Program Outline Students and their Work Teacher’s Diary and Reflection Teaching Project
Program Outline • How the students were identified • Mathematics area of the C.S.F.
How the G.L.I.M. students were identified • S.I.N.E Clinical Interview and Diagnostic Test • One of the children completed the D.A.R.T. assessment. • Identification and Assessment of Gifted Students (Bright Futures 1999)
Measuring and Estimating Level 4 At this level students focus on the identification of attributes which can be measured and select and use appropriate units when estimating, describing, comparing and measuring length, perimeter and area. They choose appropriate measuring instruments and use them accurately.
Level 5 At this level students become familiar with metric units used to measure a range of quantities and common rates and choose units appropriate to the purpose of measurement. They choose instruments to measure to the required accuracy, devising ways to extend capabilities of the instruments and using their knowledge of fractions and decimals to read scales in which not all graduations are labelled. They relate metric units to quantities in their everyday experience and use this knowledge to make reasonable estimates of other quantities as required.
Using Relationships Level 4 At this level students understand and use relationships between different attributes in measurement situations involving the perimeter and area of polygons. Level 5 At this level students investigate the relationship between the dimensions of two-dimensional shapes and three-dimensional solids and their areas and volumes. They use these relationships to determine perimeters, areas and volumes of a range of shapes and objects. They use and calculate rates in their everyday experience.
Mathematical Reasoning Level 4 At this level students continue to formulate and test conjectures in the areas of space, number, measurement and chance and data. The emphases are on providing evidence for conjectures and arguments, systematically testing these conjectures and arguments for exceptions, completeness and reasonableness and making modifications as appropriate. As well, students decide when to use mathematical models appropriate to this level and to check the reasonableness of the results obtained by using these models. They discuss and write about mathematical situations, relationships and models using mathematical terms, symbols and notations.
Level 5 At this level students continue to develop the skills and dispositions associated with the formulation and testing of conjectures in all areas of the curriculum. An emphasis is on the development of general statements in symbolic form. Students also consider the assumptions and constraints which underpin the mathematical models they use and develop. As well, they consider the reasonableness of the results obtained by using these models in relation to the context.
Students And Their Work • Students • Demographics of the Group
L B’s Profile • Age: 12 • Grade: 6 • People in family: 5 people in family. Mum and dad older brother and younger sister. • Pet: Golden Labrador called Jackie. • Favourite subject: Maths • Favourite sport: Football and tennis. • Favourite hobby: surfing. • I like to go to school see my friends and to work hard. On the weekends I play my two favourite sports tennis and footy. I play for a club. I hope to be an A.F.L footy player but I also want to go really well in school to be a vet.
L B’s Work • Amazing Arrays 1 • Amazing Arrays 2 • Find The Area Of …. • My Area Worksheet • House Plan • My Dream Bedroom • Bedroom Expenditure • Area is 100cm² and Perimeter is 66cm • Evaluation
J Cl’s Profile • Favourite Hobby: Surfing. • Favourite Subject: Maths. • Favourite Sport: Lifesaving. • Age: 12 • What you would like to be when you are older: Pilot in the Air force. • My name is Jack and I am 12 years old I enjoy Maths because I like being challenged in areas of learning. I wanted to be apart of this maths group because I thought it would be a great experience and seeing that I want to be a pilot in the air force it would help me be closer to my dream job as a pilot. I also enjoy sport and I just competed in the state Cross Country championships which I came in the place 4th. My other favourite subjects are English and History.
J Cl’s Work • Amazing Arrays 1 • Find The Area Of …. • My Area Worksheet • House Plan • My Dream Bedroom • Bedroom Expenditure • Area is 100cm² and The Perimeter is 66cms • Evaluation
J F’s Profile I am 9 years old. I am an only child. Things that I enjoy doing are maths, reading, playing games and working on the computer. The sports I enjoy are cricket, basketball, tennis, swimming and soccer. When I grow up I want to be a doctor, specializing in surgery.
J F’s Work • Amazing Arrays 1 • Amazing Arrays 2 • Find The Area Of …. • My Area Worksheet • House Plan • My Dream Bedroom • Bedroom Expenditure • Evaluation
Demographics of the Group The group consisted of nine boys – eight were year six students and one was a year four student. This student who was a new admission to our school this year. The boys are being taught in six different grades, but we met at least once a week for an hour session while working on the project.
