binary operations

1. Formula
   Observation

   The left side of the given rule is written as $a*b$.

2. Audio
   Observation

   The speaker says, "We are given a binary operation a star b".

a

First input to the binary operation *; in part (i) it is 1, and in part (ii) it is replaced by -7.

numeric inputs used in the examples

1. Formula
   Observation

   The left side of the given rule is written as $a*b$.

2. Audio
   Observation

   The speaker says, "b is equal to 3" for part (i), and later "b is now 4" for part (ii).

b

Second input to the binary operation *; in part (i) it is 3, and in part (ii) it is 4.

numeric inputs used in the examples

1. Formula
   Observation

   The board shows the custom operation symbol between a and b as *.

2. Audio
   Observation

   The speaker calls it "a binary operation" and reads expressions such as "one star three".

*

Custom binary operation symbol defined by the rule on the board; not ordinary multiplication.

binary operation on two numbers

1. Formula
   Observation

   The worked line ends with = -7.

2. Audio
   Observation

   The speaker says, "which is negative seven".

-7

Value of $1*3$ from part (i), then reused as the first input in part (ii).

integer result of the example computation

1. Formula
   Observation

   The board shows the definition $a*b = 2a - b^2$, and the presenter repeatedly identifies the first number as a.

2. Audio
   Observation

   At 01:04-01:10 the speaker says, "Once again the first number is a."

a

First operand of the custom binary operation *.

Real numbers in the worked examples; the video does not state a formal domain.

1. Formula
   Observation

   The board shows the definition $a*b = 2a - b^2$, and the presenter repeatedly identifies the second number as b.

2. Audio
   Observation

   At 01:04-01:10 the speaker says, "and the second number is b."

b

Second operand of the custom binary operation *.

Real numbers in the worked examples; the video does not state a formal domain.

1. Formula
   Observation

   The title line on the left reads "If $a*b = 2a - b^2$" and all three parts use the symbol * between two quantities.

2. Audio
   Observation

   Throughout the clip the speaker verbally calls it "star" while evaluating expressions such as $1*3$, (-7)*4, and (-8)*(-7).

*

A custom binary operation defined by $a*b = 2a - b^2$, not ordinary multiplication.

Applied to pairs of numbers appearing in the examples.

1. Formula
   Observation

   Left side shows (i) $1*3 = -7$; right side shows $1*3 = 2(1) - (3^2) = 2 - 9 = -7$.

$1*3$

The value of the custom operation with $a=1$ and $b=3$.

Numeric expression evaluated in part (i).

1. Formula
   Observation

   Left side shows (ii) ($1*3$)*$4 = -30$; right side shows (-7)*$4 = 2(-7) - (4^2) = -14 - 16 = -30$.

($1*3$)*4

Nested application of the custom operation: first compute $1*3$, then operate the result with 4.

Numeric expression evaluated in part (ii).

1. Formula
   Observation

   Left side shows (iii) (-8)*($1*3$); right side rewrites it as (-8)*(-7) and evaluates to -65.

2. Audio
   Observation

   At 01:54-01:58 the speaker says, "What is our final answer? Our final answer will be negative sixty five."

(-8)*($1*3$)

Nested application of the custom operation: first replace $1*3$ by -7, then compute (-8)*(-7).

Numeric expression evaluated in part (iii).

1. Audio
   Observation

   The speaker says, "We are given a binary operation a star b is equal to 2 multiplied by a minus b to the power 2."

2. Formula
   Observation

   The board displays If $a*b = 2a - b^2$.

The video defines a custom binary operation by the rule $a*b = 2a - b^2$. The speaker explicitly identifies this as a binary operation, meaning it takes two inputs and returns one output. In the examples, the first input is substituted for $a$ and the second input for $b$.

1. The operation is applied to two inputs.

2. In the shown examples the inputs are numbers.

1. Audio
   Observation

   The speaker says, "We are using this binary operation as our guide and we will substitute values of a and b."

2. Diagram
   Observation

   Red arrows are drawn from the symbolic rule to the numeric inputs in part (i), and later labels $a$ and $b$ are written above the numbers in part (ii).

To evaluate an expression involving $*$, the video rewrites the defining rule and substitutes the specific numbers for $a$ and $b$. This is demonstrated first with $1*3$, where $a=1$ and $b=3$, and then again with $-7*4$, where $a=-7$ and $b=4$.

