Math Lab #07

(grades 7-12)
Soft-bound, 56 page book, 20 reproducible task cards, full teaching notes.

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Great hands-on math: assemble sequences with pennies, create art based on the Fibonacci sequence, fold clever curves, travel a Mobius strip. Go on to crack bar codes, solve puzzles, experience a generalization that turns out to be false. From limits to logarithms, build both math curiosity and math confidence.

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Math Lab Starter Kit

book not included, please order separately

WE SUPPLY in quantities that serve one student or two students working in a lab pair: index card, clear tape, masking tape, hand lens, thread, straight pins, paper clips, styrofoam cups, adding machine tape, drawing compass, ruler, crayon


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  • get materials
  • teaching tips
  • objectives
  • standards

Table of Contents for #07 Math Lab:

Preparation and Support

A TOPS Model for Effective Science Teaching • Getting Ready • Gathering Materials • Sequencing Task Cards • Long Range Objectives • Review / Test Questions

Activities and Lesson Notes

  1. 1. Penny Play
  2. 2. General Term
  3. 3. The Fibonacci Sequence
  4. 4. Seashell Spiral
  5. 5. The Golden Mean
  6. 6. Circle Sequence
  7. 7. Pool Table Induction
  8. 8. To The Limit
  9. 9. Circle
  10. 10. Ellipse
  11. 11. Parabola
  12. 12. Cardioid
  13. 13. Interior Angles
  14. 14. Easy Addition
  15. 15. Prime Numbers

  17. 16. Logarithms
  18. 17. Slide and Jump
  19. 18. Mobius Twist
  20. 19. Postal Bar Codes
  21. 20. Jigsaw Puzzle

Supplementary Pages

golden rectangles • centimeter grid • number square • bar code • cardioid points • number sieve


Complete Master List for #07 Math Lab:

Key: (1st/2nd/3rd) denote needed quantities: (1st) enough for 1 student doing all activities; (2nd) enough for 30 students working in 10 lab groups all self-paced; (3rd) enough for 30 students working in 10 lab groups, all doing the same lesson. Starred* items may be purchased below.

  1. 1/10/10: rolls of pennies (or index-card paper punches)
  2. 6/180/180: sheets of lined notebook paper
  3. 1/30/30: sheets of copier paper
  4. 1/15/30: index cards
  5. 1/15/30: metric rulers
  6. 1/15/30: drawing compasses
  7. 1/20/30: scissors
  8. * 1/10/10: rolls clear tape
  9. 1/7/15: hand calculators
  10. 1/30/30: sheets dark construction paper

  11. 1/5/15: bottles white glue
  12. * 1/5/15: hand lenses (optional)
  13. 1/1/1: pencil sharpener
  14. 1/20/30: pieces 8 1/2 x 11 inch corrugated cardboard
  15. * 1/2/2: spools thread
  16. * 2/140/60: straight pins
  17. 1/20/30: dark crayons, wrappers removed
  18. 2/20/60: small styrofoam cups
  19. 1/5/15: short pieces pencil lead
  20. 4/120/120: sheets scratch paper

  21. * 1/1/1: roll masking tape
  22. * 1/2/2: boxes paper clips
  23. * 1/1/1: roll adding machine tape
  24. 3/15/15: used envelopes with bar codes.

Convenient Shopping:

Adding Machine Tape

2-1/4 inch x 100 feet

Adding machine tape.

Magnifier - hand lens

3X clear plastic hand lens

You'll find many uses for this basic tool of scientific inquiry. Very nice quality for the price. Supports #17 Light, #23 Rocks and Minerals, and #42 Focus Pocus. (One 3X hand lens is also included in each #100 Triple Magnifier Kit.)

Paper Clips

size #1, steel, box of 100

Paper clips have 1001 uses in TOPS experiments, and science in general. Feel free to use paper clips you already have, but be aware that different brands come in different sizes and weights. In experiments where uniformity is important, don't mix brands.

Straight Pins

steel, one and 1/16 inch long

Used in many TOPS experiments. Sometimes required for their magnetic properties. Don't purchase aluminum straight pins by mistake.

Tape - masking

3/4 inch x 55 yd roll

A handy science supply used in most TOPS modules.

Tape - clear

3/4 inch x 1000 inch roll

Your standard desk tape with matte write-on surface.


light duty, 25 yd spool

Just plain old thread. Used in many TOPS titles, especially in Pendulums #34.

