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24.2: Rectangles with Fractional Side Lengths

  • Page ID
    40250
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    Lesson

    Let's explore rectangles that have fractional measurements.

    Exercise \(\PageIndex{1}\): Areas of Squares

    clipboard_e311d490e1a167c5edb04509eb186d285.png
    Figure \(\PageIndex{1}\)
    1. What do you notice about the areas of the squares?
    2. Kiran says “A square with side lengths of \(\frac{1}{3}\) inch has an area of \(\frac{1}{3}\) square inches.” Do you agree? Explain or show your reasoning.

    Exercise \(\PageIndex{2}\): Areas of Squares and Rectangles

    Your teacher will give you graph paper and a ruler.

    1. On the graph paper, draw a square with side lengths of 1 inch. Inside this square, draw another square with side lengths of \(\frac{1}{4}\) inch.
      Use your drawing to answer the questions.
      1. How many squares with side lengths of \(\frac{1}{4}\) inch can fit in a square with side lengths of 1 inch?
      2. What is the area of a square with side lengths of \(\frac{1}{4}\) inch? Explain or show your reasoning.
    2. On the graph paper, draw a rectangle that is \(3\frac{1}{2}\) inches by \(2\frac{1}{4}\) inches.
      For each question, write a division expression and then find the answer.
      1. How many \(\frac{1}{4}\)-inch segments are in a length of \(3\frac{1}{2}\) inches?
      2. How many \(\frac{1}{4}\)-inch segments are in a length of \(2\frac{1}{4}\) inches?
    3. Use your drawing to show that a rectangle that is \(3\frac{1}{2}\) inches by \(2\frac{1}{4}\) inches has an area of \(7\frac{7}{8}\) square inches.

    Exercise \(\PageIndex{3}\): Areas of Rectangles

    Each of these multiplication expressions represents the area of a rectangle.

    \(2\cdot 4\qquad 2\frac{1}{2}\cdot 4\qquad 2\cdot 4\frac{3}{4}\qquad 2\frac{1}{2}\cdot 4\frac{3}{4}\)

    1. All regions shaded in light blue have the same area. Match each diagram to the expression that you think represents its area. Be prepared to explain your reasoning.
    clipboard_ee62b70a3993b2e8f07ba77076d2fcccd.png
    Figure \(\PageIndex{2}\): Four vertically oriented rectangles labeled A, B, C, D. Rectangle A has a horizontal dotted line about three quarters of the way down its width. The top portion is shaded blue. Rectangle B has a horizontal dotted line about three quarters of the way down its width and a vertical dotted line about three quarters of the way to the right of its length. The top left portion of the rectangle is shaded blue. Rectangle C is shaded blue. Rectangle D has a vertical dotted line about three quarters of the way to the right of its length. The left portion of the rectangle is shaded blue.
    1. Use the diagram that matches \(2\frac{1}{2}\cdot 4\frac{3}{4}\) to show that the value of \(2\frac{1}{2}\cdot 4\frac{3}{4}\) is \(11\frac{7}{8}\).

    Are you ready for more?

    The following rectangles are composed of squares, and each rectangle is constructed using the previous rectangle. The side length of the first square is 1 unit.

    clipboard_eb90b2b15c763a700e47a3a89e94b0a15.png
    Figure \(\PageIndex{3}\): A sequence of rectangles composed of squares. The first is a unit square. The second is composed of two unit squares, stacked vertically. The third is composed of two unit squares, stacked vertically, and a 2 by 2 unit square on the right. The fourth is composed of two unit squares, stacked vertically, a 2 by 2 unit square on the right, and a 3 by 3 unit square at the bottom. The fifth is composed of two unit squares, stacked vertically, a 2 by 2 unit square on the right, a 3 by 3 unit square at the bottom, and a 4 by 4 unit square on the left.
    1. Draw the next four rectangles that are constructed in the same way. Then complete the table with the side lengths of the rectangle and the fraction of the longer side over the shorter side.
      short side long side \(\frac{\text{long side}}{\text{short side}}\)
      \(1\)
      \(1\)
      \(2\)
      \(3\)
      Table \(\PageIndex{1}\)
    2. Describe the values of the fraction of the longer side over the shorter side. What happens to the fraction as the pattern continues?

    Exercise \(\PageIndex{4}\): How Many Would it Take? (Part 2)

    Noah would like to cover a rectangular tray with rectangular tiles. The tray has a width of \(11\frac{1}{4}\) inches and an area of \(50\frac{5}{8}\) square inches.