1. Use the same defining rule each time the operation is evaluated.

2. Identify which number occupies the first-input position and which occupies the second-input position.

1. Definition of the custom binary operation

1. Audio
   Observation

   The speaker says, "But if you look at part one we worked out one star three already and we got negative seven. So we are replacing the bracket term one star three by negative seven star four."

2. Formula
   Observation

   Part (ii) on the left is $(1*3)*4$, and the worked solution begins by rewriting it as $-7*4$.

For $(1*3)*4$, the video first evaluates the inner bracket $1*3$ to get $-7$, then replaces the bracketed subexpression by that value and continues with $-7*4$. This shows how a nested custom operation is reduced step by step to a single binary operation.

1. Evaluate the innermost bracket first.

2. Replace the completed subexpression by its numerical result before applying the outer operation.

1. Definition of the custom binary operation
2. Substitution method for evaluating the operation

1. Formula
   Observation

   The persistent board heading reads "If $a*b = 2a - b^2$ evaluate".

2. Audio
   Observation

   The whole clip consists of applying this rule to three expressions.

The video introduces a nonstandard binary operation denoted by * and defines it by the rule $a*b = 2a - b^2$. In each example, the first entry of the pair is substituted for a and the second entry for b. The operation is then evaluated arithmetically.

1. The symbol * is a defined operation, not ordinary multiplication.

2. For an expression $x*y$, substitute x for a and y for b.

3. The examples use integer inputs.

1. Formula
   Observation

   Each worked line replaces a and b directly: $1*3$ becomes $2(1)-(3^2)$, (-7)*4 becomes $2(-7)-(4^2)$, and (-8)*(-7) becomes $2(-8)-(-7)^2$.

2. Audio
   Observation

   At 01:04-01:10 the speaker explicitly states that the first number is a and the second number is b.

To evaluate any expression using this custom operation, identify the left argument as a and the right argument as b, then substitute them into $2a - b^2$. The video demonstrates this same method in all three parts.

1. Order matters: the left operand is a and the right operand is b.

2. Parentheses are used when substituting negative numbers.

1. Custom binary operation defined by $a*b = 2a - b^2$

1. Formula
   Observation

   Part (ii) uses the earlier result $1*3 = -7$ before computing (-7)*4.

2. Formula
   Observation

   Part (iii) rewrites (-8)*($1*3$) as (-8)*(-7) before applying the rule again.

3. Audio
   Observation

   At 00:44-00:53 the speaker says, "If we go back to our answer for part one, we got negative seven for one star three. So we are replacing the bracketed term by negative seven."

When the custom operation appears inside parentheses, the inner expression must be evaluated first and its numerical result substituted into the outer operation. The clip shows this in both (ii) and (iii).

1. A bracketed subexpression containing * must be simplified before the outer * is applied.

2. Previously computed results may be reused.

1. Custom binary operation defined by $a*b = 2a - b^2$
2. Substitution into the operation rule

1. Audio
   Observation

   The speaker says, "we want to work with two numbers because we are using a binary operation and binary implies two of something."

A binary operation works with two inputs.

1. The operation under discussion is called binary.

For the operation being used in this lesson.

1. Formula
   Observation

   The right side shows (i) $a*b = 2a - b^2$, then $1*3 = 2(1) - (3^2)$, then = 2 x 1 - 3 x 3, then = $2 - 9 = -7$.

2. Audio
   Observation

   The speaker says, "So 1 star 3 will be equal to ... a is equal to 1 and b is equal to 3 ... 2 times 1 minus 3 squared ... which gives 2 take away 9 which is negative 7."

1. Expression

   $$a*b = 2a - b^2$$
   Explanation

   Start from the defining rule for the custom operation.

   Justification

   Given definition on the board.

   Shown in the video
2. Expression

   $$1*3 = 2(1) - (3^2)$$
   Explanation

   Substitute $a=1$ and $b=3$ into the rule.

   Justification

   Direct substitution into the definition.

   Shown in the video
3. Expression

   $$= 2 \times 1 - 3 \times 3$$
   Explanation

   Expand the square $3^2$ as $3\times 3$.

   Justification

   Arithmetic expansion of exponent notation.

   Shown in the video
4. Expression

   $$= 2 - 9 = -7$$
   Explanation

   Compute the products and subtract.

   Justification

   Standard integer arithmetic.

   Shown in the video

The value of $1*3$ is $-7$.

1. Formula
   Observation

   The left side lists (ii) $(1*3)*4$; the worked line on the right begins with (ii) -7*$4 = 2(-7) - 4^2$.

