Matchstick Patterns
Introduction
In this task, we will practising the mathematical thinking outlined in the unit overview. In particular, we will follow the following steps when investigating patterns. Let's start with the matchstick pattern above.
Step 1: predict what the next patterns may be. One good strategy is to draw out the next few patterns.
Can you draw out the next few patterns?
Step 2: organise your findings in a systematic way. One good strategy is to put your observations in a table.
Can you summarise your findings in a table?
Hint: use the table on the right.
Step 3: use your table to come up with some general rules (term to term rule and position to term rule).
Term to term rule - looking at your table, can you find a relationship going from one term to another?
Position to term rule - can you find the relationship between the pattern number and the number of matchsticks?
This allows you to calculate the number of matchsticks needed for any pattern that you were asked.
Step 4: verify (a.k.a. check) that your rules work. This can be achieved by comparing your drawings with the answers found using your rules.
Use your rules to find how many matchsticks are used in pattern number 6. Do the answers for the term to term and position to term rules match?
Draw pattern number 6. Do the number of matchsticks match the ones found using your rules?
Step 5: justify (a.k.a explain) why the rules work.
Can you explain why your rules work? You may wish to illustrate this by using diagrams.
On the right are three diagrams explaining how the rules work.
Do they match with your explanation?
Can you explain what each diagram is showing?
Here is a suggested answer for the pattern above for your reference.
Other questions:
Use your rule to predict how many matchsticks tiles would be needed for:
The 10th pattern.
The 43rd pattern.
Use your rule to predict which pattern/ what pattern number has...
76 matchsticks?
229 matchsticks?
104 matchsticks? Explain your answer.
If you had 400 matchsticks, what is the biggest pattern that you could make?
How many matchsticks would be left over?
Repeat steps 1-5 above with the 4 patterns below:
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Now try to make some matchstick patterns that fit the following rules:
Shape number times 5, add 3
Shape number times 2, add 1
Shape number times 4
Shape number times 3, subtract 1
Shape number add 1, times 3
Further Practice
Some of the essential skills introduced in this lesson are "linear/ arithmetic sequences", "position to term rule/ nth term" and "term to term rule". The relevant skills can be found on DrFrostMaths, CorbettMaths, MyiMaths and Eedi. Watch any video and/or go through any online lesson as you see fit.
Transum
Matchstick Patterns - create a formula to describe the nth term of a sequence by examining the structure of the diagrams.
Missing Terms- can you work out which numbers are missing from these number sequences?
Arithmetic Sequences - an exercise on linear sequences including finding an expression for the nth term and the sum of n terms.
Venn Diagram of Sequences - find the formula for the nth term of sequences that belong in the given sets.
Parts of Sequences - find the formula that describes the part of the sequence that can be seen. - Only the first two levels are appropriate.
Quick Add 'Em Quiz: - find the sum of a sequence of consecutive numbers using a quick, efficient, elegant method.
For more practice, try the questions:
Extension
Transum
The following are things you will either learn in the next few lessons or are outside the scope of Year 7 Maths. You may be interested in learning about them.
Geometric Sequences: An exercise on geometric sequences including finding the nth term and the sum of any number of terms.
Iteration: Find approximate solutions to equations numerically using iteration.
Pascal's Triangle: Get to know this famous number pattern with some revealing learning activities
Fibonacci Quest: A number of self marking quizzes based on the fascinating Fibonacci Sequence.
Tower of Hanoi: Move the pieces of the tower from one place to another in the minimum number of moves.
nRich
Here are some short problems and investigations on patterns and sequences
Don Steward
Look at Don Steward's Number Sequences ideas.
Try the two following investigations: