Dual patterns

The Ministry is migrating nzmaths content to Tāhurangi.           
Relevant and up-to-date teaching resources are being moved to Tāhūrangi (tahurangi.education.govt.nz). 
When all identified resources have been successfully moved, this website will close. We expect this to be in June 2024. 
e-ako maths, e-ako Pāngarau, and e-ako PLD 360 will continue to be available. 

For more information visit https://tahurangi.education.govt.nz/updates-to-nzmaths

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
NA2-7: Generalise that whole numbers can be partitioned in many ways.
NA2-8: Find rules for the next member in a sequential pattern.
Specific Learning Outcomes
  • Create, describe and continue single-attribute and two-attribute repeating patterns with varying numbers of elements.
  • Identify and replace missing elements in a repeating pattern.
  • Identify, describe, and create composite patterns.
  • Identify the unit of repeat in a repeating pattern and apply known patterning language.
  • Describe and explain the rule for the pattern in a composite pattern.
Description of Mathematics

As students make and interpret repeating patterns, they develop understanding of the following:

  • the nature of the attributes that define or differentiate the elements in a pattern
  • how to identify the unit of repeat
  • the ordinal language used to specifically locate elements within an 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 requires careful scaffolding.

In this unit students explore such composite patterns and are 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. These should 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.

Opportunities for Adaptation and Differentiation

The learning opportunities in this unit can be differentiated by providing or removing support to students, and by varying the task requirements. Ways to support students include:

  • ensuring students understand what constitutes the start and end of a pattern
  • having students copy and continue patterns
  • allowing students to work in pairs where one makes a pattern and the other copies. This exercise can be adjusted to have a child make a pattern behind a barrier, remove the barrier and replace quickly.  The partner has a quick glimpse and recreates the pattern they saw. Consider partnering and grouping students strategically to encourage greater peer learning, scaffolding, and extension
  • removing 1, 2 or 3 elements but not leaving gaps. Look to see is students can identify what is missing
  • modifying the patterns so that they look different but have the same pattern
  • substituting objects with sound (e.g. clapping, tapping, clicking and dance)
  • allowing students to continue repeating patterns over a wider space.

The context for this unit can be adapted to suit the interests and experiences of your students by, for example:

  • teaching, as part of the introduction to the unit, a kapa haka action song, traditional folk dance or Pacific drum. Look for students to copy the pattern. Consider whether students, adults, siblings, older students, or other community members might be able to demonstrate/teach a traditional dance from their culture that has repeating patterns
  • utilising patterns in Maori stick games (tī rākau). Students could create their own patterns for others to appreciate, learn, and emulate
  • using culturally relevant patterns (e.g. tukutuku patterns) to explore and appreciate repeating patterns.

Te reo Māori kupu such as tauira (pattern) could be introduced in this unit and used throughout other mathematical learning. You could also encourage students, who speak a language other than English at home, to share the words related to patterns that they use at home.

Required Resource Materials
  • Environmental materials for patterning
  • Classroom patterning materials: for example, attribute blocks, items to go in a sorting box
  • Pencils and paper
  • Counters of at least three different colours or bottle caps
  • Art materials including paper, felts, crayons, pencils, printing ink/paint, cardboard, and newspaper
  • An item of patterned clothing (or pictures of relevant clothing items)
  • A list of local/relevant places with visible patterns.
Activity

Session 1

Activity 1

  1. Wear clothing with a pattern or present pictures of patterned clothing. Perhaps you could invite students to bring in a special, patterned item of clothing to share with the class. Discuss the patterns with the students and, if appropriate, have them share the patterns they have found in their clothing.
     
  2. Ask the students where they see repeating patterns in their lives. Record the ideas on a class chart and supplement with your own list. You will need at least one place for every two students in your class. For example: 
  • in the playground
  • in the classroom
  • on the way to school, on the bus, in the car, etc.
     
  1. Allocate a place to each pair of students by writing paired names beside each place on the list of places you have generated as a class. Make paper, felts, and crayons available to the students. Provide time for student pairs to write the place (e.g. on the bus) and to name and draw as many of the patterns that they can remember seeing in this place. You might encourage students to think of places around their immediate environment (e.g. school), and go on a walk around the school to find these patterns.
     
  2. Have them share these with another pair of students, name and describe the patterns. Elicit and encourage specific patterning language (e.g. around what repeats, first, second, third etc.). 

Activity 2

  1. Make classroom or environmental patterning materials available to student pairs. Ask them to sort the materials into three categories: environmental materials, geometric shapes, and sorting materials.
     
  2. Write the word attribute on the class chart, introducing it to the students. Discuss that it is a characteristic or something special about a particular person or thing. You might develop knowledge of this concept by having students list attributes about a thing (e.g. a square has four sides), a place (e.g. the water is cold), or person (e.g. I am friendly). 
     
