Pi DayOne constant.
Endless discovery.

Gather: Paper, pencil, three safe round objects.

1. Find three objects with circular faces and trace them.
2. Mark the curved boundary of each tracing. Compare one with an oval or a square.
3. Describe what stays the same as a circle becomes larger.

Ask next: How could you explain a circle without saying ‘round’?

Gather: A large paper circle, two colors of yarn, tape.

1. Use one color to follow the edge of the circle.
2. Lay the other across its center from edge to edge.
3. Lift both pieces and compare their lengths without stretching them.

Ask next: Which path is longer, and how can you show someone?

Gather: Paper circles, crayons; adult-prepared cutouts.

1. Fold a circle so its edges match, then unfold it.
2. Make a second fold in a different direction, again matching the edges.
3. Mark the crossing point. Draw a line through it from one edge to the other.

Ask next: Why does a line across the circle sometimes miss the center?

Gather: Paper circles of several sizes, nonstretching string, tape.

1. Choose a circle and prepare one piece of string the length of its diameter.
2. Walk that length around the edge with an adult’s help, marking each step.
3. Notice that three diameter-length steps leave a little of the edge uncovered.

Ask next: Does a bigger circle also take a little more than three steps?

Gather: Crayons, paper, the digits 3.1415926535.

1. Choose a color for each digit from 0 to 9 and draw a key.
2. Ignore the decimal point and color one square for each displayed digit.
3. Trade the key and artwork with a partner to decode.

Ask next: Where does a digit appear more than once in this short sample?

Gather: Circle drawings, labels, craft materials.

1. Make a small display of circles found in a classroom or home.
2. Add a label for the center, edge, and one line across the center.
3. Give a one-minute tour explaining one surprising observation.

Ask next: How can a visitor touch, hear, or see your explanation?

Gather: An adult-prepared paper circle with its center marked, yarn, crayons.

1. Draw paths from the marked center to different points on the edge.
2. Compare the paths with one piece of yarn, keeping it straight and unstretched.
3. Name these center-to-edge paths radii and explain what you notice about their lengths.

Ask next: Would a path starting away from the center give the same result?

Gather: Large adult-prepared cardboard disks and squares, a flat tabletop, a marker.

1. Predict how a disk and a square will move when rolled slowly on an edge.
2. With adult help, roll each shape along the table and watch a marked center point.
3. Describe when the center stays at one height and when it rises and falls.

Ask next: How does the distance from the center to the edge help explain the movement?

Gather: Two matching paper circles, one with a bright dot near its edge.

1. Place one circle on the other and turn the top one while keeping their centers together.
2. Notice that the circular outlines still match after a small turn or a big turn.
3. Add a dot to the top circle and explain how the dot lets you see the turn.

Ask next: Can the outline stay the same while a decoration changes position?

Gather: A large chalk circle in a clear space, a marked center, two colored markers.

1. Place two markers on the edge and walk the curved path between them.
2. Call that part of the edge an arc, then walk the rest of the way around.
3. Move one marker and compare the two arcs again without needing exact measurements.

Ask next: What happens to the other arc when one arc becomes longer?

Gather: Large smooth cardboard circles, ovals and squares, a cloth to cover the shapes.

1. Explore a covered shape's boundary with your fingers; keep every piece on the table.
2. Describe curves, straight parts and corners before looking at the shape.
3. Uncover it and add a drawing or spoken description of what helped you identify it.

Ask next: Is 'no corners' enough to tell a circle from an oval?

Gather: A paper circle, a paper arrow, four labels, crayons.

1. Mark the top, right, bottom and left positions on the circle.
2. Keep the arrow's tail at the center and turn it from the top to each position.
3. Describe a quarter turn, a half turn and one complete turn using the arrow.

Ask next: How many quarter turns bring the arrow back to its start?

Gather: Matching paper circles, adult-prepared halves and quarters, a circle outline.

1. Cover the outline with two matching halves and then four matching quarters.
2. Rearrange the same pieces outside the outline to make a new picture without overlaps.
3. Explain why the pieces cover the same total area even though the picture looks different.

Ask next: What changes if the pieces overlap or leave gaps?

Gather: Three circular objects, string, ruler, calculator.

1. Measure each object around its edge and across its center in the same units.
2. Divide each circumference by its diameter. Record the original measurements too.
3. Compare the three ratios and repeat the least convincing measurement.