Teaching Tips for #07 Math Lab:

We encourage improvisation - it's one of the main goals of our hands-on approach! You and your students might invent a simpler, sturdier or more accurate system; might ask a better question; might design a better extension. Hooray for ingenuity! When this occurs, we'd love to hear about it and share it with other educators. Please send ideas and photos to

Lesson by Lesson Objectives for #07 Math Lab:

  1. Lesson 1: To discover patterns in the arrangement of pennies that form number sequences.
  2. Lesson 2: To discover patterns in the arrangement of squares that form number sequences. To write a general term for each sequence.
  3. Lesson 3: To discover patterns in the Fibonacci sequence, and apply it to a study of population.
  4. Lesson 4: To draw Fibonacci squares and construct the spiral of a chambered nautilus.
  5. Lesson 5: To study the golden mean as a ratio of Fibonacci numbers. To create a work of art based on this proportion.
  6. Lesson 6: To generate a sequence of regions by joining points on the circumference of a circle. To recognize the tentative nature of predictions based on generalizations.
  7. Lesson 7: To trace the path of a ball as it is deflected across a hypothetical pool table. To recognize the tentative nature of predictions based on generalizations.
  8. Lesson 8: To approximate a curve with straight lines. To recognize that n-sided polygons approach the shape of a perfect circle as n increases.
  9. Lesson 9: To define a circle and draw it with a smooth curve. To approximate another circle with many straight fold lines.
  10. Lesson 10: To define and construct an ellipse. To form a curved figure from straight fold lines and evaluate its shape.
  11. Lesson 11: To construct a parabola using many straight fold lines. To develop a definition based on its properties.
  12. Lesson 12: To define a cardioid and draw it using circles of Styrofoam. To approximate another cardioid with straight lines.
  13. Lesson 13: To cut and tape the interior angles of polygons, and find their sums. To understand how this sum increases as the number of sides of the polygon increases.
  14. Lesson 14: To discover number patterns that simplify the addition of long sequences of numbers.
  15. Lesson 15: To list all prime integers under 100 and examine their properties.
  16. Lesson 16: To develop a table of logarithms that simplify mathematical computations. To practice four basic log operations.
  17. Lesson 17: To discover number patterns in a slide and jump game. To generalize an nth term for each pattern.
  18. Lesson 18: To be surprised by the unusual topological properties of a Mobius strip.
  19. Lesson 19: To interpret bar codes used by the U.S. postal service that automate mail sorting.
  20. Lesson 20: To cut out 12 unique 5-tile puzzle pieces and use them to construct rectangles of various dimensions.

National Science Education Standards (NRC 1996) for #07 Math Lab:

TEACHING Standards

These 20 task cards promote excellence in science teaching by these NSES criteria:
Teachers of science...
A: ...plan an inquiry-based science program. (p. 30)
B: and facilitate learning. (p. 32)
C: ...engage in ongoing assessment of their teaching and of student learning. (p. 37)
D: and manage learning environments that provide students with the time, space, and resources needed for learning science. (p. 43)

CONTENT Standards

These 20 task cards contain fundamental content as defined by these NSES guidelines (p. 109).
• Represent a central event or phenomenon in the natural world.
• Represent a central scientific idea and organizing principle.
• Have rich explanatory power.
• Guide fruitful investigations.
• Apply to situations and contexts common to everyday experiences.
• Can be linked to meaningful learning experiences.
• Are developmentally appropriate for students at the grade level specified.

Unifying Concepts and Processes

NSES Framework: Systems, order, and organization • Evidence, models and explanation • Constancy, change, and measurement • Evolution and equilibrium • Form and function
Core Concepts/Processes: Induction says that what is true for the particulars is true in general. Such claims, while reasonable, are never absolutely certain.

Science as Inquiry (content standard A)

NSES Framework: Identify questions that can be answered through scientific investigations. • Design and conduct a scientific investigation. • Use appropriate tools and techniques to gather, analyze, and interpret data. • Develop descriptions, explanations, predictions, and models using evidence. • Think critically and logically to connect evidence and explanations. • Recognize and analyze alternative explanations and predictions. • Communicate scientific procedures and explanations. • Use mathematics in all aspects of scientific inquiry.
Core Inquiries: Recognize patterns • Think inductively • Generate nth terms • Ellipses • Parabolas • Cardioids • Prime numbers • Logarithms • Solve puzzles

History and Nature of Science (content standard G)

NSES Framework: k458912 Science as a human endeavor • Nature of science • History of science
Core Content: Explore patterns and shapes of historical significance - the Fibonnacci series, Mobius strips, ellipses, parabolas, and the golden mean.