Teaching and Learning Plan: Introduction to Patterns - A Guide for Educators, Assignments of Mathematical Methods

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ALGEBRA THROUGH THE

LENS OF FUNCTIONS

JUNIOR CERTIFICATE HIGHER LEVEL

PART 1 OF 2

Introduction

To Teachers

Algebra Through the Lens of Functions Part 1 and 2 have been designed by the Maths Development Team for use by teachers of mathematics. Algebra Through the Lens of Functions Part 1 of 2 should be used prior to engaging with Part 2. Both part 1 and part 2 treat the art of teaching algebra through the lens of functions through a series of units. The material contained in the document is suitable for all levels and abilities but is particularly suited to Junior Certificate Higher Level. It includes a discussion of activities, tasks and the formation of connections suitable for classroom use. When necessary, required subject matter content is covered as well. Both Part 1 and 2 were written in response to the many teachers who attended continuing professional development courses given by the authors and were unable to find material in a single convenient source. The authors collaborated with the Maths Inspectorate of the Department of Education & Skills to provide a collection of activities and strategies for Junior Certificate mathematics classes.

Activity Book

Visualising Patterns for Linear Relationships and Visualising Patterns for Quadratic Relationships are activity workbooks which can be used to supplement the text in Part 1 and Part 2 respectively. Visualising Patterns for Linear Relationships is an activity workbook which can be used to supplement this text. It contains ten patterns that may be used for each unit of the text. Visualising Patterns for Linear Relationships can be downloaded from here. Larger versions of the images are available in PDF form here and as an interactive PowerPoint here.

Organisation, Format & Special Features of the Units

Units 1 to 10 deal with linear relationships. Unit 1 introduces linear relationships through the visual stimulus of the “Dots Activity”. The need for the skills of representing variables using letters and substitution into linear expressions should become evident to students during this Unit. At this stage, students will recognise that substitution into an expression is a more efficient strategy than continuing a table when finding outputs for a given input. The use of visual patterns as a stimulus is continued in Unit 2 when solving linear equations is introduced. During Unit 2 students will recognise that solving an equation is a more efficient strategy than continuing a table when the output is known and the corresponding input is required. Different students may see the same visual pattern slightly differently and the skills of visualising, simplifying and factorising expressions are thus required. These are explored in Units 3 and 4 and it will become evident that as visual patterns become more complex, the range and variety of expressions that can be used to represent them increases. Unit 5 focuses on the Money Box Activity and examines the equation 𝑦 = 𝑚𝑥 + 𝑐 in a variety of ways. This is an extremely important Unit as all other functions can subsequently be compared to linear functions. Unit 6 begins the process of rearranging formulae. Again, students should understand the benefits of the skill before learning it. For example, it may be more efficient to rearrange an equation initially and then use substitution into the rearranged equation rather than solving the equation repeatedly to find outputs for given inputs. Rearranging formulae for non- linear relationships will be addressed later in the document. The efficiency of adding functions to solve problems is looked at in Unit 7, which also motivates the skill of simplifying the sum of like terms. Comparing linear functions and solving simultaneous equations and inequalities is introduced using the growing patterns of four sunflowers in Unit 8. The students will solve their first simultaneous equation problem using a table and graph and then engage with the problem using algebra. The need for the algebraic method is that the use of tables and graphs becomes inefficient and inaccurate if the solution has large or non-integer values. Unit 9 introduces the elimination method for solving simultaneous equations through a problem that can also be solved using trial and improvement. A similar problem with much larger numbers could be shown to students to illustrate the need for the greater efficiency of the elimination method. Unit 10 looks at algebraic expressions and algebraic equations that contain fractions.

General Overview

The central idea in this document is that it might be better for students to understand why a skill is required before learning it. In addition to endeavouring to make functions the focus of the document, multiple representations are used throughout and some links back to what can be done when students are studying Number to help students understand algebraic concepts are made. Therefore, the learning outcomes of both Strands 4 and 5 are included in this document. The intention is that by the time students are “finished” their work on Algebra that they are also “finished” their work on Functions. The remainder of this overview looks at (a) the flow of some Units in this document, (b) the rationale for the positioning of some algebraic skills and (c) key features of functions.