2. Audio
   Observation

   The speaker says, "we are replacing the bracket term one star three by negative seven star four" and then "instead of 2a, we can see that a is now negative seven and b is now four."

1. The final numerical answer for part (ii) is not reached within the provided clip.

1. Expression

   $$(1*3)*4$$
   Explanation

   Begin with the nested expression from part (ii).

   Justification

   Problem statement on the board.

   Shown in the video
2. Expression

   $$-7*4$$
   Explanation

   Replace the inner bracket $1*3$ by its previously computed value $-7$.

   Justification

   Use the result from part (i).

   Shown in the video
3. Expression

   $$= 2(-7) - 4^2$$
   Explanation

   Apply the same defining rule with $a=-7$ and $b=4$.

   Justification

   Substitution into $a*b = 2a - b^2$.

   Shown in the video

Within this clip, $(1*3)*4$ is reduced to $-7*4$ and rewritten as $2(-7) - 4^2$; the final simplification is not shown.

1. Formula
   Observation

   Right side shows (-7)*$4 = 2(-7) - (4^2) = -14 - 16 = -30$.

2. Audio
   Observation

   From 00:00-00:22 the speaker narrates the arithmetic: 2 times negative 7, subtract 4 times 4, giving negative 14 minus 16 equals negative 30.

1. Expression

   $$1*3 = -7$$
   Explanation

   Use the already established result from part (i) as the inner value.

   Justification

   Prior worked example shown on the board.

   Shown in the video
2. Expression

   $$(1*3)*4 = (-7)*4$$
   Explanation

   Replace the bracketed term $1*3$ by -7.

   Justification

   Substitution of a previously computed value.

   Shown in the video
3. Expression

   $$(-7)*4 = 2(-7) - (4^2)$$
   Explanation

   Apply the definition $a*b = 2a - b^2$ with $a=-7$ and $b=4$.

   Justification

   Definition of the custom operation.

   Shown in the video
4. Expression

   $$2(-7) - (4^2) = -14 - 16$$
   Explanation

   Compute $2(-7) = -14$ and $4^2 = 16$.

   Justification

   Arithmetic evaluation.

   Shown in the video
5. Expression

   $$-14 - 16 = -30$$
   Explanation

   Combine the two negative terms.

   Justification

   Addition of signed numbers.

   Shown in the video

($1*3$)*$4 = -30$.

1. Formula
   Observation

   Right side shows (-8)*(-7) = $2(-8) - (-7)^2 = -16 - 49 = -65$.

2. Audio
   Observation

   From 00:39-01:58 the speaker explains replacing $1*3$ by -7, then substituting into the rule and obtaining negative sixty five.

1. Expression

   $$1*3 = -7$$
   Explanation

   Recall the result from part (i).

   Justification

   Previously computed value on the board.

   Shown in the video
2. Expression

   $$(-8)*(1*3) = (-8)*(-7)$$
   Explanation

   Replace the bracketed expression $1*3$ by -7.

   Justification

   Substitution using the inner result.

   Shown in the video
3. Expression

   $$(-8)*(-7) = 2(-8) - (-7)^2$$
   Explanation

   Apply the definition with $a=-8$ and $b=-7$.

   Justification

   Definition of the custom operation.

   Shown in the video
4. Expression

   $$2(-8) - (-7)^2 = -16 - 49$$
   Explanation

   Compute $2(-8) = -16$ and (-7)^$2 = 49$.

   Justification

   Arithmetic evaluation, including squaring a negative number.

   Shown in the video
5. Expression

   $$-16 - 49 = -65$$
   Explanation

   Combine the two negative terms.

   Justification

   Addition of signed numbers.

   Shown in the video

(-8)*($1*3$) = -65.

1. Formula
   Observation

   Left side lists (i) $1*3$; right side solves it to -7.

2. Audio
   Observation

   The speaker says, "We are asked to evaluate 1 star 3" and concludes, "which is negative 7".

Evaluate $1*3$ using the rule $a*b = 2a - b^2$.

1. $a*b = 2a - b^2$

2. First input is 1

3. Second input is 3

Find the numerical value of $1*3$.

1. Expression

   $$1*3 = 2(1) - (3^2)$$
   Explanation

   Substitute $a=1$ and $b=3$ into the definition.

   Justification

   Definition of the custom binary operation.

