EXPLORING PATTERNS THAT INVOLVE SPACE AND MEASUREMENT: SOME EXAMPLE OF NON LINEAR PATTERNS OR GROWTH PATTERNS

Transcription

1 EXPLORING PATTERNS THAT INVOLVE SPACE AND MEASUREMENT: SOME EXAMPLE OF NON LINEAR PATTERNS OR GROWTH PATTERNS KARSAN HUTCHINSON You need to work in a group of four for this learning experience. Two of your group will explore Part 1 of the experience and two of your group will explore Part 2. Part 1: Growing solids Use either one or two centimetre cubes or multilink cubes to construct a series of growing cubes. Imagine you are tracing the growth of a family of cubes. Cubes grow quite differently to humans. On the eve of a one year old cube s 2 nd birthday it goes to bed as a one year old cube but wakes up on the morning of its 2 nd birthday a complete two year old cube. Similarly, on the eve of a two year old cube s 3 rd birthday it goes to bed as a two year old cube but wakes up on the morning of its 3 rd birthday a complete three year old cube and so on. What are the patterns to this growth, if any, that will allow the parents of such cubes to plan for housing (space) and clothing (covering) their offspring? If so, what are these patterns that would be useful in making such predictions? A one year old cube is simply a single cube. high. It is one cube unit long, one cube unit wide and one cube unit two original cube units high. A two year old cube is a cube that is two original cube units long, two original cube units wide and Question 1: Predict what a three year old cube is... Use the 1 cm or 2 cm cubes or multilink cubes to construct a: 1 year old cube, 2 year old cube. 3 year old cube 4 year old cube. Question 2: Using the cubes, find: the length of each cube in cube units; the area or amount of surface for each cube in squares, and the volume or number of original cubes needed to make each cube. Arrange your information in a table in your maths book. For example: Age Length Surface Volume Area volume = =3 3 4 Search for patterns within your table that will help you predict the length, surface and volume of cubes from five years to twelve years old. A calculator may be helpful. Include your results the table.

2 Check your predictions for the five and six year old cubes by making them and measuring them. A table is a useful way of organising data to assist in searching for patterns, making predictions and making generalisations. Question 3: Describe what appears to be happening to the right hand column of the table. That is, what is happening to the ratio of the cubes volumes and areas as they grow larger? What does this suggest? Another way of representing and investigating patterns is to graphs your data. Graphs of patterns can assist students in making predictions. Question 4: Use the information from your table to construct point graphs for the following in your maths book: the cubes lengths on the horizontal axis and the cubes areas on vertical axis, and the cubes lengths on the horizontal axis and the cubes volumes on vertical axis. (You may like to use graph paper or use the chart wizard from Microsoft Excel to do this ask Mr Hutchinson if you need assistance) Question 5: Suppose cubes do not grow as described in the original story shell but grow at a constant rate. Use your graphs to predict the length, area and volume of a four and one half year old cube. Use a calculator to verify your predictions. Question 6: While counting cubes and their surfaces is useful, if a pattern exists that enables these values to be calculated rather than counted, then this pattern can be generalised. The generalisation may be described in words, using symbols and eventually as an algebraic statement. See if you can develop an algebraic statement to describe a cube s growth in terms of the space it occupies and develop an algebraic statement to describe a cube s growth in terms of the surface required to covers it. Some more possibilities What if it is rectangular prisms and not cubes that are growing. Begin say a rectangular prism that is 2 cubes long, 1 cube wide, 1 cube tall and call this is a 1 year old prism. A 2 year old prism is 3 cubes long, 2 cubes tall and 2 cubes wide. That is the prism grows one cube in length, height and width each year. Construct a 1 year old to a 5 year old prism and explore: the volume of the prism as it grows; the surface of the prism as it grows, and ways of representing any patterns and generalising these patterns even an algebraic statement. Similarly what if it is square based pyramids that are growing?

