Did you knowAE Version 2.0 11/09/18. Did you know ?? negative positive The Ancient Greeks did not...
Transcript of Did you knowAE Version 2.0 11/09/18. Did you know ?? negative positive The Ancient Greeks did not...
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AE Version 2.0 11/09/18.
Did you know ? ?
positivenegative
The Ancient Greeks did not have the notion of a co-ordinate plane but they
used similar geometric methods to develop very sophisticated algebra over
2500 years ago!
We use a 2D co-ordinate plane to
develop understanding regarding
multiplication and also a lot of algebra.
It was developed by Rene Descarte,
in 1637, when he lay in his sick bed
- watching ants crawl across a tiled
ceiling!
How can the diagram help us understand what happens when you multiply with negative numbers?
positivenegative
+
+-
-
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AE Version 2.0 11/09/18.
Dealing with Negativity
negativeEVEN
1. 4 Γ (β7) Γ 6
2. 3 Γ 9 Γ (β6)
3. 2 Γ (β3) Γ (β4)
4. 2 Γ β2 Γ β2 Γ (β5)
5. π Γ 7 Γ π
6. ππ Γ 3 Γ 6π
7. (β4π) Γ 7π Γ (β6π)
Try these quick questions
positive ODD
With an β¦β¦.. number of negative
numbers then value will be β¦β¦β¦.
With an β¦β¦.. number of negative
numbers then value will be β¦β¦β¦.
What do you
notice about
your answers?
8. Use what you have noticed
to fill in the gaps in the
sentences below
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AE Version 2.0 11/09/18.
Solutions on the next slideβ¦.
Dealing with Negativity
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AE Version 2.0 11/09/18.
Dealing with Negativity
1. 4 Γ β7 Γ 6 = β168
2. 3 Γ 9 Γ β6 = β162
3. 2 Γ β3 Γ β4 = 24
4. 2 Γ β2 Γ β2 Γ β5 = β40
5. (βπ) Γ (β 7) Γ π = 7π2
6. ππ Γ 3 Γ 6π = 18ab2
7. β4π Γ 7π Γ β6π = 168π3
Try these quick questions
With an β¦β¦.. number of negative
numbers then value will be β¦β¦β¦.
With an β¦β¦.. number of negative
numbers then value will be β¦β¦β¦.
What do you
notice about
your answers?
8. Use what you have noticed
to fill in the gaps in the
sentences below
Unsure about any of these? Search Negative numbers
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AE Version 2.0 11/09/18.
Expanding 1
1. Without doing the calculation, will the
answer to this calculation be positive or
negative ? Give a reason.
2. 24 Γ 17 is the same as which of the
following
3. Expand 3 3 β 6
4. Expand and simplify π₯ + 2 π₯ + 5
5. Expand and simplify π₯ + 6 π₯ β 2
6. Expand and simplify 2 + 3 2 + 1
7. Expand and simplify π₯2 + 2 π₯2 + 6
8. Expand and simplify π₯2 + 3 (π₯3 + 7 )
(30 - 5)(20 - 2)
2 Γ (-3) Γ (-4) Γ 6 Γ (-6) Γ (-1) Γ 7 Γ (-2)
20(10 + 7) + 4(10 + 7)
2 Γ 3 Γ 17 Γ 2 Γ 2 (20 + 4)(10 + 7)
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AE Version 2.0 11/09/18.
Solutions on the next slideβ¦.
Expanding 1
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AE Version 2.0 11/09/18.
1. Without doing the calculation, will the
answer to this calculation be positive or
negative ? Give a reason.
2. 24 Γ 17 is the same as which of the
following
3. Expand 3 3 β 6
4. Expand and simplify π₯ + 2 π₯ + 5
Expanding 1 Solutions
Negative because there are an odd
number of negative numbers
= π₯2 + 2π₯ + 5π₯ + 10= π₯2 + 7π₯ + 10
= 3 3 β 18
Unsure about any of these? Search Expanding single brackets
2 Γ 3 Γ 17 Γ 2 Γ 2 (20 + 4)(10 + 7)
(30 - 5)(20 - 2) 20(10 + 7) + 4(10 + 7)
(20 + 4)(10 + 7)
(30 - 5)(20 - 2) 20(10 + 7) + 4(10 + 7)
2 Γ (-3) Γ (-4) Γ 6 Γ (-6) Γ (-1) Γ 7 Γ (-2)
2 Γ 3 Γ 17 Γ 2 Γ 2
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AE Version 2.0 11/09/18.
Expanding 1 Solutions
5. Expand and simplify π₯ + 6 π₯ β 2
6. Expand and simplify 2 + 3 2 + 1
7. Expand and simplify π₯2 + 2 π₯2 + 6
8. Expand and simplify π₯2 + 3 (π₯3 + 7 )
= π₯2 + 6π₯ β 2π₯ β 12= π₯2 + 4π₯ β 12
= 2 + 3 2 + 2 + 3
= 4 2 + 5
= π₯4 +2π₯2 + 6π₯2 + 12= π₯4 + 8π₯2 + 12
= π₯5 +3π₯3 + 7π₯2 + 21
Unsure about any of these? Search Expanding Double Brackets
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AE Version 2.0 11/09/18.
