-5 -4 -3 -2 INTRODUCTION -1 TO ALGEBRA - II 0 PADMAPRIYA SHIRALI 1 2 3 4 5 INTRODUCTION This article is Part II of the series ‘Algebra – a language of patterns and designs.’ The approach is based on the perception of algebra as a generalisation of relationships. In Part I [http://teachersofindia.org/en/article/introduction-algebra-right-angles-pullout], we introduced the ideas of variable, constant, term and expression via numerical patterns. Various operations (addition, subtraction, multiplication) involving terms and expressions were also studied. Now in Part II, we revisit the usage of words such as variable, term and expression and concepts and operations involving terms and expressions in the context of geometric designs (line designs, 2-D designs, 3-D designs). Geometric designs are seen everywhere. Tile designs on floors, brick or stone work on walls, partitions of windows and doors, cardboard boxes (toothpaste boxes, soap boxes …) and many other everyday objects can be described in algebraic form. As always, we begin with familiar concrete objects and use algebraic language to describe them before moving progressively to abstract algebraic expressions. Ideally students should be exposed to algebra initially through the pattern approach followed by the design approach. However, the two approaches are independent of each other. By approaching algebra through different routes, we will be able to make a robust link between informal algebra at the primary stage to the more formal algebra which students encounter later. Also it will facilitate students’ fluency in the language of algebra, i.e., understanding variables and symbols and being able to use algebraic rules correctly. Prior knowledge: Students need to be familiar with notions such as line segment, length, region and area of a rectangle, area of a square, capacity, volume of a cuboid and volume of a cube. Keywords: Algebra, language, pattern, geometric design, variable, constant, term, expression, operation. 2 Azim Premji University At Right Angles, July 2018 ACTIVITY 1 Objective: Introduction to design language (in the context of line designs) and the usage of variables for different lengths. Materials: Sets of straws or straight sticks of different lengths. Dot paper A few examples of real life situations: How do we describe this ladder? How many l’s? How many s’s? Here l is the length of the long stick and s is the length of the short stick. Initially students can make designs using two different lengths and describe it as an expression. Later on they can make designs using three or four different lengths. The ladder is made up of some long sticks and some short sticks. If we take the length of the long stick as ‘l units’ and the length of the short stick as ‘s units’, we can describe the ladder design as 2l and 5s or 2l + 5s. How do we describe this house? We can express the house design as 5l and 3s. Here is a design made of three different lengths. It can be expressed as 3a + 2b + 4c. We can also express it as 5a+3b (where a stands for the length of the long stick and b stands for the length of the short stick). Students can now be given designs of a similar kind to describe using appropriate design language. Set them exercises of the reverse type as well. Give students dot paper and ask them to create designs for some given expressions. Example: 3a + 2b + 4c How do we describe this fence? Example: a + b + b + a Azim Premji University At Right Angles, July 2018 3 ACTIVITY 2 Objective: Addition and subtraction of expressions through line designs Materials: Straws or sticks, Dot paper Ask the first student to build a design to show a given expression, say, 4p + 3q. Example: 2s + 3t + u, 5s + t + 3u Ask the second student to show 2p + 1q In a similar manner one can demonstrate subtraction of expressions. Lay out the design for an expression, say, 5a + 3b + 3c. Verify that the second student has chosen the straws that correspond to the lengths p and q chosen by the first student. If not, it is an opportunity for discussing that p and q stand for specific lengths and line segments of the same length will be represented by the same letter. The students need to understand that different variables represent different numbers. Circle the sticks to be removed. Say, 3a + b + 2c. Now, how do we read these two designs together? They will be read as 6p + 4q. What is left? Ask students to record the design and expressions in a dot paper to show addition of expressions. Give students a few more sets of expressions for which they can make corresponding