Module 3 - Investigation Structure
TL;DR
Predicting more values requires you to complete more values into the table.
Describing the pattern needs you to identify the type of pattern and give the important values associated with this type of sequnce.
The first point to address is always predicting more values and recording these in the table.
This is marked under the P criteria.
There are normally 2 points for getting all the values in the table correct, and 1 point if you get some of them correct.
Sometimes the table has multiple columns of values, and then there might be a third mark.
And then you need to describe any patterns which you usually need to make the predictions.
This is marked under the D criteria, along with finding the general rule.
Describing a correct pattern usually gets you the first mark in this criteria. You also need to have a pattern described to get full marks.
The sequences in the unstructured investigation might be:
linear
quadratic
geometric
disguised
some combination of these
You need to be able to recognise them all to be successful in the unstructured investigation.
To describe a linear sequence use a phrase similar to “The values for \(L\) follow a linear pattern. It increases by \(5\) each term.”
To describe a quadratic sequence use a phrase similar to “The values for \(L\) follow a quadratic pattern. There is a constant second difference of \(2\).”
To describe a geometric sequence use a phrase similar to “The values for \(L\) follow a geometric pattern. Each term is the previous term multiplied by \(3\).”
In the cases where the sequence is hidden, make sure you incorporate the outer structure into the description.
For example, for the sequence$$2^{3},2^{8},2^{13},2^{18},2^{23},…$$we might say “The powers of \(2\) follow a linear pattern. They increase by \(5\) each term.”
If the sequence is made up of two parts, then describe the pattern for each part separately.
In some cases, the question explicitly states how many patterns you should describe. If you need more than one, often you can say something like “All the terms are even / odd / a multiple of 3”, but you should be as specific as possible if you use this.
Example
For the values in the table below, predict more values and describe the pattern in words. Check your answers by revealing them.
| Stage (\(n\)) | Cubes (\(C\)) |
|---|---|
| 1 | 12 |
| 2 | 19 |
| 3 | 26 |
| 4 | 33 |
| 5 | 40 ? |
| 6 | 47 ? |
| 7 | 54 ? |
-
The values for \(C\) follow a linear pattern. They increase by \(7\) each time.
Example
For the values in the table below, predict more values and describe the pattern in words. Check your answers by revealing them.
| Stage (\(n\)) | Count (\(C\)) |
|---|---|
| 1 | 100 |
| 2 | 93 |
| 3 | 86 |
| 4 | 79 |
| 5 | 72 ? |
| 6 | 65 ? |
| 7 | 58 ? |
-
The values for \(C\) follow a linear pattern. They decrease by \(7\) each time.
Example
For the values in the table below, predict more values and describe the pattern in words. Check your answers by revealing them.
| Stage (\(n\)) | Tiles (\(T\)) |
|---|---|
| 1 | 4 |
| 2 | 7 |
| 3 | 12 |
| 4 | 19 |
| 5 | 28 ? |
| 6 | 39 ? |
| 7 | 52 ? |
-
The values for \(T\) follow a quadratic pattern. The second difference is \(2\).
Example
For the values in the table below, predict more values and describe the pattern in words. Check your answers by revealing them.
| Stage (\(n\)) | Squares (\(S\)) |
|---|---|
| 1 | 5 |
| 2 | 10 |
| 3 | 20 |
| 4 | 40 |
| 5 | 80 ? |
| 6 | 160 ? |
| 7 | 320 ? |
-
The values for \(S\) follow a geometric pattern. Each term is the previous term multiplied by \(2\).
Example
For the values in the table below, predict more values and describe the pattern in words. Check your answers by revealing them.
| Stage (\(n\)) | Pieces (\(P\)) |
|---|---|
| 1 | 0 |
| 2 | 3 |
| 3 | 8 |
| 4 | 15 |
| 5 | 24 ? |
| 6 | 35 ? |
| 7 | 48 ? |
-
The values for \(P\) follow a quadratic pattern. The second difference is \(2\).
Example
For the values in the table below, predict more values and describe the pattern in words. Check your answers by revealing them.
| Stage (\(n\)) | Circles (\(C\)) |
|---|---|
| 1 | 10 |
| 2 | 12 |
| 3 | 16 |
| 4 | 22 |
| 5 | 30 ? |
| 6 | 40 ? |
| 7 | 52 ? |
-
The values for \(C\) follow a quadratic pattern. The second difference is \(2\).
Example
For the values in the table below, predict more values and describe the pattern in words. Check your answers by revealing them.
| Stage (\(n\)) | Length (\(L\)) |
|---|---|
| 1 | 48 |
| 2 | 24 |
| 3 | 12 |
| 4 | 6 |
| 5 | 3 ? |
| 6 | 1.5 ? |
| 7 | 0.75 ? |
-
The values for \(L\) follow a geometric pattern. Each term is the previous term multiplied by \(\frac{1}{2}\).
Example
For the values in the table below, predict more values and describe the pattern in words. Check your answers by revealing them.
| Stage (\(n\)) | Area (\(A\)) |
|---|---|
| 1 | \(\sqrt{7}\) |
| 2 | \(\sqrt{11}\) |
| 3 | \(\sqrt{15}\) |
| 4 | \(\sqrt{19}\) |
| 5 | \(\sqrt{23}\) ? |
| 6 | \(\sqrt{27}\) ? |
| 7 | \(\sqrt{31}\) ? |
-
The values for \(A\) inside the square root follow a linear pattern. They increase by \(4\) each time.
Example
For the values in the table below, predict more values and describe the pattern in words. Check your answers by revealing them.
| Stage (\(n\)) | Midpoint (\(M\)) |
|---|---|
| 1 | \(\left(2,5\right)\) |
| 2 | \(\left(2,8\right)\) |
| 3 | \(\left(2,15\right)\) |
| 4 | \(\left(2,26\right)\) |
| 5 | \(\left(2,41\right)\) ? |
| 6 | \(\left(2,60\right)\) ? |
| 7 | \(\left(2,83\right)\) ? |
-
The first value in the coordinate for \(M\) is always \(2\).
The second value in the coordinate for \(M\) follows a quadratic pattern. The second difference is \(4\).
Example
For the values in the table below, predict more values and describe the pattern in words. Check your answers by revealing them.
| Stage (\(n\)) | Ratio (\(R\)) |
|---|---|
| 1 | \(\frac{15}{3}\) |
| 2 | \(\frac{13}{9}\) |
| 3 | \(\frac{11}{27}\) |
| 4 | \(\frac{9}{81}\) |
| 5 | \(\frac{7}{243}\) ? |
| 6 | \(\frac{5}{729}\) ? |
| 7 | \(\frac{3}{2187}\) ? |
-
The numerator of \(R\) follows a linear pattern. They decrease by \(2\) each time.
The denominator of \(R\) follows a geometric pattern. Each term is the previous term multiplied by \(3\).
Predicting and Describing
For the values in the table below, predict more values and describe the pattern in words.
| Stage (\(n\)) | Value (\(L\)) |
|---|