A MONTH OF STARTERS 31 lesson starters for all abilities
Rough guide Each task can be used just as a starter, or as the starting point of an investigation. They should be accessible to most students working at level 4+ and should take around 5 minutes. A rough guide of the NC level for each task is given by the shape in the top right corner. All tasks contain extension to challenge more able students.
1. Dividing by fractions 6 ÷ 3 means how many 3’s are there in 6… so, can you work out these? 6 ÷ 1 = 6 ÷ 1/2 = 6 ÷ 1/3 = 6 ÷ 1/4 = What do you notice about your answers? Can you see a pattern? Ext 1: Try dividing by different fractions (like 2/3) and see if you can come up with a rule. Ext 2: What is 6 ÷ 0 = ?
2. Remainder 1 7 ÷ 2 = 3 remainder 1 1. Find some numbers that have remainder 1 when you divide them by 2. What do you notice about these numbers? 2. Find some numbers that have remainder 1 when you divide by 3. 3. Can you find a number that has remainder 1 when you divide it by 2 and 3? Ext 1: can you find a number that has remainder 1 when you divide it by 2, 3 and 4?? Ext 2: what about 2, 3, 4 and 5? what do you notice about remainder 1 numbers?
3. Tell the truth People in Truetown either always tell the truth or always lie… Three people from Truetown are having a conversation: A mumbles something which only B and C hear. B says ‘A said he is a liar’ C says ‘don’t listen to B, he is lying’ The question is… can you work out which of these people are telling the truth and which are lying?
4. 400 problem These two sums have the same answer 4 a b 4 0 0 - 4 0 0 - a b 4 Can you work out the value of a and b?
Difference triangles introduction Can you see how this ‘difference triangle’ is made? Starting at the bottom row, find the difference between the two numbers and put it in the box above Now try difference triangles 1… 2 1 3
5. Difference triangle 1 Can you complete this difference triangle with the numbers 1 to 6? Ext: Investigate the next triangle (1-10)… then try Difference triangle 2
6. Difference triangle 2 Note: this is possible! Can you find a method for completing it? 5 4 9
7. Centences ‘Thare are five mistukesim this centence…’ True or false? Extension: Can you make up some centences like this of your own?
8. Probably prime You may have noticed that there are 4 prime numbers between 1 and 10… (they are 2, 3, 5 and 7). • If you pick a number between 1 and 10 randomly, what is the chance it will be prime? • If you pick a number between 1 and 20, what is the chance it will be prime? Ext 1: what about 1-100? Or 1-200? What do you notice? Ext 2: What about any number – what is the probability it is prime?
9. 6 spaces I can make 6 equal spaces (rectangles) using 13 matches like this: How can you make 6 equal spaces using only 12 matches?? Extension: Is this the least number of matches you need to make 6 equal spaces?
10. 3 times table Look at the numbers in the 3 times table… 3 , 6 , 9 , 12 , 15 , 18 , 21 , 24 , Add the digits together…what do you notice? Now, which of these numbers are in the 3 times table: 39? 93? 100? 101? 102? 111? 2011? 153641? 96574281? Extension: can you find similar rules for other times tables??
11. Party I was cleaning up after a party with my friends this weekend. I found a can of drink that was 3/4 full, 3 half full ones, 2 cans that were a quarter full, one that was 1/4 empty and 3 half empty ones. How many cans did I find? Extension: can you think of a more complicated question like this?
12. M a t h s a − m = m a × m = a What numbers do a and m represent? Extension: If also, s – t = m and a x a = h …then what is m + a – t – h + s ?
13. digit-sums The sum of the digits in the number 37 is 3 + 7 = 10 1. How many 2 digit numbers have a digit-sum of 10? 2. How many 3 digit numbers have a digit-sum of 10? Extension 1: How many 4 digit numbers? Extension 2: Can you find a rule? Does your rule work for different digit-sums?
14. digit-products The product of the digits in the number 38 is 3 x 8 = 24 1. How many 2 digit numbers have a digit-product of 24? 2. How many 3 digit numbers have a digit-product of 24? Extension 1: How many 4 digit numbers? Extension 2: Can you find a rule? Does your rule work for different digit-products?
15. try-angles 1 There are three angles in a triangle: A, B and C. Angle B is 30 degrees more than angle A. Angle C is 60 degrees less than angle B. What type of triangle is it? Ext: try ‘try-angles 2’
16. try-angles 2 There are three angles in a triangle: A, B and C. Angle A is 50 degrees. The difference between Angles B and C is 30 degrees. What type of triangle is it?
