Lesson test: The rules of algebra

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The rules of algebra

1.[2p]
Which single law connects addition and multiplication?

Distributivity, $a (b + c) = a b + a c$

Correct

Commutativity of multiplication, $a b = b a$

The answer is: Distributivity, $a (b + c) = a b + a c$

Associativity of addition, $(a + b) + c = a + (b + c)$

The answer is: Distributivity, $a (b + c) = a b + a c$

The existence of the identity $1$

The answer is: Distributivity, $a (b + c) = a b + a c$

2.[2p]
Why is subtraction not on the list of laws?

It is shorthand: $a - b$ means $a + (- b)$ , so the laws of addition already cover it

Correct

It was left out by convention and could be added

The answer is: It is shorthand: $a - b$ means $a + (- b)$ , so the laws of addition already cover it

Because it fails for negative numbers

The answer is: It is shorthand: $a - b$ means $a + (- b)$ , so the laws of addition already cover it

Because it is only defined on the integers

The answer is: It is shorthand: $a - b$ means $a + (- b)$ , so the laws of addition already cover it

3.[1p]
Subtraction is associative, so $(10 - 4) - 3$ equals $10 - (4 - 3)$ .

True

The answer is: False

False

Correct

4.[2p]
Use the difference of two squares to compute $41 \times 39$ .

CorrectNot quite: 1599

5.[3p]
In the fake proof that $1 = 2$ , which step is illegal?

Cancelling the common factor $a - b$ , which is zero

Correct

Multiplying both sides by $a$

The answer is: Cancelling the common factor $a - b$ , which is zero

Subtracting $b^{2}$ from both sides

The answer is: Cancelling the common factor $a - b$ , which is zero

Factoring $a^{2} - b^{2}$ as $(a - b) (a + b)$

The answer is: Cancelling the common factor $a - b$ , which is zero

6.[3p]
What goes wrong if zero is given a multiplicative inverse?

It forces $0 = 1$ , and then every number equals zero

Correct

Nothing goes wrong, but the answer is hard to compute

The answer is: It forces $0 = 1$ , and then every number equals zero

Division stops being commutative

The answer is: It forces $0 = 1$ , and then every number equals zero

Negative numbers lose their inverses

The answer is: It forces $0 = 1$ , and then every number equals zero

7.[2p]
Factor $x^{2} + 2 x - 15$ as $(x + a) (x + b)$ with $a$ the larger. Give the pair as "a, b".

CorrectNot quite: 5, -3

8.[3p]
Which of these are consequences of the laws rather than separate assumptions?

Select all that apply

$a \times 0 = 0$ for every $a$

Correct

$(- 1) \times (- 1) = 1$

Correct

Multiplication binds more tightly than addition

The answer is: $a \times 0 = 0$ for every $a$ , $(- 1) \times (- 1) = 1$

Multiplying both sides of an inequality by a negative reverses it

The answer is: $a \times 0 = 0$ for every $a$ , $(- 1) \times (- 1) = 1$

9.[3p]
Match each move to the law that licences it.

Expanding a bracket
Reordering a sum
Writing $a - b$ as $a + (- b)$
Cancelling a non-zero common factor

the additive inverse
the multiplicative inverse
distributivity
commutativity of addition

Show the answer

Expanding a bracket: distributivity Reordering a sum: commutativity of addition Writing $a - b$ as $a + (- b)$ : the additive inverse Cancelling a non-zero common factor: the multiplicative inverse

1.[2p] Which single law connects addition and multiplication?

2.[2p] Why is subtraction not on the list of laws?

3.[1p] Subtraction is associative, so (10−4)−3 equals 10−(4−3).

4.[2p] Use the difference of two squares to compute 41×39.

5.[3p] In the fake proof that 1=2, which step is illegal?

6.[3p] What goes wrong if zero is given a multiplicative inverse?

7.[2p] Factor x2+2x−15 as (x+a)(x+b) with a the larger. Give the pair as "a, b".

8.[3p] Which of these are consequences of the laws rather than separate assumptions?

9.[3p] Match each move to the law that licences it.

1.[2p]
Which single law connects addition and multiplication?

2.[2p]
Why is subtraction not on the list of laws?

3.[1p]
Subtraction is associative, so $(10 - 4) - 3$ equals $10 - (4 - 3)$ .

4.[2p]
Use the difference of two squares to compute $41 \times 39$ .

5.[3p]
In the fake proof that $1 = 2$ , which step is illegal?

6.[3p]
What goes wrong if zero is given a multiplicative inverse?

7.[2p]
Factor $x^{2} + 2 x - 15$ as $(x + a) (x + b)$ with $a$ the larger. Give the pair as "a, b".

8.[3p]
Which of these are consequences of the laws rather than separate assumptions?

9.[3p]
Match each move to the law that licences it.