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Times 1, Times −1, And Swapping The Order

Multiplying by 1 keeps a number, multiplying by −1 gives its opposite, and swapping the two numbers never changes a product.

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Multiplying by 1 and by −1

Why 1 × a = a and −1 × a = −a for every integer.

1 × a

We know $1 \times a = a$ when $a$ is positive. Is it true for negative $a$ too?

With tokens: put $a$ into the empty bag once. If $a = -5$ , the bag holds 5 negatives, which is $-5$ . So

$1 \times a = a \quad \text{for every integer } a$

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Times −1: The Opposite-Maker

Using −1 × a = −a to answer 'what times −1 gives this?'

Problem. Find $-1 \times 27$ and $-1 \times (-31)$ . Then find the integer whose product with $(-1)$ is $27$ . (Textbook p. 42, Q3)

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Find the Integer

Which integer gives each product when multiplied by −1?

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Its product with $(-1)$ is $27$ .

Does Swapping Change the Product?

Why a × b = b × a works for all integers.

Size

The size of a product depends only on the sizes of the two numbers. Swapping two positive numbers never changes their product ( $3 \times 4 = 4 \times 3$ ), so the size stays the same.

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Checking Swapped Pairs

The book's table of swapped products, filled in.

Problem. Fill in the blanks in the book's table of swapped pairs (Textbook p. 34).

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Fill the Swap Table

Complete the blanks in the swapped-pairs table.

0 of 4 blanks filled

Textbook p. 34.

−30 × 12 =

12 × −30 =

−5 × = −70

And one more: −9 × 7 = −63, so 7 × −9 = .

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Multiplying by 1 and by −1

Why 1 × a = a and −1 × a = −a for every integer.

1 × a

We know $1 \times a = a$ when $a$ is positive. Is it true for negative $a$ too?

With tokens: put $a$ into the empty bag once. If $a = -5$ , the bag holds 5 negatives, which is $-5$ . So

$1 \times a = a \quad \text{for every integer } a$

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Times −1: The Opposite-Maker

Using −1 × a = −a to answer 'what times −1 gives this?'

Problem. Find $-1 \times 27$ and $-1 \times (-31)$ . Then find the integer whose product with $(-1)$ is $27$ . (Textbook p. 42, Q3)

1 / 5

Find the Integer

Which integer gives each product when multiplied by −1?

1 / 5

Its product with $(-1)$ is $27$ .

Does Swapping Change the Product?

Why a × b = b × a works for all integers.

Size

The size of a product depends only on the sizes of the two numbers. Swapping two positive numbers never changes their product ( $3 \times 4 = 4 \times 3$ ), so the size stays the same.

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