   Shown in the video
2. Expression

   $$= 2 \times 1 - 3 \times 3$$
   Explanation

   Rewrite the square as repeated multiplication.

   Justification

   Arithmetic notation.

   Shown in the video
3. Expression

   $$= 2 - 9 = -7$$
   Explanation

   Evaluate the products and subtract.

   Justification

   Integer arithmetic.

   Shown in the video

-7

The board and narration both state the final result as negative seven.

1. Formula
   Observation

   Left side lists (ii) ($1*3$)*4; right side rewrites it as -7*$4 = 2(-7) - 4^2$.

2. Audio
   Observation

   The speaker says, "we are replacing the bracket term one star three by negative seven star four" and then substitutes $a=-7$, $b=4$.

1. The clip stops before the arithmetic is completed, so the final value of part (ii) is not visible or audible here.

Evaluate $(1*3)*4$ using the same rule, after first finding $1*3$.

1. $a*b = 2a - b^2$

2. From part (i), $1*3 = -7$

3. Outer second input is 4

Reduce $(1*3)*4$ to a single application of the operation and substitute into the rule.

1. Expression

   $$(1*3)*4$$
   Explanation

   Start from the nested expression.

   Justification

   Problem statement on the board.

   Shown in the video
2. Expression

   $$-7*4$$
   Explanation

   Replace the inner bracket by the result from part (i).

   Justification

   Substitution of a previously computed value.

   Shown in the video
3. Expression

   $$= 2(-7) - 4^2$$
   Explanation

   Apply the defining rule with $a=-7$ and $b=4$.

   Justification

   Direct substitution into $a*b = 2a - b^2$.

   Shown in the video

The setup shown in the clip is $2(-7) - 4^2$; the final simplified value is not reached within this segment.

The visible work confirms the reduction and substitution, but no concluding arithmetic is shown before the clip ends.

1. Formula
   Observation

   Left side lists (i) $1*3 = -7$; right side shows $1*3 = 2(1) - (3^2) = 2 - 9 = -7$.

Using $a*b = 2a - b^2$, evaluate $1*3$.

1. Definition: $a*b = 2a - b^2$

2. Expression: $1*3$

Find the numerical value of $1*3$.

1. Expression

   $$1*3 = 2(1) - (3^2)$$
   Explanation

   Substitute $a=1$ and $b=3$ into the definition.

   Justification

   Direct application of the custom operation rule.

   Shown in the video
2. Expression

   $$2(1) - (3^2) = 2 - 9$$
   Explanation

   Evaluate the product and the square.

   Justification

   Basic arithmetic.

   Shown in the video
3. Expression

   $$2 - 9 = -7$$
   Explanation

   Subtract to obtain the final value.

   Justification

   Arithmetic simplification.

   Shown in the video

-7

The board writes the final result as -7 beside part (i).

1. Formula
   Observation

   Left side lists (ii) ($1*3$)*$4 = -30$; right side shows (-7)*$4 = 2(-7) - (4^2) = -14 - 16 = -30$.

Using $a*b = 2a - b^2$, evaluate ($1*3$)*4.

1. Definition: $a*b = 2a - b^2$

2. From part (i), $1*3 = -7$

3. Expression: ($1*3$)*4

Find the numerical value of ($1*3$)*4.

1. Expression

   $$(1*3)*4 = (-7)*4$$
   Explanation

   Replace the inner bracketed term by its known value -7.

   Justification

   Substitution of a previously computed result.

   Shown in the video
2. Expression

   $$(-7)*4 = 2(-7) - (4^2)$$
   Explanation

   Apply the definition with $a=-7$ and $b=4$.

   Justification

   Custom operation rule.

   Shown in the video
3. Expression

   $$2(-7) - (4^2) = -14 - 16$$
   Explanation

   Compute the product and the square.

   Justification

   Arithmetic evaluation.

   Shown in the video
4. Expression

   $$-14 - 16 = -30$$
   Explanation

   Combine the terms.

   Justification

   Signed-number addition.

   Shown in the video

-30

The board writes the final result as -30 beside part (ii).

1. Formula
   Observation

   Left side lists (iii) (-8)*($1*3$); right side rewrites it as (-8)*(-7) and evaluates to -65.

2. Audio
   Observation

   At 01:54-01:58 the speaker states the final answer is negative sixty five.

Using $a*b = 2a - b^2$, evaluate (-8)*($1*3$).

1. Definition: $a*b = 2a - b^2$

2. From part (i), $1*3 = -7$

3. Expression: (-8)*($1*3$)

Find the numerical value of (-8)*($1*3$).

