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A Progression for the Development of Equations Stage 2 and 3 Story problems are told to the class. These are recorded by the teacher as equations For example John has six marbles and loses some. If he has 4 left, how many did he lose? Recorded as 6 - = 4 Problems should include all three styles – start unknown, change unknown and result unknown Students then solve these problems by using materials Stage 4 Students work with mentally solving written box problems of the type outlined above Consideration of problems where the quantity is doubled, leading to the teacher recording + = 6 Students consider more general problems like + = 9, with a range of possible answers Transition to stage 5 Teacher recording of the + = 6 story problem as 2=6 Teacher records stories with multiplications by 2, 5, 10 as box problems Students work with mentally solving written box problems of the type outlined above D:\533562673.doc 17/02/16 Teacher introduces number machines with a single operation. Students use a single machine to work out what would happen if various numbers are input. Students practice recording story problems involving +/- as box equations, then solve them numerically Transition to stage 6 Teacher records stories with multiplications as box equations Students mentally solve problems written as this sort of box equation Number machines to show more than one operation. Using flags to show a number machine in operation. Use tables to show how a number machine can change numbers Students practice recording story problems involving any single operation as box equations, then solve them numerically Stage 6 Students record simple story problems (with either +/- or / only) as drawings, then solve them. Recording input-output pairs for a number machine as ordered pairs Reversing a number machine by reversing the flags Inverse/opposite operations Transition to stage 7 Students mentally solve box equations like 3 + 4 = 19 (whole number solutions only) D:\533562673.doc 17/02/16 Students use drawings to help solve more complex word problems (involving two operations). Students convert drawings to box equations Stage 7 Students use drawings and/or box equations to solve word problems with unknowns on both sides of the equals sign Students use flags to identify what to do first in a box equation (The BODMAS link) Transition to stage 8 Replacement of the box with an x Students mentally solve equations like x 3 + 4 = 19 (whole number solutions only) Formal solving of equations as x+2=3 x+2–2=3–2 x =1 Students start working with formal solving of simple equations involving sets of numbers other than whole numbers Removal of the multiplication sign, and writing divisions in fraction form in equations Stage 8 Students work mentally and formally with solving a range of equation types Students write equations for, and solve, word problems D:\533562673.doc 17/02/16 The Development of Equations Accompanying Notes Equations start to appear very early on in the teaching of numeracy. Earliest experiences are found within the addition/subtraction domain when moving from one-to-one counting on materials (stage 2) to one-to-one counting by imaging (stage 3). At this level, students are generally too inexperienced to be doing their own recording of the problems, but will see their teacher recording a problem like “Peter had 8 marbles in his pocket, but lost some. If he ended up with 5 marbles, how many did he lose?” as 8 - = 5. What the students are expected to do however, is solving story problems mentally (as simple number problems), without recording them first in an algebraic form. In this progression, three threads can be seen in the developing use of equations. The first of these is the use of an equation to summarise (“mathematise”) a word problem. In this thread, students are recording a word problem using the symbols and notation of the mathematical language. Associated to this is the second thread, the task of solving the equation (to answer the problem). Progressively more formal techniques for this need to be introduced so students have strategies for dealing with the more sophisticated problems that are encountered in higher levels of the curriculum. The third thread is the concept of a “number machine” (which could be talked of as a computer). This thread links equations to the work done already on mappings and patterns, and is a useful context for the consideration of inverse operations and processes, which are essential to the development of formal equation solving. When x’x and y’s are added to the number machine, they also contribute to the starting point to functions and relations, linking with the development of ordered pairs and graphs through patterning. Things to note in the use of equations 1) Use of the equals sign Within an equation, the equals sign signals that what is on one side of the equation is equivalent to that on the other side. For example 4 + 2 = 6. This is different to the interpretation of most students, for a problem like 4 + 2 = , leads them to develop the understanding that = means “work out the answer”. Consequently, the concept that the equals signs is like a balance, with equal things on either side, needs to be developed. When writing equations, only one equals sign should be written on a line, this is to make sure that the equations that are written down make sense. For example:When Tonu was explaining how he got the answer to the problem 27 + 55 he said “I first added 27 and 50 to get 77 then I added 5 to get 82”. If this is written down as it is spoken, 27 + 50 = 77 + 5 = 82, which makes it look like 27 + 50 = 82, which is obviously wrong. This form of mathematics writing is called “Running arithmetic”, and should always be discouraged. Instead the problem and its solution should be written as: 27 + 50 = 77 77 + 5 = 82 2) Start unknown, change unknown and result unknown From very early on, students need to be exposed to problems that have the unknown in a variety of places. For example 2+3= D:\533562673.doc 17/02/16 +5=6 2+=5 The main issue within such work is the place of writing an equation to solve. Pupils may find it easy to solve a word problem mentally (especially if the “story” is told to them), or even be able to solve a written equation. However, taking a word equation and writing an appropriate sentence with the unknown in the right position may take some doing, so may not be introduced (or mastered) until much later than actually being able to work out the answer of a problem 3) Development of the use of the “box” a) b) c) d) 4) Learning Experiences to support the development of equations (a) (b) 5) A “box” stands for a missing number that needs to be found, as in the examples above Two different shapes of “boxes” stand for two different numbers in a problem. For example + = 9 In this circumstance children can explore a whole family of facts Two boxes the same can stand for the same number For example + = 6 This allows students to become multiplicative, leading to 2 = 6 and even 2 = 6 Two sided problems For example + +4 = + 12 In such problems children don’t have to know what the value of the box is to be able to realise that they can take one away from each side will leave a much simpler problem. A balance scale, some counters and some (black) film canisters are effective equipment. By putting counters into the canister, and balancing these against an empty canister and some displayed counters the concept of the equals sign as the balance can be addressed, and equations written for problems modelled by the equipment. By using the balance, the nature of an equation and its solution is easy to explore. For example. In examining the equation + 3 = 9, it is easy to see that if some counters are removed from one side of the scales, then to maintain the balance, the same number of counters need to be removed from the other side of the scales. This can be recorded as follows: +3=9 +3–3=9–3 =6 which is a standard way of recording the solution of equations, and can lead to discussions of inverse operations among other things. (However, this sort of formal recording/discussion by students should wait until stage 7, when they have a good understanding of all the operations, and are starting to identify them as something that have a meaning of their own.) Number Machines Number machines are another useful approach to the development of equations, and are an alternative to box equations. However, their introduction should be delayed to later stages of the framework, as their main use relies on a clear understanding of multiplication. Below is a simplified version of a number machine. They are very useful for operating generally, as any number can be dropped into the first box. They can also be used backwards, thus D:\533562673.doc 17/02/16 introducing inverse operations, which leads nicely into topics like changing the subject and finding an inverse equation. + +3 3 2 Once formal algebra has been introduced, number machines are an effective way of considering BODMAS within the algebra setting, as each equation can be made into a number machine, which of course relies on the understanding of which piece of mathematics to do first. For example: 2x - 5 = 6 What is done first – multiplying by 2 or subtracting 5? Notice: that reversing this order then provides a neat way to sort out the order in which the equation should be undone to find out the starting number, that is to solve the equation. The Dime resources have some useful booklets that cover this approach D:\533562673.doc 17/02/16 A Progression for the Development of Equations Stage 2 and 3 Story problems are told to the class. These are recorded by the teacher as equations For example John has six marbles and loses some. If he has 4 left, how many did he lose? Recorded as 6 - = 4 Problems should include all three styles – start unknown, change unknown and result unknown Students then solve these problems by using materials Stage 4 Students work with mentally solving written box problems of the type outlined above Consideration of problems where the quantity is doubled, leading to the teacher recording + = 6 Students consider more general problems like + = 9, with a range of possible answers Transition to stage 5 Teacher recording of the + = 6 story problem as 2=6 Teacher records stories with multiplications by 2, 5, 10 as box problems Students work with mentally solving written box problems of the type outlined above Teacher introduces number machines with a single operation. Students use a single machine to work out what would happen if various numbers are input. Students practice recording story problems involving +/- as box equations, then solve them numerically Transition to stage 6 Teacher records stories with multiplications as box equations Students mentally solve problems written as this sort of box equation Number machines to show more than one operation. Using flags to show a number machine in operation. D:\533562673.doc 17/02/16 Use tables to show how a number machine can change numbers Students practice recording story problems involving any single operation as box equations, then solve them numerically Stage 6 Students record simple story problems (with either +/- or / only) as drawings, then solve them. Recording input-output pairs for a number machine as ordered pairs Reversing a number machine by reversing the flags Inverse/opposite operations Transition to stage 7 Students mentally solve box equations like 3 + 4 = 19 (whole number solutions only) Students use drawings to help solve more complex word problems (involving two operations). Students convert drawings to box equations Stage 7 Students use drawings and/or box equations to solve word problems with unknowns on both sides of the equals sign Students use flags to identify what to do first in a box equation (The BODMAS link) Transition to stage 8 Replacement of the box with an x Students mentally solve equations like x 3 + 4 = 19 (whole number solutions only) Formal solving of equations as x+2=3 x+2–2=3–2 x =1 Students start working with formal solving of simple equations involving sets of numbers other than whole numbers Removal of the multiplication sign, and writing divisions in fraction form in equations Stage 8 Students work mentally and formally with solving a range of equation types Students write equations for, and solve, word problems D:\533562673.doc 17/02/16