Ask next: What might explain a ratio far from 3.14?

Gather: A safe toy wheel or cardboard disk, tape, ruler.

1. Measure the diameter and estimate one revolution’s travel with 3.14 × diameter.
2. Mark a starting point and roll the wheel exactly one turn on a flat surface.
3. Measure its travel and compare prediction with observation.

Ask next: What changes if the wheel slips?

Gather: Paper, ruler, colored pencils.

1. Draw a circle and label its radius, diameter, and circumference.
2. Write a sentence describing how radius and diameter are related.
3. Express the relationship with symbols: d = 2r.

Ask next: Which representation helps a new learner most?

Gather: Paper circles with diameters 10 cm and 20 cm, grid paper.

1. Predict whether the larger paper pizza has twice as much area.
2. Trace both onto a grid and estimate covered squares, combining partial squares.
3. Compare the approximate areas and explain why the larger one is near four times the smaller.

Ask next: Why is ‘twice as wide’ different from ‘twice as much’?

Gather: Nonstretching string, meter stick, chalk or paper.

1. Mark a string approximately 3.14 meters long.
2. Arrange it into a circle without stretching it and predict the diameter.
3. Measure across the center in several directions, then improve the shape.

Ask next: Why might the diameters differ in a handmade circle?

Gather: Paper, pencil, calculator.

1. Read this claim: a circle with radius 5 cm has circumference 3.14 × 5 = 15.7 cm.
2. Draw and label the circle to identify the missing factor.
3. Write a corrected explanation and estimate the circumference.

Ask next: How can the drawing help someone fix the error?

Gather: Three circular cutouts, circumference clues, ruler, calculator.

1. Give a partner a circumference clue of approximately 62.8 cm without showing the matching circle.
2. Estimate the diameter by dividing the clue by 3.14 and draw a proposed circle.
3. Measure candidate cutouts and explain which one best fits the clue, allowing for measurement error.

Ask next: How could you work backward from a circumference for a different lid?

Gather: A large circular cardboard disk, string, ruler, calculator.

1. Measure the disk's diameter and estimate the string needed to go around its edge five times.
2. Wrap string carefully around the edge, counting turns and avoiding stretching it.
3. Compare your prediction with the unwrapped length; discuss overlaps, thickness and an extra tying allowance.

Ask next: Why might a real ribbon order need more than five times the ideal circumference?

Gather: Two 40 cm lengths of nonstretching string, grid paper, tape.

1. Arrange one string as a square and the other as a circle, keeping both boundaries 40 cm long.
2. Trace each shape and estimate its area by counting full grid squares and combining partial squares.
3. Compare the covered areas and explain why equal boundary lengths do not force equal areas.

Ask next: What makes the area estimate uncertain near a curved boundary?

Gather: A 12 cm by 16 cm paper rectangle, paper circles of diameter 4 cm, ruler.

1. Arrange the circles in straight rows and columns inside the rectangle without overlaps.
2. Use their diameters to predict the number of rows, columns and circles before counting them.
3. Draw the spaces left between circles and explain why multiplying the rectangle's dimensions alone does not count circles.

Ask next: What changes in this row-and-column layout if the diameter is 2 cm?

Gather: A paper circle, ruler, string, colored pencils, calculator.

1. Fold the circle into four equal parts and mark a quarter of its boundary.
2. Estimate the full circumference from its diameter, then calculate a quarter and a half of that distance.
3. Check the curved distances with string and distinguish the arc from the straight cut edges.

Ask next: Is a quarter-circle's entire boundary just one quarter of the original circumference?

Gather: Paper, ruler, the approximation 3.14159, pencil.

1. Draw a number line from 3 to 4 and mark where π belongs approximately.
2. Zoom in with a second line from 3.1 to 3.2, then a third from 3.14 to 3.15.
3. Explain why the successive marks describe the same number at different scales.

Ask next: Why should you say 'approximately here' when you use a finite decimal?

Gather: A paper circle, a paper square, crayons.

1. Fold each shape along a line that makes its two halves match exactly.
2. Try several directions through the circle's center and compare them with folds through the square.
3. Draw successful fold lines and explain why trying a few examples suggests a pattern but does not by itself prove every case.

Ask next: Why does a circle have more lines of reflection symmetry than a square?

