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Measurement and Geometry – 2D 58G Identify diagonals on convex two-dimensional shapes, recognising the endpoints as the vertices of the shape Draw all the diagonals of convex two-dimensional shapes, comparing the diagonals on different shapes Identify which of the special quadrilaterals (parallelograms (squares, rectangles and kites), rhombuses and trapeziums) have diagonals that are equal in length Identify whether any of the diagonals are also lines (axes) of symmetry of the shape Resources: protractors, rulers, pencil, paper EXPLICIT LEARNING - TRIANGLES Focuses children’s thoughts on the concept, exposing current understanding and any misconceptions Reviews constructing regular quadrilaterals – squares - with a protractor and ruler What could we do? What language could we use to explain and ask questions? Children think about, talk and listen to a friend about, then have the opportunity to share what they already know. Children construct a regular quadrilateral – a square (See Measurement and Geometry, 2D - Level 49G), for example, 4 cm 4 cm Website: http://www.alearningplace.com.au Email: [email protected] Twitter: @learn4teach ► Today brings an investigation about diagonals in twodimensional shapes. ► What do you know about diagonals in two-dimensional shapes? ► Talk about diagonals in two-dimensional shapes with a friend. ► Is anyone ready to share what they are thinking about diagonals in two-dimensional shapes? ► Let’s construct a regular quadrilateral, a square with a ruler and a protractor. 4 cm 4 cm YouTube: A Learning Place A Teaching Place Scan the QR Code Facebook: A Learning Place 1 Display the square without the angles and side measurement labelled, for example, Introduces diagonals Record, for example, diagonal Record the diagonals on the square, for example, Record, for example, diagonal = straight line Record, for example, diagonal = straight line that joins 2 vertices. Record ‘opposite’ between the words ‘2 and vertices’, for example, diagonal = straight line that joins 2 opposite vertices. Record, for example, the end point of diagonals are the vertices of the shape. Introduces the number of diagonals on a square Introduces length of the diagonals Display the square with the diagonals recorded, for example, Measure the length of the diagonals, then record, for example, equal length. Introduces Fold the square on the diagonals then record for example, axes of diagonals as lines symmetry. of symmetry Website: http://www.alearningplace.com.au Email: [email protected] Twitter: @learn4teach ► So we’ve constructed a regular quadrilateral, a square. ► And we know that this quadrilateral has sides and vertices. ► Does it also have diagonals? ► I’ve recorded diagonals on this square. ► How would you describe a diagonal? ► Is a diagonal a straight line? ► Does a diagonal join 2 vertices? ► Are the vertices adjacent or opposite one another? ► Are the vertices opposite? ► Where are the end points of the diagonals? ► Are the end points the vertices of the shape? ► Let’s investigate the diagonals on this square. ► How many diagonals on this square? ► Are there 2 diagonals? ► How could we describe the diagonals? ► Are the diagonals equal length? Let’s measure them. ► Yes, the diagonals are equal in length! ► Are the diagonals, lines of symmetry? Let’s fold the square on the diagonals to check. ► We’ve been calling these ‘lines’ of symmetry. YouTube: A Learning Place A Teaching Place Scan the QR Code Facebook: A Learning Place 2 Introduces axes of symmetry Record for example, axis of symmetry. Record, for example, axis – singular ► The mathematical term is ‘axis’ of symmetry. ► An axis is a central point, so an axis of symmetry is a central line. ► Just like vertex is singular and vertices are plural, axis is singular, but the plural is not axises! The plural is axes. ► Are the diagonal axes of symmetry? ► The diagonals are axes of symmetry! ► At what angle do the diagonals cross? ► Could we measure the angles where the diagonals cross? ► At what angle do the diagonals cross? ► Do the diagonals cross at right angles? ► Yes the diagonals do cross at right angles! ► So a square has 4 equal 90 degree vertices and 4 equal sides, 2 equal length diagonals that are axes of symmetry that cross at right angles. Record, for example, axes - plural Introduces measuring the