Triangles

Purpose

This unit is around discovering and applying the rule for area of a triangle (area equals half base times height).  Students will practice multiplication and division strategies as they calculate areas.

Specific Learning Outcomes
  • recognise that two identical right angled triangles can be joined to make a rectangle
  • recognise that a triangle has half the area of a rectangle with the same base and height lengths
  • apply the rule ‘area of triangle equals half base times height’
Description of Mathematics

Area is a two-dimensional concept related to the geometric concept of an enclosed region. It is defined in the maths curriculum as the size of a surface expressed as a number of square units. Investigations of the size of an area should begin with comparisons between different surfaces and progress to the use of non-standard, and then standard, units. The use of formulae to calculate the areas of common polygons is the final stage of the learning sequence.

When the students are able to measure efficiently and effectively using standard units, their leaning experiences can be directed to situations that encourage them to "discover" measurement formula. In area work, the students may realise as they count squares to find the area of a rectangle, that it would be quicker to find the number of squares in one row and multiply this by the number of rows. In the same way, in this unit the students find a formula for calculating the area of a triangle by seeing it as half of a rectangle.

This unit is also designed to allow students to practice their multiplicative strategies as they calculate the area of triangles.In particular it reinforces the fact that the order in which they multiply and divide is not important (commutative property of multiplication).

Required Resource Materials
Grid paper

Copymaster 1:Triangle Area Worksheet

Activity

Session 1

In this session students revise the rule for the area of a rectangle. This session may not be required if this unit is being taught following a unit on area of rectangles.

  1. Draw an unlabelled rectangle on the board.
  2. Ask students to tell you what its area is. Some discussion may be required to ensure that all students are clear about what ‘area’ means – it is likely that at least some will confuse it with perimeter.
  3. When they tell you they can’t work out the area, ask them what they would need to know to find its area.
  4. If they say “two sides” then label two opposite sides and see what they say. It is important that students realise that they need to know the base and the height, or two sides at right angles to each other, to be able to find the area of a rectangle.
  5. If students can correctly work out the area of the rectangle from its base and height measurements ask them to describe their method and explain why it works. If nobody can give an explanation then a brief discussion is in order. Drawing a grid to link the concept of area with the array model of multiplication may help to clarify students’ understanding.

5 by 7 grid

  1. Students at this level should have their tables to 10 as known facts and should be able to use a variety of methods to ‘find’ the answer to 2-digit multiplication problems. Finding areas of rectangles is a good opportunity to practice and reinforce this.
    How did you work out the area of the rectangle?
    How did you work out 5x7? [should be known fact, but this question is applicable to larger rectangles.]
  2. Ensure that students have a clear understanding that the area of a rectangle equals its base times its height. Ensure that correct units are used; if the lengths are labelled in centimetres then the area has to be given in square centimetres (cm2), if the lengths are not given units then the area should be given in square units (units2).

Session 2

In this session students divide rectangles diagonally to produce right angled triangles. They realise that the two triangles formed are equal in area.

  1. Ask students to cut a rectangle out of grid paper, ensuring that its base and height are whole square amounts. (You may want to put limits on the size of the rectangle depending on the ability of your students – eg. No more than 10 squares along each side.)
     Grid 7 by 10
  2. Ask students to work out the area of their rectangle (answers to be given in units2). Students should be encouraged to work out the area mentally and explain their method to a partner.
  3. Now ask students to rule a line from one corner of their rectangle to the diagonally opposite corner.  Ensure that they can see the two right angled triangles they have created. 
    Grid plus diagional line
  4. Ask them to find the area of each triangle. They may need to count all the squares within the triangles.
    What do you notice about the areas of the two triangles?
    Students should notice that the two triangles have the same areas and that therefore the area of each is half the area of the rectangle. If they had worked out the areas by counting, ask them to make a calculation to check their counting.
    What is the area of each triangle exactly?
    How could you work out the areas of the triangles? [Halve the area of the rectangle]
    What numbers would you need to multiply or divide? [Either base times height times half or base times height divided by two.]
    What would be the easiest strategy for you? [Initially students are likely to mistakenly think that they have to multiply the base and height before dividing - often a better strategy is to halve one of these factors before multiplying.]
  5. This can be reinforced by having students cut along the diagonal and rotate one triangle to sit on top of the other.  They will then see that not only do the triangles have the same area, but they are identical triangles.
     to half grids

    Will this work for any rectangle you can make?

