Basic Algebra and Algebraic Manipulation

Class Discussion Like and unlike terms think! Mathematics 1A, p99

Let us consider the algebraic expression 2x − 3xy + 7 + x² + 5x − 4 + 2yx.
It contains 2 variables (x and y) and 7 terms (2x, −3xy, 7, x², 5x, −4 and 2yx).

Are the following pairs of terms like or unlike terms? Explain your reasoning.

Explore Like terms with algebra discs think! Mathematics 1A, p99–100

In Chapter 2, we used number discs representing 1 (i.e. +1) and −1 as shown in Fig. 4.2(a). In this chapter, we will introduce discs representing x (i.e. +x) and −x as shown in Fig. 4.2(b).
Together, we call them ‘algebra discs’.

In primary school, we have learnt that we can add or subtract expressions such as 2x and 3x, which are like terms, e.g. 2x + 3x = 5x or 4x − x = 3x. We will now use algebra discs to illustrate this.

How the pads work: drag a disc from the row above a pad onto it, or tap the disc to add it. Tap a disc on the pad to choose it, drag it to move it, or drag it off the pad to take it away. The Value line under the pad adds up the discs; Start again puts the pad back as it began.

2x + 3y + 1

What do we get when we add or subtract unlike terms? For example, what is 2x + 3y + 1 equal to?

This pad has no Value line — deciding is your job. Try to make it show fewer terms: can any of the discs be put together, or do any of them cancel out? Then write your answer.

2x + 3y + 1 =

Investigation Addition and subtraction of like terms think! Mathematics 1A, p100–102

In primary school, we have only learnt how to work out subtractions such as 4x − x = 3x.
How do we work out subtractions such as 2x − 5x?
Or how do we perform the following operations: (−2x) + (−3x), 5x + (−2x) and 5x − (−2x)?

In Chapter 2, we have learnt how to add and subtract positive and negative real numbers:

Will the above apply to addition and subtraction involving like terms as well? Let us investigate.

What is (−2x) + (−3x) equal to?

To represent the term −2x, we put two discs as shown in Fig. 4.6(a).
To add the term −3x, we add three more discs as shown in Fig. 4.6(b).

∴ (−2x) + (−3x) = −5x

Recall: (−2) + (−3) = −5.
In other words, for (−2x) + (−3x), we add the coefficients of the two terms (i.e. −2 and −3) to get −5, which will be the coefficient of x in the sum −5x.

Investigation: Addition and subtraction of like terms

Each pad starts with the first term. To add a term, add its discs. To subtract a term, take its discs away (choose them and press Take away, or drag them off the pad). If there are not enough discs to take away, first add zero pairs: one disc and one disc together are worth 0. When you are done, press Zero pairs to find any pairs that are left, and press it again to remove them.

Explore Like terms in two variables think! Mathematics 1A, p103

How do we add or subtract linear terms involving two variables, e.g. 3x + 2y − x + (−4y)?
We can simplify this to:

3x + 2y − x + (−4y) = 3x + 2y + (−x) + (−4y) recall: 5x − 2x = 5x + (−2x)

Now we can represent the above expression using algebra discs as shown in Fig. 4.7(a).
To simplify a linear expression involving two variables, we group the like terms together first, as shown in Fig. 4.7(b).

The pad starts as Fig. 4.7(a). Rearrange it: move the disc next to the discs, and the discs next to the discs, so that like terms sit together. Then press Zero pairs to find the pairs that make 0, and press it again to remove them. This pad has no Value line: write down what is left, in its simplest form.

3x + 2y − x + (−4y) =

Explore Expansion with groups think! Mathematics 1A, p105

What does 2(x + y) mean?

To recap what we learnt in Chapter 2, the product 2 × 3 means 2 groups of 3, and it can be represented by number discs as shown in Fig. 4.8, i.e. 2 × 3 = 6.

Now, what does 2(x + y) mean?
This notation represents 2 × (x + y), just like how the notation 2x means 2 × x.