17. How much? Roughly which animal weighs around 10kg? A a mouse? B a cat? C a dog? D a person? E an elephant? Extension: make up some other questions about measure…
18. How much? Which object contains roughly 200ml when full? A a spoon? B a cup? C a pan? D a swimming pool? E a lake? Extension: make up some other questions about measure…
19. 30 years young I am 30 years old, not counting Sundays… How old am I really…? How old are you if you don’t count Sundays? My mum reckons she is 45, but she doesn’t count Saturdays or Sundays! How old is she really??
20. 2011 The digits of 2011 add up to 4. How many other years (since year 1) have the digits added up to 4? Ext: In 2012 the digits will add up to 5. How many times has this happened? Can you find a rule for working this out more quickly?
21. MMM Start with the word MMM. Suppose you are allowed to turn exactly2 of the letters in the word MMM upside down at a time. (so your next word might be WWM). How many goes will it take to make the word WWW? Ext 1: Start with the word MMMM and move 3 letters at a time. What is the least number of goes you can take to make the word WWWW? Ext 2: Investigate for other lengths of words…
22. M A T H S (again) M Ax T H S Each letter represents a different number from 1 to 5. Can you work out what the sum is? Can you think of a good method of solving this problem? Is there only one answer? Ext: Can you make up another problem like this?
23. MATHEMATICS How many letters of the word MATHEMATICS do not have any symmetry (remember, there are two types of symmetry!) Ext 1: Can you think of any other mathematical words that only contain letters that have reflective symmetry? Or rotational symmetry? Ext 2: Can you think of a mathematical word itself that is symmetrical (eg like MUM)? Or a number?
24. Cutting corners A cube has 6 faces, 8 corners and 12 edges. If I cut the corners off a cube, how many faces, corners and edges will it have now? Can you think of a quick way of working this out? Ext 1: what shape do I get if I cut the corners from the middle of all the edges? Ext 2: investigate for other shapes (like a pyramid?)
25. Broken window Someone broke a window, so 5 students who saw it were asked who did it. Sandra said: ‘It was Paul’Wendy said: ‘It wasn’t me’Paul said: ‘Ted did it’Ted said: ‘Paul is lying’Angela said: ‘It wasn’t me or Wendy’ Only one of them is telling the truth!! So… who broke the window??
26. Spiders On average, a human being swallows around 10 spiders while sleeping in his or her lifetime. Suppose people live to around 70 on average, estimate: • How many you have swallowed in your lifetime • How many your teacher has swallowed • How many have been swallowed by everyone in your class Extension: Can you estimate how many were swallowed in the UK last year?
27. Hexagon in a cube A cube can be cut to give a triangle and a square as shown… Can you find a way to cut a cube to get a hexagon? Extension: What other shapes can be made by cutting a cube?
28. Strip to cube Can you fold this strip of 7 squares to make a cube …without cutting? Ext: can you make other strips that fold to make other shapes (eg pyramid)?
29. Unattacked squares You can put 4 queens on a 4x4 board and leave one square ‘unattacked’ as shown. Can you put 5 queens on a 5x5 square and leave a square unattacked? Can you place them to leave more than 1 square unattacked? What is the most number of unattacked squares? Ext: investigate for larger boards… is there a pattern?
30. Attacked squares Can you place 3 queens on this 6x6 grid to leave nounattacked squares? Ext: How many castles would you need? Investigate for other pieces (or combinations of pieces?)
31. QWERTY Letters that appear next to each other on a normal keyboard can be put together to make a ‘string’ of letters eg QWERTY or EDCFGBV What is the longest string of letters all containing acute angles? Or obtuse angles? Or right angles? What is the longest string of letters with reflective symmetry? Or rotational symmetry?
Some NRICH starters • What do you need (number puzzle) http://nrich.maths.org/5950 • Number daisy http://nrich.maths.org/786 • Cycling squares http://nrich.maths.org/1151 • Up and down staircase http://nrich.maths.org/2283 • Make 37 http://nrich.maths.org/1885 • Mystery matrix http://nrich.maths.org/1070 • Odd squares http://nrich.maths.org/2280 • Next number http://nrich.maths.org/1021 • Ace, two, three http://nrich.maths.org/5775