drawings and sum them. 4 Azim Premji University At Right Angles, July 2018 2a + 2b + c. This operation can be recorded on the dot paper as shown. Give students some line segment designs as shown here to record the expressions and the subtraction of the given expressions. Note: Since these are concrete examples the teacher cannot, as yet, give examples of the following kind: Subtract a + c from 2a + 3b. Give students a few more sets of expressions for which they can make corresponding drawings, subtract them and give the expression for what is left, like the following: • Set up a line segment design to show subtraction of 4a + 2b + c from 7a + 5b + 3c. They will give the expression for the initial design set up, then for the removed set (lines enclosed within loop are to be removed) and finally for the remaining set. • Set up a line segment design to show subtraction of 5c + 8d + e from 10c + 8d + 3e + f. ACTIVITY 3 Objective: Design language for plane regions involving two variable terms. Materials: Rectangles of different sizes (collection of visiting cards and greeting cards will help) Prior knowledge: Familiarity with area formula for rectangles and squares. Here, again, it is good to start using examples from real life situations before moving into abstract designs. Revise the concept of area of rectangle and square using grid shapes as shown. The length and breadth of the rectangles can be named using variables. How do we describe this set up? It would be 3ab + 2cd. What will be the expression for this set up? Set up a design with rectangles of two different sizes. It would be 2ab + 4ef + gh. It is possible that different rectangles may have one common edge. Discuss the need for the usage of the same variable in such situations. This possibility is taken up in the next activity. Azim Premji University At Right Angles, July 2018 5 Give students some plane region designs as shown to draw in the dot paper and record the expressions. Similarly give some expressions for which they need to make corresponding plane designs, Example: 3ab + 2pq + mn, 2a2 + 3b2 + c2, ab + a2 + b2. ACTIVITY 4 Objective: Design language for composite figures. Materials: Rectangles and squares of different sizes which have one edge in common. Dot paper Through this, the law ab + ac = a(b + c) (i.e., the distributive law) is established. This can be further reinforced by assigning numerical values to the variable and demonstrated through dot arrays as shown. When two rectangles or a rectangle and a square have one edge in common they can be combined to form a composite figure as shown. 3 × 5 + 3 × 7 = 3 × (5 + 7) What will be the expression for this set up? Note for the teacher: This will give rise to multiplication of a two term expression by a single term (binomial by a monomial). Teacher can first place the shapes separately and state the design language as ab + ac. It would be jk + jk + jk = j(k + k + k) = 3jk. Now the shapes can be brought together and stated as a(b + c). 6 Azim Premji University At Right Angles, July 2018 What is the expression for this set up? 2ab + 2ab = 4ab. Give students some plane region designs as shown above to draw in the dot paper and record the expressions. Similarly give some expressions, as given here, for which they need to make corresponding plane designs and write the expression for the composite figure. kl + l2, 5 × 5 - 5 × 2 = 5 × (5 - 2) A few more such examples can be discussed. 3pq + 2pr + p2 pq + pr + ps At this point teacher can also discuss the design for ab - ac. It would look like the following. ab + cb + db + fb Ask students to show using dot paper: a(b + c + d) = ab + ac + ad. p2 + pq + pr = p(p + q + r). Again it can be established that ab - ac = a(b - c). This can be further reinforced by assigning numerical values to the variable and demonstrated through dot arrays as shown. 2 × 5 - 2 × 3 = 2 × (5 - 3) Azim Premji University At Right Angles, July 2018 7 ACTIVITY 5 Objective: Rules about addition, subtraction of like and unlike terms, Multiplication of expressions by a single variable Materials: Rectangles and squares of different sizes. Dot paper Vocabulary: Like rectangles, unlike rectangles, Like terms, unlike terms Ask students to pick up any two rectangles. Pose the question ‘are these rectangles alike?’ or ‘are these rectangles different?’ 5pq - 2pq = 3pq If the two rectangles have the same length and breadth they are like rectangles. Students may pick up two rectangles which have the same length but a different breadth or which have the same breadth but different lengths or which have different lengths and different breadths. How do we describe them? If two rectangles differ either in length or breadth or both they are unlike rectangles. 