  3. Brainstorm and record on the class chart the names of the materials and pattern attributes relevant to the chosen materials.
    For example:
    Environmental materials: name, colour, size, position/orientation, density (soft/hard)
    An image of environmental materials.
    Geometric shapes: shape name, colour, size, position/orientation, thickness
    An image of geometric shapes.
    Sorting material (eg. buttons); colour, size, position, number of holes
    An image of buttons.
     
  4. Have students choose a patterning material. Have them decide on one attribute and work alongside a partner to make a pattern with the chosen attribute. Have them remove or add elements.

For example: 

  • one partner creates a pattern using colour An image of a pattern created with buttons of alternating colours.   
  • the other partner creates a pattern using position An image of a button pattern created by changing the position of every second button.
     
  1. Provide time for students to look at each other’s patterns, describe them, name the attribute, continue the pattern, and remove or replace the missing elements.
  2. Have students repeat the task, creating a new pattern focused on a different attribute.

Activity 3

  1. Write ‘unit of repeat’ and ‘element’ on the class chart. As a class, explore several examples of students' patterns. Discuss and agree for each, what elements feature in the unit of repeat.
     
  2. Model a pattern with two changing attributes, eg. changing 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 etc.) Record the key patterning language on the class chart.
  3. Brainstorm and record on the class chart the names of the materials and pattern attributes relevant to the chosen materials. This time, refer to two attributes and have no more that 5 core elements in the pattern. Have students identify the two attributes in their patterns.
  4. Review key vocabulary from this session: attribute, elements, ordinal words and specific attributes. Have students draw on write a reflection of what they have done. Encourage them to use the key vocabulary in their reflection.

Session 2

Activity 1

  1. Have students share their reflections from the end of the previous session with a partner.
  2. Place a selection of coloured counters in front of the students. 
  3. Organise students into pairs. Each pair is to take a random selection of counters 
  4. Have the student pairs create, describe and extend a pattern of three repeating colours. They might need to add or remove counters from their collection. Be careful to identify the attribute, colour, and unit of repeat, three.

For example: blue, yellow, black.
This image shows a pattern of blue, yellow, and black counters.

  1. Ask: What is the second element? Third element?
  2. Have students look at the whole pattern and identify the seventh and ninth elements, and predict the colour of the tenth, twelfth, fifteenth element, etc. Have the students describe how they work out their predictions. Look for the following:
  • Are they continuing the pattern out loud, counting on their fingers in some way, or pointing to where the next counter would be?
  • 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.
     
  1. Tell students 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 move every second counter up or down. Ask students to read this two element pattern aloud (e.g. counter up, counter down etc.)
  2. 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).
  3. Ask them to decide what the core of the pattern is now.
  4. Ask, “What is the unit of repeat for the combined new pattern?”
  5. 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?
  6. Have students look at the whole pattern and identify the ninth, eleventh, and twelfth elements and predict the fourteenth, fifteenth elements etc.
    Have the students describe how they work out their predictions. Look for the following:
  • Are they continuing the pattern out loud, counting on their fingers in some way, or pointing to where the next caps would be? 
  • Are they using a pattern of adding sixes? Eg. 14th is 6 + 6 + 2
    Are they using multiplication and addition? Eg. 14th is 6 x 2 + 2 
     
  1. 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.
     
  2. Have some students share their explanations with the class. Highlight the process used to create the combined pattern.

Activity 3

  1. Make the counters and other patterning materials available to the students.
  2. Have them remain in pairs, but individually create several of their own combination patterns, following the process that was explored in Activity 2. 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.
  3. Conclude this session by reviewing key language.

Session 3 

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

Activity 1

  1. Explain that students are going 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. If possible, read a story or view art featuring graphic patterns. You might be able to invite in a local artist to talk with your class about the patterns they use in their art.
  3. Explain and demonstrate the chosen process for accomplishing this in the classroom. E.g. computer design using draw, copy and paste; stencil or 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, the level of detail they can include will be limited.
  5. 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). 
This image shows a tree pattern following the rule "3 elements in one unit of repeat".This image shows a tree pattern following the rule "12 elements in the combined unit of repeat".
3 elements in one unit of repeat 12 elements in the combined unit of repeat 
  1. Refer to the attributes listed on the class chart in Session 1.
  2. Have students consider and write down the two attributes that will comprise their design. Have them discuss and explain this to a partner.
  3. 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. Tell students to provide each other with constructive feedback.
  4. When drafts are completed and approved, the students can begin to create their frieze.
  5. 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.
  6. Conclude this session with students sharing their designs and explanations. Place an emphasis on the students’ use of appropriate and precise patterning language. Display students' work.
Add to plan

Log in or register to create plans from your planning space that include this resource.


Level Two