Teaching Project • Learning Needs To Be Targeted • Links to Other Curriculum Areas
Learning Needs To Be Targeted My focus for these children was to review and extend their understanding of the measurement strand of the Mathematics K.L.A; particularly the topic of area and perimeter. I wanted the boys to see that so many of our daily activities are influenced in some ways by measurement procedures. I aimed to provide them with rich mathematical investigation that would generate meaningful problem solving strategies that they could use in their project.
My plan was to provide explicit teaching on how to find the area and perimeter of regular and irregular polygons and then lead the children to investigate ways they could solve the area of parallelograms, quadrilaterals, triangles and circles. Throughout our sessions together my endeavor was to provide tasks that challenged and engaged the children. Many of the tasks were designed to promote discussion and lead to further inquiry and sorting out of formulas. The children needed to have a solid understanding of this topic before they could springboard into their individual projects.
The project while having parameters was aimed at providing opportunities for the children to be creative and put their individual mark on the task. The project outline was composed in consultation with the children after we had agreed what were to be basic inclusions. The use of technology was to be incorporated throughout this project.
Links To Other Curriculum Areas This work while essentially a Mathematics unit does have links with the Technology K.L.A. Throughout the project the use of software, digital imagining and programming was used to enhance and modify tasks. The boys were asked to explain and create solutions as well as plan, construct and modify their designs and report on them using different materials.
The boys were asked to investigate house plans from a variety of sources and see why houses are built in many different ways. We talked about materials used and how designs have changed over the years. This area could have been expanded and the children chosen a dwelling from another part of the world and investigated size, materials used and why, age of dwelling, design structure, permanency and if construction relates to environment, climate or availability of materials. Unfortunately time did not allow for such investigations.
Session One In this session the children came together and discussed what the project was about and where it might be leading. We discussed what the words area and perimeter meant to them as individuals; as well as linking them to everyday life. We brainstormed various occupations that might use this form of measurement in their work. Some of the jobs that were proposed were: builders, landscape gardeners, pool construction, carpet layers, interior designer, surveyor, picture framer, plumbers, glaciers, surfboard makers, architects and farmers. Following this discussion the children created individual mind maps and shared their ideas with the group.
Session Two This was an explicit teaching session where the children reviewed/discovered how to work out the perimeter and area of a regular polygon. The children brainstormed with the teacher their knowledge about how to work out the perimeter and the area of a shape. They then worked on a sheet that was designed for them to discover if there was any correlation between the sides of a regular shape and its area. Squares Following this task we discussed their findings. Every child in the group had input here. I posed the question “Is there a more efficient way to calculate the area of a square or rectangle than just counting the squares in the grid?” Feed back was automatic from all the boys and I was able to gauge that each of them had a solid understanding of the formulas for finding the perimeter of a shape and calculating the area of squares and rectangles.
Session Three As a review of the work from the last session the children were asked to calculate the area and perimeter of specific polygons using grid paper. This task was completed easily by all the group. The next aspect of this lesson was to have the children use a computer program devised by Mark Hennessey called amazing arrays. This program reinforced their understanding of how we multiply length x width to calculate the surface area of a regular polygons. The children were all successful at completing this task.
Session Four This was an explicit teaching session on how to measure the area of irregular shapes. From this point on, I did not do any specific teaching on perimeter but the feedback from the children via their work and reflection indicated to me that this area was well established. The lesson began with the children being presented with a number of irregular shapes and we discussed in twos how we might calculate their areas. Irregular Shapes activity allowed the children to test their understanding and arrive at an efficient method of calculation.
After completing the task we checked their calculations by counting the squares in the shape and using the number sentences composed by each child. All of the children choose to break each shape into regular polygons (squares and rectangles), calculate the length and width, multiply together and then add the answers together. The way they broke up the shapes varied. Shape 3 for example was broken up in the following ways. (5 x 7) + (4 x 3) + (4 x 7) 35 + 12 + 28 Answer = 75 cm2
Another answer provided by a child was (5 x 2) + (5 x 2) + (13 x 3) + (4 x 2) + (4 x 2) 10 + 10 + 39 + 8 + 8 Answer = 75 cm2
One child however chose to view the shape as a rectangle that was calculated as 13 x 7 and he then subtracted 2 x (2 x 4) Therefore he recorded the sum as 13 x 7 – 2 x (2 x 4) 91 - 16 Answer = 75 cm2
The work allowed for valuable discussion and the children were able to conclude that breaking the shape into regular polygons and calculating the answers was a much more efficient method to work out the area of a shape. They also decided that in some cases it was more efficient to calculate the outside perimeter of the shape and subtract.