4cd - cd = 3cd Also show them that unlike rectangles are referred to by using unlike terms and unlike terms cannot be added or subtracted. Show them that like rectangles are referred to using like terms and that like terms can be added only to like terms. Similarly like terms can be subtracted only from like terms. 4xy + 2ab Let students draw figures in the dot paper to demonstrate the following. ab + ab + ab = 3ab. 3mn + 5rs - mn - 3rs = 2mn + 2rs. Example: 3ab + 4ab = 7ab 8 Azim Premji University At Right Angles, July 2018 Students will now be in a position to make the transition to handle addition and subtraction of sets of abstract expressions of the following type: Add: Subtract: Multiply: 2ab + 3cd + ef 5ab + 4cd + fg a + 2b + c with d ab + 2cd + ef ab + 3cd + fg 4a2 + 6b2 + c2 2a2 + b2 + 3c2 ACTIVITY 6 Objective: Volume Materials: unit cubes, triangular. Dot paper Initially let students build cubes and cuboids of different sizes. Students can record these on triangular dot paper (also called isometric dot paper). Let them note down the dimensions in a table format (length, breadth, height, volume) to discover the formula for the volume of a cube or a cuboid. They can fill the volume column by counting the number of cubes and noticing the relationship between the length, breadth and height. Since the volume of 3D blocks is given in terms of length, breadth and height, the design language involves the use of the product of three variables. Azim Premji University At Right Angles, July 2018 9 ACTIVITY 7 Objective: Design language for sets of blocks Materials: Cuboids and cubes of different sizes and same size. Cardboard boxes of the same kind (soap boxes, toothpaste boxes) Prior knowledge: Volume of cube and cuboid Vocabulary: Like terms, unlike terms Show that cuboids having same length, breadth and height are referred to by like terms as shown. Example: 3abc + 2def Sides of unlike cuboids will be indicated by different letters (variables) and are referred to by unlike terms. Teacher can place different combinations of boxes and get the children to describe the set up using design language. ACTIVITY 8 Objective: Design language for combined blocks. Multiplication and factorisation Materials: Cuboids and cubes which have common faces can be brought together. Design language for them can be given initially considering the blocks separately and then as combined. abc + abc + abc = 3abc 10 Azim Premji University At Right Angles, July 2018 Show that pqr + tqr + sqr = (p + t + s) qr At this point students should now be ready for addition and subtraction of abstract expressions of the regular kind. For example: Add: Subtract: Multiply: Factorise: abc + 2def 5pqr + 3uvw p(p2 + pq + pr + qr) a3 + a2 b + a2c pqr + 2uvw 3abc + 4def 2a b + b c + c a 7a2b + a2c + 3a2d 3a2b + 4b2c + c2a 5a2b + a2c 2 2 2 2 2 2 a b + 2b c + c a CONCLUSION By the time we complete the activities suggested in Part I and Part II in this series, students should feel comfortable in the use of algebraic concepts and words such as variable, term and expression, and in performing various operations using them. They should be in a position to manipulate similar abstract expressions. Once the general principles of ‘like’ and ‘unlike’ terms have been understood and the rules of operations internalized, students should be ready for the take-off stage. They should be able to use the same laws and principles with higher powers and multi-variable terms and expressions. Part III of this series will be on approaches to equations. Azim Premji University At Right Angles, July 2018 11 Padmapriya Shirali is part of the Community Math Centre based in Sahyadri School (Pune) and Rishi Valley (AP), where she has worked since 1983, teaching a variety of subjects – mathematics, computer applications, geography, economics, environmental studies and Telugu. For the past few years she has been involved in teacher outreach work. At present she is working with the SCERT (AP) on curricular reform and primary level math textbooks. In the 1990s, she worked closely with the late Shri P K Srinivasan, famed mathematics educator from Chennai. She was part of the team that created the multigrade elementary learning programme of the Rishi Valley Rural Centre, known as ‘School in a Box’ Padmapriya may be contacted at padmapriya.shirali@gmail.com Padmapriya Shirali 12 Azim Premji University At Right Angles, July 2018