New 7 –10 syllabus support
A unit of work on area 2004
Curriculum K –12 Directorate Mathematics © State of New South Wales through the NSW Department of Education and Training, 2007 0

New 7 –10 syllabus support
Workshop 3 overview
The focus of this workshop is to outline a teaching sequence designed to cater for student learning needs in mixed-ability classrooms. The curriculum focus is developing formulae for the area of squares, rectangles, triangles and parallelograms (MS4.1).The implementation of the new syllabus, has brought to the fore the range of student learning in any classroom. Even streamed classes and selective schools have a range of student ability. Students in NSW classrooms are quite diverse. They come from differing cultures and have different learning needs. Acknowledging that students learn at different rates is the first step in catering for their different needs. The key features of this unit are assessment for learning, a unit overview, flexible teaching approaches focusing on concept development and flexible student grouping. This workshop provides opportunities for Mathematics head teachers and classroom teachers to begin to investigate the place of learning design in catering for the needs of all students.
Workshop 3 Outcome
Participants will:  consider a range of appropriate teaching strategies and learning sequences for the new Stage 4 mathematics syllabus to better cater for students in a mixed ability class Curriculum K –12 Directorate Mathematics © State of New South Wales through the NSW Department of Education and Training, 2007 1

New 7 –10 syllabus support Area unit overview
Lesson 1. Are we ready for area?
1 lesson
2. Developing understanding of area and area of a rectangle
1 lesson
3. Developing area formula for right-angled triangles 1 lesson 4. Developing area formula for any triangle 1 lesson Teaching strategies and learning sequence  Unit overview (whole class)  Student self-assessment, page 5. (individual)   Assessment for learning (AFL) activity Choose one or more from page 6 Why does this learning of area matter? Class discussion. Write one paragraph answering this question for homework.  Unit overview (whole class)  Homework sharing (whole class)  Small group area concept development based on readiness assessed from AFL Task 1: Rectangle area card match, page 10 and 12, and writing task, page 11 and 13. (pairs) Optional: Area of a rectangle, Learning Federation object (on CD) Task 2: Rectangle mania, an open-ended task, page 15 (small groups) Task 3: Block out, area puzzle, page 14 (individual, pairs or homework)  Return to unit overview  Unit overview – review of previous learning (whole class)  Teacher guided practical measuring activity on developing area of a triangle as half the area of the bounding rectangle, page 17 (whole class)  Task 4: Triangles 1 writing task, right- angled triangles, page 19 (pairs)  Optional: Area of triangle level 1 Learning Federation object (on CD)  Return to unit overview (whole class)  Unit overview – review of previous learning (whole class)  Teacher guided practical - area of any triangle (whole class) –developed as half the area of the bounding rectangle, page 18.  Task 5:Triangles 2 writing task moving from the explanation to developing the formula, page 20 (pairs)  Optional: Area of triangle level 2 Learning Federation object (on CD)   Optional Task 6: What’s your angle on area of triangles? see page 22 (individual, pairs or assignment) Return to unit overview  Homework – write a paragraph on how the area of a triangle and a rectangle relate Curriculum K –12 Directorate Mathematics © State of New South Wales through the NSW Department of Education and Training, 2007 2

5. Developing area formula for parallelograms 1 lesson
New 7 –10 syllabus support
   Unit overview connections (whole class) Share homework paragraphs (small group then whole class) – review previous learning and make Teacher guided practical measuring activity - area of a parallelogram - developed as half the area of the bounding rectangle. (whole class)  Task 7: Area of parallelogram writing task, page 21 (pairs)  Return to unit overview 6. Converting between metric units of area 1 lesson 7. Consolidating knowledge and understanding of area of squares, rectangles, triangles, parallelograms and combinations of these shapes 1 lesson  Unit overview (whole class)  Make a square metre from newspaper (pairs)   Task 8: Area conversion workcards, page 24. Students work in random small groups making a poster about conversions and displaying them around the room Discussion on how to convert between metric units of area (whole class)  Students answer assorted textbook questions (individual or pairs)  Return to unit overview (whole class)  Unit overview (whole class)  Students answer assorted textbook questions (individual or pairs)  Students, in pairs, compose an area question, answer it and give it to another pair to solve (pairs)  Re-do student self assessment (individual)  Assessment of learning activity 8. Do we understand how to find area of squares, rectangles triangles and parallelograms? Curriculum K –12 Directorate Mathematics © State of New South Wales through the NSW Department of Education and Training, 2007 3

