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Grade 10 · unit review

Review: Geometry: Regions, Capacity, and Probability Models

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.

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Bring the unit skills together

Use these 4 new cases to check your reasoning. Choices are temporary and are not saved to an account, sent to staff or recorded as school grades. Download your current review choices before leaving or reloading.

Unit review questions
1. A point is uniformly chosen along a segment of length 22. A target subsegment has length 7. What is the chance of landing in the target?
Need a hint?
  • A uniformly chosen position on a segment assigns probability proportional to subsegment length.
  • Uniform choice is essential; a biased selection method would not use the same ratio.
2. A 21-by-14 rectangular region has one 7-by-7 square removed. Find the remaining area.
Need a hint?
  • A composite region can be measured by taking a full enclosing rectangle and subtracting a removed square.
  • Subtract the removed area, not its perimeter.
3. A fictional uniform block has mass 189 grams and volume 21 cm³. Find its density.
Need a hint?
  • Density compares total mass with occupied volume.
  • Dividing volume by mass gives a reciprocal unit and a different quantity.
4. A rectangular floor is 14 m by 16 m. Identical 2 m by 2 m square mats fit in a grid without gaps. How many fit?
Need a hint?
  • When rectangular dimensions are exact multiples of square-mat side length, dividing each direction gives whole grid counts.
  • Area division alone can hide layout problems when dimensions do not fit.

The download contains your current choices and any checked feedback. Keep the separate transfer-task work in your own document; downloading its instructions does not mark it complete.

Use the skills in a new task

Create a four-part mathematical solution sheet for these supplied cases. 1. A point is uniformly chosen along a segment of length 28. A target subsegment has length 9. What is the chance of landing in the target? Candidate answers: (A) 19/28 (B) 28/9 (C) 9/19 (D) 9/28. 2. A 27-by-18 rectangular region has one 9-by-9 square removed. Find the remaining area. Candidate answers: (A) 404 square units (B) 405 square units (C) 406 square units (D) 810 square units. 3. A fictional uniform block has mass 297 grams and volume 27 cm³. Find its density. Candidate answers: (A) 1/11 g/cm³ (B) 11 g/cm³ (C) 297 g/cm³ (D) 8019 g/cm³. 4. 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.

Work on paper or in your own document. Explain the choices you made.

  • State each requested quantity or relationship and its input restrictions.
  • Show intermediate calculations or the applicable theorem with matching correspondences.
  • Give all four results, including exact values or requested rounding and units.
  • Check a result by substitution, reversal, dimensions, a representation or a counterexample as appropriate.
Compare with a model response

1. 9/28. Under uniform selection, probability is target length/total length=9/28=9/28. Continuous endpoints have zero probability. 2. 405 square units. The full rectangle has area 27×18=486. Subtract the nonoverlapping removed square’s area 81: remaining area=405. 3. 11 g/cm³. Density=mass/volume=297/27=11 g/cm³. The ratio names mass in one cubic centimeter, not the block’s total mass. 4. 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.

A different response can also meet the criteria. Compare the evidence and reasoning, rather than copying the wording.