1. Expression

   $$(-8)*(1*3) = (-8)*(-7)$$
   Explanation

   Replace the bracketed term $1*3$ by -7.

   Justification

   Use the result from part (i).

   Shown in the video
2. Expression

   $$(-8)*(-7) = 2(-8) - (-7)^2$$
   Explanation

   Apply the definition with $a=-8$ and $b=-7$.

   Justification

   Custom operation rule.

   Shown in the video
3. Expression

   $$2(-8) - (-7)^2 = -16 - 49$$
   Explanation

   Compute $2(-8)$ and (-7)^2.

   Justification

   Arithmetic, including squaring a negative number.

   Shown in the video
4. Expression

   $$-16 - 49 = -65$$
   Explanation

   Combine the two negative terms.

   Justification

   Signed-number addition.

   Shown in the video

-65

The board writes the final result as -65 beside part (iii).

1. Diagram
   Observation

   The whiteboard is split by a vertical line, with the problem statement on the left and worked solutions on the right.

1. left problem column

2. right solution column

3. vertical divider line

1. The right column gains handwritten solution steps over time while the left column remains the problem list.

1. The defining rule and the three listed parts stay visible on the left throughout the clip.

The layout separates the given task from the step-by-step working, making the substitution process easier to follow.

1. Diagram
   Observation

   Red arrows are drawn from the symbolic rule toward the numbers in $1*3$ while the speaker identifies $a=1$ and $b=3$.

1. rule $a*b = 2a - b^2$

2. expression $1*3$

3. red arrows

1. Arrows appear to connect the abstract variables to the concrete numbers.

1. The underlying formula remains unchanged.

The visual annotation emphasizes positional substitution: the first number corresponds to $a$ and the second to $b$.

1. Diagram
   Observation

   The letters $a$ and $b$ are written above the numbers in $-7*4$ during the explanation of part (ii).

1. expression $-7*4$

2. labels $a$ and $b$

1. The previously solved inner result becomes the new first input, and the labels are placed above the numbers.

1. The same defining rule is still being applied.

This visual step shows that after reducing the bracket, the outer expression is again just a binary operation with a first and second input.

1. Diagram
   Observation

   The whiteboard is split by a vertical black line: the left side lists the problem statement and final answers for (i), (ii), and (iii), while the right side shows the step-by-step working.

1. Left column with problem statement and answers

2. Right column with worked solutions

3. Vertical divider line

1. As the clip progresses, the right column gains the detailed computations for parts (ii) and (iii).

2. The left column is completed with the final answers -30 and -65.

1. The definition $a*b = 2a - b^2$ remains visible at the top throughout.

2. The three requested expressions remain listed on the left throughout.

The layout separates the task statement from the solution process, making the substitution steps easy to track.

1. Diagram
   Observation

   Red handwritten labels a and b appear above operands in the worked examples, including the line for (-8)*(-7).

1. Red letter a

2. Red letter b

3. Blue worked expressions

1. The labels are used to indicate which number plays the role of a and which plays the role of b in each application of the rule.

1. The first position is consistently marked as a and the second as b.

The color coding visually reinforces the substitution rule $a*b = 2a - b^2$ and the importance of order.

1. Formula
   Observation

   The board uses $*$ inside the custom rule $a*b = 2a - b^2$, while ordinary multiplication is written separately as $2 \times 1$ and $3 \times 3$ in the worked lines.

Because the symbol $*$ appears, a learner may treat $a*b$ as standard multiplication.

Here $*$ names a special binary operation defined by the formula $2a-b^2$; ordinary multiplication is handled separately in the arithmetic steps.

1. Audio
   Observation

   The speaker stresses that a binary operation uses two numbers and then identifies which number is $a$ and which is $b$ in each example.

2. Formula
   Observation

   Part (ii) treats $-7$ as the first input and $4$ as the second input in $-7*4$.

One might think the two numbers can be inserted into the formula in either order.

The video consistently assigns the left operand to $a$ and the right operand to $b$, so $-7*4$ means $a=-7$ and $b=4$, not the reverse.

1. Audio
   Observation

   At 01:04-01:10 the speaker emphasizes, "Once again the first number is a and the second number is b."

2. Diagram
   Observation

   Red labels a and b are written over the first and second operands in the worked lines.

One might treat $a*b$ like ordinary multiplication and think the order of the two inputs does not matter.