    1. Find the length of the tray in inches.
    2. If the tiles are \(\frac{3}{4}\) inch by \(\frac{9}{16}\) inch, how many would Noah need to cover the tray completely, without gaps or overlaps? Explain or show your reasoning.
    3. Draw a diagram to show how Noah could lay the tiles. Your diagram should show how many tiles would be needed to cover the length and width of the tray, but does not need to show every tile.

    Summary

    If a rectangle has side lengths \(a\) units and \(b\) units, the area is \(a\cdot b\) square units. For example, if we have a rectangle with \(\frac{1}{2}\)-inch side lengths, its area is \(\frac{1}{2}\cdot\frac{1}{2}\) or \(\frac{1}{4}\) square inches.

    clipboard_ef131bb7b81c480aa9e8b71b31ad4bd02.png
    Figure \(\PageIndex{4}\): A large square evenly divided into 4 smaller squares. The large square has bottom horizontal side length of 1 inch. Of the four smaller squares, the top left square is shaded blue. It has side lengths labeled one half inch.

    This means that if we know the area and one side length of a rectangle, we can divide to find the other side length.

    clipboard_e0c229650bc2ccde698923d87b986f627.png
    Figure \(\PageIndex{5}\)

    If one side length of a rectangle is \(10\frac{1}{2}\) in and its area is \(89\frac{1}{4}\) in2, we can write this equation to show their relationship: \(10\frac{1}{2}\cdot ?=89\frac{1}{4}\)

    Then, we can find the other side length, in inches, using division: \(89\frac{1}{4}\div 10\frac{1}{2}=?\)

    Practice

    Exercise \(\PageIndex{5}\)

    1. Find the unknown side length of the rectangle if its area is 11 m2. Show your reasoning.
    clipboard_e784c05d2b238c257e3b9e949f072b0b9.png
    Figure \(\PageIndex{6}\): A rectangle that has area labeled 11 meters squared. The side length of one side of the rectangle is labeled three and two thirds meters and the side length of the other side is labeled with a question mark. Each angle has a right angle symbol indicated.
    1. Check your answer by multiplying it by the given side length (\(3\frac{2}{3}\)). Is the resulting product 11? If not, revise your previous work.

    Exercise \(\PageIndex{6}\)

    A worker is tiling the floor of a rectangular room that is 12 feet by 15 feet. The tiles are square with side lengths \(1\frac{1}{3}\) feet. How many tiles are needed to cover the entire floor? Show your reasoning.

    Exercise \(\PageIndex{7}\)

    A television screen has length \(16\frac{1}{2}\) inches, width \(w\) inches, and area \(462\) square inches. Select all the equations that represent the relationship of the side lengths and area of the television.

    1. \(w\cdot 462=16\frac{1}{2}\)
    2. \(16\frac{1}{2}\cdot w=462\)
    3. \(462\div 16\frac{1}{2}=w\)
    4. \(462\div w=16\frac{1}{2}\)
    5. \(16\frac{1}{2}\cdot 462=w\)

    Exercise \(\PageIndex{8}\)

    The area of a rectangle is \(17\frac{1}{2}\) in2 and its shorter side is \(3\frac{1}{2}\) in. Draw a diagram that shows this information. What is the length of the longer side?

    Exercise \(\PageIndex{9}\)

    A bookshelf is 42 inches long.

    1. How many books of length \(1\frac{1}{2}\) inches will fit on the bookshelf? Explain your reasoning.
    2. A bookcase has 5 of these bookshelves. How many feet of shelf space is there? Explain your reasoning.

    (From Unit 4.4.1)

    Exercise \(\PageIndex{10}\)

    Find the value of \(\frac{5}{32}\div\frac{25}{4}\). Show your reasoning.

    (From Unit 4.3.2)

    Exercise \(\PageIndex{11}\)

    How many groups of \(1\frac{2}{3}\) are in each of these quantities?

    1. \(1\frac{5}{6}\)
    2. \(4\frac{1}{3}\)
    3. \(\frac{5}{6}\)

    (From Unit 4.2.3)

    Exercise \(\PageIndex{12}\)

    It takes \(1\frac{1}{4}\) minutes to fill a 3-gallon bucket of water with a hose. At this rate, how long does it take to fill a 50-gallon tub? If you get stuck, consider using a table.

    (From Unit 2.4.4)


    24.2: Rectangles with Fractional Side Lengths is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by LibreTexts.

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