Lesson 5.02: Algebra

Algebra is an incredibly powerful tool for solving problems—so powerful that it often feels like it's the only way we’re allowed to do math. But sometimes, jumping straight into an equation feels like trying to speak a new language without knowing the alphabet. Today, we are going to learn how to use a strategy you already use—guess-and-check—to help you build perfect algebraic equations every time

Function Machines

are informal diagrams that help us visualize the relationships within a problem. Unlike the formal functions you might have encountered in advanced math courses, these function machines are more flexible and have fewer rules. They are designed to help us break down and understand complex problems by clearly laying out their logical structure

Algebra

on the other hand, is a familiar topic from your math courses. However, in Thinking Skills, our focus is on expressing problems in algebraic terms. This requires creativity and imagination, as it involves framing the problem rather than just manipulating algebraic symbols. While we will touch on some basic algebraic manipulations, our main goal is to use algebra as a tool to aid in problem-solving.

Function machines are just diagrams which encode the different moving parts of a problem and how they relate. They use boxes and arrows to represent objects and processes, but they are meant to be adaptable. You should aim to develop a few good habits which will allow your diagramming to work, and be willing to break those habits if the problem needs it.

Sample Question

Try this problem yourself, before reading the analysis

Charles’ journey from his home to his school takes a varying amount of time depending on when he leaves, the volume of traffic on the roads, the weather, and how late the bus is when it arrives. The longest bus journey time is 35 minutes, the average 25 minutes, and the shortest 20 minutes. The bus, which is timetabled to run every 15 minutes, can be as much as 12 minutes late.

He has to arrive at school no later than 08:00 to ensure he is able to go on a school trip.

Given that his walk to the bus stop takes just under two minutes, at what time must he leave home to make sure he is able to go on the school trip?

HINT

you can always use more than one strategy to solve problems. For example, add a diagram to your function machine!

The question involves an unknown value (the times he must leave) subject to a number of varying constraints. This can be easily represented as a function machine, according to the basic principles:

• Start with the answer, if it is unclear. In this case, the time he must leave home.

• Box the number along the way. In this case, the times he will be at the different points in his journey.

• Write the processes by the connecting arrows. In this case, the relevant times that each event could take.

This gives something like the diagram above.

Some of these values take care and thought — and you may leave a couple of options. For example, is the maximum time until the first bus 14 minutes or 15 minutes? Do we need the average time for a bus journey or the longest time?

Once the information has been codified it is worth checking that all the relevant restrictions and times have been selected.

It is possible to work backwards to the answer (diagram below).

This gives the answer: the latest time he can leave to be sure he is in time for the trip is 06:56.

Make sure you write down the processes carefully, so that you are able to 'reverse them' to find missing values. Your default approach to the diagramming of function machines should be to use boxes for the values (which may change as you work towards a solution) and arrows for the processes (which normally do not change).

Don't be afraid to restart your diagram from a different perspective if you are struggling to draw it.

Be flexible. Different problems may require you to consider multiple inputs combining to produce outputs which are subject to various conditions, with processing values that change; they may require you to change a value in the middle of the diagram, in order to find another one which is obscurely linked to the first. The key principle for diagramming a function machine is that you should try to represent all relevant connections and processes, even if the resulting network is a mess. And that you should represent any missing values with a '?'

As with function machines, algebra attempts to capture the relationship described in a problem. The structure is much more rigid and prescriptive, but the mechanisms for extracting the answer are more predictable. If one is to use algebra to solve a problem, it is vital to:

define your variables. Always begin with a sentence like: 'Let x = the new distance travelled.' If you introduce other variables, define these too.
make sure your equations have an equals sign.
separate out the equations you create
state your answer as a sentence, in the context of the question, at the end.

Sample Question

Try this problem yourself, before reading the analysis

Two different types of cake are on sale at prices of 30$? and 40$? each. The cakes that are being sold for 30$? cost 20$? to make and the ones for sale at 40f cost 25$? to make. No cakes can be kept to be sold the next day, so all of the cakes arc reduced to half price 2 hours before the sales finish.

All of the 30$? cakes sold out before any of the prices were reduced, and all of the 40$? were eventually sold, even though only half had been sold when the price was reduced. The overall profit at the end of the day was $30, but it could have been S40 if all of the cakes had sold before the prices were reduced.

How many of each type of cake were there in the sale?

The profits in dollars have been converted into cents, in order to make all the units correspond. This is easy to forget when phrasing relationships algebraically.

With all the information encoded into two equations, the problem solver needs to reflect on how to solve the pair simultaneously. This requires simplification of the algebraic relationships, and strategic decisions about how to eliminate one of the variables. The following shows one way to do this.

The answer: 100 400 cakes were for sale, and 250 300 cakes. This can be checked by trying those numbers in the two original statements.

Phrasing a problem algebraically depends upon you being very clear about what any letters you use stand for. It is vital that you state briefly what they mean, for instance 'H' = contact hours during study leave', and it is a good habit to do this with absolute precision, for example, stating what units you are using: 'Let H= the number of contact hours per week during study leave.'

One of the benefits of stating a problem algebraically is that it is normally possible to see whether an equation is solvable. As a rule of thumb, you can find a unique answer to one equation involving one unknown, two equations involving two unknowns and three equations involving three unknowns. Anything more than that gets messy.

Why Practice This?

Even if you can solve a problem just by guessing, you should still practice writing the algebraic version. Why?

Deep Understanding: Using guess-and-check helps you understand the "why" behind the equation, making algebra feel more accessible and less like a series of random rules.
Future-Proofing: One day, you might need to use math on the job. Even if you forget your specific algebra formulas, you can use guess-and-check to help you remember how to set up equations for difficult problems, especially those where the answer isn't a simple whole number.

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