New 7 –10 syllabus support
Curriculum K –12 Directorate Mathematics © State of New South Wales through the NSW Department of Education and Training, 2007 4

New 7 –10 syllabus support Name: ___________________________ What do I already know about area of squares, rectangles, triangles and parallelograms and what do I need to learn?
Place a tick (√) in the box against the statement that describes what you CAN do.
Knowledge I can always do this I can do this sometimes I don’t know how to do this
1. I know what squares and rectangles look like. 2. I can draw squares and rectangles. 3. I can draw a triangle. 4. I can label the perpendicular height on a triangle 5. I know what a parallelogram looks like. 6. I can label the perpendicular height on a parallelogram. 7. I can explain what the area of a shape is. 8. I can find the length and breadth of a rectangle. 9. I can find the area of squares and rectangles by counting square units. 10. I can calculate the area of squares and rectangles by using the area formula. 11. I can find the area of triangles by counting square units. 12. I can calculate the area of any triangle by using the area formula. 13. I can calculate the area of any parallelogram by using the formula. 14. I can find the area of shapes that can be cut into rectangles and triangles. 15. I can convert square millimetres to square centimetres. Curriculum K –12 Directorate Mathematics © State of New South Wales through the NSW Department of Education and Training, 2007 5

New 7 –10 syllabus support
Assessment for learning
Activities include:  Using the grid worksheet, page 7, and half a coloured sheet of A4 paper, students construct a rectangle (using whole centimetre dimensions). Cut the rectangle out and paste it onto the grid. Alternatively, cut a post-it note to 5cm x 3cm, or similar size, and each student places it on the grid
Explain how you found the area of your shape and the shape printed on the grid.
 Small group brainstorm or mindmap (each student in group uses a different coloured pencil or crayon).
Tell me all you know about area. You may include diagrams and examples of types of questions to show what you know.
 Write down the hardest question you can answer on area and write the solution to it.  Using the Diagnosing student readiness, interest and learning profile for area, page 8-9. Curriculum K –12 Directorate Mathematics © State of New South Wales through the NSW Department of Education and Training, 2007 6

New 7 –10 syllabus support
Name: ___________________________
Curriculum K –12 Directorate Mathematics © State of New South Wales through the NSW Department of Education and Training, 2007 7

New 7 –10 syllabus support Diagnosing student readiness, interest and learning profile
Interest questionnaire Name:________________
What do you want to learn about area?
These are some of the key ideas we will be learning about in this unit on area. We want to know what you are interested to learn about. Number your choices from 1 to 6. Make sure 1 is your favourite and 6 is your least favourite. ___ What is area? ___ ___ ___ Area of squares and rectangles Area of triangles Area of parallelograms ___ ___ Area of composite shapes Conversion of area units
Readiness questionnaire
What can you tell us about area?
1. What is area? 2. Why do we need to learn about area? 3. Give some examples of different area questions you can do. 4. What is the formula for the area of a triangle and how do you use it? Give an example. Curriculum K –12 Directorate Mathematics © State of New South Wales through the NSW Department of Education and Training, 2007 8

New 7 –10 syllabus support
Learning profile questionnaire
How do you like to learn?
1. I learn best when it is quiet.
2. I am able to ignore the noise of other people talking while I am working. 3. I like to work at a table or desk. 4. I like to work on the floor. 5. I work hard for myself. 6. I work hard for my parents or teacher. 7. I will work hard on an assignment until I finish it. 8. Sometimes I get frustrated with my work and do not finish it. 9. When my teacher gives an assignment, I like to have exact instructions on how to complete it. 10. When my teacher gives an assignment, I like to create my own steps on how to complete it. 11. 12. I like to work by myself. I like to work in pairs or in groups. 13. I like to have an unlimited amount of time to work on an assignment 14. I like to have a certain amount of time to work on an assignment. 15. I like to learn by moving and doing. 16. I like to learn while sitting at my desk. Adapted from Carol Ann Tomlinson, How to differentiate instruction in mixed- ability classrooms, Association for Supervision and Curriculum Development (ASCD), USA, 2001. Curriculum K –12 Directorate Mathematics © State of New South Wales through the NSW Department of Education and Training, 2007 9

