Areas of Rectangles

Purpose

In this unit students learn to use the multiplication formula to find the area of a rectangle. Using proportional reasoning students explore what happens to the area when the length and/or height of a rectangle is doubled.

Achievement Objectives
GM3-2: Find areas of rectangles and volumes of cuboids by applying multiplication.
Specific Learning Outcomes
  • Use multiplication to calculate the area of a rectangle.
  • Measure the length of a side using a ruler.
  • Use proportional reasoning to find the area of a rectangle.
Description of Mathematics

Area is the amount of flat surface enclosed within a shape. Commonly used standard units for area are cm2 (square centimetres), m2 (square metres), and km2 (square kilometres). Squares are used because they iterate, that is they fit together in two dimensions with no gaps or overlaps.

Rectangles are the easiest shapes to find the area of because the array structure of iterating units is most obvious. Consider this rectangle filled with square units of area:

The units are arranged in three rows of five squares. The total number of units can be found by multiplication, 3 x 5 = 15. Similarly, the rectangle contains five columns of three squares, so 5 x 3 = 15 also gives the total area.

    Opportunities for Adaptation and Differentiation

    The learning opportunities in this unit can be differentiated by providing or removing support to students, or by varying the task requirements. Use these strategies to support students:

    • Manipulate the side lengths of the rectangles you use. Rectangles with smaller side lengths make drawing and counting solutions more accessible. In general, use rectangles with smaller side lengths when introducing the concept of arrays and how the arrangement of rows and columns connects to multiplication equations. However, increasing side lengths promotes the need for more efficient ways to find area. In that way, students see the efficiency of multiplicative methods.
    • Use diagrams and physical models to support student recognise arrays within the boundaries of rectangle. Gradually withdraw the diagrammatic and physical support to encourage imaging and thinking with established results. Refer to pages 11-13 of Teaching number through measurement, geometry, algebra and statistics for further ideas.
    • Allow access to calculators where calculation is not the primary purpose of the lesson. For example, finding all rectangles with areas of 72cm2 offers opportunities to apply multiplicative thinking and systematic reasoning. Those opportunities may be lost if students are preoccupied with finding products mentally.

    The context for this unit can be adapted to suit the interests and experiences of your students. Students could be challenged to find the area of a room in their own home, or the area of a space of interest in your local community, such as a sports ground, skate park, marae or similar. A diagram with measurements could be provided if the area is not readily accessible during school time.

    Required Resource Materials
    • 10cm by 10cm squares of paper or card, e.g. memo pads
    • Large pieces of paper, e.g. butchers’ paper
    • Tape measures, trundle wheels, metre rulers (whatever is available)
    • Newspaper, recycled cardboard, scissors and tape
    • 1cm square grid paper
    • Copymasters One and Two
    Activity

    Session 1

    In this session students are introduced to the idea of using multiplication to find the area of a rectangle.

    1. Show the students a large rectangular piece of paper measuring 30cm by 60cm and a pile of smaller squares each measuring 10cm by 10cm (like memo squares). Tell the students you want to know how many of these small squares are needed to cover the large paper rectangle. You can set a context such as this is the back garden and these are the concrete tiles we will be using to cover it. 
      How many square tiles will cover this area?
    2. Let students briefly discuss how they might estimate an answer, then share the ideas. Look for students to explain two main processes:
      • Iteration – repeated copying of the unit of measurement (memo square) along a side, with no gaps or overlaps.
      • Equi-partitioning – equally splitting a side until the divisions are about the same length as the sides of the memo square.
    3. Ask about how the units will be arranged. Introduce the terms, rows (across) and columns (down) if students are not familiar with those words. 
      Do students recognise the array (rows and columns) structure in the arrangement of units?
    4. Ask a volunteer to place the squares side by side on the rectangle. Blutac can help to secure the units in place.
    5. Ask the students for ways to work out the total number of units. One by one counting, or skip counting/repeated addition (6, 12, 18 or 6 + 6 + 6 = 18) are legitimate strategies given the small number of units. Explain that the area of the rectangle is 18 squares. (Actually, 10cm x 10cm squares are one dm2, one square decimetre)
      Can we count the squares even more efficiently?
    6. Record 3 x 6 = 18 and ask student where the six and three reside in the model (rows and columns). Ask where the 18 is found (total number of square units).
    7. Model the same process with different sized rectangles, e.g. 20cm x 80cm, 50cm x 40cm, 100cm x 100 cm (A square is a special rectangle). The rectangles might be cut out of paper, drawn on the whiteboard, or drawn on the carpet/concrete with chalk.
      Look for students to:
      • Recognise the array structure.
      • Use multiplication as an efficient method to calculate the area.
    8. Provide the students with copies of Copymaster One. Tell them to work with a partner to find out the area of each rectangle in small squares. As students work look for their calculation strategies. Are they using additive or multiplicative methods?
      Recognise that much will depend on their knowledge of multiplication facts and strategies.
    9. Gather the class and share solutions. It is interesting that Rectangle E, a square has the greatest area, though other rectangles may look larger.
      Answers: A (3 x 7 = 21), B (7 x 5 = 35), C (4 x 11 = 44), D (12 x 3 = 36), E (7 x 7 = 49), F (8 x 6 = 48), G (11 x 2 = 22).
      What do the answers tell us about these rectangles?
      How big are the little squares?
      Students might measure with a ruler to check that the units are square centimetres.
      Ask students to include the unit in their answers, e.g. 21cm2. Recording the notation for each rectangle is good practice.

