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Grade 8 · Roots and approximation

Number-line puzzle: Bracket a principal square root

Number-line puzzle investigation: bracket a principal square root. Change the quantities, follow the motion, and explain the result using the displayed mathematical relationship.

Predict what will happen, press Play, then pause and explain what changed. Every control also works without playback.

Watch the relationship

Paused
Number-line puzzle: Bracket a principal square root. Bisections completed: 0. Lower bound (rounded down): 0. Upper bound (rounded up): 47. Unrounded interval width: 47Keep the root between two bounds011.7523.535.25470² ≤ 47 ≤ 47²Bisections: 0 · bounds rounded outward
The radicand is positive. This method approximates the principal nonnegative root only; solving x²=n would also require its negative counterpart. The number-line window adapts to n and stays fixed during playback.

Starts paused. Play once, pause anywhere, or use Step to inspect the mathematics. Playback stops when this panel leaves the screen.

Make it your experiment

Change one value. Notice what follows.

The controls adjust the model. Numbers below describe the current frame. Decimals are rounded.

Bisections completed
0
Lower bound (rounded down)
0
Upper bound (rounded up)
47
Unrounded interval width
47

HD animation studio

From experiment to screen.

Present a crisp Canvas scene, save a full-HD image, or capture your model as a silent video.

Understand what you are seeing

The idea behind the motion.

A positive square root can be trapped between numbers whose squares lie below and above the target. Bisection halves this interval repeatedly. A short decimal or narrow interval is an approximation; it does not prove that a non-square target has a rational square root. This investigation begins with radicand n = 47; number of bisections = 6. Treat the named setting as a constructed classroom model, then explain which relationships would still hold if its numbers changed.

A relationship to keep

L² ≤ n ≤ U²; midpoint M=(L+U)/2

Read the symbols alongside the explanation. A diagram shows the relationship; the assumptions tell you when it applies.

  1. STEP 1

    Represent the given quantities

    Begin with the safe bounds zero and max(1,n), then mark their midpoint.

  2. STEP 2

    Follow the changing model

    At each whole step compare the midpoint's square with n and retain the half containing the nonnegative root.

  3. STEP 3

    Check and explain the relationship

    Read the final interval width as an error bound. Both endpoint squares must still bracket n.

Your turn to explain

Make a prediction. Test your reasoning.

Number-line puzzle: After 6 bisections starting with [0,47], which bounds enclose √47?

Use the values specified in the question. Reset restores the initial values for this investigation.

Compare your explanation

[6.609, 7.344], with unrounded interval width 0.734. Displayed endpoints are rounded outward to preserve the bound.

Work through a full lesson

Connect the animation to a worked example and practice questions.

Grade 8 · Algebra

Locate an irrational square root between decimals

Comparing squares gives reliable bounds on a positive square root without pretending a decimal approximation is exact.