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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:

Achievement Objective: NA1-6: Create and continue sequential patterns.
AO elaboration and other teaching resources
Achievement Objective: NA1-1: Use a range of counting, grouping, and equal-sharing strategies with whole numbers and fractions.
AO elaboration and other teaching resources

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.

But this exercise has two other links. First it links Algebra and Statistics by using a bar chart to represent the skip-counting pattern. It is worth remembering that there are many links in mathematics. Being able to see and use such links is a powerful tool to have.

And second, skip-counting patterns are also called Arithmetic Progressions. In secondary school these are considered again and expressions for both the general term of the progression and the sum of all of the numbers in the progression are found. These are both reasonably simple algebraic expressions.

Links to Numeracy

This unit provides an opportunity to develop number knowledge in the area of Number Sequence and Order, in particular development of knowledge of skip counting patterns. It can also be used to focus on the development of strategies to solve multiplication problems.

Once students have created bar graphs of the relationships, help them to focus on the number patterns involved by creating tables. For example:

           Number of Beetle Wheels
Number of Beetles
Number of Wheels
1
4
2
8
3
12
4
16
5
20

As students create tables focus their attention on the patterns that are emerging and pose questions about the continuation of the patterns. 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. Patterns of 2, 5 and 10 are a good place to start but for students that are coping well you can make it more difficult by using larger numbers. For example, if there were 7 friends in each beetle how many people would there be in 2 beetles? 3 beetles? What about 10 beetles?

Working with larger numbers of beetles (or other items) will help students develop strategies to solve multiplication and division problems. Encourage students to talk about the way they are solving these problems. Are they using materials, repeated addition or can they derive some of the answers from known multiplication facts?

Questions to develop knowledge / strategy use:
Which number comes next in this pattern?
How do you know?
Which number will be before 24 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 did you work it out?
How many wheels will there be in 5 beetles? 10 beetles? How did you work it out?
If there were 48 wheels in a car park how many beetles would there be? How did you work it out?

 

Required Resource Materials: 
counters
cubes
squared paper for graphing
picture of VW beetle
pictures of objects for exploration
Key Vocabulary: 

patterns, next, before, largest, how many?  table, graph

Activity: 

Getting Started

Today we explore the pattern of 4’s 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.
  3. Using counters begin to develop a chart of the number of wheels to the number of cars.
    wheels.1 car
  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 be to count on from 4 to find the answer and many would 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. As 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 hundred’s chart.

  1.  As a class look at the chart to show hands (5 fingers). Skip count together in 5’s shading the counts on a hundred’s 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.