angle at which the diagonals cross Use a protractor to measure the angles where the diagonals cross, for example, Website: http://www.alearningplace.com.au Email: [email protected] Twitter: @learn4teach YouTube: A Learning Place A Teaching Place Scan the QR Code Facebook: A Learning Place 3 Introduces recording diagonals of a square in a table Record, for example, Shape Square Vertices 4 equal ► Sides 4 equal Diagonals length 2 equal Diagonals axes of symmetry? Diagonals cross at right angles? yes yes Reviews constructing irregular Children construct an irregular quadrilateral – a rectangle that is not a quadrilaterals – square, (See Measurement and Geometry, 2D - Level 49G) for rectangles that example, are not squares 4 cm with a protractor 6 cm and ruler 4 cm Website: http://www.alearningplace.com.au Email: [email protected] Twitter: @learn4teach 6 cm ► Could we record this in a table? Let’s construct an irregular quadrilateral – a rectangle that is not a square - using a protractor and a ruler! ► Is this a regular quadrilateral? ► How could we check? ► Could we measure the length of the sides? ► Are the sides equal or unequal? ► Are opposite sides equal? ► Could we measure the size of the angles? ► Are the angles equal or unequal? ► Because the sides are unequal while the angles are equal, is this an irregular quadrilateral? ► Because opposite sides are equal, and all angles are right angles, is this irregular quadrilateral, a rectangle? YouTube: A Learning Place A Teaching Place Scan the QR Code Facebook: A Learning Place 4 Reviews diagonals Display the irregular quadrilateral, a rectangle, without the angle sizes and side lengths labelled, for example, ► So we know that this quadrilateral has both sides and vertices. ► Does it also have diagonals? ► How could we draw diagonals on this rectangle? Record, for example, diagonal ► How would you describe a diagonal? Record the diagonals on the rectangle, for example, ► Is a diagonal a straight line? ► Does a diagonal join 2 vertices? Record, for example, diagonal = straight line ► Are the 2 vertices adjacent or opposite one another? Record, for example, diagonal = straight line that joins 2 vertices. ► Are the vertices opposite? Record ‘opposite’ between the words ‘2 and vertices’, for example, diagonal = straight line that joins 2 opposite vertices. ► Where are the end points of the diagonals? ► Are the end points the vertices of the shape? Reviews the number of diagonals on a rectangle that is not square Record, for example, the end point of diagonals are the vertices of the shape. ► How many diagonals on this rectangle? ► Are there 2 diagonals? Reviews length of the diagonals Measure the length of the diagonals, then record, for example, equal length. ► How could we describe the diagonals? ► Are the diagonals equal length? Let’s measure them. ► Yes, the diagonals are equal in length! ► Are the diagonals axes of symmetry? Let’s fold the square on the diagonals to check. ► No, the diagonals are not axes of symmetry! Fold the rectangle on the diagonals then record for example, not axes Reviews diagonals axes of of symmetry. symmetry Website: http://www.alearningplace.com.au Email: [email protected] Twitter: @learn4teach YouTube: A Learning Place A Teaching Place Scan the QR Code Facebook: A Learning Place 5 Introduces measuring the angle at which the diagonals cross Use a protractor to measure the angles where the diagonals cross, for example, Record, for example, Introduces recording Shape Vertices Sides diagonals of a rectangle that is not a square, in a Square 4 equal 4 equal table Rectangle Reviews differentiating the investigation for children as they demonstrate understanding 4 equal 4, 2 opposite pairs equal Diagonals length Diagonals axes of symmetry? Diagonals cross at right angles? 