representations will Get you pretty far into function understanding.
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