Hexagon Number Sense
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About this ebook
Understanding how numbers go together and come apart, how number operations are related to each other: number sense, is key to students' comprehension of higher levels of mathematics. Using these materials will help students see the interconnectedness of numbers, as well as giving them practice manipulating numbers.
For each hexagon, any two numbers can be given, and the other numbers can be found. The simplest case is to give the x and y number and then students follow the operations (see thumbnail for examples). However, this can become more difficult if only x or y is given, plus one other hexagon. The most difficult case is giving any two hexagons except x or y. In addition, the hexagon can be used for whole numbers, integers, decimals, fractions, and even algebra.
These materials can be used from third grade through middle school, depending on the level of difficulty and the type of numbers involved. There is consistency throughout though because any relationships between the hexagons exist for all levels and all types of numbers.
You can use these as homework dittos, classroom activities, or a one a day warm up problem on the board or overhead.
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Hexagon Number Sense - Christopher Casey
No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form, or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior consent of the publisher.
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Worksheets may be copied for your own classroom use only. If you have questions, comments, concerns, email me at:
cccmathpro@yahoo.com
Understanding how numbers go together and come apart, how number operations are related to each other: number sense, is key to students’ comprehension of higher levels of mathematics. Using these materials will help students see the interconnectedness of numbers, as well as giving them practice manipulating numbers.
For each hexagon, any two numbers can be given, and the other numbers can be found. The simplest case is to give the x and y number and then students follow the operations. However, this can become more difficult if only x or y is given, plus one other hexagon. The most difficult case is giving any two hexagons except x or y. In addition, the
hexagon can be used for whole numbers, integers, decimals, fractions, and even algebra.
These materials can be used from third grade through middle school, depending on the level of difficulty and the type of numbers involved. There is consistency throughout though because any relationships between the hexagons exist for all levels and all types of numbers.
On the primary level, students will learn that they can solve by using inverse operations. Given xy = 12, and y = 2, 12/2 will be the solution for x. Students will see these relationships as they work through the dittos and apply the same relationships to other numbers.
Aside from the inverse operation connections between addition and subtraction, and multiplication and division, there are several other relationships between the hexagons that students will gradually see (and/or you can help them to see). All of the relationships can be explained using algebra, and in fact at the middle school level, using algebra to solve would be helpful.
First, we’ll replace each hexagon with a letter to make it easier to refer to a specific hexagon. Again, there are obvious (to us)
relationships such as a/x = y, and yb = x which have been mentioned.
Along the same lines, students will notice that a/x = x/b = y. Students will wonder if it matters if x and y are reversed. This depends on which numbers are given. If a and c are given, x and y are interchangeable. Students will see that if x and y are whole numbers and y > x, then 0 < b < 1 and d < 0. If y < x, then b > 1 and d > 0. Students can experiment by reversing x and y and seeing the effect it has on the other numbers.
Students will also notice that c + d = 2x and c – d = 2y. In addition, ab = x²and a/b = y². It is important that students see these relationships when working with whole numbers because they are easier to see and the same relationships will exist for all numbers.
Students will be curious why these relationships exist. Challenge them to discover this on their own. Again, algebra can be used:
c + d = 2x,
(x + y) + (x - y) = 2x,
but at lower grades students can use other methods.
The worksheets are ordered from easiest to hardest. The worksheets begin with whole numbers, then move on to integers, then decimals, and finally fractions. Within each number type first x and y are given, the just x or y, and finally neither x nor y. As you look through the worksheets, you will see how the difficulty increases. Students need a firm foundation in operations for each type of number to be able to use these materials. Again, the purpose isn’t to teach number operations, but to see how they are connected,
and for all types of numbers.
If the worksheets are used for middle school