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Pede Patterns

Keywords:
Purpose: 

This unit is about generating number patterns for certain ‘insects’ from the mythical planet of Elsinore. Each ‘Pede’ is made up of square parts and has a number of feet. The patterns range from counting by 2s and 3s to being the number of feet plus three.

Achievement Objectives:

Achievement Objective: NA2-6: Communicate and interpret simple additive strategies, using words, diagrams (pictures), and symbols.
AO elaboration and other teaching resources
Achievement Objective: NA2-8: Find rules for the next member in a sequential pattern.
AO elaboration and other teaching resources

Specific Learning Outcomes: 
  • continue a simple pattern
  • generalise the pattern
Description of mathematics: 

This unit is the first in a sequence of units on patterns. In developing these units we were conscious of the progression that is shown in the New Zealand Curriculum Exemplar task in Algebra. We give the six steps of this progression below.

  1. copy a pattern and create the next element
  2. continue the pattern with systematic counting
  3. predict values using relations between successive elements
  4. predict values using rules
  5. (i) find an algebraic expression for a relation
    (ii) solve linear equations related to patterns

The numbers on the left above, represent the approximate curriculum level of that step of the progression. Hence a student who can copy a pattern and create the next element is operating at Level 1 unless she can also perform the Level 2 stage.

Pattern is at the basis of much mathematics. There is always a need to find a link between this variable and that variable. This unit provides an introduction to pattern in the context of ‘insects’. The students are given practice in finding the next insect in a sequence. This leads to the main aim, which is for the students to begin to see the link between the number of feet that certain insects have and the number of squares that make them up.

One of the things that is deliberately attempted here is for them to see the link above in both directions. So not only do they get practice in linking squares to feet but they also are asked to try to find the number of feet that an insect with a certain number of squares has.

This unit provides an opportunity to develop number knowledge in the area of Number Sequence and Order, in particular development of knowledge of the Forward Number Word Sequence and skip counting patterns.

As each pede is developed help students focus on the number patterns involved by creating tables as below. Similar tables can be drawn for each type of pede.

Humped Back Pede
Number of Feet
Number of squares
1
2
2
4
3
6
4
8
5
10

Use of a hundreds chart will help students visualise the number patterns more easily and help them to predict which numbers will be part of the patterns.

The conclusion of each session is an ideal time to focus on the number patterns involved. Questions to develop number knowledge include:

Which number comes next in the number pattern for this pede? How do you know?
Which number will be before 20 in this pattern? (or another number as appropriate)
How do you know?
What is the largest number you can think of in this pattern? How do you know?
Could a pede with 20 squares be a Spotted Pede? Why / Why not?
Could a pede with 32 squares be a 2-pede? Why / Why not?
Are there any numbers that could be Spotted Pedes or Humped Back Pedes? What are they? How did you work that out?

Required Resource Materials: 
(magnetic) tiles
squares of coloured paper
Activity: 

Session 1

Here we explore a couple of number patterns related to the mythical insects that live on the planet Elsinore. The patterns involve skip counting by 2s.

On the planet Elsinore there live a strange collection of insects. There is the Humped-Back Pede. The Humped Back 1-pede looks like this. Can you see his eye? And the Humped Back 2-pede looks like this. He has an eye too. (Show them the pictures below.) Ask the students to work individually or in pairs to make a Humped Back 3-pede with the green tiles.
Can you work our how many squares a Humped Back 4-pede has?

diagram

Gather the students together to talk about the insects that they drew. Explore the number pattern of counting in twos that comes from the Humped-Back Pedes. Also ask them questions like:
Can you tell me how many green squares a Humped Back 5-pede will have?
Can you tell me how many green squares a Humped Back 7-pede will have?
Can you tell me how many green squares a Humped Back 10-pede will have?
How many feet has a Humped-Back Pede with 12 squares?
How many feet has a Humped-Back Pede with 18 squares?
How many feet has a Humped-Back Pede with 20 squares?
Can you tell me how to get the number of squares that a Humped-Back Pede with a particular number of feet has?
Can you tell me how to get the number of feet that a Humped-Back Pede with a particular number of squares has?

Session 2

Here we investigate some more of the mythical insects that live on the planet Elsinore. The patterns here involve skip counting by 3s.

  1. There are other insects on the planet Elsinore. They look as if they have been made up from squares. The ones with one foot are called 1-pedes. The ones with two feet are called 2-pedes and the ones with 3 feet are called 3-pedes. (Show the students the picture below.)

    diagram of pede.