3 Part 2: Growing Figures Use either square grid paper or square flip tiles to construct growing squares. As with part 1, imagine you are tracing the growth of a family of squares. Squares grow quite differently to humans. On the eve of a one year old square s 2 nd birthday it goes to bed as a one year old square but wakes up on the morning of its 2 nd birthday a complete two year old square. Similarly, on the eve of a one year old square s 3 rd birthday it goes to bed as a two year old square but wakes up on the morning of its 3 rd birthday a complete three year old square and so on. What are the patterns to this growth, if any, that allow the parents of such squares to plan for housing and clothing (covering) their offspring? If so, what are these patterns that could be useful in making such predictions? A one year old square is simply a single square. It is one square unit long and one square unit wide. A two year old square is a square that is two original square units long and two original square units wide. Question 1: A three year old square is... Use the 1 cm or 2 cm grid paper or square flip tiles to construct a: 1 year old square, 2 year old square. 3 year old square 4 year square. Question 2: Using the cubes, find: the length of each square in square units; the area or amount of surface for each square in as the number of original squares, and the perimeter or number of original square lengths needed to surround square. Arrange your information in a table in your maths book. For example: Age Length Surface (area) Perimeter Perimeter Area = =2 3 4 Question 3: Search for patterns within your table that will help you predict the length, surface and perimeter of squares from five years to twenty years old. A calculator may be helpful. Include your results the table. Question 4: Describe what appears to be happening to the right hand column of the table. That is the ratio of the squares perimeters and areas as they grow larger? Another way of representing and investigating patterns is to graph your data. Graphs of patterns can assist students in making predictions.

4 Question 4: Use the information from your table to construct point graphs for the following in your maths book: the squares lengths on the horizontal axis and the squares areas on vertical axis, and the squares lengths on the horizontal axis and the squares perimeters on vertical axis. (You may like to use graph paper or use the chart wizard from Microsoft Excel to do this ask Mr Hutchinson if you need assistance) Question 5: Suppose squares do not grow as described in the original story shell but grow at a constant rate. Use your graphs to predict the length, area and perimeter of a five and one half year old square. Use a calculator to verify your predictions. Question 6: While counting squares and their surfaces is useful, if a pattern exists that enables these values to be calculated rather than counted, then this pattern can be generalised. The generalisation may be described in words, using symbols and eventually as an algebraic statement. See if you can develop an algebraic statement to describe a square s growth in terms of the perimeter and develop an algebraic statement to describe a square s growth in terms of the surface required to cover it. Some more possibilities to explore as a group What if it was other polygons growing rather than squares? For example, growing triangles, rhombuses, trapeziums or growing hexagons. Use pattern blocks and grid paper to explore what happens to the perimeters and areas of at least one of the above growing polygons. As with the growing squares, the perimeter is described in terms of the length of one side of a one year old polygon and the surface (area) described in terms of the number of original or 1 year old polygons. For example with growing triangles: perimeter of 3 units and 6 units respectively and the surface of 1 triangle and 4 triangles respectively. Construct a 1 year old to a 5 year old polygon and explore: the perimeter of the polygon as it grows; the surface of the polygon as it grows, and ways of representing any patterns and generalising these patterns. Some other possibilities with growth patterns In your pair choose two of these arrangements in turn and for each: copy the first three steps; continue the pattern for at least the 4 th and 5 th steps; put a number to the each of the five steps; search for a pattern with the number sequence and predict what the numbers would be for 6 th to 10 th steps, and draft a generalised statement that would enable you predict the number for any step, either in words, using symbols or as an algebraic statement. or

5 or or So what does this mean? We believe that the key ideas of pattern, relationship and generalising are central to the pattern workshops and to working mathematically. by relationship we mean, an association between two or more attributes, eg. How does the area of a cube increase as its length increases? by generalising we mean, drawing broad conclusions from particular instances, describing patterns and relationships. In mathematics and science the conclusions can be described in words, diagrams, graphs and using algebraic statements. In your group of four share your findings and ideas about the two different starting points and record questions that your explorations have raised for you.