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Dual patterns

Keywords:
Purpose: 

This unit develops pattern-based thinking through the exploration of a pattern that has more than one attribute and more than one unit of repeat.

Achievement Objectives:

Achievement Objective: NA2-7: Generalise that whole numbers can be partitioned in many ways.
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: 
  • Create, describe and continue a single-attribute repeating pattern.
  • Identify and describe the composite pattern..
  • Create an original composite pattern with a unit of repeat of more than 6 elements (3 and 2) but no more than 12 (3 and 4).
  • Describe and explain the rule for the pattern in the composite pattern.
Description of mathematics: 

As students have been making and interpreting repeating patterns, they have been developing understanding of the nature of the attributes that define or differentiate the elements in a pattern, of how to identify the unit of repeat, and of the ordinal language to specifically locate elements within and extended pattern. These are important early skills upon which the understanding of more complex patterning depends.

One form of complexity that some patterns present is the presence of two or more units of repeat within one pattern. These combine, in a multiplicative way, to create an additional overarching unit of repeat that integrates the discrete single-feature units of repeat. For example, if a pattern with a unit of repeat of three is combined with a pattern with a unit of repeat of two, the new ‘composite’ pattern has a unit of repeat of six. To come to understand how the composite unit of repeat is comprised takes careful scaffolding.

In this unit of three lessons the students explore such composite patterns and are then challenged to create and describe their own. Because there is a deceptive complexity in combining repeating patterns, the first lesson focuses on recognising and consolidating the important language and key understandings associated with repeating patterns with a straight forward single unit of repeat. It is assumed that these consolidate and build upon already familiar patterning experiences.

The activities suggested in this series of three lessons can form the basis of independent practice tasks. It is also assumed that throughout the school day, all class members, students and teacher alike, will look for and take opportunities to apply learning included in this unit of work.

Links to the Number Framework
Advanced counting (Stage 4)
Early additive (Stage 5)
Early multiplicative (Stage 6)

This unit of work supports the development of the content ideas in student e-ako 8 in particular.

Required Resource Materials: 
Environmental materials for patterning
Classroom patterning materials: for example, attribute blocks, sorting box contents
Pencils and paper
Bottle caps
Art materials including: paper, printing ink/paint, cardboard, newspaper
Activity: 

Session 1

SLOs:

  • Create, describe and continue a single-attribute repeating pattern.
  • Identify and replace missing elements in a repeating pattern.
  • Identify the unit of repeat in a repeating pattern and apply known patterning language.
  • Create and describe a two-attribute repeating pattern.

Activity 1

  1. Introduction: Wear clothing with a pattern. Discuss this with the students. Have them find patterns on their own clothes.
     
  2. Ask the students where they see repeating patterns in their lives.
    List these generic places on the class chart/modeling book:
    For example: in the playground, in the classroom, on our clothes, at the park, on TV, on the computer/iPad, (in the street) on the way to school, on the bus, in the car, etc.
    Allocate a place to each pair of students by writing paired names beside each place on the list. ii. Make available to the students, paper, felts, and crayons. Give them a time limit to write the place (eg. on the bus) and to name and draw as many of the patterns that they can remember seeing in this place.
    Have them share these with another pair of students, name and describe the patterns. Elicit and encourage specific patterning language. Display the work.

Activity 2

  1. Make classroom or environmental patterning material available to student pairs.
    Write the word attribute on the class chart, introducing it to the students. Develop through discussion the understanding that it is a characteristic or something special about a particular person or thing.
    Brainstorm and list on the class chart/modelling book, the names of the materials and attributes relevant to the chosen materials.
    For example:
    Environmental materials: name, colour, size, position/orientation, density (soft/hard)

    Geometric shapes: shape name, colour, size, position/orientation, thickness

    Sorting material (eg. buttons); colour, size, position, number of holes

     
  2. Have students choose pattern material. Have them decide on one attribute. Have them work alongside a partner as they make their individual patterns related to that attribute.
    For example: one partner (colour)    the other (position) 
    Students should look at each other’s patterns, describe them, naming the attribute, continue each other’s pattern, and remove and replace missing elements.
     
  3. Students should be given several more opportunities to repeat this.

Activity 3

  1. Write ‘unit of repeat’ and ‘element’ on the class chart/ modeling book. As a class, explore several student pattern examples. Discuss and agree for each, what elements feature in the unit of repeat.
     
  2. Model a pattern with two changing attributes, eg. (size and colour) big red square, little blue square, big yellow circle, little green circle. As a class, name the attributes, describe the pattern, continue it, remove and replace missing elements, all the while eliciting and modelling attribute and patterning language, including ordinal language: first, second, third, fourth….
    Record in the class book key patterning language.

Activity 4

Repeat Activity 2, Step 2 above this time with two attributes and no more that 5 core elements in the pattern. Have students identify the two attributes in their patterns.

Activity 5

Make available to the students, paper, felts and crayons.
On the class chart/modeling book, highlight key words from this session (including ‘attribute’, ‘elements’, ordinal words and specific attributes).
Conclude the session by having the students draw and write about what they have done.
Explain to students that they are to include some of the highlighted words in their patterning explanation.

Session 2

SLOs:

  • Combine a single-attribute repeating pattern with three elements, with a single-attribute repeating pattern with two elements.
  • Identify and describe the composite pattern.
  • Create an original composite pattern.

Activity 1

Begin by having the students pair share their writing from the previous session.

