Beetle Wheels

Purpose

In this unit of work we link the development of skip-counting patterns to bars on a relationship graph. We also plot our skip-counting patterns on a hundreds board.

Achievement Objectives
NA1-1: Use a range of counting, grouping, and equal-sharing strategies with whole numbers and fractions.
NA1-6: Create and continue sequential patterns.
Specific Learning Outcomes
  • Continue a skip-counting pattern.
  • Describe skip-counting patterns.
  • Use graphs to illustrate skip-counting patterns.
Description of Mathematics

In this unit we look at skip-counting patterns. These are patterns obtained by adding the same, constant, number to make the next number every time. So the difference between any two terms in a skip-counting pattern is the same. This is a good exercise to help reinforce the various concepts relating to pattern. In particular, it helps us to understand the idea of a recurrence relation between consecutive terms.

Skip-counting patterns are also called arithmetic progressions. In secondary school, expressions for both the general term of an arithmetic progression and the sum of all of the numbers in the progression are found. 

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 differentiate the tasks include:  

  • Varying the difficulty of the skip.  For example, reduce the complexity by using skips of 2, 5 or 10.  Increase the complexity by using odd numbers or numbers larger than 5. 
  • Increase the difficulty by asking the students to extend the number of skips in the pattern.  For example, in the Exploring part of the unit ask the students to extend the pattern to 10 or more skips.

The contexts for this skip patterns used in this unit can be adapted to suit the interests and experiences of your students. For example in the Exploring part of the unit:

  • involve the students in finding or suggesting the skip counting images
  • use images of spiders (8 legs), lizards (4 legs), frogs (4 legs) and butterflies (2 wings) that are native to New Zealand
  • use images of animals, plants and insects that are found in the local area.
Required Resource Materials
  • Counters
  • Cubes
  • Squared paper for graphing
  • Picture of VW beetle
  • Pictures of objects for exploration
Activity

Getting Started

Today we explore the pattern of 4s by counting the number of wheels on cars.  We then use this information to build a relationship graph.

  1. Ask: How many wheels does a beetle have?
  2. Share ideas. Hopefully someone will link the beetle to the Volkswagen car rather than the insect or you may have to give a few more hints. Show students a picture of a VW beetle and discuss why it got this nickname (it is shaped like a beetle).
  3. Using counters begin to develop a chart of the number of wheels to the number of cars.
  4. Ask: How many wheels are there on 2 beetles?
    How did you work that out?
  5. It is useful for the students to listen to the strategies that others use. More advanced Level 1 students will be able to count on from 4 to find the answer and many may have 4 + 4 as a known fact.
  6. Repeat the process with 3 and 4 VW beetles. Each time continue to add the information to the chart.
  7. Ask the students to work out how many wheels there would be on 6 beetles. If some of the students find the answer quickly, ask them to find the answer using another strategy.
  8. Share solutions. These may include:
    • skip-counting with or without the calculator
    • counting on using a number line or hundred’s board
    • using counters to find 6 groups of 4.
  9. Ask the class to complete the chart up to 6 cars.
    wheels.
  10. Ask: What can you tell me about this chart?
    Share ideas. Encourage the students to focus on the relationship between the number of cars and the number of wheels.
  11. Ask the students how they could record this information using grid paper.

Exploring

Over the next 2-3 days, the students work in pairs to explore the number patterns of other skip-counts. At the end of each session the students share their charts with the rest of the class.

  1. Place pictures of items that the students are to investigate in a “hat”. Ask each pair to draw one out and then investigate the pattern up to at least 6. Encourage the more able to students to extend the pattern beyond 6.
  2. Pictures could include:
    • tricycles (3 wheels)
    • bicycles (2 wheels)
    • hands (5 fingers)
    • spiders (8 legs)
    • glasses (2 lenses)
    • frog (4 limbs)
    • stool (3 legs)
  3. Remind the students that they are to record their explorations on a chart.
  4. At the end of each session share and discuss charts and number patterns.  Ask the students to identify the patterns that are the same.

Reflecting

In today’s session we use calculators to extend our skip-counting into the hundreds.  We record our patterns on a hundreds chart.

  1.  As a class look at the chart to show hands (5 fingers). Skip count together in 5s, shading the counts on a hundreds chart.
    Hundreds chart.
  2. As the chart is shaded ask questions which encourage the students to look for patterns in the numbers as they make their predictions.
    Which number will be next?
    How do you know?
  3. Give the students (in pairs) a hundred’s chart and ask them to shade in one of the skip counting patterns that they had charted on the previous days.
  4. Display, share and discuss at the end of the session.
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Level One