    Did you know that ‘pede’ means ‘foot’?
    How many feet would a 4-pede have? What about a 5-pede?
    Can you tell me how many squares a 1-pede has?
    How many squares does a 2-pede have?
    What about a 3-pede?

    (Put the numbers of squares beside the insects as the students answer the questions.)

  2. Can someone tell me what a 4-pede looks like?
    How many feet will it have?
    How many squares will it have?
    Can someone make one for me with these square tiles?
    Does everyone agree with that?

    (Write 11 under the 4-pede.)
  3. Let’s have a look at the number of squares that these Pedes have. Count them out. 2, 5, 8, 11.
    I wonder what sort of Pede comes next? (A 5-pede.)
    How many squares does a 5-pede have? (14.)
    I wonder what sort of Pede comes next? (A 6-pede.)
    How many squares does a 6-pede have? (17)
    Is there any pattern in the number of squares that Pedes have?
    (Add on 3 for each extra foot. Talk about skip counting by threes.)

Session 3

Here we explore patterns further. Here we are particularly interested in linking the number of squares on an insect and the number of feet it has.

  1. Here we are going to explore the Spotted Pedes. This is what they look like. Talk about the number of squares they have and the number of blue and red squares.  Record these beside each of the Spotted Pedes.
                Spotted Pede.    Spotted Pede.    Spotted Pede.
     
  2. Now ask the students to draw the Spotted 4-pedes? As they work ask them the following questions:
    How many red squares does a Spotted 4-pede have?
    How many blue squares does a Spotted 4-pede have?
    How many squares all together? How did you work that out?
    Why are there more blue squares than red squares? How many more?
  3. Repeat with Spotted 5-pedes and Spotted 6-pedes.
    How many red squares does a Spotted 5-pede have?
    Can you tell me how many blue squares a Spotted 5-pede has?
    Draw it.
    How many red squares does a Spotted 6-pede have?
    How many more blue squares does a Spotted 6-pede have?
    Draw it.
  4. For those students who need challenging ask them about the Spotted 10-pede?
    How many red squares does a Spotted 10-pede have?
    How many blue squares does a Spotted 10-pede have?
  5. At the end of the session spend time sharing findings as a class. Ask them
    What did you find out about the Spotted Pedes?
    What patterns did you find?

    (Try to get them to see that they have as many red squares as they have feet. This means that it is very easy to find out how many red squares they have.)

Session 4

Here we look at more patterns, this time related to the Big Headed Pedes. Here the number pattern involves starting with 4 and adding 1 each time to get the next number of squares.

  1. Here we are going to explore the Big-Headed Pedes. This is what they look like.

                Big-Headed Pede.    Big-Headed Pede.    Big-Headed Pede.
  2. Ask the students to work with a partner to draw the next three Big-Headed Pedes (4, 5, 6). As they work ask them the following questions:
    How many yellow squares does a Big-Headed Pede 4-pede have?
    How many yellow squares does a Big-Headed Pede 5-pede have?
    How many yellow squares does a Big-Headed Pede 6-pede have?
    Do you need to draw the insects to work out how many squares they have? Why or why not?
    Could you work out how many yellow squares a Big-Headed Pede 10-pede would have?

  3. At the end of the session spend time sharing findings as a class. Ask them
    What did you find out about the Big-Headed Pedes?
    What patterns did you find?

    (Try to get them to see that they have three more yellow squares than they have feet. This means that it is very easy to find out how many yellow squares they have.)
    I saw a Big-Headed Pede with 16 yellow squares. How many feet did she have?

Session 5

The students now work at making up their own Pedes.

  1. Can you tell me what kinds of Pedes we have been talking about this week?
    Discuss the Pedes, the Humped-Back Pedes, the Spotted Pedes and the Big-Headed Pedes.

  2. Tell them that you want them to work with their partners to invent a Pede of their own. Ask the students to record the first three pedes on one piece of paper and the other three on a second sheet. Ask them to also invent a name for their insect.

  3. Pairs could then swap the first three Pedes to see if they can work out the next three Pedes for each other’s insect. They can then check with each other to see if they arrived at the same Pedes for the 4, 5 and 6 insects.

  4. As time allows get the students to swap pedes with other pairs.

  5. As the pairs work ask them to discuss the various patterns that they have produced. Ask them questions such as:
    How many squares does one of your 5-pedes have?
    Can you tell me the number of squares an X Pede with 10 feet (or some other relatively large number) has?
    If I had 15 squares what is the Pede with the largest number of feet that I could draw?

  6. Collate the classes Pedes into a book of Pede problems that can be worked on by the students during any “free” or “choosing” time.