Gather: Five circular objects, string, ruler, graph paper.

1. Measure circumference and diameter for each object using the same units.
2. Plot diameter horizontally and circumference vertically.
3. Draw a sensible line near the data and estimate its rise divided by run.

Ask next: Why should an ideal model pass through the origin?

Gather: Paper, calculator, a design brief.

1. Plan a circular bed with radius 2 m. Calculate its edging length and planting area.
2. Price edging at 8 currency units per meter and soil cover at 5 per square meter.
3. Show a budget with units and state whether waste or overlap is included.

Ask next: Which costs change if the radius doubles?

Gather: Calculator and a table of diameters: 1, 10, 100 units.

1. For each diameter calculate circumference using 3, 3.14, and 22/7.
2. Compare each with the calculator’s π result and record absolute differences.
3. Decide which approximation suits an allowed error of 0.1 unit.

Ask next: Does one approximation work equally well at every scale?

Gather: The polygon lab, paper, pencil.

1. Record the displayed lower and upper values for 6 sides.
2. Repeat for 12, 24, 48, and 96 sides.
3. Compare interval widths and describe what each polygon contributes.

Ask next: Why does a lower estimate alone not establish the full interval?

Gather: The sampling lab and a recording sheet.

1. Add 100 points and record the estimate. Add 1,000 more and record again.
2. Reset and repeat the experiment three times.
3. Compare runs and mark any step that moved away from π.

Ask next: Why does more data not guarantee a better estimate at every step?

Gather: Calculator, paper circles or the circle lab.

1. Choose a radius and calculate C = 2πr and A = πr².
2. Double the radius and calculate both again.
3. Use a drawing and ratios to explain the changes.

Ask next: What happens for a scale factor of three instead of two?

Gather: Paper, ruler or compass, calculator.

1. Model two circular running paths with radii 20 m and 21 m.
2. Calculate each circumference and find the extra distance in one lap on the outer path.
3. Express the difference as 2π times the radius difference and explain a fair starting adjustment for this simplified model.

Ask next: Would the extra distance change for radii 40 m and 41 m?

Gather: Paper, protractor, ruler or compass, calculator.

1. Draw a 90-degree sector with radius 12 cm and calculate its arc length and area.
2. Include both straight radii when finding the total boundary length for a trim allowance.
3. Repeat the plan for a 120-degree sector and compare materials needed.

Ask next: Which quantities scale with the angle, and which straight edges stay the same length?

Gather: Graph paper, ruler or compass, calculator.

1. Draw a ring with outer radius 6 cm and inner radius 4 cm.
2. Calculate the outer disk's area, subtract the inner disk's area and keep an exact answer in terms of π.
3. Check the scale of your answer with a grid estimate and explain why π times the ring's width squared would be incorrect.

Ask next: What happens to the ring's area if both radii double?

Gather: Two matching paper circles, ruler, adult-supervised scissors, colored pencils.

1. Divide one circle into eight equal sectors and the other into sixteen; color alternating sectors.
2. Arrange each set in alternating up-and-down order to make a shape that resembles a parallelogram.
3. Relate its approximate height to r and its long edge to half the circumference, then explain why more sectors improve the model.

Ask next: Why does this suggest πr² while a finite paper model still has curved edges?

Gather: A large circular object, two strings, ruler, a results table.

1. Measure the same diameter and circumference three times, recording each result before calculating ratios.
2. Compare common problems: a string pulled too tightly, a ruler missing the center, or measurements in mixed units.
3. Choose one improvement and repeat the measurements, reporting the spread rather than selecting only your favorite ratio.

Ask next: Can repeated measurements agree with each other and still be inaccurate?

Gather: Grid paper, ruler or compass, calculator, two colors.

1. Draw a circle of radius 5 cm inside a snug 10 cm square.
2. Calculate the circle's area as a fraction of the square's area and estimate the fraction by counting grid squares.
3. Connect that fraction to the sampling lab's multiplication by four, keeping area evidence separate from random-sample results.

Ask next: Why is the ideal fraction the same for a larger circle in a snug square?

Gather: Paper, trigonometry tools, the polygon lab.

1. Draw a unit circle and split an inscribed regular n-gon into triangles.
2. Use the half central angle to derive an inner half-perimeter n sin(π/n).
3. Derive the outer expression n tan(π/n) and compare values as n increases.