Activity 2

  1. Place in front of the students a selection of coloured bottle caps with their coloured tops facing up.

    Have student pairs take and share a selection of these.
    Have the student pair together create, describe and extend a pattern of three repeating colours. Be careful to identify the attribute, colour, and unit of repeat, three.
    For example: blue, yellow, black.

    Ask: What is the second? Third element?
    Have them look at the whole pattern and identify the seventh, ninth, and predict the colour of the tenth, twelfth, fifteenth element, etc.
    Have the students describe how they work out their predictions.
    Are they continuing the pattern out loud and counting on their fingers in some way or pointing to where the next caps would be, saying the repeating pattern?
    Are they using a pattern of adding threes?
    Eg. 14th is 3 + 3 + 3 + 3 + 2 so it’s yellow.
    Are they using multiplication and addition? Eg. 14th is 4 x 3 + 2 so it’s yellow.
    It is important to have students share and explain their ideas.
     
  2. Tell them they are going to add a second attribute, position to their pattern, but that it will have just two elements and a unit repeat of two.
    Have them downturn every second cap, so that the inside of the outside top of the cap faces down.

    Ask students to read aloud this two element pattern: cap top up, cap top down, cap top up, cap top down, cap top up, cap top down….
    Have student pairs look at and discuss what has happened to the whole pattern. (It now has a unit of repeat of 6 and begins to repeat at the 7th element in the whole pattern, not at the 4th element as previously).
    Ask them to decide what the core of the pattern is now.
    Ask, “What is the unit of repeat for the combined new pattern?”
     
  3. Discuss that there are 6 elements in the combined pattern. Name each of these by reading the pattern aloud.
    Ask: What is the third? Fourth? Fifth? Sixth element?
    Have them look at the whole pattern and identify the ninth, eleventh, twelfth elements and predict the fourteenth, fifteenth elements etc.
    Have the students describe how they work out their predictions.
    Are they continuing the pattern out loud and counting on their fingers in some way or pointing to where the next caps would be, saying the repeating pattern?
    Are they using a pattern of adding sixes? Eg. 14th is 6 + 6 + 2 so it’s (yellow) cap top down.
    Are they using multiplication and addition? Eg. 14th is 6 x 2 + 2 so it’s (yellow) cap top down.
    It is important to have students share and explain their ideas.
    Make available paper and pencils.
     
  4. Ask: Why is the combined unit of repeat now six?
    Have the students discuss this in pairs. Then have them individually write and show their thinking using patterning words in their descriptions.
     
  5. Have some students share their explanations with the class. Highlight the process used to create the combined pattern.

Activity 3

  1. Make available to the students, the caps and other patterning material.
    Have them remain in pairs, but individually create several of their own combination patterns using the two steps modeled in Activity 2, Steps 1 and 2 above. Begin by limiting the units of repeat to two and three.
    Students should take turns to look at the other’s pattern, identify the attributes, describe the pattern, continue/extend it, replace missing elements that the pattern creator has removed, and state the units of repeat.

Activity 4

Conclude this session by reviewing key language, highlighting its importance in facilitating communication about patterns.

Session 3

SLOs:

  • Create an original composite pattern with a unit of repeat of more than 6 elements (3 and 2) but no more than 12 (3 and 4).
  • Describe and explain the rule for the pattern in the composite pattern.

If possible, have students design and create their pattern electronically.
The lesson will also refer to the manual printing process using potato or stencil prints, however it will not detail the art printing processes themselves.

Activity 1

  1. Explain that the task and challenge is to design and make a patterned wall frieze for the classroom. It will be comprised of individual sections made and designed by each class member.
     
  2. Read The Finishing Touch by Kenn Benn. Connected 2, 2007, in which a child designs and makes a patterned wall frieze. Discuss the way in which the repeating pattern of stenciled sea creatures of different colours is designed and executed by Carrie and her mother.

     
  3. Explain and demonstrate the chosen process for accomplishing this in the classroom: computer design using draw, copy and paste; stencil or potato print design using card, paint or printing ink, long paper strips.
     
  4. Brainstorm and list student ideas for the subject matter of their own frieze. Consider topical and cultural school or classroom contexts. Give consideration to the level of simplicity demanded by the practical constraints of the process. For example, if the students are cutting stencils or potatoes, the level of detail they can include will be limited.

Activity 2

  1. Make available to the students, draft design paper and coloured felts/pencils/crayons.
     
  2. Have students decide the subject of their own design. Explain to the students that they should limit their design to two attributes, with different units of repeat of no more than three and four elements.
    Provide an example such as:
    A 3 element size pattern (small, middle size, big) combined with a four element colour pattern (pale green, light green, dark green, black). 
      
    3 elements in one unit of repeat 12 elements in the combined unit of repeat 
    Refer to the attributes listed on the class chart/modeling book in Session 1, Activity 2, Step 1.
     
  3. Have individual students consider and write down the two attributes that will comprise their design. Have them discuss and explain this to a partner. ii. Have individual students draw/plan a simple draft of their first repeating pattern, then add their second attribute and unit of repeat. They should clearly identify the new combined unit of repeat.
    Have them share their draft designs with a partner, who should be encouraged to give a constructive critique.
     
  4. When drafts are completed and approved, the students can begin to create their frieze.

Activity 3

As students complete their wall friezes, have them write a description of the process for their design, including a description of the two units of repeat and the number of elements in their final design. Their explanation should include diagrams and equations.

Activity 4

Conclude this session with students sharing their designs and explanations. Place an emphasis on the students’ use of appropriate and precise patterning language. Display both.