Justifying mathematical reasoning

Question 1

Which explanation correctly shows why multiplying any even number by any whole number always gives an even number?

Accepted Answer

An even number can be written as 2 × k. When you multiply it by another whole number n, you get (2 × k) × n = 2 × (k × n), which still has a factor of 2, so the product is even.

Answer

If you multiply an even number by an odd number, the odd number changes the result and makes the product odd.

Answer

Multiplying is the same as adding the number many times, and adding an even number to an odd number always makes an odd number.

Answer

Only when you multiply an even number by another even number do you get an even result; multiplying by an odd number can give an odd result.

Question 2

Which explanation correctly shows why adding 5 to any odd number always gives an even number?

Accepted Answer

An odd number ends with 1, 3, 5, 7 or 9. Adding 5 to each of those gives 6, 8, 10, 12, or 14. In every case the result ends with an even digit (or 0 with a carry), so the whole number is even.

Answer

Since 5 is an odd number, adding it to another odd number must keep the sum odd.

Answer

When you add 5, you only need to look at the tens place; the ones place doesn't affect whether the total is even or odd.

Answer

The rule works only for odd numbers smaller than 10; larger odd numbers can give an odd result when 5 is added.

Question 3

Which statement correctly explains why the sum of any two even numbers is always even?

Accepted Answer

An even number can be written as 2 × something. Adding 2 × a and 2 × b gives 2 × (a + b), which is also a multiple of 2, so the result is even.

Answer

When we add two even numbers the answer always ends with the digit 5, and numbers that end with 5 are even.

Answer

Even numbers are the numbers that are bigger than 10, so adding two of them must make a bigger even number.

Answer

If we count the fingers on both hands we get an odd number, so adding two even numbers must give an odd number.

Try three questions

Which explanation correctly shows why multiplying any even number by any whole number always gives an even number?

Which explanation correctly shows why adding 5 to any odd number always gives an even number?

Which statement correctly explains why the sum of any two even numbers is always even?

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