1. 2. 3. 4.
New 7 –10 syllabus support
Rectangle diagram Explanation of how to find the area of the rectangle Using the grid background you count the number of squares across the top of the rectangle, 4. Then count the number of squares down the side of the rectangle, 3. There are 3 rows of 4 squares in the rectangle. So the area of the rectangle is 12 square units. Using the grid background you count the number of squares across the top of the rectangle, 3. Then count the number of squares down the side of the rectangle, 4. There are 4 rows of 3 squares in the rectangle. So the area of the rectangle is 12 square units. Using the grid background you count the number of squares across the top of the rectangle, 4. Then count the number of squares down the side of the rectangle, 2. There are 2 rows of 4 squares in the rectangle. So the area of the rectangle is 8 square units. Using the grid background you count the number of squares across the top of the rectangle, 2. Then count the number of squares down the side of the rectangle, 4. There are 4 rows of 2 squares in the rectangle. So the area of the rectangle is 8 square units. Curriculum K –12 Directorate Mathematics © State of New South Wales through the NSW Department of Education and Training, 2007 10

1.
New 7 –10 syllabus support
Rectangle diagram Explanation of how to find the area of the rectangle 2. 3. 4. Curriculum K –12 Directorate Mathematics © State of New South Wales through the NSW Department of Education and Training, 2007 11

1. 2. 3. 4.
New 7 –10 syllabus support
Rectangle diagram Explanation of how to find the area of the rectangle Using the grid background you count the number of squares across the top of the rectangle, 4. Then count the number of squares down the side of the rectangle, 3. There are 3 rows of 4 squares in the rectangle. So the area of the rectangle is 12 square units. Using the grid background you count the number of squares across the top of the rectangle, 3. Then count the number of squares down the side of the rectangle, 4. There are 4 rows of 3 squares in the rectangle. So the area of the rectangle is 12 square units. Using the grid background you count the number of squares across the top of the rectangle, 4. Then count the number of squares down the side of the rectangle, 2. There are 2 rows of 4 squares in the rectangle. So the area of the rectangle is 8 square units. Using the grid background you count the number of squares across the top of the rectangle, 2. Then count the number of squares down the side of the rectangle, 4. There are 4 rows of 2 squares in the rectangle. So the area of the rectangle is 8 square units. Curriculum K –12 Directorate Mathematics © State of New South Wales through the NSW Department of Education and Training, 2007 12

1.
New 7 –10 syllabus support
Rectangle diagram Explanation of how to find the area of the rectangle 2. 3. 4. Curriculum K –12 Directorate Mathematics © State of New South Wales through the NSW Department of Education and Training, 2007 13

New 7 –10 syllabus support
Can you make up another puzzle like this one? Curriculum K –12 Directorate Mathematics © State of New South Wales through the NSW Department of Education and Training, 2007 14

New 7 –10 syllabus support
2


2



Curriculum K –12 Directorate Mathematics © State of New South Wales through the NSW Department of Education and Training, 2007 15

New 7 –10 syllabus support
Name: ___________________________
Curriculum K –12 Directorate Mathematics © State of New South Wales through the NSW Department of Education and Training, 2007 16

New 7 –10 syllabus support
1. On half an A4 coloured sheet of paper (not grid paper), students are asked to draw, using pencil and ruler, any rectangle of their choice with whole centimetre dimensions. 2. Students find the area of their rectangle. Pair check solutions. 3. Class discusses properties of a rectangle. Students check accuracy of their shape on worksheet provided. Teacher models technique at the OHP.
How can you check if your rectangle has four right angles? How do builder’s check a wall frame is square?
4. Ask students to fold their rectangle along a diagonal. Rule the diagonal line first then use a ruler to fold accurately from corner to corner. 5. Discuss shapes formed. Model some at OHP. Possible questions include:
What types of triangles are formed? Why? Could you have a scalene triangle? Could you have an isosceles triangle? Could you have an obtuse-angled triangle?
6. Cut the rectangle along the diagonal carefully. 7. Paste one of the two triangles onto the grid worksheet, page 16. 8. Students find the area of their triangle and write down how they worked it out. 9. Students share their work with a partner. 10. Students continue working with their partner to discuss the question: Why did we start with a rectangle if we are trying to work out the area of a
triangle?
11. Class discussion on question. 12. Focus on bounding rectangle. Model at OHP. Students draw in their bounding rectangle. Make wall poster of worksheets. 13. Students complete, in pairs, Triangle 1 worksheet, page 19, on drawing in bounding rectangles and writing about how they found the area of the shaded triangles. Curriculum K –12 Directorate Mathematics © State of New South Wales through the NSW Department of Education and Training, 2007 17