  6. Students should be given the opportunity to experiment with a few different rectangles so that they can see that the rule holds true for any rectangle.

Session 3

In this session students draw right angled triangles, complete the rectangle and calculate the area of the original triangle.

  1. Draw a right angled triangle on the board; label its base 10cm and its height 5cm. Triangle 5 by 10
  2. Ask students whether they can tell you its area (insist on units – cm2).
  3. Discuss suggestions for how you could work out its area.
    Do we have enough information to work out the area?
    Could we work out the area of another shape that would help?
    Remember what we discovered about rectangles last maths lesson – could that help?
  4. Ask students to draw a triangle on their grid paper so that two of its sides are along lines of the grid paper. (You may want to put limits on the size of the triangle depending on the ability of your students – eg. No more than 10 squares along each side.)grid plus triangle
  5. Now get them to draw the matching triangle that makes a rectangle.  Draw the triangle to make a rectangle on your diagram on the board to illustrate.
  6. Now challenge students to work out the area of their completed rectangle and then that of the triangle.
  7. Get each student to complete several triangles to ensure they are doing it correctly.
  8. Return together as a class to discuss:
    Does this work for every right angled triangle?
    Can you describe a rule for the area of a right angled triangle?

    Are there any clever tricks to make the maths easier? [Divide one of the sides by two before multiplying]
  9. Get students to record a rule for the area of right angled triangles in their own words. Hopefully the students will see that the area of any right angled triangle is equal to the area of the rectangle with the same base and height divided by two. Get them to record the statement “For right angled triangles, area equals half base times height.” Ensure that they can see that this means the same thing.

Session 4

In this session students draw non right angled triangles, and experiment with finding their area.

  1. Draw an unlabelled non right angled triangle on the board. Draw one of its sides horizontal. 
    Triangle
  2. Ask students what information they will need to be able to work out its area.  It is likely that students will try to apply their learning from the previous session and tell you that they need to know the length of two sides.
  3. Ask students to draw a triangle of their own on grid paper, so that all three corners are on grid intersections, but only one of its sides is along a line of the grid.
  4. Now challenge them to find its area by measuring two sides.
  5. If they apply their rule “area equals half base times height”, ask them to draw the rectangle to illustrate. They will be unable to.
  6. Bring the class back together and discuss why it does not work. Hopefully at least one of your students will recognise that in a non-right angled triangle the height is not equal to either of the sides. Discuss this then send students to try to find a rectangle that will work for their triangle.
  7. When most students have identified that the height of the rectangle needs to be at right angles to the base since rectangles have all right angles bring the class back together.
  8. Draw the diagram below on the board and ask students whether the rectangle is twice the size of the triangle. 
    Triangle on grid with box
  9. If students cannot see that the two smaller triangles join to make the larger triangle, add an extra line as illustrated below.
    What is the area of the left hand rectangle?
    What is the area of the left hand part of the right angled triangle?
    What is the area of the right hand rectangle?
    What is the area of the right hand part of the right angled triangle?

    Triangle on grid with other lines
  10. Give students time to draw some of their own triangles with only one side along a grid line, and work out their area. Ensure that they are including units in their answers.
  11. Return together as a class to discuss:
    Does this work for every triangle?
    Can you describe a rule for the area of any triangle?
  12. Get students to record a rule for the area of right angled triangles in their own words. Hopefully the students will see that the area of any triangle is equal to the area of the rectangle with the same base and height divided by two. Get them to record the statement “For all triangles, area equals half base times height.” Ensure that they can see that this means the same thing.

Session 5

In this session students state a rule for the area of a triangle and use it to find the area of some triangles.

  1. Begin the lesson by asking students to tell you the rule for the area of a triangle. All going well they should be able to answer easily!
  2. Have students complete the triangle area worksheet individually (Copymaster 1). This should give you a good idea of any students who are still struggling with the concepts.
  3. Challenge students to find triangles in the classroom and make the measurements required to calculate their area. Provide measuring tapes and rulers as required. Insist on correct units.
  4. Ask students to challenge each other with triangles to calculate the areas of. Compare answers and strategies for measuring and calculating and discuss differences.

 

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