شرح فهم القيمة المكانية للأعداد العشرية

Worked example

1. 1

   $4072.36$ — One number, six digits. The point sits between the ones digit (2) and the tenths digit (3).

2. 2

   $7\text{is in the tens}$ — The 7 is two places left of the point, the tens place. So it is worth 7 × 10 = 70.

3. 3

   $3\text{is in the tenths}$ — The 3 is the first place after the point, the tenths. So it is worth 3 tenths, which is 0.3.

4. 4

   $0\text{holds the hundreds place}$ — There are no hundreds, so the 0 is worth nothing. But it keeps the 4 in the thousands: without it the number would be 472.36.

Worked example

1. 1

   $158000$ — The question: the value of the 5. First find which place the 5 sits in.

2. 2

   $\text{place: ten thousands}$ — Count places from the ones, moving left: ones, tens, hundreds, thousands, ten thousands. The 5 is fifth, so it is in the ten thousands.

3. 3

   $5\times 10000=50000$ — The value is the digit times its place. So the value is 50 000: a 5 followed by four zeros.

Worked example

1. 1

   $1.549$ — The question: the value of the 9. It is after the point, so its place is a fraction.

2. 2

   $\text{third place after the point}$ — Count from the point: tenths, hundredths, thousandths. The 9 is third, so it is in the thousandths.

3. 3

   $9\text{thousandths}=\frac{9}{1000}$ — So the value is 9 thousandths. You may also write it as 0.009; all three mean the same.

Worked example

1. 1

   $30408$ — Split the digits into groups of three from the right: 30 and 408. The left group counts thousands.

2. 2

   $30\to \text{thirty thousand}$ — Read the thousands group as a normal number, thirty, then say its name, thousand.

3. 3

   $408\to \text{four hundred and eight}$ — Read the last three digits on their own. The 0 means no tens, so you say nothing for it.

4. 4

   $\text{thirty thousand, four hundred and eight}$ — Join the two parts. This is the full answer.

Worked example

1. 1

   $215060$ — A bigger one. Split into groups of three from the right: 215 and 060.

2. 2

   $215\to \text{two hundred and fifteen thousand}$ — Read the thousands group, two hundred and fifteen, then say thousand.

3. 3

   $060\to \text{sixty}$ — The last group has no hundreds, so the 0 at the front is silent. Just say “and sixty”.

4. 4

   $\text{two hundred and fifteen thousand and sixty}$ — Join the two parts. This is the full answer.

Worked example

1. 1

   $\text{sixty thousand and nine}$ — The words. “Sixty thousand” is the thousands group; “nine” is what comes after it.

2. 2

   $60\square \square \square$ — Write 60 for sixty thousand. After the word thousand there must be exactly three more places.

3. 3

   $60009$ — Fill the three places: no hundreds, no tens, 9 ones. The two zeros hold the empty places.

Worked example

1. 1

   $\text{four hundred thousand, five hundred and seven}$ — The words. Everything before the word thousand is the thousands group.

2. 2

   $400\square \square \square$ — Write 400 for four hundred thousand, then leave three places for the rest.

3. 3

   $400507$ — Fill the three places: 5 hundreds, no tens, 7 ones. The 0 holds the empty tens place.

Worked example

1. 1

   $4.3\text{million}$ — The question: write it in figures. The 4 is whole millions; the 3 is 3 tenths of a million.

2. 2

   $4.3\times 1000000$ — Write “million” as × 1 000 000. It has six zeros, so the digits move six places left.

3. 3

   $4300000$ — The 4 lands in the millions and the 3 in the hundred thousands. Zeros fill every empty place.

4. 4

   $4|300|000$ — Check with groups of three: the millions, then two full groups of three. Seven digits in all.

Worked example

1. 1

   $\text{half a million}$ — Half is 0.5, so this is 0.5 million.

2. 2

   $0.5\times 1000000=500000$ — Move the digits of 0.5 six places left. The 5 lands in the hundred thousands.

Worked example

1. 1

   $48.2÷\square =0.0482$ — The question. The digits 4, 8, 2 stay the same; only their places change.

2. 2

   $48.2\to 4.82\to 0.482\to 0.0482$ — Move the digits one place right at a time. Each move is one ÷ 10.

3. 3

   $3\text{places}$ — Count the moves: the digits moved three places in total.

4. 4

   $\square =1000$ — Three moves of ÷ 10 is ÷ 1000, because 10 × 10 × 10 = 1000.

Worked example

1. 1

   $0.004\times \square =12$ — A harder box. The answer 12 has different digits from 0.004, so moving digits alone will not do it.

2. 2

   $0.004\times 1000=4$ — First multiply by 1000: the digits move three places left and 0.004 becomes the whole number 4.

3. 3

   $4\times 3=12$ — Now 4 still has to be multiplied by 3 to reach 12.

4. 4

   $\square =1000\times 3=3000$ — You multiplied by 1000 and then by 3, so the box is 3000. Check: 0.004 × 3000 = 12.

Worked example

1. 1

   $36\times 52=1872$ — The fact you are given. Your answer will use the same digits, 1872.

2. 2

   $3.6=36÷10$ — Compare the numbers: 3.6 is ten times smaller than 36.

3. 3

   $5.2=52÷10$ — 5.2 is also ten times smaller than 52.

4. 4

   $3.6\times 5.2=1872÷100$ — Both numbers were divided by 10, so the answer is divided by 10 × 10 = 100.

5. 5

   $=18.72$ — Move the digits of 1872 two places right. Check: 3.6 × 5.2 is about 4 × 5 = 20, so 18.72 makes sense.

Worked example

1. 1

   $1872÷52=36$ — Turn the fact round: if 36 × 52 = 1872, then 1872 ÷ 52 = 36.

2. 2

   $5.2=52÷10$ — Now you divide by 5.2, a number ten times smaller than 52.

3. 3

   $1872÷5.2=36\times 10$ — Dividing by a number ten times smaller gives an answer ten times bigger.

4. 4

   $=360$ — Check: 1872 ÷ 5.2 is about 2000 ÷ 5 = 400, so 360 makes sense.

Worked example

1. 1

   $\text{largest: }8630$ — Put the biggest digit in the biggest place. So order the cards from largest to smallest: 8, 6, 3, 0.

2. 2

   $\text{smallest: }3068$ — Order from smallest, but 0 cannot go first. So start with the next smallest card, 3, then 0, 6, 8.

3. 3

   $\text{largest odd: }8603$ — An odd number must end in an odd digit. The only odd card is 3, so it goes last. Put 8, 6, 0 in front, largest first.

Worked example

1. 1

   $\text{try just above and just below }6000$ — The closest number is either the smallest one above 6000 or the largest one below it. Find both.

2. 2

   $\text{above: }6259$ — Start with 6, then make the rest as small as possible: 2, 5, 9.

3. 3

   $\text{below: }5962$ — Start with 5, then make the rest as large as possible: 9, 6, 2.

4. 4

   $6259-6000=259$ — How far the number above is from 6000. Write this subtraction down: it earns the method mark.

5. 5

   $6000-5962=38$ — How far the number below is. 38 is less than 259.

6. 6

   $5962$ — So 5962 is the closest, with a difference of 38.