Whatβs gone wrong?
Here is a studentβs work on expanding brackets.
Take a look and decide if they have done the work correctly or not.
If they have made a mistake can you say why ?
What are the correct answers?
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AE Version 2.0 11/09/18.
Solutions on the next slideβ¦.
Whatβs gone wrong?
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AE Version 2.0 11/09/18.
+2π₯ + 3π₯ +2 Γ 3
= π₯2 +5π₯ + 6
Whatβs gone wrong? Solutions
+4π₯ β 5π₯
= π₯2 βπ₯ β 20Correct
Here is a studentβs work on expanding brackets.
Take a look and decide if they have done the work correctly or not.
If they have made a mistake can you say why ?
What are the correct answers?
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AE Version 2.0 11/09/18.
Whatβs gone wrong? Solutions
Here is a studentβs work on expanding brackets.
Take a look and decide if they have done the work correctly or not.
If they have made a mistake can you say why ?
What are the correct answers?
2 Γ 22
+3 Γ β3β9
+3 2 β 3 20
= 2 + 0 + β9= β7
3π₯
=2π₯2 + 3π₯ + 12
4π₯
= π₯ + 2 π₯ + 2= π₯2 +2π₯ + 2π₯ + 4= π₯2 +4π₯ + 4
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AE Version 2.0 11/09/18.
Expand the expressions on the left of the page and find the matching expression
in the grid on the right of the page.
When completed there should be four answers unmatched.
Find the sum of these four expressions and simplify it
1. π₯ + 3 2
2. π₯ + 4 π₯ + 3
3. π₯ β 4 2 β 2
4. π₯ β 3 π₯ + 4
5. π₯ + 5 2 + 3
6. π₯ π₯ + 4 + 2 π₯ + 4
7. 3 β π₯ 3 + π₯
8. π₯ π₯ β 8 β (π₯ β 8)
Expand and Simplify
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AE Version 2.0 11/09/18.
Solutions on the next slideβ¦.
Expand and Simplify
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AE Version 2.0 11/09/18.
1. π₯ + 3 2
2. π₯ + 4 π₯ + 3
3. π₯ β 4 2 β 2
4. π₯ β 3 π₯ + 4
5. π₯ + 5 2 + 3
6. π₯ π₯ + 4 + 2 π₯ + 4
7. 3 β π₯ 3 + π₯
8. π₯ π₯ β 8 β (π₯ β 8)
The four expressions left simplify to 2π₯2 + 7π₯ + 41
61
2 8 3
7 5 4
Expand the expressions on the left of the page and find the matching expression
in the grid on the right of the page.
When completed there should be four answers unmatched.
Find the sum of these four expressions and simplify it
Expand and Simplify
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AE Version 2.0 11/09/18.
Quadratic Puzzles ?
π₯2 3π₯
4π₯ 12
These are multiplication grids
We can use these to expand quadratics such as π₯ + 3 (π₯ + 4 )
Γ π₯
π₯
+4
+3
π₯ + 3
(π₯+4)
On the next page fill in the blanks in the multiplication grids
What do you notice?
π₯2 + 3π₯ + 4π₯ + 12
π₯2 + 7π₯ + 12Now we can simplify by
collecting like terms to get this
There are 4 terms after
expanding the brackets. Thatβs
why these expressions are
called quadratics β as βquadβ
means four.
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AE Version 2.0 11/09/18.
Γ
Γ Γ Γ
Γ Γ
π₯ β 4
π₯β4
2π₯ + 1 3π₯ β 5
2π₯ + 3 3π₯ + 4 5π₯ β 2π₯+2
π₯+3
2π₯β4
2π₯β3
5π₯+2
π₯2 + 16 2π₯2
π₯
π₯
π₯
2π₯ 3π₯
π₯
2π₯
β4
β4 +2β4π₯ +16 +2
Quadratic Puzzles
+9π₯
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AE Version 2.0 11/09/18.
Solutions on the next slideβ¦.
Quadratic Puzzles
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AE Version 2.0 11/09/18.
Γ
Γ Γ Γ
Γ Γ
π₯ β 4
π₯β4
2π₯ + 1 3π₯ β 5
2π₯ + 3 3π₯ + 4 5π₯ β 2π₯+2
π₯+3
2π₯β4
2π₯β3
5π₯+2
π₯2 β 8π₯ + 16 2π₯2 + 5π₯ + 3
π₯
π₯
π₯
2π₯ 3π₯
π₯
2π₯
β4
β4 +2β4π₯ +16 +2
Quadratic Puzzles Solutions
+1
2π₯2 +π₯
+4π₯
π₯2 β4π₯
+3
β5
3π₯2
4π₯2 6π₯2 25π₯2
+9π₯
β5π₯
β8π₯
2π₯
+6π₯
+3
β4 β12
3π₯ 5π₯
2π₯
+4
β3 β9π₯
+8π₯
β12
5π₯
β10π₯
β10π₯
β2
β2
β4
β15
3π₯2 + 4π₯ β 15
4π₯2 β 2π₯ β 12 6π₯2 β π₯ β 12 25π₯2 β 4
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AE Version 2.0 11/09/18.
