Bracket a Square Root between Integer Squares

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: Between which consecutive whole numbers is √5? Candidate answers: (A) 1 and 2 (B) 2 and 3 (C) 3 and 4 (D) 4 and 5.
• Independent check inputs: Between which consecutive whole numbers is √10? Candidate answers: (A) 2 and 3 (B) 3 and 4 (C) 4 and 5 (D) 9 and 10.

Learn the idea

Roots reverse powers, but their exact values and decimal approximations must be distinguished. Integer squares provide reliable bounds, and odd roots retain negative signs while principal square roots are nonnegative. To locate a positive square root, compare its radicand with nearby perfect squares. Squaring is increasing on nonnegative numbers, so those square comparisons determine root bounds.

Here is the method on the supplied teaching case. 2²=4<5, while 3²=9>5. The positive square root therefore lies strictly between 2 and 3.

Compare squares of candidate endpoints, not endpoints directly with the radicand. Strictly between consecutive squares gives strict root bounds; an exact perfect square would sit on an endpoint instead.

1. Write or sketch the supplied teaching case, labeling its quantities and restrictions.
2. Apply the method shown in the worked solution, keeping each intermediate equality or justified inference.
3. Solve the independent check with its changed inputs before revealing feedback.
4. Complete the three practice cases; explain which mathematical rule rules out one distractor in each case.

Check your reasoning

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Take it further

Changed-input task: Between which consecutive whole numbers is √82? Candidate answers: (A) 10 and 11 (B) 8 and 9 (C) 81 and 82 (D) 9 and 10. Show the mathematical reason, not only the selected answer. Expected result: 9 and 10. 9²=81<82, while 10²=100>82. The positive square root therefore lies strictly between 9 and 10.

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