Number Properties in Venn Diagrams Stimulus question 79 Teaching notes Key question/task Learners are presented with five descriptions of sets of numbers, and four Venn diagrams to which to match them. Supplementary questions: • Could you begin to write a list of the numbers in each set to help you? • Are there any numbers that are multiples of 4 but not multiples of 2? • Is the set of multiples of 4 completely contained within the set of multiples of 2? How could this be shown in the Venn diagram? • What is the difference between an integer and a whole number? • Is zero a whole number? • Is a number a multiple of itself? Resources: • Question sheet; • Paper, pencils and/or mini-­‐‑
whiteboards and pens to explore ideas. Reasoning: questions to discuss and explore • We note that the set of multiples of 4 are completely contained within the set of multiples of 2. Can you name other pairs of numbers that would work in this way? • We know that the number 1 is both a square and a cube number. Are there any other possibilities? Consolidation • Learners could experiment on paper/whiteboards marking in actual members of each set, to help them reach solutions. Possible extension • Devise your own descriptions of sets of numbers that would fit into each of these Venn diagrams. Commentary/notes: This activity should be suitable for all tiers. It combines the use of Venn diagrams with developing a more precise understanding of mathematical vocabulary and number properties, but it does not deal with probability. This activity links with Question 37: ‘Venn diagrams’ (which deals with probability) and Task 18: ‘Wales’. Note there is an optional solution to the final question, involving the idea of zero being a whole number. A commonly expressed view is that whole numbers differ from the natural or counting numbers by including zero, but there are differences of opinion on this. Here, we suggest that zero should be included in the whole numbers, and therefore the solution is shown in a bold black line. If the view is taken that zero is not included, then the solution is marked by a dashed line. This should provide a worthwhile discussion with the class. 1 Solutions Ɛ = {Whole numbers}
Ɛ
a = {Multiples of 2}
a
b
b = {Multiples of 4}
Ɛ = {Integers}
a = {Square numbers}
b = {Cube numbers}
Ɛ
a
b
Ɛ = {Whole numbers}
a = {Primes}
b = {Multiples of 5}
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Ɛ = {Factors of 1000}
a
b
a = {Factors of 20}
b = {Factors of 100}
Ɛ = {Integers}
a = {The number zero}
b = {Whole numbers}
Ɛ
a
b
2 GCSE Subject Content Foundation Intermediate Higher Using the common properties of numbers, including odd, even, multiples, factors, primes. Using the terms square, square root and cube. Understanding and using Venn diagrams to solve problems. Learner Outcomes and Assessment (to aid comment-­‐‑only marking) Reasoning strand – Learners are able to: • Identify, measure or obtain required information to complete the task; • Select appropriate mathematics and techniques to use; • Explain results and procedures precisely using appropriate mathematical language; • Interpret mathematical information; use diagrams to draw conclusions. Assessment guidance – Can learners: • Start to find members of sets to help organise them into the diagrams? • Find squares, cubes, primes, multiples and factors? • Identify integers and whole numbers? • Use the vocabulary of factors, multiples, primes, squares, cubes, integers and whole numbers precisely? • Link the descriptions of the sets with the Venn diagrams? 3