# Painted Rods

### Task 152 ... Years 4 - 10

#### Summary

The story shell is that a company produces rods like those shown in the photo and has to paint them all over. The square cross section is the unit. The challenge is:
• If I tell you any length rod can you tell me the total number of squares of area that are painted?

#### Materials

• Cuisenaire rods in six different lengths

#### Content

• basic operations
• patterns in tables
• linear algebra including:
• concept of a variable
• generalisation
• substitution
• solving linear equations
• domain and range
• equivalent algebraic expressions
• #### Iceberg

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

Some students might need the experience of stamping the square end of one rod along the length of another. For some it might be appropriate to reinforce the concept of area (surface area too) as they do so. The solutions to the questions on the card are:

 Length of Rod (L) 1 2 3 4 5 6 Squares Painted (S) 6 10 14 18 22 26

By exploring the rods and examining this table, students will begin to see a generalisation. Encourage them to express how they 'see' the rods in order to calculate the number or painted squares. Here are some sample ways to explain the generalisation:

• The total is 4 times the long side plus the 2 ends.
• You paint 6 squares on the shortest rod, then every length after than only needs 4 squares painted.
• You could paint one end, up along the length, then down the other end. Then you would have three more lengths to paint.
...and there may be more.

Generalisations such as these are what students will use to calculate the result for 10 and 100, however each visualisation will lead to a different calculation:

• S = 4L + 2
S = 4 x 10 + 2 = 42
S = 4 x 100 + 2 = 402
• S = 6 + 4(L - 1)
S = 6 + 4(10 - 1) = 6 + 4 x 9 = 6 + 36 = 42
S = 6 + 4(100 - 1) = 6 + 4 x 99 = 6 + 396 = 402
• S = (1 + L + 1) +3L
S = (1 + 10 + 1) + 3 x 10 = 12 + 30 = 42
S = (1 + 100 + 1) + 3 x 100 = 102 + 300 = 402
Each of these are equally valid ways to calculate the total number of squares painted and in this form they represent examples of equivalent algebraic representations of the same problem. Using the generalisation to find the number of squares for any length is substituting into equations.

#### Extensions

• Ask backwards questions: The number of squares painted was 62. How long was the rod? This is solving equations.
• Select several lengths and calculate the matching number of squares. These make pairs, in fact, ordered pairs, and these can be graphed.
• What does it mean if we join the dots on the graph? Can we imagine a rod perhaps 2.5 units long? Can we calculate the number of squares of area that would be painted? Would this point lie on the joined up graph.
• Why is the graph a straight line?
• What happens if the company creates a special type of rod by first gluing two of the same lengths side by side. If I tell you any length of this double rod, can you tell me its painted area?
• What happens if the company creates a special type of rod by completely surrounding a length of rod by others of the same length glued to each of the 'long' sides. If I tell you any length of this multi-rod, can you tell me its painted area?
• Build a unit of work in Pattern & Algebra with this task and others such as Garden Beds, 4 Arm Shapes, Crossing The River 1, Thirty-One, Match Triangles, Making Monuments, Painted Cubes
Or, order Maths With Attitude kits to receive tasks grouped in content strands with prepared units of work.

#### 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.

 To activate the whole class lesson life of this task you really need sets of Cuisenaire Rods. One between 4 is enough. Begin by asking students to compare different colour rods to find out what is common to all of them. They will soon discover that all have the same cross section, but perhaps not so easily discover that the cross section can be 'stamped' along the side of any rod a whole number of times. • So if we can use the square end section as a measuring unit. It can count the length of each rod and it can count the number of squares of paint that would cover a rod if all of its surfaces were painted.
Ask the students to make these count for any three of the rods in the box. To check the results, ask a group which lengths they chose and what results they got for the number of painted squares. Record their results as given to you and then ask the next group their results and record those too. Do this two or three more times to get several results on the board and to confirm that the displayed results are correct.

Note that there is no suggestion here of drawing a table or choosing symbols for length and squares painted. The students are learning to work like a mathematician, so they have to learn to make those decisions for themselves.

• Okay, we're all happy with this data now, so your first challenge for the day is this. If I tell you any length of rod, can you tell me the number of squares that will be painted?
Continue the lesson from here using the ideas and extensions above. Encourage recording in journals and, perhaps, publishing for others to understand. In the debrief of the lesson highlight connections with measurement of area and the concept of total surface area, as well as reviewing the lesson against the Working Mathematically Process.
• How have we worked like a mathematician today?

For more ideas and discussion about this investigation, open a new browser tab (or page) and visit Maths300 Lesson 39, Painted Rods, which also includes an Investigation Guide.

#### 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 Painted Rods task is an integral part of:

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

The Painted Rods lesson is an integral part of:

• MWA Pattern & Algebra Years 5 & 6
• MWA Pattern & Algebra Years 7 & 8 