(a) The Flow of Some Units in this Document

Broadly speaking the flow below is used in many Units in this document to show students the need for the algebraic skills they are about to learn prior to learning the skill. Throughout the document multiple solution strategies are used. In many cases students can find the answer to various questions by analysing a table or by interpreting a graph before engaging with the problem using an algebraic method. This means that the algebraic method should make sense to the students as they already know what the answer should be. In many cases too the need for an algebraic method is obvious as other methods become be too tedious when the numbers are large or prove inaccurate when the answer is not a whole number. Students can bring their own thinking to many of the problems before they are introduced to the formal algebraic approach. By doing so, the students get a greater sense of what they are doing and why they are doing it, can recognise the value of thoughtful engagement with problem solving and appreciate that algebraic techniques offer them incredibly powerful ways of tackling problems. Students should then have a greater appreciation of where algebra fits into mathematics as a whole and how understanding algebraic relationships and techniques is worthwhile.

1. Students engage with a problem

Students engage with a problem through (i) whole class discussion led by teacher questioning, (ii)

working in groups, (iii) individual work or (iv) a combination of some or all of the above. Following

this, the students implement strategies such as drawing diagrams, analysing a table, trial and

improvement or interpreting a graph. The effectiveness of these strategies are then compared by

the students through discussion.

2. Students see the need for a new strategy

The students engage with a follow-on problem where the limitations of earlier strategies become

apparent and the need for a new strategy is obvious.

3. Students are guided by the teacher to learn the new strategy

Students are guided by the teacher to learn the new strategy. In this document the new strategy

will always incorporate algebraic solutions.

4. Students compare the new strategy with the previous strategies

Students compare the new strategy with the previous ones to see the advantages of the algebraic

approach and to recognise that algebraic solutions can be checked by using other methods.

(b) Rationale for the position of some algebraic skills

It is acknowledged that students may have prior knowledge (of using the laws of indices when multiplying numbers with a common base or using the distributive law when working in number, for example), but it is envisaged that moving from expanding expressions of the type 3 (𝑥 + 2 ) to more complex examples like 𝑥(𝑥 + 2 ), (𝑥 + 2 )(𝑥 + 3 ) or (𝑥 + 2 )(𝑥^2 + 3 𝑥 + 4 ) is delayed until the need for such expansions is obvious. For this reason, work on linear relationships is treated separately from that on quadratics and cubics. This also brings the additional benefit that when a student looks at (𝑥 + 2 )(𝑥 + 3 ) for the first time they will see it from a number of perspectives and not just as an algebraic skill.

Once a skill is learned in one Unit it can then be used in all subsequent Units. For example, substitution is learned in the “Dots Activity” in Unit 1 and while it might not be explicit in the Units that follow it should be seen as a key skill in developing understanding in those Units too.

(c) Key Features of Functions

Throughout the document the key features of functions are referred to. These features can be used when analysing functions so students can use the same criteria for analysing the various functions they encounter as they progress from first year through to sixth year. The key features of functions are:

1. The domain and range
2. Where the graph of the function meets the axes.
3. What is constant and what varies in the function?
4. The behaviour of the graph of the function
5. The rate of change of the function Note: Average rates of change can be used to begin the discussion about how rates of change can be positive, negative or zero and how this can be used to decide if the function is increasing, decreasing or neither. This work can be continued at senior cycle when the slope of the tangent to the function can be dealt with.

How to use this document

Throughout the document there are hyperlinks to useful resources, for example, booklets of visual patterns, matching activities, Teacher Resource Booklets from the various workshops and Teaching and Learning Plans. Clicking on a hyperlink will bring you to the resource. The document also contains boxes entitled “Number Work” which provide ideas that should be used when students are studying number, per se, but that can also help students see properties of number that are important for algebra. They are included in this document so that they can be revisited when students encounter the related concept in algebra. Each Unit contains a number of Sample Problems that can be used with students.

𝑠^2 − (𝑠 − 1)^2 is possible but might not be the most important thing at this stage with a first-year class.

Next, Near, Far, Any

The phrase “next, near, far, any” is a useful one to keep in mind when working with visual patterns. Progressing through “next”, “near”, “far”, “any” should enable students to see the advantages of generalising. In this context “next” is the next (or 5th^ ) stage. There are 9 dots in the 5th^ stage. This answer can be found using many strategies, for example, drawing the stage or continuing a table. In this context “near” is the something like the 10th^ stage. There are 19 dots in the 10th^ stage. This answer can be found using many strategies, for example, drawing it or continuing a table. However, drawing all the intervening stages is time consuming. In this context “far” is something like the 100th^ stage. There are 199 dots in the 100th^ stage. Drawing the intervening stages or continuing the table to the 100th^ stage will be too time consuming. Asking how many dots there are in a far stage encourages students to choose to generalise themselves. Asking students how many dots there are in “any” stage is explicitly asking them to generalise. It is hoped that they would have chosen to generalise when asked how many dots are there in a “far” stage.