Build it yourself: tap and to make one (x + y) in the top row. Press Repeat top row to make the second group underneath, then press Group all. Write down what 2(x + y) is equal to and press Check.

2(x + y) =

Explore The negative of an expression think! Mathematics 1A, p105–106

What does −(x + y) mean?

Now, what does −(x + y) mean?
−(x + y) means the negative of (x + y), and it can be represented by algebra discs as shown in Fig. 4.12, i.e. −(x + y) = −x + (−y) = −x − y.

The red wedge on the left of a group means “the negative of”. Tap the group to choose it, then press Flip: every disc turns over. Then press Check.

What does −2(x + y) mean?

To recap what we have learnt in Chapter 2, the product (−3) × 2 means the negative of ‘3 groups of 2’, and it can be represented by algebra discs as shown in Fig. 4.13, i.e. (−3) × 2 = −(3 × 2) = −6.

Now, what does −2(x + y) mean?
−2(x + y) means the negative of ‘2 groups of (x + y)’, and it can be represented by algebra discs as shown in Fig. 4.14, i.e. −2(x + y) = −2x + (−2y) = −2x − 2y.

Flip this group too, then press Check.

Investigation Expansion of linear expressions think! Mathematics 1A, p106–107

Work out the answers to each of the following, using algebra discs as shown above. Write down the final answer for each part.

Hint: For part (b), 3(x − y) = 3[x + (−y)]. Why?

On each pad: tap discs to make what is inside the brackets in the top row. Press Repeat top row until there is one row for each group, then press Group all. If the number in front of the brackets is negative, press Negative to put the wedge on, then Flip. Read the value, type the final answer and press Check.

Explore The Distributive Law think! Mathematics 1A, p107

The Distributive Law can be visualised using the rectangle PQRS (see Fig 4.15).

The Distributive Law is the conclusion of the Investigation above — open Show the conclusion there once you have worked out parts (a) to (h).

Move the sliders to change the lengths a, b and c. Rectangle PTUS is shaded blue and rectangle TQRU orange; each small square of the grid is 1 square unit, so you can count the squares to check the areas.

Explore Factorisation: the reverse of expansion think! Mathematics 1A, p108–109

In Section 4.3A, we have learnt how to expand linear expressions, e.g. 2(x + y) = 2x + 2y.
Now, let us learn how to carry out the reverse process, e.g. 2x + 2y = 2(x + y).
This process is called factorisation because we factorise 2x + 2y into its two factors, 2 and (x + y), to become 2(x + y).
Fig. 4.16 illustrates the processes of expansion and factorisation as the reverse of each other.

Extracting positive common factors

To factorise 2x + 2y, we need to identify common factor(s) of the two terms 2x and 2y. In this case, the common factor is 2. Then we extract the common factor 2 from both terms, 2x and 2y, to obtain 2(x + y).

Factorising with discs is sharing them out: type a number of groups and press Share. If every kind of disc can be shared out equally, the number of groups is a common factor. Only the biggest common factor — the HCF of the numbers — completes the factorisation: then the discs are laid out one row per group. Write the factorised expression and press Check.

Sometimes, there is more than one common factor, e.g. in 6xy − 9x, the common factors are 3 (which is the HCF of 6 and 9) and x. We extract 3x from both terms, 6xy and 9x, to obtain 3x(2y − 3).
The factorisation is incomplete if we only extract one common factor, e.g. 3(2xy − 3x).

Worked Example 6 (p109) is in the textbook — it is not on this page.

Practise Now 6  Factorise each of the following expressions completely.

Parts (a) and (b) of Practise Now 6 — the two whose terms can be made with algebra discs.

Extracting negative common factors

In Section 4.3A, we learnt how to expand −(x + y) to get −x − y.
Now, we are going to do the reverse by factorising −x − y to get −(x + y).

Make the pad show −(x + y): a group of and with the wedge on it. Use Flip, Group all and Negative — and keep an eye on the value: it must still be −x − y at the end.