Ask next: Why is our browser illustration not an independent computation of π?

Gather: A programming environment with a random-number function.

1. Sample x and y uniformly between −1 and 1.
2. Count points with x² + y² ≤ 1, then calculate 4 × inside ÷ total.
3. Run several sample sizes and compare independent trials at each size.

Ask next: How would biased point generation distort the result?

Gather: Calculator or spreadsheet.

1. Set a circular track’s diameter to exactly 100 m for this model.
2. Compare circumference using π rounded to 2, 4, and 6 decimal places.
3. Choose the fewest of these decimal places that keeps rounding error below 1 mm.

Ask next: What if the measured diameter itself is uncertain by 1 cm?

Gather: A circle, string, protractor.

1. Cut or mark a string one radius long.
2. Lay it along the circle’s arc and connect the arc’s endpoints to the center.
3. Estimate that angle in degrees and relate a full turn to 2π radians.

Ask next: Why does the same radius-to-arc relationship work for larger circles?

Gather: Spreadsheet or calculator, graph paper.

1. Model a closed cylinder with volume 1,000 cm³. Choose several positive radii.
2. For each, calculate h = 1,000/(πr²) and surface area 2πr² + 2πrh.
3. Compare material area and discuss why a real design might favor another shape.

Ask next: Which manufacturing constraints are absent from the ideal model?

Gather: Poster supplies, source access, measurements or code results.

1. Choose one claim about π and support it with a derivation, experiment, or reliable historical source.
2. Clearly distinguish proof, measurement, simulation, and historical evidence.
3. Invite a peer to challenge your explanation and revise it for a public audience.

Ask next: What would make your claim convincing to someone who has not taken your class?

Gather: A spreadsheet or coding environment with a calculator reference for π.

1. Calculate 4 × (1 − 1/3 + 1/5 − 1/7 + …) using 10, 100 and 1,000 terms.
2. Compare each partial sum with the reference value and graph the absolute error against the number of terms.
3. Use the alternating-series remainder bound 4/(2N + 1) after N terms to discuss the cost of a requested error tolerance.

Ask next: How can a correct formula still be an inefficient way to obtain many decimal places?

Gather: Graph paper, a spreadsheet or coding environment, calculator.

1. Model the upper unit semicircle with y = √(1 − x²) on the interval from −1 to 1.
2. Use midpoint rectangles with N = 10, 100 and 1,000 equal subintervals to approximate its area.
3. Double each area estimate, compare it with π and explain the role of rectangle width in the sum.

Ask next: How would you justify the limiting value independently of the numerical experiment?

Gather: Unit-circle diagram, graph paper or a spreadsheet, trigonometric calculator.

1. Choose angles from 0 to 2π radians and record the vertical coordinate sin(θ) of a point on the unit circle.
2. Plot angle horizontally and vertical position vertically, then connect the points smoothly.
3. Label one full period and explain what one full rotation means in both pictures.

Ask next: How would doubling angular speed change a plot against time instead of angle?

Gather: Calculator or spreadsheet and a measurement brief.

1. Treat a measured radius of 10.0 cm as lying anywhere from 9.9 to 10.1 cm.
2. Calculate the smallest and largest possible areas using the interval endpoints and retain sensible units.
3. Compare the resulting uncertainty with the change caused by replacing π with 3.14 and explain which source matters more here.

Ask next: Why can displaying more calculator digits give a misleading impression of accuracy?

Gather: Calculator, a sphere diagram, a table of radii.

1. For sphere radii 2, 4 and 6 units, calculate surface area 4πr² and volume (4/3)πr³.
2. Compare the ratios when the radius doubles and triples, then calculate surface area divided by volume.
3. Explain why a scaled model's surface-to-volume ratio changes and identify assumptions before using the model for a real object.

Ask next: Which ratio contains π in both numerator and denominator so that it cancels?

Gather: A coding environment, graphing tools, the sampling-lab method.

1. Estimate π using points with x and y independently uniform from −1 to 1, recording multiple trials of the same size.
2. Create a deliberately biased comparison by multiplying each coordinate by 0.5 before testing x² + y² ≤ 1.
3. Plot both point clouds and explain why the biased version gives 4 even with a very large sample.

Ask next: Why can more data make a biased method look stable without making it correct?