Measure Area by Counting Squares - 3rd Grade Math

1. Two figures are covered with unit squares. What is the combined area of the two figures, in square centimeters?
   Answer: 20 Figure A is 3 squares across and 4 squares up, or 12 square centimeters. Figure B splits into a 3-by-2 rectangle and a 1-by-2 rectangle, which is 6 + 2 = 8 square centimeters. Together they cover 12 + 8 = 20 square centimeters. Choice A is the difference between the two areas, and choices B and C each give only one figure.
2. A deck would be 8 squares long and 5 squares wide if it were a full rectangle, but a corner section 3 squares by 2 squares is missing. Each square covers 1 square foot. What is the area of the deck, in square feet?
   Answer: 34 A full rectangle would cover 8 times 5, or 40 square feet. The missing corner covers 3 times 2, or 6 square feet. The deck covers 40 - 6 = 34 square feet. Choice D forgets the missing corner, choice A is the missing corner alone, and choice B counts only part of the deck.
3. Refer to the figure. Two children measured the area of the same rectangle using DIFFERENT sized paper squares. Anna used small squares (1 sq in each) and got an area of 24 square inches. Ben used larger squares as his unit. If Ben's larger squares are each 4 times the area of Anna's squares, how many of Ben's larger squares fit in the same rectangle?
   Answer: 6 squares If each of Ben's squares covers 4 of Anna's squares, then $24 \div 4 = 6$ of Ben's larger squares fit in the rectangle. Choice A confuses the answer with the factor 4. Choice C incorrectly uses $24 \div 3$. Choice D divides by 2 instead of 4, treating Ben's squares as only twice as large.
4. A rug is 6 squares long and 4 squares wide. A mat is 5 squares long and 6 squares wide. Each square covers 1 square foot. How many more square feet does the mat cover than the rug?
   Answer: 6 The rug covers 6 times 4, or 24 square feet. The mat covers 5 times 6, or 30 square feet. The mat covers 30 - 24 = 6 more square feet. Choice B is the area of the rug, choice C is the area of the mat, and choice D adds the two areas.
5. Refer to the figure. Maya covered a rectangle with unit squares, but a few squares in the middle are missing from the drawing. She knows the rectangle is completely filled with the same-size squares. How many square units is the total area?
   Answer: 24 square units The rectangle is 4 rows by 6 columns, so the total area is $4 \times 6 = 24$ square units, even though some middle squares are not drawn. Choice A (18) counts only the visible squares. Choice B (20) miscounts one row. Choice D (15) counts only one section of the rectangle.
6. A quilt pattern is made of unit squares arranged in 5 rows and 6 columns. The pattern will be extended by adding one more complete row at the bottom and one more complete column on the right side. What will be the area of the extended pattern?
   Answer: 42 square units The original quilt is 5 rows by 6 columns, so adding a row and a column makes it 6 rows by 7 columns. That gives 6 times 7, which is 42 square units, so choice B is correct. Choice A only extends the columns and forgets the extra row. Choices C and D grow the pattern by more than one row or column, which is more than the problem describes.
7. A hallway floor is covered with squares that each measure 1 square foot. The long part has 9 rows of 2 squares. The short part has 3 rows of 4 squares. What is the total area of the hallway floor, in square feet?
   Answer: 30 The long part covers 9 times 2, or 18 square feet. The short part covers 3 times 4, or 12 square feet. The floor covers 18 + 12 = 30 square feet. Choice C gives only the long part, choice B only the short part, and choice A is the difference between them.
8. Refer to the figure. Lily used sticky notes to cover a poster board. Each sticky note is 1 square unit. She covered the entire poster board without overlapping. What is the area of the poster board?
   Answer: 20 square units The poster is covered by 4 rows of 5 sticky notes: $4 \times 5 = 20$ square units. Choice A counts only 4x4. Choice B is an off-by-one error on one dimension. Choice D uses 4x6 by miscounting.
9. Mrs. Okafor tiles a floor with squares that each cover 1 square meter. The floor is 8 squares long and 4 squares wide. She leaves a section 2 squares by 3 squares untiled for a planter. How many square meters does she tile?
   Answer: 26 The whole floor covers 8 times 4, or 32 square meters. The planter section covers 2 times 3, or 6 square meters. The tiled part is 32 - 6 = 26 square meters. Choice C forgets to remove the planter, choice A is the planter alone, and choice D adds the planter instead of removing it.
10. Refer to the figure. A garden is shaped like the figure shown, drawn on a grid where each square is 1 square meter. What is the area of the garden?
