Investigating Division Further Years 2 - 8

Summary Poly Plug are used to represent numbers. A chosen number is divided into equal rows and left overs are allowed. This physical division can be explored with the calculator and recorded in symbols. Suitable for threading. The activity can be used with The Big One which is a calculator game built around the property that any number divided by itself equals 1. Materials One calculator each One Poly Plug each

Acknowledgement

The original activity was contributed by Kath Shearer and Rhonda Woods of Margate Primary School, Tasmania. The division discussions in this version have been inspired by Ulla Öberg, Malmö, Sweden .

Procedure The Free Tour version of this activity suggests: Choose a number of the day, say 18. Children use that many plugs and find as many ways as possible to arrange them in equal rows, with left overs if necessary. Colour can be used to highlight the arrangement. Children record in their journal (Poly Plug Paper may be useful) and check their recording with a calculator. For example, this one illustrates: 18 ÷ 7 = 2 rem 4 (2 x 7) + 4 = 18 7 + 7 + 4 = 18 18 - 7 - 7 = 4

Content decimal calculations decimal interpretation decimal representation of a fraction division equations: creating/solving exploring large numbers multiples, factors & primes multiplication - array model multiplication operations - whole number order of operations recording - calculator recording - written times tables visual and kinaesthetic representation of number

Approaching this by asking the mathematician's questions:

• How many ways can you arrange this number in equal rows?
• How do you know when you have found them all?

• 18 ÷ 1 = 18 ... 18 x 1 = 18
• 18 ÷ 2 = 9 ... 9 x 2 = 18
• 18 ÷ 3 = 6 ... 6 x 3 = 18
• 18 ÷ 4 = 4 + 2 rem ... 4 x 4 + 2 = 18
• 18 ÷ 5 = 3 + 3 rem ... 3 x 5 + 3 = 18
• 18 ÷ 6 = 3 ... 3 x 6 = 18
• 18 ÷ 7 = 2 + 4 rem ... 2 x 7 + 4 = 18
• etc.

• Can I check this another way?

• 18
• 9
• 6
• 4·5
• 3·6
• 3
• 2·5714285
• etc.

What happens if 18 is divided into rows of half plugs?

Division Discussions

• Activity A
  Each person secretly enters a number on their calculator and also secretly writes it on a sticky-note which is attached to the rim of the table. The objective is for the group to devise a set of questions that will result in knowing the order of the numbers. Obviously, no one is allowed to ask for any of the actual numbers.

  When the group has an hypothesis about the order of their numbers, they stand in the chosen order then show their calculators to check. Repeat the process with new numbers to refine the set of questions.

  Arrange a class discussion to share and record each group's strategies and growing knowledge.

• Activity B
  Start as for Activity A but each person divides their number by 29 (say) and then records. Repeat the process of asking good questions and checking the ordering hypothesis.

  • What happens if we divide by a different number?
  • What happens if the divisor is a decimal?

  Arrange a class discussion to share and record each group's strategies and growing knowledge.

• Activity C
  Children each have a blank card on which they write a 1 or 2 digit number - they must all be different. Place them on the table in order. Each person writes their number on the calculator, then divides it by 5. Predict and check the order of the calculator numbers.

  Now each person starts with 5 on the calculator and divides it by their card number. Predict and check the order of the calculator numbers.

  Try again with a number different from 5.

  Arrange a class discussion to share and record each group's strategies and growing knowledge.

• Activity D
  Children write any number on their calculator then sort the calculators into those that give a whole number answer when divided by 2 and those that don't. Check by dividing. Do you notice anything in the answers?

  Find numbers that can be divided by 4 and:

  • ...give a whole number answer
  • ...give an answer with decimal point then 5 on the end
  • ...give an answer with decimal point then 25 on the end
  • ...give an answer with decimal point then 75 on the end

  What happens if we divide by 5, 10, 8, 3, 6, 7?
  Why does dividing whole numbers by 2 give two types of answers, dividing by 3 give three types of answers, dividing by 4 give 4 types of answers,...
  What happens if we divide decimal numbers by 1, 2, 3, ..., 9, 10?

  Arrange a class discussion to share and record each group's strategies and growing knowledge.

• Activity E
  Each child writes a number on a card and on their turn each of the other children has to find a number to divide it by so that the answer is a whole number. Each child's response is expected to be different.

  Repeat so that the division will give a remainder of 1, ...of 2, ...of 3 etc.

  Arrange a class discussion to share and record each group's strategies and growing knowledge.

• Activity F
  All of these are good answers: 4¹/₄, 4 rem 1, 4·25, 4, 5
  • What was the question?
  • When is each one a good answer?

  Arrange a class discussion to share and record each group's strategies and growing knowledge.

• Activity G
  Each child in the group takes a turn to write a number on a piece of paper and place it in the centre of the table. For each number the children have to find a way to divide to get:
  • ...an answer about 100
  • ...an answer less than 1
  • ...an answer about 10
  • ...an answer between 5 and 10 (or other limits)
  • ...very close to 1
  Another form of this activity is to start with a number in the 700s (for example) and divide to get:
  • ...an answer about 7
  • ...an answer about 70
  • ...an answer about 700
  • ...an answer about 7000
  • ...

  Arrange a class discussion to share and record each group's strategies and growing knowledge.

• Activity H
  Investigate division by 9, 99 and 999.
  Investigate division by 1, 11 and 111.

  Arrange a class discussion to share and record each group's strategies and growing knowledge.

Task 210, Division Boxes, is an investigation related to divisibility tests. This is also explored in Maths300 Lesson 146, Division Boxes, where it is extended by companion software.

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