2 equal yes yes 2 equal no no ► Could we measure the angles where the diagonals cross? ► At what angle do the diagonals cross? ► Do the diagonals cross at right angles? ► No, the diagonals do not cross at right angles. ► So a rectangle that is not a square, has 4 sides with opposite sides equal, 4 equal angles, 2 equal length diagonals that are axes of symmetry that do not cross at right angles. ► Could we record this in a table? Allow children time now to engage in guided and independent investigation (at the end of this teaching plan) of describing diagonals in quadrilaterals. A child who has not demonstrated understanding of measuring and constructing angles will continue to investigate this before progressing to describing diagonals in quadrilaterals. A child could be sitting next to a child who is investigating at a different level. They will explain their current levels of understanding to one another as they investigate. This is a research-based way to accelerate learning for children at all levels. Children move to Guided and Independent Investigation now to investigate the concept at increasing levels of understanding over many learning sessions Website: http://www.alearningplace.com.au Email: [email protected] Twitter: @learn4teach YouTube: A Learning Place A Teaching Place Scan the QR Code Facebook: A Learning Place 6 GUIDED INVESTIGATION INDEPENDENT INVESTIGATION REFLECTION Resources: protractors, rulers, pencil, paper What could we do? Children: 1. construct quadrilaterals using a protractor and ruler, identifying the properties of the shape as guided by the teacher 2. identify and describe the diagonals What language could we use to ask questions and explain? What could we do? How could we use the protractor and ruler to construct a quadrilateral? Children: 1. sit in pairs 2. construct triangles and What are the properties of the quadrilaterals using a quadrilateral? protractor and ruler, identifying the How many diagonals? properties of the shape, Are the diagonals equal in length? that are neither too Do the diagonals cross at right easy nor too angles? challenging Are the diagonals axes of 3. identify and describe symmetry? the diagonals What language could we use to explain? I constructed a quadrilateral using a protractor and ruler, by making the sum of the angles 360 degrees. The properties of my quadrilateral are 4 sides and 4 angles, … equal sides, … equal angles, … pairs of parallel sides, … axes of symmetry. I drew the diagonals. There are … diagonals. The diagonals are / are not equal length. The diagonals do / do not cross at right angles. The diagonals are / are not axes of symmetry. What questions could children discuss and record a response to? How can we construct a quadrilateral with a protractor and ruler? What is a diagonal? How can we draw diagonals on quadrilaterals? Which quadrilaterals have diagonals that are equal in length? Which quadrilaterals have diagonals that are axes of symmetry? Which quadrilaterals have diagonals that cross at right angles? Children may be investigating concepts at a level that varies from other children. In one class, there may be children investigating the concept at Level 1 while another child is investigating the concept at Level 4, Level 12 or even higher. Regardless of the child's current grade, children need to investigate concepts at the level of their current understanding. This means that a child in a given grade, who has current understanding at Level 5, will investigate at Level 6, then Level 7 etc. If this makes you worried that they are investigating at a level much lower than their grade level, consider this: If the child is made to try to investigate at a higher level than their current level of understanding, they will be building on an unstable knowledge base with gaps, and will continue to use inefficient strategies often based on misconceptions, guaranteeing that their level of understanding will be the same at the end of the year as it was at the beginning of the year. If the child is allowed to investigate the concept at their current level of understanding, they will correct misconceptions, fill gaps in their understanding and build a firm knowledge base, as they move through the levels, investigating at a higher level by the end of the year. Website: http://www.alearningplace.com.au Email: [email protected] Twitter: @learn4teach YouTube: A Learning Place A Teaching Place Scan the QR Code Facebook: A Learning Place 7 OTHER LEARNING ACTIVITIES MEASUREMENT AND GEOMETRY – 2D 58G IDENTIFY DIAGONALS ON CONVEX TWO-DIMENSIONAL SHAPES, RECOGNISING THE ENDPOINTS AS THE VERTICES OF THE SHAPE, DRAW ALL THE DIAGONALS OF CONVEX TWO-DIMENSIONAL SHAPES, COMPARING THE DIAGONALS ON DIFFERENT SHAPES, IDENTIFY WHICH OF THE SPECIAL QUADRILATERALS (PARALLELOGRAMS (SQUARES, RECTANGLES AND KITES), RHOMBUSES AND TRAPEZIUMS) HAVE DIAGONALS THAT ARE EQUAL IN LENGTH, IDENTIFY WHETHER ANY OF THE DIAGONALS ARE ALSO LINES (AXES) OF SYMMETRY OF THE SHAPE: OTHER LEARNING ACTIVITIES These learning activities allow children to investigate and explain the concept in various situations. ‘Doing’ mathematics is not enough and is not a good indicator of understanding. As Einstein said, ‘If you can’t explain it simply, you don’t understand it’! Investigation takes time as children develop both the capacity and meta-language to explain mathematical concepts at their current level of understanding. Differentiate learning for children working at all levels of the concept, including those requiring extension, and allow children to differentiate their own learning, by varying the range and size of numbers investigated. In pairs, children construct special quadrilaterals, including regular quadrilaterals – squares, irregular quadrilaterals – rectangles that are not squares, rhombuses that are not squares, kites that are not rhombuses or squares, trapeziums (See Measurement and Geometry, 2D - Level 49G). They describe the side and angle properties and draw in diagonals, recording their data in a table, for example, Shape Vertices Sides Diagonals length Diagonals axes of symmetry? Diagonals cross at right angles? Square 4 equal 4 equal 2 equal yes yes Irregular Rectangle 4 equal 4, 2 opposite pairs equal 2 equal no no Irregular Rhombus 4, 2 opposite pairs equal 4, 1 opposite pair equal, 1 opposite pair unequal 4 equal 2 unequal yes yes 4, 1 adjacent pair equal, 1 opposite pair unequal 2, unequal yes yes 4, 2 adjacent pairs equal 4, 3 adjacent equal, 1 twice as long 2, equal no no 4, zero equal 4, zero equal 2, unequal no no Irregular Kite that is not a rhombus Isosceles Trapezium Non-isosceles Trapezium Children review the data in the table, identifying that quadrilaterals with ► 2 equal length diagonals also have 3 or 4 sides and angles equal ► diagonals that are axes of symmetry also have 2 or 4 adjacent sides equal ► diagonals that cross at right angles also have 2 or 4 adjacent sides equal Website: http://www.alearningplace.com.au Email: [email protected] Twitter: @learn4teach YouTube: A Learning Place A Teaching Place Scan the QR Code Facebook: A Learning Place 8 Children construct different quadrilaterals using a protractor and a ruler. They cut the quadrilateral into 4 parts with an angle is each part. They rotate the parts to place the angles together, identifying that the 4 angles create a rotation, which is 360 degrees, for example, In pairs, children each construct a quadrilateral from a description of its angles, for example 4 equal vertices, 4 equal sides, 2 equal diagonals, diagonals cross at right angles and diagonals are equal length They share and classify their quadrilateral. Children could also investigate concave quadrilaterals, for example, Children draw the diagonals, identifying that in a concave quadrilateral, a diagonal will lie outside the shape, for example, In small groups, children have 8 metres of string tied into a loop, and an extra length of string with which to construct and compare diagonals. Children determine a strategy to construct specified quadrilaterals using only the loop of string. They justify their shape by describing its side and angle properties. Children use the extra string to construct and compare diagonals of their quadrilateral. In pairs children have 2 craft sticks of the same length. They cross sticks at right angles (perpendicular) to create diagonals. They draw shapes using the ends of the sticks as the vertices of the shape. They describe the shape drawn based on the diagonals. In pairs children have 2 craft sticks of different lengths. They cross sticks at right angles (perpendicular) to create diagonals. They draw shapes using the ends of the sticks as the vertices of the shape. They describe the shape drawn based on the diagonals. In pairs children have 2 craft sticks of the same length. They cross sticks at non-right angles (non-perpendicular) to create diagonals. They draw shapes using the ends of the sticks as the vertices of the shape. They describe the shape drawn based on the diagonals. In pairs children have 2 craft sticks of different lengths. They cross sticks at non-right angles (non-perpendicular) to create diagonals. They draw shapes using the ends of the sticks as the vertices of the shape. They describe the shape drawn based on the diagonals. Website: http://www.alearningplace.com.au Email: [email protected] Twitter: @learn4teach YouTube: A Learning Place A Teaching Place Scan the QR Code Facebook: A Learning Place 9 In pairs, children make quadrilaterals with and without diagonals using chenille sticks in straws. They test the rigidity of quadrilaterals without diagonals and with diagonals by pushing with equal force on all 4 vertices at the same time. EXTENSION: In pairs children could investigate the relationship between the number of vertices and the number of diagonals. For example, a quadrilateral has 2 diagonals and 4 vertices. Each vertex is the meeting point of 1 diagonal and 2 sides, for example, So each vertex has 1 diagonal, but that would mean counting each diagonal twice – once at each end. In a quadrilateral, every vertex (4) has a diagonal to every vertex except itself (1) and the 2 adjacent vertices (2). So in a quadrilateral, 4 vertices have diagonals to 4 minus 1 minus 2 vertices. 4 minus 1 minus 2 = 1. So in a quadrilateral each vertex has 1 diagonal. Because each end of a diagonal ends in a vertex, that means we have counted every diagonal twice – once at each end. So we need to halve the number of diagonals = 2. (The formula in Year 7 is Number of vertices – 3 divided by 2, which means the number of vertices, minus the 3 vertices that each vertex does not have diagonal to - itself and the 2 adjacent vertices -, divided by 2 because we counted each diagonal at both ends - in both vertices) In pairs children investigate whether triangles have diagonals. EXTENSION: In pairs children could investigate diagonals in other two-dimensional shapes, pentagons, hexagons, octagons. They could investigate the relationship that they found between the number of vertices and the number of diagonals in quadrilaterals, in pentagons, hexagons, octagons. Website: http://www.alearningplace.com.au Email: [email protected] Twitter: @learn4teach YouTube: A Learning Place A Teaching Place Scan the QR Code Facebook: A Learning Place 10 PROBLEM SOLVING MEASUREMENT AND GEOMETRY – 2D 49G MEASURE LENGTHS OF SIDES AND SIZES OF ANGLES TO IDENTIFY THE SIDE AND ANGLE PROPERTIES OF TRIANGLES AND QUADRILATERALS, CONSTRUCT AND CLASSIFY TWO-DIMENSIONAL SHAPES, INCLUDING TRIANGLES AND QUADRILATERALS, FROM A DESCRIPTION OF THEIR SIDE AND ANGLE PROPERTIES, ENLARGE TWO-DIMENSIONAL SHAPES, COMPARING SIDE PROPORTIONS AND ANGLES AND IDENTIFYING ONLY THE AREA HAS CHANGED: PROBLEM SOLVING Problems allow children to investigate concepts in various situations. Any problem worth solving takes time and effort – that’s why they’re called problems! Problems are designed to develop and use higher order thinking. Allowing children to grapple with problems, providing minimal support by asking strategic questions, is key. Differentiating problems allows children to solve simpler problems, before solving more complex problems on a concept. Problems may not always be solved the first time they are presented. Returning to a problem after further learning, develops both resilience and increased confidence as children take the necessary time and input the necessary effort. As Einstein said, ‘It’s not that I’m so smart – I just stay with problems longer’. The problem solving steps may be followed to solve problems. Mark constructed a shape that had 4 equal vertices, 4 equal sides, 2 equal diagonals, diagonals cross at right angles and diagonals are equal length. What could the shape have looked like? Which one of these shapes has diagonals that cross at right-angles? Which one of these shapes has diagonals that are axes of symmetry? Which 2 of these shapes has diagonals that are unequal lengths? Website: http://www.alearningplace.com.au Email: [email protected] Twitter: @learn4teach YouTube: A Learning Place A Teaching Place Scan the QR Code Facebook: A Learning Place 11 Problem Solving Steps (back to Problems) 1. Read 2. Understand 3. Choose a strategy Read the part that is asking you to find out. Read the information you need to find it out. Think about what you could do to work it out. 4. Work it out 5. Check 6. Share Use your strategy to work it out. Read the part that asked you to find out. Share and compare your strategy and answer with a friend’s strategy and answer. Did you find it out? Website: http://www.alearningplace.com.au Email: [email protected] Twitter: @learn4teach YouTube: A Learning Place A Teaching Place Scan the QR Code Facebook: A Learning Place 12