    Answer: 13 square meters Counting all the unit squares inside the L-shape gives 13 square meters. Choice A misses the corner section. Choice B is a common off-by-one miscount. Choice D counts the full bounding rectangle instead of only the shaded shape.
11. Two figures are covered with unit squares. How many more square units does Figure A cover than Figure B?
    Answer: 7 Figure A is a 5-by-4 rectangle with a 2-by-2 corner missing, so it covers 20 - 4 = 16 square units. Figure B is 3 squares across and 3 squares up, or 9 square units. The difference is 16 - 9 = 7 square units. Choice B is the area of Figure B, choice C is the area of Figure A, and choice D adds the two areas.
12. A floor plan is covered with unit squares. What is the total area of the floor, in square feet?
    Answer: 22 The taller section is 4 squares across and 3 squares up, or 12 square feet. The lower section is 5 squares across and 2 squares up, or 10 square feet. The total is 12 + 10 = 22 square feet. Choices A and B each give only one of the two sections, and choice D is the area of the whole 9-by-3 rectangle around the plan.
13. A rug is covered with unit squares. A square opening has been cut out of the middle. What is the area of the rug, in square feet?
    Answer: 20 If the opening were filled in, the rug would be 6 squares across and 4 squares up, or 24 square feet. The opening is 2 squares across and 2 squares up, or 4 square feet. So the rug covers 24 - 4 = 20 square feet. Choice C forgets to remove the opening, choice A is the area of the opening alone, and choice D adds the opening instead of removing it.
14. The rectangle below is covered with unit squares. Some squares are gray and the rest are white. How many more white square units are there than gray square units?
    Answer: 6 The whole rectangle covers 6 times 4, or 24 square units. The gray part is 3 squares across and 3 squares up, or 9 square units, so the white part covers 24 - 9 = 15 square units. The white area exceeds the gray area by 15 - 9 = 6 square units. Choice B is the gray area, choice C is the white area, and choice D is the area of the whole rectangle.
15. Two garden beds are covered with unit squares. What is the combined area of the two beds, in square meters?
    Answer: 21 Figure A is 4 squares across and 3 squares up, or 12 square meters. Figure B is 3 squares across and 3 squares up, or 9 square meters. Together the beds cover 12 + 9 = 21 square meters. Choice A is the difference between the two areas rather than the total, and choices B and C each give only one bed.
16. The figure below is covered with unit squares. What is the area of the figure, in square inches?
    Answer: 18 Split the figure into two rectangles. The top bar is 6 squares across and 2 squares up, or 12 squares. The stem below it is 2 squares across and 3 squares up, or 6 squares. The area is 12 + 6 = 18 square inches. Choice B counts only the top bar, choice A counts only the stem, and choice D is the area of the full 6-by-5 rectangle around the figure.
17. Jamal's garden is made of 1-meter squares arranged in 2 rows of 7 squares each. How many square meters is the garden?
    Answer: 14 square meters (sq m) The garden has 2 rows of 7 squares, and 2 times 7 equals 14 square meters. Choice B (10) does not match the number of rows and columns given. Choice C (12) is close but still miscounts the total squares. Choice D (15) overcounts by one square.
18. The rectangle below is covered with unit squares. Some of the squares are gray. How many square units are NOT gray?
    Answer: 13 The whole rectangle is 5 squares across and 4 squares up, so it covers 20 square units. Counting the gray squares gives 7. That leaves 20 - 7 = 13 square units that are not gray. Choice A is the number of gray squares, choice C is the area of the whole rectangle, and choice D adds the two counts instead of subtracting.
19. Use the grid to answer the question. Sara colored part of a grid to make a rectangle. Each unit square represents 1 square centimeter. If she now colors 5 more unit squares to make the rectangle larger, what is the new total area?
    Answer: 17 square centimeters The original rectangle has 12 unit squares ($3 \times 4$), and adding 5 more gives $12 + 5 = 17$ square centimeters. Choice A is the original area only. Choice B subtracts instead of adds. Choice D adds too many.
20. Kenji covers a rectangle with unit squares. He fits 6 squares in each row and makes 4 rows. He then shades 5 of the squares. How many of the squares are NOT shaded?
    Answer: 19 Six squares in each of 4 rows is 24 squares in all. Removing the 5 shaded squares leaves 24 - 5 = 19 squares that are not shaded. Choice A is the number shaded, choice C is the total number of squares, and choice D adds the 5 instead of subtracting.

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