Justifying mathematical reasoning (age 8+)

Question 1

Sam tried to subtract 742 – 386 using column subtraction and wrote the steps shown below: Step 1: In the ones column he wrote 2 – 6 = 4. Step 2: In the tens column he wrote 4 – 8 = 5. Step 3: In the hundreds column he wrote 7 – 3 = 3. He recorded the answer as 356. Which statement correctly identifies the error in Sam’s work?

Accepted Answer

Sam forgot to borrow from the tens column before subtracting the ones.

Answer

Sam correctly borrowed from the tens column before subtracting the ones.

Answer

Sam borrowed from the hundreds column when he should have borrowed from the tens column.

Answer

Sam subtracted the tens column incorrectly; 8 should be subtracted from 4.

Question 2

Sam's friend Maya tried to subtract 805 – 467 using column subtraction and wrote the work like this: Ones column: 5 – 7 = 2 Tens column: 0 – 6 = 4 Hundreds column: 8 – 4 = 3 Which statement correctly identifies the error in Maya's work and explains why it is wrong?

Accepted Answer

Maya subtracted the ones place without borrowing. Because 5 is smaller than 7, she must borrow 1 ten (making the ones 15) and reduce the tens column before subtracting, so the ones column should be 15 – 7 = 8.

Answer

Maya wrote 0 – 6 = 4, but the correct result is –6, so she should have added 10 to the tens column first.

Answer

Maya correctly borrowed from the hundreds column, but she forgot to change the hundreds digit after borrowing, so the hundreds column should be 7 – 4 = 3.

Answer

Maya swapped the numbers in the subtraction, so she actually computed 467 – 805.

Question 3

Which explanation correctly shows why 1/3 is greater than 1/5?

Accepted Answer

When a whole is divided into more equal parts, each part gets smaller, so one of five parts is smaller than one of three parts; therefore 1/3 is larger.

Answer

Because 3 is a bigger number than 5, dividing 1 by 3 gives a bigger result than dividing 1 by 5.

Answer

Since the numerator 1 is the same, the fraction with the larger denominator must be larger.

Answer

If you add the denominators, 3 + 5 = 8, and 1/8 is between them, so 1/3 must be the larger fraction.

Try three questions

Sam's friend Maya tried to subtract 805 – 467 using column subtraction and wrote the work like this: Ones column: 5 – 7 = 2 Tens column: 0 – 6 = 4 Hundreds column: 8 – 4 = 3 Which statement correctly identifies the error in Maya's work and explains why it is wrong?

Which explanation correctly shows why 1/3 is greater than 1/5?

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