Addition 5
Problems, patterns, rules & predictions
Objectives

Add several small numbers

Solve mathematical problems

Spot patterns and rules

Make and test predictions

Explain their reasoning
For this unit you will need:
1-20 number cards
Watch out for pupils who:

when they spot a pattern, do not test out their theory with further
examples;

do not look for quick ways to add several numbers.
HSNP © Hamilton 2014
Shining Term 3
Addition 5
Session 1
Objectives: Add several small numbers; Solve mathematical problems
Teacher input with whole class
 Explain that a person has a set of 1 to 20 cards and has divided them
into six uneven piles, i.e. the piles do not have the same number of
cards in each. BUT the total of the numbers on the cards in each pile is
the same!
Paired pupil work
 Pupils work in pairs to discuss how they might go about solving this
problem, and in particular what the total of each pile of cards might
be.
Teacher input with whole class
 After a while take feedback and draw out that the total must be a sixth
of the total of all the 1 to 20 cards.
 Show pupils a quick way of doing this. Write the numbers 1 to 20 in
order from least to greatest, and then from greatest to least under the
first row:
1 2 3 4 5 … 20
20 19 18 17 16 ... 1
 Ask pupils to discuss in pairs what they see. What is the total of each
column? How many columns? Pupils discuss how to use this to find the
total (20 × 21 × ½). Pupils then divide 210 by 6 to give 35.
Paired pupil work
 Pupils then continue working in pairs to move cards around so that the
total in each of six piles of 35.
 You can find solutions for this problem at:
http://nrich.maths.org/1047/solution.
Teaching input with whole class
 Take feedback from pupils about how they went about the problem.
HSNP © Hamilton 2014
Shining Term 3
Addition 5
Session 2
Objectives: Add several small numbers; Spot patterns and rules; Make
and test predictions
Paired pupil work
 Ask pupils to add pairs, and then trios of consecutive numbers, e.g. 5 +
6, then 5 + 6 + 7. They discuss in pairs what they notice.
Teacher input with whole class
 Take feedback. Draw out that the sum of any pair of consecutive
numbers is odd (even + odd).
 Ask pupils to discuss why the sum of any three consecutive numbers is
a multiple of 3. (The 1st and 3rd numbers are 1 less and 1 more than the
middle number so the sum is 3 times the middle number, i.e. a
multiple of 3.)
 Draw a table on the board with numbers 2 to 15 in the left column. Ask
pupils to think of how they could make some of these numbers by
adding consecutive numbers. Write a few suggestions in the table:
2
3
1+2
4
5
2+3
6
1+2+3
7
3+4
8
9
4 + 5 or 2 + 3 + 4
Paired pupil work
 Pupils work in pairs to copy and complete the table, continuing it
further if they have time. They work out which numbers they can make
by adding consecutive numbers, which they can’t and which can be
made in more than one way.
Teacher input with whole class
 Pupils share their findings, e.g. every odd number can be made (by
adding pairs of consecutive numbers: 1 + 2, 2 + 3…) and every multiple
of 3 (1 + 2, then 1 + 2 + 3, 2 + 3 + 4…) but not all even numbers.
HSNP © Hamilton 2014
Shining Term 3
Addition 5
Session 3
Objectives: Spot patterns and rules; make and test predictions; explain
their reasoning
Teacher input with whole class
 Together make a list of the first 12 prime numbers on the board: 2, 3,
5, 7, 11, 13, 17, 19, 23, 29, 31, 37.
Paired pupil work
 Ask pupils to work in pairs to find the difference between each
neighbouring pair of prime numbers and discuss what they notice.
 They then add neighbouring pairs and discuss what they notice.
Teacher input with whole class
 Take feedback on what pupils found. For example they may have
discovered that other than the difference between 2 and 3, all
differences are even. Discuss why this might be the case (all prime
numbers are odd apart from 2, therefore all differences are even).
 All the sums are even as each sum is the sum of two odd numbers
which is always even. The exception is 2 and 3, which still gives an odd
total.
Paired pupil work
 Pupils find the digital root of each prime number. They list which
numbers appear and which don’t appear.
Teacher input with whole class
 Take feedback. Which numbers do not appear in the list of digital
roots? (6 and 9, and 3 only once.) Discuss why this might be the case.
Remind pupils of the tests of divisibility! (The test of divisibility for
multiples of 3 is if the digital root is 3, 6 or 9, the number is a multiple
of 3. So any number with a digital root of 3, 6, or 9 is a multiple of 3
and therefore not prime, with the exception of 3 itself.)
 They may also have noted that digital roots of prime numbers are not
all prime themselves.
HSNP © Hamilton 2014
Shining Term 3