We use concrete materials and manipulatives in the classroom in the lower primary years but often move away from them as students get older, giving students the impression that they are ‘babyish’ and that it is more ‘grown-up’ to manage without them. This is a misguided approach and manipulatives should be used all the way through school.

Students often experience difficulty with Algebra and the notation it employs. Algebra tiles are a manipulative that can help develop student’s understanding and confidence with algebra, at many different levels.

The tiles employ the area model of multiplication and this needs to be explored and understood if the students are to get the maximum understanding and benefit from their use. See the references at the end of the document for more detail on the area model.

This lesson sequence is not meant to replace a textbook but is an introduction to the use of algebra tiles. Use your usual textbook and classroom resources for exercises and practice.

The activity has students collecting their own information from a well-known Australia car buying website and using this to find an estimated relationship between the ‘asking price’ for a used car and the distance it has traveled in kilometres.

Students then examine the equation that expresses this linear relationship, using both graphical and algebraic information to help analyse and predict expected values.

The resource provides both a Teacher Guide and Student Workbook, as well as some background reading. there is also a Microsoft Excel spreadsheet with which to collect data, create a scatterplot graph and generate a linear relationship within the collected data.

Teachers are provided with a background introductory lesson and activity for the whole class, a guide to using and setting up the spreadsheet and a full set of sample worked solutions for the resource, with further enrichment links to material on linear relationships, bivariate data and regression analysis.

*This Unit was designed in collaboration with Ms Natalie Hammond from Bayside P-12 College, Victoria.*

Reading: Approximate-linear-relationships

Year-9-10-Linear-Task-Teacher-Guide

Year-9-10-Linear-Task-Student-Workbook

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Voronoi diagrams are mathematical constructs that provide useful applications in a variety of different disciplines. They were named after 19^{th} century Ukrainian mathematician Georgy Voronoi, however informal use of such diagrams can be traced back to Descartes as early as 1644.

In short, Voronoi diagrams, also known as tessellations, decompositions or partitions, are proximity diagrams that partition a plane into regions based on the distance to points in the specific subset of that plane. Each of these subsets or ‘cells’ are constructed with a given point or ‘seed’ in mind, to determine an area of all points closest to that seed, than any other in the plane. These regions, called Voronoi cells can be seen in Figure 1.

Voronoi Diagrams have been used in a multitude of areas such as environmental studies, cell biology, crystallography, transport planning, and communications theory. One of the first and best-known applications of Voronoi diagrams was by 19^{th} century British physician Dr. John Snow during a serious Cholera outbreak in 1854. During this time, Snow was busy gathering statistics on the number of victims and location of outbreaks. Snow had theorised that cholera was being spread by an agent in contaminated water and not through pollution or “bad air” as previously thought. He attempted to prove his theory by mapping the locations of individual water pumps and then generated cells which represented all the points on his map which were closest to each pump. Using this map, he was able to determine that almost all the fatalities were in houses supplied by a single pump on Broad Street, Soho. Once the handle of the pump was removed, death rates greatly diminished or ceased altogether. An interactive map of this outbreak can be viewed using the GeoGebra file at the link: **https://www.geogebra.org/m/deqcfxzp . **GeoGebra is a Dynamic Mathematical Software that has various useful applications in many areas of mathematics. GeoGebra Classic 5 can be downloaded directly onto any computer for free and can be accessed at the following site: **https://www.geogebra.org/download.**

The once popular television series ‘Numb3rs’ also discusses Voronoi maps and their uses in season 2 episode 10, called ‘Bones of Contention’. In the episode, the protagonists try to recreate a Native American settlement-pattern analysis in order to solve a case. They discuss how these diagrams have been used in mathematics and give an example of finding the closest cheeseburger restaurant from any given location.

Voronoi diagrams can be constructed by hand or using computer imaging software. To accurately construct a Voronoi diagram, a map called a *Delaunay Triangulation* must first be created. These triangulations can be constructed using either circumcircles or linear geometry. For the purposes of the investigations within this module, we will be using the later method, however, you may first need to walk your students through some basic knowledge of coordinate geometry and straight lines. This information can be accessed in the ‘Teacher Notes’ section of this module.

**Bob Anderssen** – January 17, 2018

“We think that many students …. have difficulty in following abstract arguments, not on account of *incapacity*, but because they *need to see the point* before their interest can be aroused.”

Jeffreys and Jeffreys

Preface to “Methods of Mathematical Physics”

This insightful comment by Jeffreys and Jeffreys, mathematicians at Cambridge University, UK, over many decades, applies equally well to all levels of mathematics and, in particular, to activating the mathematical circuitry in the brains of children, to stimulate their awareness of and confidence with numbers. The challenge, as highlighted in the quotation, is the identification of ways to assist children to see the point.

