Numbers and the number system
Rounding Challenge
Solving mathematical problems
You will need: • set of 1–9 digit cards
Shuffle a set of 1–9 digit cards. Choose the top four cards. Use these cards to make different 2-, 3- and 4-digit numbers. Then round each: • • •
2-digit number to the nearest ten 3-digit number to the nearest ten and hundred 4-digit number to the nearest ten, hundred and thousand.
4 7 2 9
Numbers and the number system
Factors Challenge Investigate which numbers less than 100 have proper factors that are only even numbers. Investigate which numbers less than 100 have proper factors that are only odd numbers.
Think about … For the ‘Challenge’, make sure that you use the four digit cards to create at least six 2-digit numbers, six 3-digit numbers and six 4-digit numbers.
For the ‘What if?’ task, make sure that you use the four digit cards to create at least three decimals with 1 decimal place (tenths), and three decimals with 2 decimal places (hundredths)
What if ? What if you use the four digit cards to make different decimal numbers for tenths and hundredths? Multiply each of the decimal numbers by 10 and 100. Divide each of the decimal numbers by 10. Round each decimal with: • • 8
1 decimal place to the nearest whole number 2 decimal places to the nearest tenth and whole number.
9∙2 × 10 = 9∙2 × 100 = 9∙2 ÷ 10 = 9∙2 rounded to the nearest tenth is …
9∙2 27∙4 47∙29 7∙42
Solving mathematical problems
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Think about … A factor is a whole number that divides exactly into another whole number without a remainder.
Proper factors of a number are all its factors except 1 and the number itself.
A prime number is a whole number greater than 1 that can’t be divided by another whole number, except for itself and 1.
A factor of a whole number that is also a prime number is called a prime factor.
What if ? 20 : 1, 2, 4, 5, 10 and 20 20 has two prime factors: 2 and 5.
When you’v e finished, tur n to page 80.
Investigate other numbers less than 100 that have two prime factors.
When you’v e finished, tur n to page 80. 9
Unit fractions
Solving mathematical problems
Challenge
Numbers and the number system
Equivalences Challenge
1 2
2, 3
Apart from the Ancient Egyptians only wrote unit fractions. This is how they might have expressed 34 and 38:
1 3
You will need:
1 4
1 5
1 6
1 7
1 8
1 9
• calculator
…
Convert some unit fractions to decimals. Then sort the decimals into those that:
3 4
= 12 + 14
3 8
= 14 + 18
• convert to tenths • convert to hundredths · ·· • are recurring decimals, for example 0·33333 or 0·090909 … • have anything different. What observations and generalisations can you make?
What non-unit ke using the fractions can you make Ancient Egyptian method?
Think about …
Think about … A unit fraction is a fraction that has 1 as its numerator, for example: 12, 14, 15.
A non-unit fraction is a fraction that has as its numerator any number other than 1, for example: 2, 3 or 4. 3 4 5
Use diagrams to help you add or subtract unit fractions, for example: + 1 2
+
= 1 4
=
3 4
For each of the unit fractions you converted to equivalent decimals in the ‘Challenge’, work out the equivalent percentages.
How can you make the non-unit fractions you have created using the smallest possible number of fractions?
10
= 12 + 14
Remember, a unit fraction is a fraction that has 1 as its numerator, for example 12, 14, 15.
For the ‘What if?’ tasks, round each percentage to the nearest whole per cent.
What if ?
What if ?
3 4
Solving mathematical problems
Collins: Is this written correctly - should it be 0.09 with dots over the 09 after the decimal pt? Or 0.0909...
Numbers and the number system
rather than
3 4
= 14 +
1 4
+
1 4
When you’v e finished, tur n to page 80.
Convert some non-unit fractions such as: 2, 3, 2, 3, 4, 5 ... to decimals and work out the 3 4 5 5 5 6 equivalent percentages. What observations and generalisations can you make?
When you’v e finished, tur n to page 80. 11
Numbers and the number system
Percentages Challenge
Solving mathematical problems
You will need:
0 1 2 3 4 5 6 7 8 9 How many of these percentage statements can you make using five of the 0–9 digit cards?
% ×
• set of 0–9 digit cards
Domino ratios Challenge
Solving mathematical problems
You will need: • set of 6 × 6 dominoes
Look at a set of dominoes with the double blank removed, that is: Count the total number of dots on each domino.
25% × 16 = 4 12% × 50 = 6
3+6=9
What fraction of the dominoes have totals that are an even number? What fraction of the dominoes have totals that are an odd number? What is the ratio of even totals to odd totals?
=
Think about …
You can’t use the same digit twice in a statement.
Think about … Start with percentages such as 10%, 20%, 25%, 50% and 75%.
Think about equivalent fractions and decimals to help you write different statements.
Numbers and the number system
Be systematic in the way you record the totals, differences and products.
What if ? Look at a set of dominoes with the doubles removed, that is:
Look at a set of dominoes with the dominoes with a blank removed, that is:
What if ? What if you use four of the 0–9 digit cards? What if you find the difference between the number of dots on
How many of these percentage statements can you make?
% ×
=
each side?
What fraction of the dominoes have differences that are an even number?
What if you use six of the 0–9 digit cards? How many of these percentage statements can you make?
% × 12
=
6–3=3
When you’v e finished, tur n to page 80.
What fraction of the dominoes have differences that are an odd number? What is the ratio of even differences to odd differences?
What if you multiply together the number of dots on each side? 3 × 6 = 18 What fraction of the dominoes have products that are an even number? What fraction of the dominoes have products that are an odd number? What is the ratio of even products to odd products?
When you’v e finished, tur n to page 80. 13