Question

the calculation of the difference \\(30,005 - 1097\\) might look like this to your class. carefully discuss the exchanges indicated.

here, 1 ten thousand is exchanged for 10 thousands, leaving 2 ten thousands. then one thousand is exchanged for 10 hundreds, leaving 9 thousands. next, 1 hundred is exchanged for 10 tens, leaving 9 hundreds. finally, 1 ten is exchanged for 10 ones, for a total of 8 ones. with these exchanges, it is now possible to perform the subtraction.

Explanation:

Identify the missing values in the explanation

We need to fill in the blanks in the explanation of the subtraction \(30,005 - 1097\).
The text reads:
"Here, 1 ten thousand is exchanged for [blank 1] thousands, leaving [blank 2] ten thousands. Then one thousand is exchanged for [blank 3] hundreds, leaving 9 thousands. Next, 1 hundred is exchanged for [blank 4] tens, leaving 9 hundreds. Finally, 1 ten is exchanged for [blank 5] ones, for a total of [blank 6] ones. With these exchanges, it is now possible to perform the subtraction."

Analyze the first exchange

We start with \(30,005\).
The ten-thousands digit is \(3\).
We take \(1\) ten thousand, leaving \(2\) ten thousands.
\(1\) ten thousand is exchanged for \(10\) thousands.

• Blank 1: \(10\)
• Blank 2: \(2\)

Analyze the second exchange

We now have \(10\) thousands.
We take \(1\) thousand (leaving \(9\) thousands) and exchange it for \(10\) hundreds.

• Blank 3: \(10\)

Analyze the third exchange

We now have \(10\) hundreds.
We take \(1\) hundred (leaving \(9\) hundreds) and exchange it for \(10\) tens.

• Blank 4: \(10\)

Analyze the final exchange

We now have \(10\) tens.
We take \(1\) ten (leaving \(9\) tens) and exchange it for \(10\) ones.

• Blank 5: \(10\)

Since we already had \(5\) ones in the units place of \(30,005\), adding the \(10\) ones gives a total of \(10 + 5 = 15\) ones.

• Blank 6: \(15\)

Answer:

Here, 1 ten thousand is exchanged for <blank>10</blank> thousands, leaving <blank>2</blank> ten thousands. Then one thousand is exchanged for <blank>10</blank> hundreds, leaving 9 thousands. Next, 1 hundred is exchanged for <blank>10</blank> tens, leaving 9 hundreds. Finally, 1 ten is exchanged for <blank>10</blank> ones, for a total of <blank>15</blank> ones. With these exchanges, it is now possible to perform the subtraction.