A child's introduction to geometric shapes begins in infancy with mobiles, books, blocks, puzzles, sorting toys, and segments on various television programs (Hannibal, 1999, p. 354). Children begin forming concepts of shape long before they enter school (Clements & Sarama, 2000, p. 82). They may recognise a shape to be a rectangle because it looks like a door, or a shape to be a circle because it looks like a ball. These relationships fall under the mathematics strand of geometry. Geometry explores the properties of shapes (Origo, 2008, p. 47). Geometry is an essential component of mathematics instruction as it helps us represent and describe in an orderly manner the world in which we live (Oberdorf & Taylor-Cox, 1999, p. 340).
Children will not develop concepts of shape merely by being exposed to shape. Experiences must be directed specifically to shape in space. Through these tutorials, we were introduced to activities and learning experiences that will assist in assisting students to develop their concepts of shape.
Initially, students must begin by learning how to identify and represent shapes. An simple learning experience to begin to develop sight recognition of shapes would be to ask students to draw a shape with a certain attribute, e.g. 4 straight sides and 2 of the sides have the same length. Students may draw something similar to the example below.
Students can now be asked to state what shape they have drawn. We know this shape to be a rectangle. Another example may involve asking students to cut out a 3 sided shape, with straight sides from coloured paper. Students may construct something similar to the example below.
\ Students can now be asked to state what they have drawn. We know this shape to be a triangle. Exercises such as these will begin to develop students ability to recognise different shapes. In discussions with peers, we looked at how this activity can be used over different year levels when introducing new shapes.
Once students have the ability to recognise shapes, they can being to visualise and analyse different shapes. A learning experience that would greatly benefit students would be finding shapes within shapes, or, what shapes together make up a shape? (known as tangrams). Students may use Mira's in this activity (Mira: a geometric tool that has the reflective quality of a mirror, as well as a transparent quality to see through. Used predominately in activities where looking for lines of symmetry).
By placing the Mira on a square, and dissecting the shape from top left corner to bottom right corner, students can investigate what shapes they can see. In the example below, the square has been dissected, and has revealed that it can be made up of 2 right angled triangles. Alternatively, the square could be dissected straight down the middle to reveal it is made up of 2 rectangles, both of equal size.
Once students have the ability to recognise shapes, they can being to visualise and analyse different shapes. A learning experience that would greatly benefit students would be finding shapes within shapes, or, what shapes together make up a shape? (known as tangrams). Students may use Mira's in this activity (Mira: a geometric tool that has the reflective quality of a mirror, as well as a transparent quality to see through. Used predominately in activities where looking for lines of symmetry).
By placing the Mira on a square, and dissecting the shape from top left corner to bottom right corner, students can investigate what shapes they can see. In the example below, the square has been dissected, and has revealed that it can be made up of 2 right angled triangles. Alternatively, the square could be dissected straight down the middle to reveal it is made up of 2 rectangles, both of equal size.
What about if we now dissected the square using 2 lines, 1 from the top left corner to the bottom right corner, and 1 from the top right corner to the bottom left corner. What new shapes have we found? In the example below we can see the square is now made up of 4 isosceles triangles. What about if we dissected the square right down the centre vertically, and right across the centre horizontally. What new shapes have we found? We can see in the example below that the square is now made up of 4 smaller squares, each the same size.
Online programs such as "Tangram Puzzles" can be used. The website allows the user to fill in the outlines of different shapes with a range of smaller shapes.
Students could also be introduced the flat nets of 3D shapes. For example, what does a cube look like if all the sides are folded out? How many different shapes is it made of? What shapes can you see. This can be illustrated in an the example below where a cube with the height, length, and depth of 6cm has been folded out. Students can explore the shapes and lines that make up the cube.
In a tutorial activity we constructed a cubic meter using plastic rods and connectors. Through discussions with peers, we looked at how we could use the cubic meter in investigations into shape. As seen in the photo below, students could use meter rulers, paper streamers, cardboard, or even body parts to explore line and shape of the cubic meter. Similarly, other 3D shapes could be constructed and explored in the same manner.
Students could now begin to analyse shapes: describe properties of different shapes, describe critical attributes of shapes, and describe relationships between shapes. An example of an appropriate learning experience for younger grades would be to present the students with 4 shapes, ensuring 3 have a similar attribute, and another 3 have a similar attribute (as seen in the example below). Ask the students, "Which one doesn't belong?". Students must use their own skills of analysis to view the properties, attributes and relationships between shapes to find the odd man out. As we can see in the example below, there are 3 red shapes, and 1 green shape. Students can analyse the attribute of colour to find which doesn't belong. However, if we look at the properties of the shapes, and count sides, we find 3 shapes with 4 sides, and 1 shape with 5 sides, making the pentagon the odd one out. This example allows students to use their developed skills of analysis to describe both attributes and properties of a shape.
A simple activity to introduce students to different shapes is by organising a shape walk. As a class, or in small groups with teacher or teacher aide supervision, students can walk around the school grounds and look for different shapes in their environment. Students could take photos of the different shapes they find and present them to the class. On a computer, the teacher can insert lines into the photo to find what smaller shapes make up the larger shapes, how many shapes altogether make up the shape, what attributes and properties do the shapes each have, and what is the relationship between each shape? This learning experience gives students the opportunity to use the inquiry model of learning to find and describe their own shapes.