Session Five This session began with a review of what we had learnt about calculating the area of a regular polygon and irregular shapes. They completed a task called ‘New Boundaries’. In this task the children were asked to divide mainland Australia into six states of equal area and redraw the new state boundaries. (Examples of the children’s work is attached) This was also to be an explicit teaching session on calculating the area of a parallelogram and a triangle. The children shared their understandings of what constituted a parallelogram and triangle. They drew the shapes to show a visual representation of their understanding.
The children were then given a sheet and asked to find the area of these shapes. Parallelograms The discussion and solutions presented indicated the various ways each child arrived at a solution to the areas of the shape. Some cut out the shape and made into a regular polygon, others shaded and calculated the area; they all used the grid squares to affirm their calculations.
We then took this further and I asked them to look at a regular polygon I had drawn and asked them to calculate its area. All completed easily by multiplying length x width. I then handed out the following sheet Parallelogram 2 and asked then to answer the sheet and write about their findings. We spoke of new terminology – base and height. It was a great way for them to discover that the area of a quadrilateral is calculated using a formula similar to a regular polygon – base x height
Some of the drawn diagrams of the children showed that they had constructed triangles in their effort to calculate the area of a parallelogram. I lead a discussion about how we could calculate the area of a triangle. Two of the children said that their triangles were right-angled triangles and half of the new polygon that they had drawn. After some teacher input the children came up with a formula for working out the area of a triangle. Area of a Triangle = base x height or ½ (base x height) 2
The question was posed – ‘Does this formula work for all triangles?’ - Isosceles triangles - Equilateral triangles - Scalene triangles - Right-angled triangles. Children tested this discovery using grid paper and formulas using the following task. Triangles
While most children counted the squares one child coloured the parts of squares to prove the formula. Another child cut up the part squares and rearranged into whole squares.
Session Six The session began with the children going on a computer and opening a word document and formatting a table. The children were to go to clip art and insert a picture. The others in the group where challenged to try and work out the area of that picture. Area Pictures Informal lesson on how to measure a circle. Working On Area Pictures
The ‘House Project’ was introduced to the children. They were given a design outline of what was expected to be included in their house plan which was negotiated with the group. House Project
Session Seven When the children first heard that their homes were to be 42 squares, all of them imagined 42 square metres. They had all made newspaper square metres as part of their homeroom class lessons. The Square Metre The task for the children in this lesson was to make a house square that equalled 9.6 square metres. The children loved this task and were amazed with the size. House Square
Session Eight This session began with a lesson on how to work out the area of a circle. Reference Curriculum @ Work. The children then had the following problem put forward to them ‘Design a shape that has an area of 100cm² and a perimeter of 66cms.’ The children began by inserting a 10 by 10 array on a grid and then moved parts to increase the outside perimeter from 40cms to 66cms. Teacher Examples. The children’s work showed that solutions to this problem are not limited to just one answer.
The rest of the time was devoted to the children designing their house plans. There are two groups and four individual projects. The children looked at plans from newspapers as a guide to the set out of a plan and symbols used to note fixtures, doors etc.
One of the groups wanted to double their house size to be 84 squares. This was allowed but it could cause problems when blocks of land are given out, so house design needs to be considered. We used this session also to talk about areas that are included in the plans like verandas, garages, carports. We spoke about a staircase and what type of staircase would take up the most room. Position of rooms is also important; while we are not doing plumbing and gas plumbing quotes, have all your wet areas on one side of a house cuts costs. Children reflected where in their homes, wet areas are located.
Session Nine This session began with the allocation of house blocks for their plans. House blocks are not measured in house squares but in square metres. We had a walk around the local area and using a trundle wheel measured house blocks to gauge the average size of blocks. Measuring A House Block Our school is situated in an older area but has a new housing estate close by. The measuring that took place allowed the children to see that not all blocks are the same, or regular in their shape and that newer homes while being bigger have less free land. The children opted to construct their homes on blocks found in an older area.
Session Ten Working To Scale – In this session the children’s plans were discussed and they were presented with a 6 x 7 grid. This grid was to be cut up and rearranged by the grid to represent their plans. No piece was to be discarded in the completion of this task. Photo