New 7 –10 syllabus support
1. On half an A4 coloured sheet of paper (not grid paper), students are asked to draw, using pencil and ruler, any triangle of their choice with whole centimetre base length and whole centimetre altitude. Model at OHP. 2. Students draw in the altitude and fold along the line. Use as a hinge effect to see two bounding rectangles when placed on the grid worksheet, page 16. Use a sheet of paper for cover effect. Model at OHP. 3. Students paste the triangle onto the grid and draw in 2 bounding rectangles.
4. Students write down how they found the area of the triangle.
5. Students complete, in pairs, Triangle 2 worksheet, page 20, on drawing in bounding rectangles and explaining how they found the area of the shaded triangles. 6. Class discussion on finding a more efficient way of calculating the area of any triangle. Lead to area formula for a triangle. Curriculum K –12 Directorate Mathematics © State of New South Wales through the NSW Department of Education and Training, 2007 18

New 7 –10 syllabus support Name: ________________________ Pair check triangles 1
1. Draw in the bounding rectangle and write down how you found the area of the triangle. 2. 3. 4. Draw in the bounding rectangle and write down how you found the area of the triangle.
Draw in the bounding rectangle and write down how you found the area of the triangle.
Draw in the bounding rectangle and write down how you found the area of the triangle.
Curriculum K –12 Directorate Mathematics © State of New South Wales through the NSW Department of Education and Training, 2007 19

New 7 –10 syllabus support Name: ________________________ Pair check triangles 2
1. Draw in the bounding rectangles and write down how you found the area of the triangle. 2. 3. 4. Draw in the bounding rectangles and write down how you found the area of the triangle.
Draw in the bounding rectangles and write down how you found the area of the triangle.
Draw in the bounding rectangles and write down how you found the area of the triangle.
Curriculum K –12 Directorate Mathematics © State of New South Wales through the NSW Department of Education and Training, 2007 20

New 7 –10 syllabus support Name: ________________________ Pair check parallelogram 1
1. Draw in the bounding rectangle and write down how you found the area of the parallelogram. 2. 3. 4. Draw in the bounding rectangle and write down how you found the area of the parallelogram.
Draw in the bounding rectangle and write down how you found the area of the parallelogram.
Draw in the bounding rectangle and write down how you found the area of the parallelogram.
Curriculum K –12 Directorate Mathematics © State of New South Wales through the NSW Department of Education and Training, 2007 21

New 7 –10 syllabus support
Investigate and research some methods used to find the area of a triangle other than Area = ½ x base x height.

Below are some possibilities worth investigating: Pick’s formula

Heron’s formula

Area of circumscribed rectangle You may:

work individually or in pairs

present your work as
o o o o
a powerpoint presentation a poster a model with a written explanation a video
Curriculum K –12 Directorate Mathematics © State of New South Wales through the NSW Department of Education and Training, 2007 22

New 7 –10 syllabus support
Area conversion workcards
Make a poster explaining this conversion: 10 cm 2 = 1000 mm 2 You may use grid paper, diagrams and words. Choose a conversion of your own and add it to the poster. Make a poster explaining this conversion: 17 cm 2 = 1700 mm 2 You may use grid paper, diagrams and words Choose a conversion of your own and add it to the poster. Make a poster explaining this conversion: 23 cm 2 = 2300 mm 2 You may use grid paper, diagrams and words. Choose a conversion of your own and add it to the poster. Make a poster explaining this conversion: 3 cm 2 = 300 mm 2 You may use grid paper, diagrams and words. Choose a conversion of your own and add it to the poster. Curriculum K –12 Directorate Mathematics © State of New South Wales through the NSW Department of Education and Training, 2007 23