    Sessions 2 and 3

    1. Discuss the idea of a formula. You might find a funny video online about someone using a formula to make something. A recipe is a type of formula.
      What do we mean by a formula?
      Do students explain that a formula is like an algorithm for getting the same result each time? 
      Record W x L = A. This is a mathematical formula written as an equation. 
      I wonder what the letters W, L and A might represent?
    2. Apply the formula to the examples students worked on in the previous lesson (Copymaster One). 
      For example, Rectangle B had seven rows of five squares.
      The row gives the length of the rectangle. In the case of B length equals 5. (rub off L in the formula and write 5 in it’s place)
      The number of rows gives the width of the rectangle. In the case of B width equals 7. (rub off W in the formula and write 7 in it’s place).
      The formula now reads 7 x 5 = A. I wonder what A equals. What value for area makes the equation true and matches the formula?
    3. Ask students to use the examples from Copymaster One and rehearse starting with the formula, and substituting the values of length, width, and area for each rectangle.
    4. Provide students with a group worthy task to work on collaboratively (see Copymaster Two). Students might be given 1cm grid paper or work in their exercise books.
    5. Look for students to apply the W x L = A formula to construct appropriate rectangles. For example, if they choose an area of 72cm2 they will need to consider all the factors of 72. Encourage students to find those factors systematically. Some students may need to support of a multiplication basic facts poster.
      A systematic approach involves starting with 1 as a factor then increasing the smallest factor by one and testing 72 for divisibility.
      1 x 72, 2 x 36 (72 ÷ 2 = 36), 3 x 24 (72 ÷ 3 = 24), 4 x 18 (72 ÷ 4 = 18), 5 x (72 is not divisible by 5), 6 x 12 (72 ÷ 6 = 12), 7 x (72 is not divisible by 7), 8 x 9 (72 ÷ 8 = 9).
      If the process continues the factors will appear in reverse order, e.g. 9 x 8 = 72. 8 x 9 and 9 x 8 are essentially the same rectangle though they may appear differently if the direction of the label is considered.
    6. Gather the class to discuss solutions and look at real sized diagrams of the possible labels. Some options are mathematically correct but unworkable as a label option.
      Discuss criteria for eliminating labels. For example, a label with width of less than 5cm might be considered too ‘skinny.’
      Discuss the best options, cut them out at real size, then use a real jam jar to consider how well each label will work.
    7. Students might write a letter to Karly outlining how they investigated her problem and giving their recommendations.
    8. Another good investigation is to tile a large rectangular area with 1m2 carpet tiles. A hall or gymnasium is an ideal area though a classroom is also viable. Tiles of that size are commonly found at hardware stores. You will find an advertisement easily online. 
    9. Get students to construct a unit square using newspaper or recycled boxes. They can use the unit to get a sense of the scale of 1m2 and make estimates of area of the space before they calculate.
    10. Ask students to work in small teams to calculate the number of tiles that will be needed for the rectangular space. Look for them to measure the side lengths of the rectangular area using tape measures, trundle wheels, or metre rulers.
      Do they apply the W x L = A formula?
    11. Students can find the area of composite shapes by finding the area of the rectangles. For example: 
      Composite shape
    12. This shape can be seen to be comprised of two 2cm by 4 cm rectangles, or a 2cm by 6cm rectangle and a 2 cm by 2 cm rectangle, or 4 cm by 6 cm rectangle with a 4 by 2 rectangle missing. There are different ways to solve composite shapes.

    Session 4

    In this session students explore using proportional reasoning to find areas of rectangles.

    1. Pose the problem: Sam’s family was shopping for a ground sheet to take camping. The first one they looked at measured 2 by 3m. Sam said if they wanted one with an area twice as big they should get the 4 by 6m size. Is Sam right?
    2. Ask the students to draw pictures of the ground sheets and to help them decide if Sam is correct.
    3. Work with students to establish that doubling the area only involves doubling one side of the rectangle. Doubling both sides of the rectangle increases the area by four times.
    4. Using this proportional reasoning students will be able to solve problems without recalculating from side lengths. Here are some example problems:
      • The recipe made enough icing to cover the top of a 20cm by 20cm cake, what size cake can you ice if you double the amount of icing?
      • The birthday card had a front cover measuring 15cm by 10cm, what is area of the piece of cardboard used to make it?
      • The marae had two areas that needed paving. Each area measured 5m by 8m. What is the total area to be paved?
      • The gardener charged his customers by the area of their lawn. If the bill was $20 to mow a lawn that was 6m by 20m, what should the bill be for a 20m by 12m lawn?

    Session 5

    In the session students demonstrate their ability to apply measurement of area independently.

    The following links provide pages from Figure It Out books that are suitable:

    Students might also create a mat design and provided the dimensions and areas of the rectangular pieces that compose it. An example is given below:

    Colourful  rectange

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    Level Three