Quadratic Puzzles Solutions
What did you notice?
3π₯2 β5π₯
+9π₯ β15
Γ 3π₯
π₯
+3
β5
3π₯ β 5
(π₯+3)
3π₯2 + 9π₯ β 5π₯ β 15
3π₯2 + 4π₯ β 15
The products of the diagonals
are identical expressions
The sum of these terms make
the middle term in the simplified
expression
3π₯2 Γ β15 = β45π₯2
9π₯ Γ β5π₯ = β45π₯2
You might want to go back and
check this with other quadratics
yourself - this might be useful
to know later on
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AE Version 2.0 11/09/18.
Expanding 2
1. Expand and simplify 2π₯ + 3 (π₯ β 2)
2. Expand and simplify 3π₯ π₯ + 3 + 4 (π₯ + 3)
3. Expand and simplify π₯ + 6 2 + π₯ β 3 2
4. Expand and simplify 2 β 32
5. Simplify 2
π₯+3+
π₯β3
π₯
6. Expand and simplify π₯3 β 7 (π₯3 + 7)
7. Expand and simplify 3π₯ + 2 ( 4π₯2 + 2π₯ β 3)
8. Simplify 2π₯ β2
π₯+2β
π₯β2
3π₯
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AE Version 2.0 11/09/18.
Solutions on the next slideβ¦.
Expanding 2
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AE Version 2.0 11/09/18.
Expanding 2 Solutions
1. Expand and simplify 2π₯ + 3 (π₯ β 2)
2. Expand and simplify 3π₯ π₯ + 3 + 4 (π₯ + 3)
3. Expand and simplify π₯ + 6 2 + π₯ β 3 2
4. Expand and simplify 2 β 32
= 2π₯2 + 3π₯ β 4π₯ β 6= 2π₯2 β π₯ β 6
= π₯ + 6 π₯ + 6 + π₯ β 3 (π₯ β 3)= π₯2 +12π₯ + 36 + π₯2 β 6π₯ + 9= 2π₯2 +6π₯ + 45
= 3π₯2 +9π₯ + 4π₯ + 12= 3π₯2 +13π₯ + 12
= (2 β 3)(2 β 3)
= 4 β 2 3 β 2 3 + 3
= 7 β 4β3
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AE Version 2.0 11/09/18.
5. Simplify 2
π₯+3+
π₯β3
π₯
6. Expand and simplify π₯3 β 7 (π₯3 + 7)
7. Expand and simplify 3π₯ + 2 ( 4π₯2 + 2π₯ β 3)
8. Simplify 2π₯ β2
π₯+2β
π₯β2
3π₯
Expanding 2 Solutions
=2
π₯+3+
π₯β3
π₯β
2π₯
π₯ π₯+3+
(π₯β3)(π₯+3)
π₯(π₯+3)
=2π₯
π₯ π₯+3+
π₯2 β9
π₯(π₯+3)β
π₯2+2π₯β9
π₯ π₯+3
= π₯6 β7π₯3 + 7π₯3 β 49= π₯6 β49
= 3π₯ 4π₯2 + 2π₯ β 3 + 2 4π₯2 + 2π₯ β 3= 12π₯3 + 6π₯2 β 9π₯ + 8π₯2 + 4π₯ β 6= 12π₯3 + 14π₯2 β 5π₯ β 6
=3π₯(2π₯β2)
3π₯ π₯+2β
(π₯β2)(π₯+2)
3π₯(π₯+2)β
6π₯2β6π₯
3π₯ π₯+2β
π₯2β4
3π₯(π₯+2)
=5π₯2 β 6π₯ + 4
3π₯ π₯ + 2
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AE Version 2.0 11/09/18.
Write some digits in a circle
54
8 6
3
The sum of the squares of the two-digit numbers read clockwise is
542 + 462 + 682 + 832 + 352 = 17770
The sum of the squares of the two-digit numbers read anticlockwise is
532 + 382 + 862 + 642 + 452 = 17770
Prove that the two sums will always be equal for any circle of digits
e.g.
Prove it!
Hint Solution
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AE Version 2.0 11/09/18.
Read more about how algebra was developed thousands of
years ago and how visualisations were used even then!
Watch this video to find how there are actually patterns in
prime numbers and how simple algebra can show this β
with brackets of course!
Discover the history of negative numbers and how they
were thought of as making dark of mathematics!
Still want more?
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AE Version 2.0 11/09/18.
Contact the AMSP
01225 716 492
amsp.org.uk
Advanced_Maths