Count an Exact Grid Layout

Supplied materials

• Paper or a browser scratch area; a calculator is optional unless rounding is requested.
• Supplied teaching case: A rectangular floor is 4 m by 6 m. Identical 2 m by 2 m square mats fit in a grid without gaps. How many fit? Candidate answers: (A) 12 mats (B) 5 mats (C) 6 mats (D) 7 mats.
• Independent check inputs: A rectangular floor is 6 m by 8 m. Identical 2 m by 2 m square mats fit in a grid without gaps. How many fit? Candidate answers: (A) 11 mats (B) 12 mats (C) 13 mats (D) 24 mats.

Learn the idea

Geometry can support probability, material, density and layout models when the assumptions are explicit. Ratios and decompositions must use nonoverlapping regions, consistent dimensions and the correct uniformity or exact-fit conditions. When rectangular dimensions are exact multiples of square-mat side length, dividing each direction gives whole grid counts. Multiplying the two counts gives the complete nonoverlapping arrangement.

Here is the method on the supplied teaching case. 4/2=2 mats fit along one direction and 6/2=3 along the other. Their product is 6. The stated exact grid fit avoids layout and leftover-space uncertainty.

Area division alone can hide layout problems when dimensions do not fit. This supplied model states an exact grid fit; a real room with obstacles or leftover strips would need more constraints.

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: A rectangular floor is 18 m by 20 m. Identical 2 m by 2 m square mats fit in a grid without gaps. How many fit? Candidate answers: (A) 180 mats (B) 89 mats (C) 90 mats (D) 91 mats. Show the mathematical reason, not only the selected answer. Expected result: 90 mats. 18/2=9 mats fit along one direction and 20/2=10 along the other. Their product is 90. The stated exact grid fit avoids layout and leftover-space uncertainty.

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