Bailey Bridges

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Purpose

This is a level 4 algebra strand activity from the Figure It Out series.

A PDF of the student activity is included.

Achievement Objectives
NA4-7: Form and solve simple linear equations.
NA4-9: Use graphs, tables, and rules to describe linear relationships found in number and spatial patterns.
Student Activity

  

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Specific Learning Outcomes

use a table to find a numbr pattern

use an equation to describe a patter

Required Resource Materials

FIO, Level 4, Algebra, Book Two, Bailey Bridges, pages 6-7

sticks

Activity

In this activity, students work with four different short cuts for counting the sticks needed to make Bailey bridges. he short cuts arise from different ways of visualising the structure of Bailey bridges. The students explain the hort cuts and use them to predict the number of sticks in Bailey bridges with any number of triangles.
While some students will be able to relate the short cuts directly to the bridge diagrams, others may need to use sticks to build the models for themselves to help them make the connections between the models and the short cuts.
As an example, this Bailey bridge has 7 triangles:

triangles.
The structure of the bridge may be visualised in several ways. The four different ways explored in the students’ activities are shown in the table below.

table.
Note, for example, how the short cut 7 x 2 + 1 generalises to the algebraic rule y = x x 2 + 1 or more simply, y = 2x + 1. An important point is that, while the rules may be different, the outcome is the same. A bridge that has 7 triangles uses 15 sticks, no matter which rule is used. In questions 1–4, the students explain how each short cut works. These explanations are, in effect, the rules or generalisations arising from the short cuts. The students may need support and time to work with other students in refining their rule explanations, which are the key to this work in algebraic thinking.
While it is not intended that students at this level work with algebraic rules expressed in symbolic form, some students may benefit from such activity. It is likely, however, that even the most able students will only benefit in situations where their teachers are confident in their own understanding of algebra.
The simplest algebraic rule for the number of sticks in Bailey bridges is y = 2x + 1. The other rules all reduce to this rule. So y = 2(x – 1) + 3 becomes y = 2 x x – 2 x 1 + 3, which further reduces to y = 2x – 2 + 3 and then to y = 2x + 1. The algebraic manipulations needed to produce y = 2x + 1 from the other two rules are:
equations.
Note how the second subtraction sign in y = 3x – (x – 1) becomes an addition sign (because a negative minus a negative is a positive), so that y = 3x – x + 1. If it didn’t change like this, the simplified form would be y = 2x – 1, which is not correct.

Answers to Activity
1. a.

triangles.
b. A bridge with 5 triangles has 5 sets of 2 sticks and an extra stick to complete the final triangle.

triangles.
A bridge with 9 triangles has 9 sets of 2 sticks and an extra stick to complete the final triangle.

triangles.
c.

table.

2. a.

triangles.
b. 6 x 2 + 3
c. A bridge with 5 triangles has 4 sets of 2 sticks and then 3 sticks for the fifth triangle. For a bridge with 7 triangles, there are 6 sets of 2 sticks and then 3 sticks for the seventh triangle.
d. 101 sticks. The short cut is 49 x 2 + 3 = 98 + 3.
3. a. 3 + 11 x 2 = 25

b.

table.
c. In Rory’s short cut, the last triangle has 3 sticks. In Sali’s short cut, the first triangle has 3 sticks. Using either short cut will make no difference to the total.
4. a. The inclusion of 6 additional sticks helps us to see 7 triangles, each with 3 sticks.

triangles.
So, altogether, there are now 7 sets of 3 or 7 x 3 sticks. But the 6 coloured sticks are not part of the bridge and so must be removed. The short cut is then 7 x 3 – 6.
b. 10 x 3 – 9 = 21

c.

table.

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Level Four