1. See if you can complete the argument about grouping by units, tens, hundreds, … to justify why the addition algorithm works.
2. How would you modify the algorithm to add two decimal fractions?
- 1. See if you can complete the argument about grouping by units, tens, hundreds, … to justify why the addition algorithm works.
- 2. How would you modify the algorithm to add two decimal fractions?
Step-by-step solution
Idea: Place value: a number is units + tens + hundreds + …. Adding the same groups together does not change the total, and 10 of one group always equals 1 of the next group. For decimals, the decimal point fixes the place values, so it is the point (not the last digit) that must be aligned.
1. See if you can complete the argument about grouping by units, tens, hundreds, … to justify why the addition algorithm works.
- Write each number as groups. For example, 473 = 4 hundreds + 7 tens + 3 units and 695 = 6 hundreds + 9 tens + 5 units.½ mark
- Addition can be done in any order and grouping, so we may add the units together, the tens together and the hundreds together:
473 + 695 = (4 + 6) hundreds + (7 + 9) tens + (3 + 5) units = 10 hundreds + 16 tens + 8 units.
Aligning the numbers from the right puts exactly these groups in the same column.½ mark - A group may now hold 10 or more. Since 10 units = 1 ten, 10 tens = 1 hundred, and so on, we exchange ten of a group for one of the next group. This is the carry. Here 16 tens = 1 hundred + 6 tens, so 6 stays in the tens place and 1 is carried to the hundreds: 10 + 1 = 11 hundreds = 1 thousand + 1 hundred.½ mark
- After each exchange a group holds less than 10, so it is a single digit in its place. The exchange never changes the total, only how it is grouped. So the digits written down, 1 thousand + 1 hundred + 6 tens + 8 units = 1168, are exactly the sum. Working from right to left lets each carry be added to the next group before that group is written. The same reasoning works for any two numbers, so the algorithm always gives the sum.½ mark
2. How would you modify the algorithm to add two decimal fractions?
- In a decimal fraction the place values are fixed by the decimal point: tenths, hundredths, … to its right; units, tens, … to its left. So change Step 1: write the numbers one below the other with the decimal points in a line (not the rightmost digits).½ mark
- Fill in zeros on the right so both numbers have the same number of decimal places (12.5 = 12.50). This does not change the value, and every column now has two digits.½ mark
- Steps 2 to 5 stay the same, with “more than 10” read as “10 or more”: add from right to left with carries (a carry from tenths goes into the units, because 10 tenths = 1 unit).½ mark
- Add one last step: put the decimal point in the answer directly below the other decimal points.
Example: 12.5 + 3.75 → 12.50 + 03.75: 0 + 5 = 5; 5 + 7 = 12, write 2, carry 1; 2 + 3 + 1 = 6; 1 + 0 = 1. Answer 16.25.½ mark
Check: 16.25 − 3.75 = 12.5 ✓. 1168 − 695 = 473 ✓.
Answer to write in the exam
1.
473 = 4 H + 7 T + 3 U, 695 = 6 H + 9 T + 5 U
473 + 695 = (4 + 6) H + (7 + 9) T + (3 + 5) U (adding like groups)
= 10 H + 16 T + 8 U
16 T = 1 H + 6 T (10 tens = 1 hundred: carry 1)
10 H + 1 H = 11 H = 1 Th + 1 H (10 hundreds = 1 thousand: carry 1)
= 1 Th + 1 H + 6 T + 8 U = 1168
∴ Adding column by column with carries only regroups the same total, so the algorithm gives the sum.
2.
Step 1 changed: write the numbers with the decimal points one below the other.
Fill in zeros on the right so both have the same number of decimal places.
Add right to left with carries, as before (carry when a column is 10 or more).
Put the decimal point in the answer below the other points.
e.g. 12.50 + 3.75: 5; 12 → 2, carry 1; 2 + 3 + 1 = 6; 1 → 16.25
∴ 12.5 + 3.75 = 16.25
Common mistakes that cost marks
- Aligning decimals by the last digit, as with whole numbers. In 12.5 + 3.75 that puts the 5 tenths of 12.5 under the 5 hundredths of 3.75 and gives a wrong sum. The decimal points must be in a line.
- Forgetting that a carry from the tenths column goes into the units column.
- In the argument, saying “we carry 1” without saying why: 10 of one place value equals 1 of the next place value.
How this can come in the exam
7.6 + 0.85 =
- 8.45
- 0.161
- 1.61
- 8.41
Show answer
(A) 8.45
Align the points and pad: 7.60 + 0.85. 0 + 5 = 5; 6 + 8 = 14, write 4, carry 1; 7 + 0 + 1 = 8. Answer 8.45.
Assertion (A): In 58 + 67, the 1 carried from the units column is added to the tens column.
Reason (R): 10 units make 1 ten.
- Both A and R are true, and R is the correct explanation of A.
- Both A and R are true, but R is not the correct explanation of A.
- A is true but R is false.
- A is false but R is true.
Show answer
(A) Both A and R are true, and R is the correct explanation of A.
8 + 7 = 15 units = 1 ten + 5 units, so the 1 ten joins the tens column: 5 + 6 + 1 = 12 tens. Answer 125. R explains A.
Try one yourself
Add 46.08 + 9.957 by the column method.
Show answer
Align the points and pad: 46.080 + 09.957. 0 + 7 = 7; 8 + 5 = 13, write 3, carry 1; 0 + 9 + 1 = 10, write 0, carry 1; 6 + 9 + 1 = 16, write 6, carry 1; 4 + 0 + 1 = 5. Answer 56.037.
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