Matilda's Waltz with the 5 Strip Box

Purpose

This unit allows the class to work on two different pattern themes. It lays the foundations for Algebra at Level 5 by requiring them to articulate the patterns.

Specific Learning Outcomes
  • to develop, justify and use rules to solve problems that involve number strips
  • to identify and clearly articulate patterns, and make generalisations based on these
Description of Mathematics

This unit focuses on students detecting and articulating patterns, and describing their rules. It involves the students in making sense of each other’s rules, expressing these in coherent and reasoned ways, using discussion, writing, drawing, and demonstrations, where necessary. It also involves them using their ‘refined’ sense of the situation provided, to construct new rules or generalisations. At this stage there is not an emphasis on algebra as symbols to be manipulated, but rather a clear focus on the development of algebraic thinking. An important aspect of algebraic thinking is engaging students in articulating and making sense of relational aspects that contribute to pattern formation and ultimately, a generalised rule. It is a small step from here to writing algebraic expressions for patterns and other relationships.

Links to Numeracy

This unit provides an opportunity to focus on the strategies students are using to solve number problems and develop student knowledge in the area of factors, multiples and divisibility rules.

When students are identifying and describing rules for Matilda’s number strips encourage them to describe the strategies they are using to predict and calculate numbers in the pattern. Continue the pattern with much larger numbers to enhance strategy use.

Which number comes next in this pattern?
How did you work that out?

Where patterns involve multiples of numbers this can be used to develop knowledge of basic facts, in particular factors, multiples and divisibility rules.

What is the biggest number you can think of to fit this pattern?
How do you know it will fit?
Will the number 156 be in this sequence?(or other number as appropriate)
Why / Why not?
How could you check which of these numbers fits the pattern?
What is the same about all the numbers in this sequence?
Can you see an easy way to check whether numbers will be in this sequence?

Required Resource Materials
hundred's board.

Session 3 copymasters

Session 2 copymasters

Session 1 copymasters

Key Vocabulary

 sequential patterns, generalisations, divisibility rules, factors, multiples, rules

Activity

Session 1

In this first session, the students investigate number sequences (patterns) using Matilda’s number strips and her special rule.

  1. Tell the class that Matilda has made a line of 5 boxes and placed numbers in the first 2 squares. She then filled her number strip using a rule that she has made up.
    Have a look at some of Matilda’s number strips.
    Can you find the rule she is using?
    We’ll call this Matilda’s Rule.

    2

    5

    7

    12

    19

     

    0

    1

    1

    2

    3

     

    4

    1

    5

    6

    11

  2. Here are some incomplete number strips that Matilda has partially filled using her rule.
    Can you fill in the missing squares?

    0

    5

     

    10

     
     

    3

     

    12

       
     

    1

       

    5

     

Session 2

  1. Matilda had in her bag a sheet of paper that had some number strips she had already made using her Rule. Unfortunately, ink from her felt pens had spilt over some of the numbers in her strips.
    Can you find the missing numbers?

    1. 3

         

      13

      21


    2. 2

           

      16


    3. 2

           

      76


    4. 1

           

      26


    5. 1

           

      11


    6. 21

         

      5

      18

 

  1. What are the missing numbers for number strip (a)?
  2. Matilda decided to use a bit of number sense to complete some of the other number strips. She reasoned that for number strips (b), (d) and (e), the 2nd number couldn’t be too big, and so it would be worth guessing a ‘smallish’ number – some number that was fairly close to the number in the 1st square. However, for number strip (c) she reasoned that the number in the 2nd square would probably be much bigger than that in the 1st square. She thought that strip (f) might need some negative numbers somewhere.
    Explain Matilda’s reasoning. Make up some number strip examples of your own to illustrate her reasoning.
  3. Using Matilda’s ideas to help you, find the missing numbers for each of the number strips (b) to (e).
  4. Matilda wondered if there is a method that she could use to find the missing numbers in any number strip that used her Rule, where only the 1st and the last squares had numbers in them.
    Investigate the following number strips and see if you can find a method that will help Matilda.

    1

         

    8

    4

         

    5

    2

         

    7

    10

         

    20

    3

         

    27

    2

         

    28

Session 3

  1. Matilda thinks that she can analyse her number strips best if she records them like this:

    1

    5

    1 + 5

    5 + (1 + 5)

    (1 + 5) + (5 + 1 + 5)

     

    3

    2

    3 + 2

    2 + (3 + 2)

    (3 + 2) + (2 + 3 + 2)

  2. She tidies up her recording by now writing

    1

    5

    1 + 5

    5 + (1 + 5)

    2 lots of 1 + 3 lots of 5

     

    3

    2

    3 + 2

    3 + 2 + 2

    2 lots of 3 + 3 lots of 2

     
  3. Use Matilda’s method to analyse 3 different number strips that you have looked at so far.
  4. How can the first and last square be combined to make a relationship that is the same for all of the number strips you have analysed using Matilda’s method?
  5. Select any one of the number strips you have analysed. Look for any link between the relationship you have just found and any other square(s) in the strip.
    Is this new relationship the same for the other number strips you have analysed?
  6. Now, write a method that will allow you to solve the following

    1. 3

         

      13

      21


    2. 2

           

      16


    3. 2

           

      78


    4. 1

           

      26


    5. 1

           

      11


    6. 21

         

      5

      18

  7. Here are some empty number strips. Put numbers in the 1st and last squares so that each number strip can be solved using Matilda’s Rule. Notice that you can’t just put any numbers in the first and last squares and hope that Matilda’s Rule will work.
             

     

             

     

             

     

             

     

             

     

             

Session 4

  1. Explore the following number strips taken from a hundred’s board.
  2. Write down a relationship between the 3 numbers in the various vertical strips. (Add on 10 each time.)
    Does your relationship work for any other vertical strip of 3?
    Check by selecting another vertical number strip of 3.

diagram.

  1. What if you chose a horizontal strip of 3? Are the same relationships evident?
    What if you were to a diagonal strip of 3?
    Check these by selecting appropriately number strips of 3 from the hundred’s board.

hundreds board.

  1. How would you write your relationship as a rule about any strip of 3 numbers chosen from a hundred’s board.
  2. Shown below is a selection of 9 numbers that form a square shape called a 9-square. Using ideas developed in the previous task, write down some interesting relationships that link the various numbers together. (There are a lot of possibilities here. Get the class to discuss what they have found.)
    Hundreds board.
  3. Check that the relationships you have found work for:

hundreds board.

Session 5

  1. In the 9-square grid below there is a relationship between the numbers in the outer squares and the number in the central square. (The average of the 4 outer square equals the central square.) Using this relationship what are some possible 9-squares that can be made if the middle number is 36?

     

     

     

     

    36

     

     

     

     

  2. Look carefully at the outer numbers. There seems to be some symmetry between the difference of 2 neighbouring numbers and other pairs on the ‘outside’.
    Record and illustrate some of your findings.
    Are the same patterns evident in other 9-squares?
    Try for example a 9-square with 50 in the middle square.

     

     

     

     

    50

     

     

     

     

  3. Investigate other forms of number strips, for example, the 2 x 3 number strips below:
    hundreds board.
  4. Can you find a quick way of finding the sum of all of the numbers that belong to each 2 x 3 number strip?
    Explain/demonstrate/illustrate your rule for adding.
  5. What similar patterns can you find in a 2 x 4 number strip?
    Explain/demonstrate/illustrate your rule for adding.
  6. At this stage either extend the number strips horizontally and/or vertically and look for patterns.
  7. Discuss the results with the class.
    Get them to each to write down one or two of the patterns that interested them most.
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