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9.1: Introducing Ratios and Ratio Language

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    39920

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    Lesson

    Let's describe two quantities at the same time.

    Exercise \(\PageIndex{1}\): What Kind and How Many?

    clipboard_e1f6c028cd7f41cb035bf42b75b1a32ca.png
    Figure \(\PageIndex{1}\)

    Think of different ways you could sort these figures. What categories could you use? How many groups would you have?

    Exercise \(\PageIndex{2}\): The Teacher's Collection

    1. Think of a way to sort your teacher’s collection into two or three categories. Count the items in each category, and record the information in the table.
    category name
    category amount
    Table \(\PageIndex{1}\)

    Pause here so your teacher can review your work.

    1. Write at least two sentences that describe ratios in the collection. Remember, there are many ways to write a ratio:
      • The ratio of one category to another category is ________ to ________.
      • The ratio of one category to another category is ________ : ________.
      • There are _______ of one category for every _______ of another category.

    Exercise \(\PageIndex{3}\): The Student's Collection

    1. Sort your collection into three categories. You can experiment with different ways of arranging these categories. Then, count the items in each category, and record the information in the table.
    category name
    category amount
    Table \(\PageIndex{2}\)
    1. Write at least two sentences that describe ratios in the collection. Remember, there are many ways to write a ratio:
      • The ratio of one category to another category is ________ to ________.
      • The ratio of one category to another category is ________ : ________.
      • There are _______ of one category for every _______ of another category.
        Pause here so your teacher can review your sentences.
    2. Make a visual display of your items that clearly shows one of your statements. Be prepared to share your display with the class.

    Are you ready for more?

    1. Use two colors to shade the rectangle so there are 2 square units of one color for every 1 square unit of the other color.
    2. The rectangle you just colored has an area of 24 square units. Draw a different shape that does not have an area of 24 square units, but that can also be shaded with two colors in a \(2:1\) ratio. Shade your new shape using two colors.
    clipboard_e5d31bd9b2286b9ffd85d7acf38286a64.png
    Figure \(\PageIndex{2}\)

    Summary

    A ratio is an association between two or more quantities. There are many ways to describe a situation in terms of ratios. For example, look at this collection:

    clipboard_eceb067c595560a5a43983032350e82b8.png
    Figure \(\PageIndex{3}\)

    Here are some correct ways to describe the collection:

    • The ratio of squares to circles is \(6:3\).
    • The ratio of circles to squares is 3 to 6.

    Notice that the shapes can be arranged in equal groups, which allow us to describe the shapes using other numbers.

    clipboard_e4939c16153e7757e1dcc9ce3defcdf7e.png
    Figure \(\PageIndex{4}\)
    • There are 2 squares for every 1 circle.
    • There is 1 circle for every 2 squares.

    Glossary Entries

    Definition: Ratio

    A ratio is an association between two or more quantities.

    For example, the ratio \(3:2\) could describe a recipe that uses 3 cups of flour for every 2 eggs, or a boat that moves 3 meters every 2 seconds. One way to represent the ratio \(3:2\) is with a diagram that has 3 blue squares for every 2 green squares.

    clipboard_ea9b143595346d0fb6df303cb0916cf87.png
    Figure \(\PageIndex{5}\)

    Practice

    Exercise \(\PageIndex{4}\)

    In a fruit basket there are 9 bananas, 4 apples, and 3 plums.

    1. The ratio of bananas to apples is ________ : ________.
    2. The ratio of plums to apples is ________ to ________.
    3. For every ________ apples, there are ________ plums.
    4. For every 3 bananas there is one ________.

    Exercise \(\PageIndex{5}\)

    Complete the sentences to describe this picture.

    clipboard_ec4dece36cf919203bb70288c73318ee4.png
    Figure \(\PageIndex{6}\)
    1. The ratio of dogs to cats is ________.
    2. For every ________ dogs, there are ________ cats.

    Exercise \(\PageIndex{6}\)

    Write two different sentences that use ratios to describe the number of eyes and legs in this picture.

    clipboard_ef6d468647fac64ea5ed92620345dadda.png
    Figure \(\PageIndex{7}\)

    Exercise \(\PageIndex{7}\)

    Choose an appropriate unit of measurement for each quantity.

    1. area of a rectangle
    2. volume of a prism
    3. side of a square
    4. area of a square
    5. volume of a cube
    • cm
    • cm3
    • cm2

    (From Unit 1.6.1)

    Exercise \(\PageIndex{8}\)

    Find the volume and surface area of each prism.

    1. Prism A: 3 cm by 3 cm by 3 cm
    clipboard_e1b924edd0c492c66326b7b9995b1e6fb.png
    Figure \(\PageIndex{8}\)
    1. Prism B: 5 cm by 5 cm by 1 cm
    clipboard_ecb876dbdd5e400ff6f11e61bd69a8ebc.png
    Figure \(\PageIndex{9}\)
    1. Compare the volumes of the prisms and then their surface areas. Does the prism with the greater volume also have the greater surface area?

    (From Unit 1.5.5)

    Exercise \(\PageIndex{9}\)

    Which figure is a triangular prism? Select all that apply.

    clipboard_ebbba13f20ed4b1e3a38b0ff5a423d77a.png
    Figure \(\PageIndex{10}\): Five shapes. Shape A has a square flat bottom and 2 triangular sides connected by rectangular sides. Shape B has a hexagon flat bottom, a hexagon flat top, and five flat rectangular sides. Shape C has a triangle flat bottom, a triangle flat top, and three rectangular sides. Shape D has a rectangle flat bottom and two sides shaped like a triangle connected by rectangular sides. Shape E has a rectangle flat bottom and four triangle sides that connect at a top point.
    1. A
    2. B
    3. C
    4. D
    5. E

    (From Unit 1.5.2)

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