Making Monuments

Task 180 ... Years 4 - 8

Summary

Set in a story shell resembling real life, the students build tower monuments in the middle of a park and construct pathways leading to the monument from the four arms of the compass. These monuments and paths are special though because:
  • the length of each path arm is the same as the height of the tower.
  • the height of the tower is measured by the number of square surfacing tiles up one side.
  • there is a capping tile the same size on top of the monument.
The essence of the challenge is:
If I tell you any height of a tower can you tell me how many tiles are needed to surface the paths and the monument?

This cameo has a From The Classroom section which shows how it can lead into an area of the Year 11 curriculum.

 

Materials

Content

  • basic number skills
  • seeking & seeing patterns
  • generalisation
  • equivalent algebraic expressions
  • symbolic representation
  • substituting into equations
  • solving equations
  • graphing ordered pairs
  • relationship to gradient and y intercept
  • mapping and functions
  • domain and range (codomain)
  • hybrid functions
Making Monuments

Iceberg

A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.
   

Answers are:

Monument Size (S) 1 2 3 4 5 ... 10 ... 100
Number of tiles (T) 9 17 25 33 41 ... 81 ... 801

and students should keep this record in their journal.

The iceberg begins to open with the Challenge section which asks students to explain how they worked out these results, then pushes the communication aspect of a mathematician's work further by challenging the students to explain in another way. Students might reason in the following ways:

  • Generalisation A
    It's a bit like a shadow. Each side is 2 times the height (because it's height plus arm). That happens four times. Then you add the top tile.
  • Generalisation B
    There are four sides of the monument and one tile on top. Then there's four arms the same length as the height.
  • Generalisation C
    Height and arms are the same length and there are four path arms and four sides to the tower, so that's eight things the same and there's one tile on top.
and there are undoubtedly more ways to see the construction. Encourage students to explain orally and then record in words as here. The written words become the genesis of symbolic representation as an equation. To find the number of tiles:
  • Generalisation A ... T = 4 x (2 x S) + 1
  • Generalisation B ... T = 4S + 1 + 4S
  • Generalisation C ... T = 8S + 1
Note that each equation is a direct translation of its words above and each is equally valid. Generalisation C might be the simplest algebraic expression, but it may not be the most meaningful if you first see the construction a different way. However, developing these different expressions does introduce the idea of equivalent algebraic expressions and, in essence, does so through application of the mathematician's question: Can I check it another way?. Other task which highlight this type of visual algebra and equivalent expressions are:

Extend Making Monuments further with questions such as:

  • Do these different ways of seeing the pattern give the same answers for Sizes 6, 17, 26?
  • Suppose I tell you a number of tiles I have. Can you tell me the height of the tallest monument with paths that I could build?
  • Can I tell you any number for the number of tiles? Discuss.
  • Choose any five numbers for the Size. Work out the number of tiles in each case and make pairs of numbers like this (size, tiles). If these pairs were plotted on a graph what would you expect to see. Plot them to check your hypothesis.
  • If you joined up these dots with a pencil line, how could you measure the slope (gradient) of the line? Which number does it go through on the vertical axis?
  • What happens if we change the tiling pattern?
(At this point, especially if you teach Years 11/12 or university, it is worth considering the From The Classroom section below from Damian Howison.)

Whole Class Investigation

Tasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.
   

If you intend to have every pair or group in a position to build towers with matching paths, then you will need cubes and tiles of the same size, for example, 2cm squares as in the task. However, 2cm cubes are reasonably easy to obtain, but 2cm tiles are not - which is why the tiles in the task have been specially made. So an alternative approach to developing the whole class lesson is to use the task equipment in a fish bowl situation to set up the problem. Most students can visualise the structure of the problem from here. Perhaps set up the Size 2 situation as on the card with student assistance first. Then use your electronic whiteboard to show how this can be recorded on isometric paper. Handing out the Recording Sheet at this stage and asking students to make their own copy of Size 2 will help to confirm the structure. Encourage drawing Sizes 1 and 3 to discover (or confirm) the number of tiles. The model is always in the fish bowl to check. From here, develop the lesson along the lines suggested above.

At this stage, Making Monuments does not have a matching lesson on Maths300.

Is it in Maths With Attitude?

Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.
   

The 4 Arm Shapes task is an integral part of:

  • MWA Pattern & Algebra Years 3 & 4
  • MWA Pattern & Algebra Years 7 & 8

From The Classroom

Mackillop College
Swan Hill

Damian Howison
Year 11
Damian asked the question: What happens if the monument is placed at a specific place inside a courtyard walled on three sides?. The mathematical model which develops is perhaps more connected to real life, since the task assumes the area of the park is infinite, whereas in real life such a park must be bounded.

G'day Doug,
In our emails earlier in the year we were discussing Making Monuments. Now that things have slowed down a bit I thought I'd share what has happened with this task. At the time it had occurred to me to use it with a Year 12 class who needed to understand what a hybrid function is and I used an extension of the problem I had thought of while using Making Monuments in a Year 7 Replacement Unit the previous year.

I used the extension again with a Year 11 group this term. My motivation for using it was to have a context in which to explain how functions are defined, written and understood in terms of the notation used, and also to introduce hybrid functions - how they work and why we might need them. The MM task really worked in this respect. I've attached the IWB slides from the lesson.

We dealt with only a specific version of the problem - 2 units, 5 units and 9 units distance from each of the respective walls. But once you look at the solution, the way it is defined, it is not too hard to see a general solution for placing the monument anywhere - A, B and C units, (or A, B, C and D units for that matter - if the courtyard was fully enclosed).

Cheers,
Damian

Making Monuments

Damian's slides are a very clear development of the thoughts opposite. They are highly recommended, especially to teachers of senior high school and university.

Damian's Slides
This is a PDF file (~3Mb).
Ctrl L will show the slides in full screen.

Green Line
Follow this link to Task Centre Home page.