1. Grade 10 Academic (MPM2D) Unit 3: Algebra & Quadratic Models Finite Differences Investigations
Mr. Choi
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2. Finite Differences
First Difference: Values that are calculated by subtracting consecutive y – values in a table of values that has a constant difference between the x – values.
Finite Differences: Differences between the y – values in a table of values in which the x – values increase by the same amount.
 Relation: A description of how two variables are connected.
Linear Relation: A relation whose graph is a straight line and whose first differences are constant.
Non-linear Relation: A relation whose graph is a not straight line and whose first differences are not constant.
Quadratic Relation: A relation whose graph is a parabolic shape and whose second differences are constant.
Finite Differences Investigations
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3. Example 1: Patterns
The first four diagrams in a pattern show number of apples are contained in each basket.
 a) Complete the table according to the given diagram. Guess the
number of apples in the baskets 5 & 6.
Basket #
Number of apples
First Differences 1 2 3 4 5 6 5 8 11 14
8 – 5 = 3
11 – 8 = 3
14 – 11 = 3 17 20 3
Finite Differences Investigations
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4. Example 1: Patterns - Continued
Use finite differences to
determine whether the
function is a linear or
quadratic.
Basket #
Number of apples
First Differences 1 5 2 8 3 11 4 14 17 6 20
Linear
Determine an equation of
the relation.
How many apples will there be in the 10th basket? the 35th
basket?
There will be 32 apples in the 10th basket.
There will be 107 apples in the 35th basket.
Finite Differences Investigations
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5. Example 1: Patterns - Continued
Which basket contains 128 apples?
(Assume the basket can hold infinite apples!! )
Recall:
The 42nd basket contains 128 apples.
Finite Differences Investigations
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6. Example 2: Patterns
The first four diagrams in a pattern show number of worms grow during the first 4 weeks
 a) Complete the table according to the given diagram. Guess the
number of worms in the weeks 5 & 6.
Week #
Number of Worms
First Differences
Second Differences 1 2 3 4 5 6 2 5 10 17
5 – 2 = 3
10 – 5 = 5
5 – 3 = 2
17 – 10 = 7
7 – 5 = 2 26 37 9 11 2
Finite Differences Investigations
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7. Example 2: Patterns - Continued
Use finite differences to
determine whether the
function is a linear or
quadratic.
Week #
Number of Worms
First Differences
Second Differences 1 2 5 3 10 4 17 7 26 9 6 37 11
Non- Linear
 Quadratic
Determine an equation of
the relation.
(By observation)
How many worms will there be in the 9th week? the 20th
week?
There will be 82 worms in the 9th week.
There will be 401 worms in the 20th week.
Finite Differences Investigations
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8. Example 2: Patterns - Continued
Which week will have
1025 worms?
Week #
Number of Worms
First Differences
Second Differences 1 2 5 3 10 4 17 7 26 9 6 37 11
Week 32 will have 1025 worms.
f)  Use the table to create a
quadratic relation.
Finite Differences Investigations
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9. Example 3: Patterns
Use the finite differences to determine the relation is linear or quadratic or neither, then determine the relation. x y
First Differences
Second Differences 1 -2 2
-11 3
-26 4
-47 -3 -9 -6
-15
-21
Finite Differences Investigations
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10. Homework Work sheet: Finite Differences Investigations Text:
P. 254 #2, 4, 13, 14
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Finite Differences Investigations
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11. End of lesson Finite Differences Investigations
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