
Exploring Functions through the Use of Manipulatives
 © '94, '95, '97, '98, '00, '04, '12 Agnes Azzolino (asquared@mathnstuff.com)

 1) PRETEST Answer these now,
 mentally if you wish.
 1. Sketch the graphs of y = 0 and x = 0.
 2. Write the equation for this function.
 3. What effect will the k have in the function y = f(x) + k.
 4. For what numbers is the square of the number greater than on more than the number?
 5. Sketch y = x · sin(x)
 HOMEWORK
 6. Before graphing y = x·e^{x}, predict what the
graph should look like. Mentally rehearse an explanation, addressed to
another student, which explains or justifies the graph of the function.

 2) The Pretest focuses on the content of this session.
 It's what
the session is about. Please:
 1. Complete the pretest, mentally or in some other manner.
 2. Introduce yourselves to each other and find a partner
 3. Discuss the pretest with your partner.
 [Discuss this with your partner (DTWYP).]

 3) The materials you will be using today must be returned.
 As you leave, please return your materials. If you leave
early, please return your materials.
 At the end of the session, you will receive a copy of the
text of most of the transparencies and a large coordinate plane.

 4) Please:
 a) Leave the printed coordinate plane in the pocket.
 b) Examine and handle the other things in the plastic pocket.
 c) Consider how they might be used in mathematics classes.
 d) Discuss this with your partner (DTWYP).

 5) Consider each question and DTWYP.
 What functions have been included?
 identity, reciprocal, opposite, absolute value, square,
 dilations of the absolute value
 What things have not been included but might be?
 square root, exponentiallog, sinecosine, halfplane, reciprocal,
 halfcircles, tangent cotangent, secantcosecant
 What kinds of things might one do with these manipulatives?

 Identify (through pointplotting),
 Graph,
 Sketch,
 Find the inverse function,
 Estimate the slope,
 Describe,
 Discuss,
 Write an expression,
 Solve,
 Predict the graph of,
 Explain behavior,
 Show functions with your hands and body.

 6) Graph by placement on the coordinate plane. DTWYP.
 a. the identity function
 b. the opposite function
 c. the reciprocal function
 d.
 e.
 f.
 g.
 h.
 i. y = (x  4)²
 j. y =  (x  2)² + 3
 k. x = y²
 m.
 n. the distance between a number and 3

 7) Sketch each of the above on paper to debrief and DTWYP.
 Debriefing can not be emphasized enough.
 Do not leave students
"thinking in the concrete."
(www.mathnstuff.com/papers/langu/page9.htm)
 Force a summary in the more abstract
written
activity of sketching and writing notes, and discussing.
(www.mathnstuff.com/papers/langu/page6.htm)

 Use of the Slopemeter
 Be crude with your slopegenerated curve.
 [Repeat the slope: Don't use a ruler. Think m=3/4 is also 3/(4), 6/8.]

 8) Solve the statements by graphing 2 functions on the same plane:
 [a.] x  3 = 4
 [b.] x  3 = 0
 [c.] x  3 = 4
 [d.] x  3 = x + 2
 [e.] x  3 = ( x + 2 )² + 1

 9) Use the identity function to compare, describe, then discuss.
 A) Graph the identify function and a specific function
 on the same coordinate plane,
 B) Compare these two functions.
 C) Use these thoughts to describe the function. DTWYP.

 10) We are not through for the day but we have finished with the manipulatives.
 Please place your materials back in the pockets.
 Defrief by listing some ideas we considered.
 DTWYP as the manipulatives are collected.

 11) Before we continue, I'd like to answer any quick Questions.

 12)
Math "Exercises"(www.mathnstuff.com/papers/langu/exerc.htm)
 a) Use your body, arms, and hands to represent the graph of:
 1. y = 0
 2. x = 0
 3. y = x
 4. y = x
 5. y = x²
 6. x = y²
 7. y = 2x²
 8. y =  x
 9. y = 2x
 10. y = x  2
 11. y = x + 2
 12. y = 1/x
 b) Please be seated.


 13) Dilation by a nonconstant function is wonderful.
 Use the composition of functions to sketch an unknown function.
 a) What does a cubic look like and why? DTWYP.
 y= x^{3} = x(x²)
 b) What does a rational function look like?
 y = (x +3)(x4)/(x2)
 c) Why does the tangent look that way?
 Tangent to the unit circle,
 y = tan x = (sin x)/(cos x)
 d) What are envelope functions?
 How do they look?
 y = e^{x}(sin x)
 Consider the graphs of
 sin(x), 4sin(x), x · sin(x),
 x² · sin(x), ln(x) · sin(x),
 e^{x} · sin(x)
 Other toys:

Sine Law Ambiguous Case & Sketchpad trig stuff,

Polynomial &
Rational Function through Dilation
 Spreadsheets for Dialations Creating
Polynomials &
Rationals


 14) Words to the Wise and Wrap Up.
Words to the Wise
 Manipulatives must be available in order to be used.
 Expect breakage and loss of material.
 Plan free play before work.
 Start with short activities and build on these.
 Always close with an abstraction activity.
 Selfevaluate after each lesson.


 15) We must teach our students to:
 Play
 Experiment
 Suppose
 Create
 Sketch
 Draw
 Write Notes
 Write Sentences
 Debrief
 Dream
 Think and
 Visualize

 You've sampled the power of the manipulative and
seen it used as it should be in conjunction with introspection, communication,
and technology. I hope you learned or relearned a little math, might consider
using a strategy demonstrated here today, and had a little fun. Thank you for coming.
Math Class Languages (www.mathnstuff.com/papers/langu/page4.htm)
Mother Tongue Other Tongue(s)

VERBAL
 formal spoken mathematics
 informal spoken mathematics
 spoken symbol
 symbol speak
 calculatoreze/computereze
 web speak
 
WRITTEN
 written word
 written symbol
 semisymbolic
 calculator symbol


PICTORIAL
 picture
 numeral
 graph
 nonverbal body language


CONCRETE
 object
 model
 manipulative/token

The above graphic may be found at www.mathnstuff.com/gif/mciplane.gif