Notes:

1. In addressing questions of the type “Find a relationship between the stage number and the number of dots”. It is important to emphasise that a relationship should contain; =, <, >, , or  (or a verbalised version of these). For example, students might say the relationship is 𝑠 + 𝑠 − 1, where 𝑠 is the stage number. Strictly speaking, the relationship in the “Dots Activity” should resemble 𝑡 = 𝑠 + 𝑠 − 1, where 𝑠 is the stage number and 𝑡 is the total number of dots in each stage.
2. While the students will come up with an expression it should take the form of an equation , for example, Number of Dots in each stage = Twice(𝑠)– 1
3. If the students do not organise their thoughts with a table then they should be shown a table of inputs and outputs. Plotting points is part of the Common Introductory Course (CIC) and should be addressed before introducing the students to algebra. This will enable students to plot each input and corresponding output as a set of points and to see the relationship in another representation. Information and communication technology (ICT) can be used to create many sets of inputs and outputs and plot the sets of points.
4. Variables: 𝑠, which represents the stage number in the relationship is a variable, as the stage number varies. 𝑡, which represents the total number of dots in each stage is also a variable. This topic opens up the possibility of in-depth collaboration with the science department at some stage as the concepts of independent and dependent variables will be met by those students studying junior certificate science.
5. Constants: The “–1” in the expressions or relationships is constant.

A Challenge for Homework

The following problem could be posed to students: “Which stage contains 47 dots?” Formally solving problems like this will be the focus of the next Unit but it is useful for students to engage with this type of problem in an informal way before engaging with it more formally. Trial and Improvement: Some students in the class should be able solve the problem by using trial and improvement. For example, stage 20 has 20+20–1 dots which is too few, stage 30 has way too many dots, stage 25 is getting very close and stage 24 has 24+24–1 dots which is just right. Using a Table: Some students could make out a table and continue the table to stage 24. Interpreting a Graph: Some students could make out a graph of points and see that the output is 47 when the input is 24. “Undoing the Relationship”: Some students might try to "undo the relationship”. They might do this incorrectly by reducing the 47 by one and dividing by two to get stage 23 or they might do it correctly by increasing the 47 by one and dividing by two to get stage 24. Both approaches should form the focus of subsequent whole-class discussion. Note: The description above focuses on one part of the relationship, the 47. It would be beneficial to share with students a solution that encapsulates the whole relationship. Twice the unknown stage number minus one is 47. This means twice the unknown stage number is 48. Therefore the unknown stage number is 24.

Notes

1. A more difficult equation might also be introduced to justify the need for the procedure.
2. Unknown: There is an opportunity here to talk about the difference between 𝑛 as a variable in the expression 3𝑛 + 2 compared to the relationship 𝑡 = 3𝑛 + 2 where 𝑛 is an unknown and 𝑡 is a variable. A table or a graph can also be used to show that 𝑛 can take on many values but when we are solving we are only interested in the value(s) that satisfy the equation in question.
3. It is a good idea to use a table and/or a graph the first time students formally solve an equation so that they can see the solution in these representations and can understand what they have just done. The “Challenge for Homework” from the previous Unit could be returned to now with the new knowledge and thus solving the equation 2𝑛 − 1 = 47 formally.

Inputs and Outputs and Informally Looking at Some Concepts of Functions

The phrases input and output can be used by both the teacher and students from quite an early stage for activities similar to those in Units 1 and 2. Substitution is used to find the output for any given input. Solving equations provides an input from a given output. Independent and dependent variables can also be discussed. It is worth noting that the concepts of independent and dependent variables will be met by those students studying junior certificate science.

Questioning can be used in laying the foundation for the concepts of domain , co-domain and range. Three examples of the type of questioning that could be considered are given below. This type of questioning could also be used to revise the different number systems and make a link to discrete and continuous data. The questions are designed to lay the foundation for the concepts of domain , co-domain and range. Once the concept is understood the terminology can be introduced at the appropriate time. This could be at any stage from first to third year.

