Download Year 10 Task – Algebraic modelling with difference

Year 10 Task – Algebraic modelling with difference tables.
Part 1 – Warren Bridges
Warren bridges are a type of truss bridge used throughout the world. Favoured for their
strength, durability, the simple design allows prefabrication in preparation for
construction.
The Warren Bridge, when simplified is made of beams that are organised into triangles
as shown below:
1 triangle is made using 3 beams.
2 triangles require 5 beams.
3 triangles require 7 beams.
1. Fill in the table for the number of beams needed for the following bridges:
2. What do you notice about the pattern?
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Linear relationships, when fully simplified, can be represented using the general
equation for a linear relationship:
y = mx + c
Where m and c are real numbers that define the relationship.
The general equation can be put into a table to find the value of the ‘difference’:
Inserting the general
formula into a table:
Substituting the value for x Simplifying and calculating the
for each row:
‘difference’
x
y
x
y
1
2
3
4
5
6
mx + c
mx + c
mx + c
mx + c
mx + c
mx + c
1
2
3
4
5
6
m(1) + c
m(2) + c
m(3) + c
m(4) + c
m(5) + c
m(6) + c
Comparing this table to our Warren Bridge pattern:
3. Comparing the tables, what is the value of m for the Warren Bridge equation?
m = ___ (Circled in red)
4. Examine the corresponding value of y when x = 1.
m + c = ___ (Circled in black)
5. Substitute the value of m found in Question 3 to find the value of c.
6. Given that linear relationships can be represented in fully simplified form as:
y = mx + c
What is the simplest equation that represents the Warren
Bridge pattern?
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Part 2 – Squares of Squares
Consider the pattern below, made by arranging beams into squares.
7. When the size of the grid is 4×4 squares, how many beams are required?
The first difference between each step of the pattern is not constant – this is not a linear
relationship.
This particular pattern requires a second difference to be calculated, indicating that it is
a quadratic relationship.
8. Fill in the table for the number of beams required for 4×4, 5×5, and 6×6 square
grids, and the 1st and 2nd differences for the rest of the table.
9. Suggest how someone could distinguish between a linear relationship and a
quadratic relationship.
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Part 3 – The General Quadratic
Quadratic relationships, when fully simplified, can be represented using the general
equation for a quadratic relationship:
y = ax2 + bx + c
Where a, b, and c are real numbers that define the
relationship.
Again, this can be inserted into a table in order to find the values for a, b, and c:
Inserting the general
formula into a table:
Substituting the value for
x for each row:
Simplifying:
x
y
x
y
x
y
1
2
3
4
5
6
ax2 + bx + c
ax2 + bx + c
ax2 + bx + c
ax2 + bx + c
ax2 + bx + c
ax2 + bx + c
1
2
3
4
5
6
a(1)2 + b(1) + c
a(2)2 + b(2) + c
a(3)2 + b(3) + c
a(4)2 + b(4) + c
a(5)2 + b(5) + c
a(6)2 + b(6) + c
1
2
3
4
5
6
a+b+c
4a+ 2b + c
9a + 3b + c
16a + 4b + c
25a + 5b + c
36a + 6b + c
Compare the values of the general quadratic pattern above to the values in the table for
the ‘Squares of Squares’ pattern in Question 8.
10. Use the 2nd difference to calculate the value of a for the ‘Squares of Squares’
pattern.
11. Use the 1st difference to calculate the value of b for the ‘Squares of Squares’
pattern.
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12. Find the value of c for the ‘Squares of Squares’ pattern.
13. Using your values of a, b, and c give the equation that represents the ‘Squares of
Squares’ pattern.
You should define the variables involved and verify that your equation is correct.
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Part 3 – Stepped Pyramids
You can use blocks to build pyramids such as those shown above.
For 1 level, there is 1 block.
For 2 levels, there are 5 blocks.
For 3 levels, there are 14 blocks; and so on.
14. Complete the table showing the number of layers in each pyramid and the number
of blocks needed to build it. All pyramids are solid with no empty space.
Layers (n)
1
2
3
4
5
6
Blocks (b)
15. Use finite differences to find which difference is constant.
16. Write an equation for this relationship.
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17. Use your model to predict the number of blocks needed to build a pyramid 8 layers
high.
You should define the variables involved and verify that your equation is correct.
Note: the general formula for a cubic relationship is:
y = ax3 + bx2 + cx + d
Where a, b, c, and d are real numbers that define
the relationship.
The following tables may be of assistance to you:
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