Addition And Subtraction Of Integers Investigation

Grade 7 Investigation: Adding and Subtracting Integers. In this investigation you are going to investigate the rules for adding and subracting integers.

Sets of numbers: The numbers 1, 2, 3, 4, …. are called NATURAL numbers (the counting numbers). This set of numbers is often represented by the symbol N and can be written using the following notation N = {1, 2,3, 4,.........} The numbers 0, 1, 2, 3, 4, …. are called WHOLE numbers. This set of numbers is often represented by the symbol W and can be written using the following notation W = {0,1,2,3,4,.........}

These numbers are shown on the number line below

If we extend the number line to the left we get;

This larger set of numbers is called INTEGERS. The number 1, 2, 3, 4, …. are the POSITIVE INTEGERS and the numbers ….,-4, -3, -2, -1 are the NEGATIVE INTEGERS . Notice that he number 0 is neither positive nor negative but is still an integer. This set can be written using the symbol Z for integers as,

Z = {..., −4, −3, −2, −1, 0,1, 2, 3, 4,...}

Part 1: A - Adding and subtracting integers – with the help of a number line Representing number sentences on a number line. Example 1: To represent the number sentence 3 + 2 = 5, we find 3 on the number line and simply move 2 more places to the right. Example 2: To represent the number sentence 4 – 3 = 1, we find 4 on the number line and simply move 3 places to the left. adding, we move further to the right

+2

-3

subtracting, we move further to the left

Of course, the above ideas should already be very familiar to you!

B - Adding and subtracting integers – with the help of the Chinese “Tai ji” or Yin – yan symbol – see below.

The Chinese Yin – yan consists of two parts: light (yin) and shadow (yan). The light part represents the warm and bright sides of nature while the shadow part represents the cold and dark sides. Therefore, the light part can be treated as positive and the shadow part as negative. These two parts, when grouped together, have a meaning of balance and harmony.

How can this help us with adding and subtracting?

+1 -1

If we use +1 to represent light and -1 to represent shadow, the whole diagram now represents the number zero.

note: in this diagram the half with the black dot represents shadow

Example 1: To calculate 3 + 2, is the same as,

+1

+1

and

+1

+1

+1

So altogether we have a total of five positive symbols and so the answer to 3 + 2 is 5, and therefore 3 + 2 = 5 Example 2: To calculate 3 - 1, we can think of this as the same as three remove one,

+1

+1

removing one leaves us with

+1

+1

+1

So altogether we have a total of two positive symbols and a zero, and so, 3 -1 = 2.

One more example and then it’s your turn Example 3: To calculate -1 + (-2), is the same as,

-1

and

-1

-1

So altogether we have a total of three positive symbols and so the answer to -1 + (-2) is -3, and therefore -1 + (-2) = -3

Qu 1 - Show how you would represent -1 + (-2) using a number line. Qu 2 - Illustrate each of the following addition problems b. using the Yin – yan symbols

a. with number lines

i. -2 + (-4)

ii. -3 + (-2)

iii. -1 + (-4)

Qu 3 - Finish the following sentence: i. When I add two positive numbers I always get a positive number however, when I add two negative numbers I always get ____________________. ii. Explain in your own words why this happens – use any diagrams you need to help with your explanation.

Example 4: To calculate 2 + (-3), is the same as

+1

and

+1

or

+1

-1

and

+1 -1

-1

-1

-1

-1

So altogether we have a total of two zeros and a negative 1, and so, 2 + (-3) = -1.

Qu 4 - Illustrate each of the following addition problems a. with number lines

i. -1 + 3

ii. -3 + 2

b. using the Yin – yan symbols

iii. 3 + (-5)

iv. 1 + (-3)

Qu 5 - Explain what happens when you add a positive and a negative number together. (think carefully) – again, use any diagrams you need to help with your explanation.

Part B – Subtracting integers

In both the examples below, subtracting is shown as removing symbols.

Example 5: To calculate -3 - (-2), is the same as removing two negative from three negative. We can show this as follows

-1

-1

-1

removing two negatives gives

-1

So altogether we have a total of one negative , and so, -3 - (-2) = -1.

Example 6: To calculate 1 – (-2), we can do the following,

+1

and

+1

+1 -1

+1

and

+1

removing two negatives gives

-1

+1

So altogether we have a total of three positive 1’s, and so, 1 - (-2) = 3.

Qu 6: Look carefully at example 6 and explain how it makes sense to you.

Qu7: How would you show that using a number line?

i. -3 - (-2) = -1

and

ii. 1 - (-2) = 3

Qu 8: Using either the Yin –yan symbols, or a number line, find solutions to the following, i. (-2) – 1

ii. 3 – (-1)

iii. 2 – (-4)

iv. (-2) – (-2)

v. (-4) – 2

vi. 1 – (-3)

vii. 4 – (-1)

vii. (-3) – (-2)

Qu 9: Look at the examples you have just found solutions for, what patterns can you describe regarding subtracting integers?

Part C - Summarising the patterns you have found in this task for adding and subtracting integers.

Qu 10: Summarise what you have found – the following two tables may help you organise your findings: note: (+ve) stands for a positive integer and (-ve) stands for a negative integer

Carefully describe the patterns you noticed

Adding (+ve) + (+ve)

(-ve) + (-ve)

(+ve) + (-ve)

(-ve) + (+ve)

Give one example to help explain the pattern you noticed in each situation.

Subtracting

Carefully describe the patterns you noticed

(+ve) – (+ve)

(-ve) – (+ ve)

(+ve) – (-ve)

(-ve) – (-ve)

Give one example to help explain the pattern you noticed in each situation.

Assessment

Due __________

Criterion B – patterns in mathematics Level Descriptor 0

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You failed to submit any work or your work does not reach any of the standards described below.

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You were unable, even with help, to use mathematical problem-solving techniques and so could not describe any patterns for addition or subtraction.

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With some help you used some basic mathematical problem-solving techniques. You tried to describe the simpler patterns for addition and subtraction.

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You used mathematical problem-solving techniques to help you see patterns for addition and subtraction. You were able to describe the simpler patterns for addition and subtraction.

You used mathematical problem-solving techniques to help see patterns for both addition and subtraction. You were able to describe the simpler patterns and attempted to describe the more difficult ones. You tried to summarise your findings by writing rules.

You used mathematical problem-solving techniques to help see patterns for both addition and subtraction. You were able to describe the simpler patterns and had some success in describing the more difficult ones. You had some success in summarising your findings by writing rules. You tried to use diagrams to explain why your rules worked

You used mathematical problem-solving techniques to help see patterns for both addition and subtraction. You were able to describe the simpler patterns and were successful in describing the more difficult ones. You successfully summarised your findings by writing rules. You used diagrams to explain why your rules worked

Self assessment – use the rubric above to assess your work I think my grade will be

Criterion C: communication in mathematics

Level 0

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You failed to submit any work to be assessed.

You submitted the work but it did not reach any of the standards stated below.

You tried to use words, tables, diagrams and correct mathematical language to explain your work. You presented some information clearly.

You used words, tables, diagrams and correct mathematical language to explain your work. Your explanations were not always clear, or tables and diagrams were not always well organized.

You clearly explained your work, by using words, tables and diagrams, as well as mathematical language. Your explanations were clear and diagrams and tables were all very well organized and presented.

Self assessment – use the rubric above to assess your work

I think my grade will be