In this post it is argued that this can be achieved by asking children to perform arithmetic tasks which highlight the patterns in numbers and the role that numbers play in the world around them. I am a grandfather, and I have had hours of fun with my grandchildren counting and playing with mathematical puzzles whilst driving in the car on the way home from school.

The basis for these tasks comes from simple things around the home that parents and grandparents can easily and repeatedly do. This relates to the saying: *“Simple things need to be said over and over again before they are fully understood.” *

I was also inspired by an article in AMSI’s The Update by Sally Saviane (*https://amsi.org.au/wp-content/uploads/2017/06/the_update_5th-ed_web.pdf)* entitled ** FUTURE PROOFING OUR KIDS: Why parents are important in conquering maths anxiety.** Interestingly, Sally argues that “the anxiety arises as something that is a reflection of the attitude of the parent rather something that is the essence of a child.”

It might be said that grandparents potentially have both opportunity and sound reasons to assist. Along with the fact that parent have jobs and family responsibilities to manage, it is well known that, though children listen to their parents, it is hearing the same from an independent source that brings acknowledgment (sometimes grudgingly), recognition and engagement.

Engaging students early and often with mathematical thinking has an impact on later mathematical development. Recent research has established that mathematical facility in later life correlates with the use of finger counting to solve addition and subtraction problems in childhood [2]. It is even being concluded that the arithmetic system in the brain is a type of finger counting process [1, 3]. Interestingly, academic success has also been correlated with the number of books in the family home.

There is a universality of the importance of the simpler underpinning the greater tasks in mathematical endeavours. Even in deep mathematics, the whole argument often turns on a simple step made at a crucial point.

**Mathematical awareness and confidence comes through familiarity with the simple patterns hidden in numbers**

Mathematics is patterns. Numbers are the starting point. The strategy is that the patterns are observed and discussed over and over again before they are fully understood. One of the great mistakes is the view that understanding is achieved through confronting children with difficult problems. In that case, the problem ownership is transferred from the child to the proposer. We want to encourage the small child to propose and investigate their own problems, by starting off simply.

Here are the things that I have been doing with the grandsons (now age 10 and 12) after I pick them up from school:

- Count to 100 in 10s. It gives them a buzz. The challenge is to have them recognize (conceptualize) that they are still only counting 1, 2, 3, 4, 5, …..
- Count to 100 in 5s. The pattern now to conceptualize is the repeating pattern 0, 5, 0, 5, 0, …. .
- Count to 50 in 2s. The repeating pattern now is 0, 2, 4, 6, 8, 0, …..
- Count to exactly 51 in 2s starting at 1. The repeating pattern becomes 1, 3, 5, 7, 9, 1,
- Count in 3s, arriving at exactly 60. The pattern is not 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, ….. Initially, it is a little challenging.
- For the more advanced children: Count arriving at exactly 70 in 7s.
- Discuss the prime number sieve. Here is a link: https://en.wikipedia.org/wiki/Sieve_of_Eratosthenes
- The special properties of the number 1089 as explained in the following link: https://en.wikipedia.org/wiki/1089
- Count the petals on flowers to see the Fibonacci numbers 2, 3, 5, 8, 13, … .
- Odd and even numbers Count in steps to two, steps of 3, etc from arbitrary starting points.
- Number patterns support learning about multiplication. It is a challenge for primary school teachers with respect to some students to help them to know and understand multiplication tables by the time they get to high school. Yes, multiplication tables are still very important.

*Bob Anderssen is a mathematician who works at CSIRO. Bob is the Chair of the AMSI Education Advisory Committee and a proud grandfather.*

- S. Anderssen, CSIRO Data61, GPO Box 664, Canberra, ACT 2601

**REFERENCES**

[1] V. Crollen and M.-P. Noël. The role of fingers in the development of counting and arithmetic skills. *Acta psychologica*, 156:37-44, 2015.

[2] K. Moeller, L. Martignon, S. Wessolowski, J. Engel, and H.-C. Nuerk. Effects of finger counting on numerical development-the opposing views of neurocognition and mathematics education. *Frontiers in psychology*, 2:1-5, 2011.

[3] N. Tschentscher, O. Hauk, M. H. Fischer, and F. Pulvermüller. You can count on the motor cortex: finger counting habits modulate motor cortex activation evoked by numbers. Neuroimage, 59:3139-3148, 2012.