The next strand, measurement, determines the extent of an attribute, usually in comparison to a standard example (Origo, 2008, p. 65). In primary grades, measurement focuses on the spatial attributes of length, area, and volume (Curry, Mitchelmore & Outhred, 2006, p. 377). Through tutorial activities, we focused primarily on length, area, volume, mass and time, and both standard, and non-standard units for measuring each. For each, activities were conducted, and appropriate learning experiences were developed to assist in developing students early sense of measurement. We discovered the order of difficulty between the 3 major components: length, area and volume, related in differences in dimensionality and computational complexity (Curry, 2006, p. 383).
We first explored non-standard units. Valuing the usefulness of measuring length is a crucial element that drives children's learning (Schwartz, 1995, p. 413). An initial activity involved creating a measuring device which compared the length of objects to the length of a paperclip. Students should be given the opportunity to create their own device using a long strip of paper ribbon, a paperclip and a pen or pencil. By laying the ribbon out flat and marking a point (0), the student can lay the paperclip down, ensuring one end is resting on the mark, and draw another mark on the top of the paperclip. By repeating this process, they will eventually be left with a strip of ribbon, with equal markings spaced out the length of a paperclip. The student will then need to number each mark in a sequential order. The measuring device can then be used to measure different objects. For example, if a students measured the length of the side of a desk, their answer may come to 24. The units would be paperclips. Therefore the answer would be 24 paperclips long. Other units may be used, for example, length of a hand from wrist to tip of middle finger, plastic drinking straws, or pencil sharpeners to name a few. With larger measuring units, students can observe the relationships between 2 units. An example of this would be constructing a measuring device using paperclips, and a device using drinking straws. Students can see that the larger the unit, the smaller the measurement, or vice versa. For example, that same desk may be 24 paperclips long, but may only be 7 drinking straws long.
Secondly, we looked at standard units of measurement. Standard units of measurement assist students in gaining precision in measurement. One of a young child's greatest challenges is to achieve accuracy in linear measurement (Schwartz, 1995, p. 413). Students may continue on from the activity conducted for non-standard units to gain an understanding of standard units. Once again, a measuring device may be constructed using a paper ribbon, or a measuring tool such as a ruler could be used. Standard measurements on the ruler may be millimetres (mm), or centimetres (cm). Students may begin by finding the relationship between the 2 measurements. 10mm is equal to 1cm. A ruler may be 30cm long, how may mm is this equal to? Students can measure length of objects using their standard measuring tools. How long is the length of their desk in cm? mm? What other object in the classroom can they find that is the same length? Here students begin to compare objects using measurement. An activity we took part in during this tutorial was comparing parts of our own bodies using a length of paper ribbon. In this activity we weren't using a standard or non-standard markings on the ribbon. Students may begin my measuring the length of their foot. By placing one end of the ribbon at the end of their longest toe, and placing a mark at the back of their heel, they have found the length of their foot. Next, they may be instructed to find the length of their forearm. By placing one end of the ribbon on their wrist, and placing a mark at the join on the back of their elbow, they have found the length of their forearm. Now compare the 2 lengths of ribbon. What have you found? In most cases, these lengths of ribbon will be equal. What other body parts can we now measure and compare? Students can measure their height and compare this to their arm span; measure the height of their faces from top of the forehead to the bottom of the chin; measure the length of their thumbs, noses, around wrists, waists, heads etc, ensuring the mark each ribbon with what the measurement is take off. What they will be left with is a number of lengths of ribbons. Students should line the ribbons up on a straight line, comparing all lengths of ribbon. What will they find?
We next looked at area. Area is the amount of surface contained within the perimeter of a closed 2D shape (Origo, 2008, p. 5). Students may begin by taking part in activities where they are to compare the area of 2 shapes using standard units. For example a square and a triangle. What do you think may be the relationship between the 2 shapes? Both shapes have equal height and equal width. Students may look at different aspects such as measurements of height and width, shape or angles to compare the 2 shapes. From there, students may begin to look at measuring area using non-standard units. An activity we took part in involved covering an A4 sheet of paper with 1 type of pattern block. We explored this activity in a small group of 5, where each group member used a different pattern block. In the example below, we can see how many yellow hexagons cover a sheet of A4 paper. Once we all covered our sheet with the different shapes, we compared our answers to find relationships between shapes. Before revealing our answers to the rest of the group, we discussed what we thought the relationship between each shape might be. Since the yellow hexagons were exactly twice the size of the red trapeziums, it was presumed the trapeziums would cover the area using twice as many blocks as the hexagons. This activity will allow students an opportunity to compare and explore relationships between different shapes, and practice measurement with non-standard units.
Finally, we looked at measuring volume using non-standard units. Volume appears to be the most difficult for students to comprehend, as the relative similarity between length and area greatly differs to the substantial gap between area and volume (Curry, 2006, p. 383). We used a a standard MAB block as the non standard unit for measuring volume, and a set of scales in this activity. By placing an object in one of the baskets of the scales, we were able to keep adding MAB blocks to the other basket until we balanced the scales. Once balanced, MAB blocks could be counted, and a volume in MAB units could be given. For example, we weighed a small mobile phone to find it's volume in MAB blocks. By adding blocks into the basket, we found that the phone weighed 18 MAB blocks. This now gave us another unit to use when finding volume. We knew that 1 MAB block was equal to 1 MAB block, and that 1 mobile phone was equal to 18 MAB blocks. Next we wanted to find the volume of a water bottle. In the photo below, we can see that the mobile phone was also used in this comparison. Since the phone was worth 18 MAB blocks, we could use it in place of the blocks.
This activity will give students the opportunity of furthering their understanding of measurement, non-standard units and volume, and developing their comparison skills.
