Expression trees: the structure of terms

What is it about?

An expression (term) is a meaningful calculation made of numbers, variables, operators and brackets, e.g. $(12 - 4) \cdot 3 + 6 : 2$ . The expression tree shows how it is built: the numbers are at the top, the operations below. Each operation joins two parts; at the very bottom is the last operation carried out – it gives the value of the expression.

Order of operations:

Work out what is in brackets first (with several brackets, the innermost first).
Powers before $\cdot$ and $:$ before $+$ and $-$ .
Otherwise work from left to right: $20 - 8 - 2 = 12 - 2 = 10$ .

Type of expression: An expression is named after its last operation:

last operation	type	parts
$+$	sum	first addend $+$ second addend
$-$	difference	minuend $-$ subtrahend
$\cdot$	product	first factor $\cdot$ second factor
$:$	quotient	dividend $:$ divisor
raising to a power	power	base $^{\text{exponent}}$

So $(12 + 4) - 3 \cdot 2$ is a difference: the difference of the sum of $12$ and $4$ and the product of $3$ and $2$ .

Careful with signs and powers: $-3^2$ is the opposite of $3^2$ , so $-9$ . But $(-3)^2 = (-3) \cdot (-3) = 9$ . In the same way, $2x^2$ means $2 \cdot x^2$ – not $(2x)^2$ .

Variables: If you substitute a number for a variable, you get a value of the expression. Negative numbers are put in brackets: for $x = -2$ , $3x + 1 = 3 \cdot (-2) + 1 = -5$ .

Try it

Compare the trees of 3 + 4 · 5 and (3 + 4) · 5. Which operation is at the bottom? What type of expression is it?
Press Next step and watch the values flow through the tree. The line above the tree shows the working.
In 4 · 5 + 2³ tap the multiplication first. The simulation allows it – why does the result not change? Then tap the addition before everything above it has been worked out.
−3² or (−3)²? Where is the sign in the tree?
Compare 2x² and (2x)². Move the slider for $x$ .
Calculate with fractions: the simulation works exactly, $\frac{1}{2} + \frac{1}{3} = \frac{5}{6}$ .
Type your own expressions. What does the simulation say for $3 + {}$ , for $5 \cdot -3$ or for $5 : (3 - 3)$ ?

Tasks

Task 1: Type of expression

Load task Find the type of $(12 + 4) - 3 \cdot 2$ , name its two parts and work out its value. Then check with the tree.

Show solution

The last operation is a subtraction, so the expression is a difference. Minuend: the sum $12 + 4$ , subtrahend: the product $3 \cdot 2$ .

$(12 + 4) - 3 \cdot 2 = 16 - 6 = 10$

Task 2: Order of operations

Work out step by step: $2 \cdot [3 + (4 - 1)] - 12 : (2 + 4) + 5^2$ . What type of expression is it? Load task

Show solution

Innermost bracket first, then the other brackets, then the power, then $\cdot$ and $:$ before $+$ and $-$ , finally from left to right:

$2 \cdot [3 + 3] - 12 : 6 + 25 = 2 \cdot 6 - 2 + 25 = 12 - 2 + 25 = 35$

The last operation is an addition – the expression is a sum.

Task 3: From words to an expression

Write as an expression and work it out: “the difference of the product of 7 and 8 and the sum of 15 and 9”.

Show solution

$7 \cdot 8 - (15 + 9) = 56 - 24 = 32$ . The brackets are needed because the whole sum $15 + 9$ is subtracted. Show tree

Task 4: Inserting brackets

Insert brackets into $2 + 3 \cdot 4 - 1$ so that the value is $19$ , $11$ or $15$ . How does the tree change each time? Without brackets

Show solution

$(2 + 3) \cdot 4 - 1 = 19$ – a difference
$2 + 3 \cdot (4 - 1) = 11$ – a sum
$(2 + 3) \cdot (4 - 1) = 15$ – a product

Without brackets the value is $2 + 12 - 1 = 13$ .

Task 5: Values

Work out the value of $2x^2 - 3x + 1$ for $x = -2$ and for $x = 0.5$ . Load task

Show solution

$x = -2$ : $2 \cdot (-2)^2 - 3 \cdot (-2) + 1 = 8 + 6 + 1 = 15$

$x = 0.5$ : $2 \cdot 0.25 - 1.5 + 1 = 0$ . So for $x = 0.5$ the expression has the value $0$ . The table of values below the tree shows more values.

Task 6: Find the mistakes

Tim calculates: $-3^2 = 9$ and $20 - 8 - 2 = 20 - 6 = 14$ . Explain his mistakes using the expression tree.

Show solution

$-3^2$ : First square, then take the opposite: $-3^2 = -9$ . Only $(-3)^2 = 9$ .
$20 - 8 - 2$ : Operations of the same rank are done from left to right: $20 - 8 - 2 = 12 - 2 = 10$ . Tim calculated $20 - (8 - 2)$ – that is a different tree.

Ideas for the classroom

Use: About grade 5 (structure of expressions, multiplication and division before addition and subtraction), grade 6 (structure of expressions with rational numbers) and grade 7 (expressions with variables, values, naming the structure).
Input: * or · for multiplication, : for division, ^ or ² for powers, brackets ( ) and [ ]. A slash between two whole numbers (3/4) stands for a fraction (one number), otherwise for a division. Allowed variables: $a, b, c, n, x, y, z$ . Your own expressions can be shared as a link or QR code with Share.
Display: As in many textbooks, the numbers are at the top and the result at the bottom; under “Display” you can turn the tree upside down. The operations have different colours (addition and subtraction blue, multiplication and division green, powers orange).
Order and structure: “Next step” follows the rules. If you tap another possible operation yourself, it is carried out and commented on: independent parts may be worked out in any order – the rules fix the structure, not the order within independent parts.
Calculation: exact with fractions; decimals are shown as decimals if they occur in the expression or in a variable. A sign directly after an operator is not accepted (as in class): $5 \cdot (-3)$ instead of $5 \cdot -3$ . $0^0$ and division by $0$ are not defined.
Common stumbling blocks:
- Working strictly from left to right: $3 + 4 \cdot 5 = 35$ .
- $-3^2 = 9$ and $2x^2 = (2x)^2$ .
- Naming the type after the first operator instead of the last operation carried out.
Task mode: With names=0 and _hide=1 (see Task 1) the types and results are hidden at first.