*Please note: This lesson could easily take more than one session to investigate fully*

- To investigate measurement (and pattern) in a problem solving context

**https://www.australiancurriculum.edu.au/f-10-curriculum/mathematics/structure/**

*Content will vary according to the age of the students and the investigations they choose. *

- Number and algebra are developed together, as each enriches the study of the other. Students apply number sense and strategies for counting and representing numbers. They explore the magnitude and properties of numbers. They apply a range of strategies for computation and understand the connections between operations. They recognise patterns and understand the concepts of variable and function. They build on their understanding of the number system to describe relationships and formulate generalisations.
**They recognise equivalence and solve equations and inequalities. They apply their number and algebra skills to conduct investigations, solve problems and communicate their reasoning.**

- Measurement and geometry are presented together to emphasise their relationship to each other, enhancing their practical relevance. Students develop an increasingly sophisticated
**understanding of size, shape, relative position and movement of two-dimensional figures in the plane and three-dimensional objects in space.**They investigate properties and apply their understanding of them to define, compare and construct figures and objects. They learn to develop geometric arguments.**They make meaningful measurements of quantities, choosing appropriate metric units of measurement.**They build an understanding of the connections between units and calculate derived measures such as area, speed and density.

- Statistics and probability initially develop in parallel and the curriculum then progressively builds the links between them. Students recognise and analyse data and draw inferences. They represent, summarise and interpret data and
**undertake purposeful investigations involving the collection and interpretation of data**. They assess likelihood and assign probabilities using experimental and theoretical approaches. They develop an increasingly sophisticated ability to critically evaluate chance and data concepts and make reasoned judgements and decisions, as well as building skills to critically evaluate statistical information and develop intuitions about data.

- What is the length of ribbon needed to make a bow?
- Number of loops needed to make a bow
- Pattern of loops in the circles (1, 5, 9??)
- How do you make a 3D bow from a flat ribbon?
- Could I make a bow with 1 m of ribbon?
- How big is the bow?
- How many bows would fit on my maths book?
- What is the shortest/longest length of ribbon to make a bow with 15 loops?

**Second picture of multiple ribbons:**

- Which one doesn’t belong?
- What patterns could I make?
- Which bow needs the longest ribbon?
- How do you make the points?
- What angle are the points?

*For more information, please download the attached lesson plan*.

- To investigate odd and even numbers

- Represent and solve simple addition and subtraction problems using a range of strategies including counting on, partitioning and rearranging parts (ACMNA015)

- Solve simple addition and subtraction problems using a range of efficient mental and written strategies (ACMNA030)

- Investigate the conditions required for a number to be odd or even and identify odd and even numbers (ACMNA051)

- What does it mean if a number is odd or even?
- How can you work out if a number is odd or even?
- What did you learn today about odd and even numbers?
- Can you think of a “rule” to decide if a number is odd or even?

*For more information, please download the attached lesson plan*.

Use bead strings to model numbers in different ways

Students ‘trust’ the count and can explain their thinking, i.e. they do not need to count all the beads to represent the number

- Use beads of equal size, such as Pony Beads
- For the cord use a bead string or a shoe lace for easy threading
- Choose two colours (bead strings of many different colours can confuse children)
- Place groups of 5 or 10 beads together before changing colours – this helps children to count the beads
- Start with at least 30 beads on the string
- Tie a loop in the end for easy storage or display

To use beads to model numbers

- Each student holds their bead string in front of them
- Teacher names a number, for example, “Show me 7”
- Students use their bead strings to model the number
- Teacher asks students to explain how they know they are showing that number
- Student replies, “I know 5 and 2 makes 7”
- Teacher prompts students to show the number another way

- Students stand in a circle with their bead string
- Teacher begins with a statement like, “I have 10 beads”
- All students show 10 beads on their string
- Teacher now says, “I wish I had 9” – all students change their bead stings to show the new number
- The next student in the circle continues the game “I have 9 beads, I wish I had 12”
- All students model the new number
- This process of identifying and changing the number of beads continues
- A discussion about the various methods students use to make and move between the different numbers can take place during and after the activity

*More detailed instructions and activity booklet about this game are available in the PDF.*

The first activity is used to see if students have any misconceptions regarding the use of common chance terms. The second task is encouraging students to begin to record all the possible outcomes of a chance experiment and use this information to begin to predict the likelihood of different outcomes.