Question 1. How could the inputs be described? Possible student answers might include:

1. The inputs are positive whole numbers.
2. The natural numbers
3. {1, 2, 3, 4 …} The concept of the domain is introduced here in a meaningful way as there is a context to which to the concept can be linked. Students should be able to see that neither stage – 32 nor stage 2¾ make any sense. Over time it might seem unwieldy to keep saying the set of inputs so there becomes a need for the term domain. When students are ready, they can be told that the set of inputs in a relationship is called the domain and this should be then used interchangeably with the phrase “set of inputs”.

Question 2. How might the outputs be described? The students might respond that the outputs are all positive whole numbers. While this answer is broadly correct it lacks rigour and opens the possibility of discussing the idea of the co-domain. The outputs are indeed all positive numbers, but all the positive whole numbers are not included in the set.

Question 3. Can you define the outputs more specifically? The students may describe the output as the set {5, 8, 11…} or the whole numbers starting with five and increasing constantly by three thereafter, etc. The discussion that ensues allows for the definition of the range , as the set of outputs, to be introduced and explored

Thinking ahead Examples 1 and 2, below, indicate that this type of approach can be revisited for continuous linear and quadratic relationships: Example 1: A sunflower with a starting height of 2 cm and a growth rate of 3 cm per day could be modelled by a function that has a domain of 𝑥ℝ, 𝑥0, a co-domain of 𝑦ℝ, 𝑦0 and a range of 𝑦ℝ, 𝑦2. Example 2: A quadratic like 𝑥^2 − 9 could have a domain of ℝ, a co-domain of ℝ and a range of 𝑦ℝ,

𝑦 – 9

E

E

E

E

W

Multiply 𝑛 by three, then add 4.

W

Divide 𝑛 by two, then add 6. W Multiply 𝑛 by two, then add 6.

W

Multiply 𝑛 by two, then add 12. W Add three to 𝑛, then multiply by two.

W

Multiply 𝑛 by three, then add 12. W Add six to 𝑛, then divide by two.

W

Add four to 𝑛, then multiply by three.

The solutions to the above activity are: A1 E7 W A2 E1, E8 W4, W A3 E2 W A4 E4 W A5 E5 W A6 E3, E6 W6, W

An extension to this activity would be to ask students to create their own expressions and represent them using words, area etc.

Note: Matching 2(𝑛 + 3) and 2𝑛 + 6 with the diagram below could be the first time students will have seen the distributive law containing variables.

Number Work Understanding factors incorporating prime, composite and square numbers would all help with the previous and later activities, including, for example, simple factorising, factorising by grouping and factorising quadratics. By building all the possible rectangles from whole numbers of unit squares (for example 2, 3, 4, 5, 6, 7, etc.) students will see that some numbers only have a very limited choice in how a rectangle can be built, for example, 2, 3, 5, 7 etc. ( prime numbers ).

Some numbers have more choice in how a rectangle can be built, for example, 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20 etc. ( composite numbers ). Some of this second set of numbers can be arranged into a square, for example, 4, 9, 16, 25 etc. ( square numbers ).

A discussion can be had about how the number 6 can be represented as 3 groups of 2 and also as 2 groups of 3 to illustrate the commutative law.

B. Drawing The aim of this activity is to visualise like and unlike terms, the distributive law and equivalent expressions. A similar activity to this focussing on the quadratic, was used in the Workshop 5 Teacher Resource Booklet. This document, however, focuses on linear expressions for now. Students are asked to draw the arrays to represent: 𝒙, 𝒚, 𝟐𝒙, 𝟐𝒚, 𝟐𝒙 + 𝟐𝒚, 𝟐(𝒙 + 𝒚), 𝟑𝒙 + 𝟑𝒚, 𝟑(𝒙 + 𝒚) where 𝒙 ≠ 𝒚. Students should be encouraged to use squared paper for this activity. Prompting students to see 𝑥 as 1𝑥 or +1𝑥 and 𝑦 as 1𝑦 or +1𝑦 may also prove useful.