The video explicitly tracks the first number as a and the second as b, because the rule $a*b = 2a - b^2$ depends on their positions.

1. Audio
   Observation

   At 00:44-00:53 the speaker says they go back to part one and replace the bracketed term by negative seven before continuing.

2. Formula
   Observation

   Part (iii) is rewritten as (-8)*(-7) before applying the definition again.

One might try to apply the definition directly to (-8)*($1*3$) without simplifying the parenthesis first.

The worked solution first computes $1*3 = -7$ and only then evaluates (-8)*(-7).

1. Audio
   Observation

   After stating the rule, the speaker says they will substitute values of $a$ and $b$.

2. Formula
   Observation

   The worked line directly transforms $a*b = 2a - b^2$ into $1*3 = 2(1) - (3^2)$.

The substitution method is the practical way to apply the defined binary operation to concrete inputs.

1. Audio
   Observation

   The speaker explains that because $1*3$ was already worked out, the bracket term can be replaced by $-7$ before continuing.

2. Formula
   Observation

   The solution rewrites $(1*3)*4$ as $-7*4$ and then applies the rule again.

Nested evaluation depends on first using substitution on the inner expression, then substituting that result into the outer expression.

1. Audio
   Observation

   The speaker says, "binary implies two of something" while discussing why the expression must be handled as two numbers.

The claim about binary operations supports why the rule is applied to an ordered pair of inputs in each example.

1. Formula
   Observation

   Every worked line begins by turning an expression $x*y$ into $2x - y^2$.

The substitution method is the direct procedural use of the defined operation $a*b = 2a - b^2$.

1. Formula
   Observation

   Parts (ii) and (iii) both require an inner result before the outer substitution can be made.

Nested evaluation relies on first knowing how to substitute values into the operation rule.

1. Formula
   Observation

   Part (ii) uses the value -7 obtained in part (i) as the new left operand.

The computation of ($1*3$)*4 depends on the earlier result $1*3 = -7$.

1. Formula
   Observation

   Part (iii) rewrites (-8)*($1*3$) as (-8)*(-7) using the result from part (i).

The computation of (-8)*($1*3$) also depends on the previously found value $1*3 = -7$.

1. Audio
   Observation

   Opening narration states the given binary operation and formula.

2. Formula
   Observation

   Board shows If $a*b = 2a - b^2$.

1. Definition of the custom binary operation

1. Formula
   Observation

   Worked solution for part (i) ends at -7.

2. Audio
   Observation

   Speaker narrates the substitution and arithmetic for $1*3$.

1. Definition of the custom binary operation
2. Substitution method for evaluating the operation
3. Evaluation of $1*3$
4. Example (i): evaluate $1*3$

1. Audio
   Observation

   Speaker explains replacing the bracket term $1*3$ by $-7$ in $(1*3)*4$.

2. Formula
   Observation

   Right side shows $(ii)\ -7*4 = 2(-7) - 4^2$.

1. Reducing a nested expression by substituting an inner result
2. Setup for evaluating ($1*3$)*4
3. Example (ii): set up ($1*3$)*4

1. Audio
   Observation

   The speaker says binary implies two of something.

1. Binary operation uses two inputs
2. Definition of the custom binary operation

1. Formula
   Observation

   The custom rule uses $*$ while ordinary multiplication is written with $\times$ in the arithmetic lines.

1. Definition of the custom binary operation
2. Do not confuse the custom operator with ordinary multiplication

1. Formula
   Observation

   The board heading states "If $a*b = 2a - b^2$".

1. Custom binary operation defined by $a*b = 2a - b^2$

1. Audio
   Observation

   The speaker says the first number is a and the second number is b.

1. Substitution into the operation rule
2. Do not swap the operands in a custom binary operation

1. Audio
   Observation

   The speaker replaces the bracketed term $1*3$ by -7 before continuing.

1. Evaluate inside parentheses first in nested operations
2. Evaluate the inner bracket before the outer operation

1. Formula
   Observation

   The board shows (ii) ($1*3$)*$4 = -30$ with the full derivation on the right.

1. Example (ii): evaluate ($1*3$)*4
2. Derivation of part (ii): ($1*3$)*4

1. Formula
   Observation

   The board shows (-8)*(-7) = $2(-8) - (-7)^2 = -16 - 49 = -65$.

1. Example (iii): evaluate (-8)*($1*3$)
2. Derivation of part (iii): (-8)*($1*3$)