Chapter 25 · Defined Operations

Evaluate Nested Operations One Layer at a Time

Parentheses are a route map. Start with the deepest custom operation, replace only that symbol with its definition, save the result, and carry it outward one layer at a time.

Inside → outsideOperation treesExact fractionsSymmetryNo assumed associativity

One safe routine

read the deepest parentheses → substitute → simplify → move outward

A custom operation is a small machine. In a nested expression, the output of one machine becomes an input to another.

Do not flatten the expression. Unless a property has been proved, (a*b)*c and a*(b*c) are different routes.

Read the route map

Find the deepest operation before calculating

Not complete

outside *

↑

inside 2*3

space

Evaluate 2*3.
It is deepest inside the parentheses.

Save the inner output.
It becomes the left input of the outside operation.

Evaluate the outside operation.
Use the same current definition again.

Route-map checkpoint

First operation in (2*3)*5

The inner output becomes…

May we change it to 2*(3*5) without proof?

Need a hint?

Start at the deepest parentheses. Do not move them.

original Guided Practice 1

Carry one inner output into the next layer

Not complete

a*b=(b+a)×b

Inner node

2*3

(3+2)×3=15

→

Outside node

15*5

(5+15)×5=100

The number 15 is not an extra term. It replaces the complete inner expression (2*3).

Complete the worked example practice.

Inner result 2*3

Outside left input

Final result

Need a hint?

The inner output replaces all of (2*3), not just one of its numbers.

Parentheses choose the route

Compare left-grouped and right-grouped expressions

Not complete

Current rule: a*b=(a+b)×b. Inputs here are nonnegative integers.

Left grouped

(2*3)*5

100

≠

Right grouped

2*(3*5)

1680

Associativity means that regrouping three inputs preserves the result for every allowed triple. It is not automatic. Parentheses are part of the problem’s data. Keep them until the operation has been proved associative.

For this checkpoint use the fixed inputs 2,3,5, regardless of the current explorer settings.

(2*3)*5

2*(3*5)

Are the results equal?

Need a hint?

Build a separate inner result for each parenthesization.

Worked example 6

Use symmetry to reuse an inner result

Not complete

a○b= ab + ba + 12

Swapping the inputs only swaps the first two fractions, so:

a○b=b○a

First positive integer input

Second positive integer input

2008○10043

1004○20083

→

equal inner results combined5/2

Once both inner nodes are 3, the outside operation is simply 3○3=1+1+1/2.

Complete the worked example.

2008○1004

1004○2008

Final value

Need a hint?

Swapping the two inputs exchanges the first two fractions; their sum stays the same.

Interactive operation tree

Evaluate any two-layer expression exactly

Not complete

Laboratory input rule

This laboratory accepts whole-number inputs in the displayed ranges. It carries any fractional intermediate results exactly. Decimal entries are rejected rather than rounded. If a denominator becomes zero, that node and the outside result are undefined; change the inputs before continuing.

Checkpoint preset: a◆b=(a+b)÷4 and 2◆(3◆5).

Inner result 3◆5

Final result

In the checkpoint expression 2◆(3◆5), where is the first node?

Need a hint?

Follow the selected tree. The output of the inner node becomes one complete outer input.

Exercise 6

Separate custom operations from an ordinary plus sign

Not complete

A*B=3A+4B

One nested chain

(4*5)*6

4*5=32

32*6=120

↔

Two separate custom nodes

(1*5)+(2*4)

1*5=23, 2*4=22

The middle + is ordinary addition: 23+22=45.

Custom symbols are replaced by their definitions. Ordinary symbols keep their ordinary meanings.

Complete both original calculations.

4*5

(4*5)*6

(1*5)+(2*4)

Need a hint?

Finish each custom node before the ordinary plus sign.

Exercise 7

Keep fractions exact while moving outward

Not complete

x*y=(x+y)÷4

Direct

13*17=(13+17)÷4=30÷4=15/2

Nested

3*5=2

2*2=1

Backward check

a*16=10

(a+16)÷4=10

Do not round 15/2 unless the problem asks for a decimal. Exact fractions prevent errors in later layers.

Complete all three original parts.

13*17

2*(3*5)

a when a*16=10

Need a hint?

Keep each result as a fraction until you reach the outside node.

original transfer studio

Apply the same inside-out routine to different rules

Not complete

A · Linear rule with fractions

a*b=3a−2b

(5/3*4/5)*3/4

5/3*4/5=17/5

87/10

B · Nested rule inside an equation

a*b=3a−2b

x*(4*1)=7

4*1=10, 3x−20=7

x=9

C · Product-over-sum rule

x*y=xy÷(x+y)

10*(10*10)

10*10=5

10/3

D · Half-product rule

P*Q=P×Q÷2

3*(6*8)

6*8=24

E · Selection operators

Let a△b choose the smaller input and a▽b choose the larger input. Here x may be any number.

5▽(4▽(x△4))

x△4≤4, so 4▽(x△4)=4, then 5▽4=5.

Only one possible value

Enter the five results in order.

B: x

E: how many different output values are possible as x varies?

Need a hint?

Read each card’s definition afresh and start with its deepest operation.

Independent practice

Nested-operation workshop

Not complete

Correct all eight questions to complete the workshop. Fractions may be entered as p/q or exact equivalent decimals. Use a fraction when its decimal does not terminate; a rounded decimal is not exact.

1. If a*b=(a+b)b, find (1*4)*2.

2. Under the same rule, find 1*(4*2).

3. If a○b=a/b+b/a+1/2, find 6○3.

4. Find (6○3)○(3○6).

5. If A*B=3A+4B, find (2*1)*3.

6. If x*y=(x+y)/4, find 6*(2*6).

7. If x*y=xy/(x+y), find 6*(3*3).

8. May you assume (a*b)*c=a*(b*c) for a newly defined operation?

Need a hint?

Answers: 44;156;3;5/2;42;2;6/5;no. The inner outputs for Questions1,2,5,6,7 are20,12,10,2,3/2 respectively. Question4 has two equal inner results3, so the outside value is1+1+1/2=5/2.

Five-question check

Exit ticket and certificate

Not complete

1. Which part is evaluated first in (a*b)*c?

2. If a*b=(a+b)b, find (2*3)*5.

3. Under the same rule, find 2*(3*5).

4. If a○b=a/b+b/a+1/2, find (2008○1004)○(1004○2008).

5. If A*B=3A+4B, find (1*5)+(2*4).

Need a hint?

Answers: a*b;100;1680;5/2;45. For the two groupings, compute inner outputs15 and40 separately. Symmetry makes the two circle inputs equal, so the final value is5/2. Ordinary addition joins23 and22 to give45.

Optional reflection — not automatically gradedThis response is saved but does not affect mission completion.

Chapter 25 Achievement

Nested-Operation Navigator

This certifies that the learner can read operation trees, evaluate the deepest layer first, preserve exact values, use symmetry, and respect parentheses in newly defined operations.

Lesson 25.2 complete