Unit 4: Algebraic Expression Skills

In this Unit students will:  factorise  rearrange areas into rectangular shapes so the area can be expressed as a product of factors  draw an area representation when given a linear expression expressed in words or symbols  visualise some transformational activities e.g. collecting like terms, simplifying expression, expanding and factoring.  add and subtract simple algebraic expressions of the form (𝑎𝑥 + 𝑏𝑦 + 𝑐) ± (𝑑𝑥 + 𝑒𝑦 + 𝑓)

Some informal work on simplifying the sum and difference of like terms took place in previous units,

for example, in the “Dots Activity” students may have come up with expressions such as “twice 𝑠–1”,

“𝑠+𝑠–1” and “1+𝑠–1+𝑠–1” are all equivalent. This Unit focusses on formally simplifying the sum and

difference of like terms and simple factorising using areas of rectangular shapes. Students will

sketch and understand expressions of the form 𝑎(𝑏𝑥 + 𝑐).

Sample Problem 1

(i) Describe the relationship between the stage number and the total number of tiles. (ii) Describe the relationship between the stage number and the number of red tiles.

Prompting students to draw the first three stages themselves and to draw the next stage in the pattern might help students understand the pattern. Questions like “Where can you see three tiles in stage three” can also help students.

Colour is used so students can see the relationships in a variety of ways: Blue Some students will see two blue tiles in each row and add up three twos i.e. 2+2+2 = 6. Other students will see two blue tiles in each row and see three rows and multiply three by two i.e. 3(2) = 6. Other students will see a total of six blue tiles in each stage. Red Some students will see three red tiles in the first stage, six in the second stage and nine in the thirds stage meaning the total number of reds in each stage is three times the stage number i.e. 3𝑥. Some students will see a stage number of red tiles in each of the three rows of the pattern i.e. 𝑥 + 𝑥 + 𝑥. Total The relationship between the total number of tiles, 𝑡 and the stage number, 𝑥, can be written as 𝑡 = 𝑥 + 2 + 𝑥 + 2 + 𝑥 + 2 or as 𝑡 = 3(𝑥 + 2) or as 𝑡 = 3𝑥 + 6.

Notes:

1. A table and a graph could be used to represent the inputs and outputs at this stage. Alternatively, this could be deferred until the next Unit where these representations will be looked at in detail.
2. Many students, from their work with the order of operations in number, might want to “do” the brackets first if they were asked to expand 3(𝑥 + 2) i.e. they might want to combine the

𝑥 + 2 into something like 2𝑥 and then triple the result to get 6𝑥. Work with area representations should show that 𝑥 + 2 and 2𝑥 are not equivalent and that 3(𝑥 + 2)^ is different from 6𝑥.

Factorising Sample Problem 2 The green rectangles below have dimensions 𝑥 and 1 and have an area of 𝑥 units squared. The yellow squares have side lengths of 1 unit and thus an area of 1 unit squared. How can all the individual areas in 6𝑥 + 3 be arranged into the form of a rectangle? What are the dimensions of this rectangle? Rearranging the objects reveals that the dimensions are 3 and 2𝑥 + 1 and ultimately to the realisation that the factors of 6𝑥 + 3 are 3 and 2𝑥 + 1 giving the factorised expression as 3(2𝑥 + 1).

Notes:

1. Acknowledge 1(6𝑥 + 3) is a legitimate way of factoring the expression, but as 6𝑥 and 3 are not mutually prime, further factorisation is possible. In fact it is always the highest common factor of both terms that is used in factoring such expressions.
2. This would be an opportune time for students to practice a range of questions involving expanding and factorising expressions, for example 4𝑥 + 8 = 4(𝑥 + 2) or 2(2𝑥 + 4) = 1(4𝑥 + 8) etc.
3. To solidify the concept of factorising it could be beneficial when, students multiply two expressions, that they are then asked to state the factors of the expression formed.
4. The context of area can be used throughout for many application questions.
5. Questions like simplifying 3(2𝑥 + 1) − 2(𝑥 + 2) and 𝑎(𝑏𝑥 + 𝑐𝑦 + 𝑑) + 𝑒(𝑓𝑥 + 𝑔𝑦 + ℎ) in a purely mathematical context can also be done.

Attention should be drawn to the €3, in all representations above. The €3 occurs in the story. The €3 is the amount of money when the time elapsed is zero in the table. The €3 is the 𝑦-intercept of the graph. €3 is the amount before a number of €2 is added on in the other table. The €3 can also be seen in the relationship expressed in words and the relationship express in symbols. The constant rate of change, the €2 per day, should be made obvious in all of the representations above. The €2 occurs in the story. The €2 is the change in the outputs between each day in the table. The €2 is the slope of the dotted line in the graph. €2 is the amount added on each day in the other table. The €2 can also be seen in the relationship expressed in words and the relationship express in symbols.