- Order chance terms and events according to their likelihood
- Match familiar chance events to their likelihood
- List all the possible outcomes from a simple chance experiment
- Predict the likelihood of an experiment based on the possible outcomes
- Use chance terms to describe the likelihood of simple chance experiments
- Explain the variation in results (i.e. compare predicted to observed results)

- Conduct chance experiments, identify and describe possible outcomes and recognise variation in results (VCMSP147)

- Describe possible everyday events and order their chances of occurring (VCMSP175)

- List outcomes of chance experiments involving equally likely outcomes and represent probabilities of those outcomes using fractions (VCMSP203)
- Recognise that probabilities range from 0 to 1 (VCMSP204)

- Can you order the chance terms in order of likelihood?
- Can you match the events to the appropriate chance term?
- Can you think of any other events that match each term?
- Can you assign a probability to each event (i.e. a number value)?
- Which horse do you believe is the most likely to win? Why?
- What outcomes can you get from rolling 2-dice?
- How can we list all the outcomes in a systematic way?
- Can we use numbers to describe these outcomes?
- Which number is most likely to be the total? Why?
- Why does the mathematical (or theoretical) chance not always match the results?

*For more information, please download the attached lesson plan*.

The focus of this activity is to discover what students know shapes, including their features and properties. What language are students using to describe and sort shapes? How can we as teachers help students increase their shape vocabulary?

- Use materials to create 2D shapes
- Identify 2D shapes and 3D objects
- Use everyday language to describe features of shapes
- Sort 2D shapes and 3D objects
- Justify reason for classifying different shapes

Sort, describe and name familiar two-dimensional shapes in the environment (ACMMG009)

- identify, represent and name circles, triangles, squares and rectangles presented in different orientations, e.g.
- identify circles, triangles, squares and rectangles in pictures and the environment, including in Aboriginal art (Problem Solving)
- ask and respond to questions that help identify a particular shape (Communicating, Problem Solving)
- sort two-dimensional shapes according to features such as size and shape
- recognise and explain how a group of two-dimensional shapes has been sorted (Communicating, Reasoning)
- manipulate circles, triangles, squares and rectangles, and describe their features using everyday language, e.g. ‘A square has four sides’
- turn two-dimensional shapes to fit into or match a given space (Problem Solving)
- make representations of two-dimensional shapes using a variety of materials, including paint, paper, body movements and computer drawing tools
- make pictures and designs using a selection of shapes, e.g. make a house from a square and a triangle (Communicating)
- draw a two-dimensional shape by tracing around one face of a three-dimensional object
- identify and draw straight and curved lines
- compare and describe closed shapes and open lines
- draw closed two-dimensional shapes without tracing
- recognise and explain the importance of closing the shape when drawing a shape (Communicating, Reasoning)

Sort, describe and name familiar three-dimensional objects in the environment (ACMMG009)

- describe the features of familiar three-dimensional objects, such as local landmarks including Aboriginal landmarks, using everyday language, e.g. flat, round, curved
- describe the difference between three-dimensional objects and two-dimensional shapes using everyday language (Communicating)
- sort three-dimensional objects and explain the attributes used to sort them, e.g. colour, size, shape, function
- recognise how a group of objects has been sorted, e.g. ‘These objects are all pointy’ (Communicating, Reasoning)
- recognise and use informal names for three-dimensional objects, e.g. box, ball
- manipulate and describe a variety of objects found in the environment
- manipulate and describe an object hidden from view using everyday language, e.g. describe an object hidden in a ‘mystery bag’ (Communicating)
- predict and describe the movement of objects, e.g. ‘This will roll because it is round’
- use a plank or board to determine which objects roll and which objects slide (Problem Solving)
- make models using a variety of three-dimensional objects and describe the models, e.g. ‘I made a model of a person using a ball and some blocks’
- predict the building and stacking capabilities of various three-dimensional objects (Reasoning)

- What shape have you made?
- How do you know?
- Can you make any other shapes?
- What shapes have you chosen?
- Which shape doesn’t belong?
- Is there another way to sort the shapes?
- Can we use something other than colour (size, material, features, properties, etc.)
- What are some of the features of shapes?
- What are the different properties of shapes?
- Does turning the shape change it?

*For more information, please download the attached lesson plan*.

The focus of this activity is to discover what students know about coins and money. Some students will be familiar with coins and will have no trouble sharing their knowledge. Other students will have had limited exposure to money and what it looks like and how it is used.

- Identify the Australian coins
- Recognise and describe features of the Australian coins
- Identify situations that involve money
- Understand that different coins represent different amounts
- Use the coins to purchase goods in a play shop

- Represent simple, everyday financial situations involving money (VCMNA075)

- Recognise, describe and order Australian coins according to their value (VCMNA092)

- What are the names of the Australian coins?
- What other features do you notice about the coins?
- Who is the person on the coins?
- What animals are on the coins?
- Why do some coins have different images?
- What is the difference between the gold and silver coins?
- Can you combine coins to make different amounts?
- Can you use the coins to buy something in the play shop?
- How do you know if you have the right amount?

*For more information, please download the attached lesson plan*.