Notes:

1. It is important to note that in this instance the time elapsed is measured in increments of one day. Students will encounter linear relationships with other increments as they progress through their studies.
2. This function in this question does not map the reals to the reals. It maps an arithmetic progression to an arithmetic progression with the ratio of the common difference of the outputs to that of the input being the rate of change.

Work can continue on substitution and solving equations using problems similar to the ones below. Sample Problem 2 (Substitution) How much money does John have in his money box 4 days after he got the box? The answer can be found by (i) analysing the table, (ii) identifying what the output is when the input is the 4th^ day on the graph, (iii) substituting the 4 into the word expression or (iv) substituting the 4 into the algebraic expression 3 + 2𝑑 and/or the function 𝑓(𝑑) = 3 + 2𝑑. “How much money does John have in his money box 100 days after he got the box?” While the answer can be found by using any of the four methods outlined immediately above, it is unlikely that the table or graphical methods merit consideration as they are inefficient given the size of the numbers involved.

Sample Problem 3 (Solving an Equation) John wants to buy a new book. The book costs €13. What is the minimum number of days John will have to save so that he has enough money to buy the book? The answer can be found by (i) analysing the table, (ii) identifying what day has an output of €13 on the graph or (iii) solving 3 + 2𝑑 = 13.

Sample Problem 4 (Solving an Equation) John wants to buy a new computer game. The game costs €69. What is the minimum number of days John will have to save so that he has enough money to buy the computer game? Again the answer can be found by using any of the three methods immediately above however it is unlikely that analysing the table should be considered as it is inefficient given the size of the numbers involved.

The Key Features of Functions including the Domain, Range, etc. should be explored and discussed as an integral part this work. Therefore for the function in Sample Problem 1 above for example, the students will be guided to recognise that:

1. The domain is the set of non-negative integers and the range is the set of multiples of two that are greater than or equal four.
2. The function has a 𝑦-intercept of 3 and the function does not have an 𝑥-intercept.
3. The starting value of three and rate of change of two are constant while the number of days elapsed, 𝑑, and the amount of money in the money box, 𝐴, are varying.
4. The outputs of the function are always positive and the function is ever-increasing.

5. The average rate of change between outputs when the input is increased by one is constant and is €2 per day.

Notes:

1. It is not feasible to look at the instantaneous rate of change of the function as it does not map the reals to the reals (and also the slope of the tangent to the function is reserved for the senior cycle).
2. Activities from Modular Course 3 that utilised different stimuli could also be used. Some of these activities start with a story, others start with a graph while others start with a table. If students can see the 𝑦-intercept (𝑐) and (𝑚) in many representations it should mean we can assess their understanding by seeing if they are able to sketch equations/functions of the form 𝑦 = 𝑚𝑥 + 𝑐 and move from a sketch to 𝑦 = 𝑚𝑥 + 𝑐. Questions with negative slopes and fractional slopes should be done too.
3. If it has not occurred naturally already in the questions you use after the “Dots Activity” students should be shown how to go from (1, 5) (2, 8) (3, 11) to (1, 5), (2, 5+3), (3, 5+3+3) and to (1, 5),(2, 5+1(3)),(3, 5+2(3)) and finally to (𝑛, 5+(𝑛–1)(3)). Inputs Outputs 1 5 5 5+0(3) 2 8 5+3 5+1(3) 3 11 5+3+3 5+2(3) 4 14 5+3+3+3 5+3(3) 5 17 5+3+3+3+3 5+4(3)

Unit 6: Rearranging Linear Formulae

In this Unit students will:  understand why it is useful to be able to rearrange formulae  rearrange linear formulae

Repeatedly solving for 𝑥 when you have an equation in the form 𝑦 = 𝑚𝑥 + 𝑐 is inefficient. The purpose of the Sample Problem below is to show students that there is a need to find a more efficient way of finding inputs when we are given outputs.

Sample Problem John receives a gift of a money box containing €3 for his birthday. John decides he will save €2 every day, beginning the day immediately following his birthday. After how many days will John have (i) €23 (ii) €45 and (iii) €125 in his money box?

Rearranging the formula 𝑦 = 2𝑥 + 3 provides a more efficient approach here.

For 𝑦 = 2𝑥 + 3 if we could isolate 𝑥 then we would have something really useful. Students should be able to verbalise the steps required to do this i.e. taking 3 from both sides and dividing both sides by 2.

𝑦− 3 2