CSER Number - Content In Action

Resources

Bar model with ratios
Pythagoras Theorem to calculate distances - a nice little system
Prime factorisation
Rational triangles
Standard index form matching

Introduction#

Numbers - mental constructs dealing with: correspondence; estimation and magnitude; and order.
Useful for: counting; measuring, and quantifying situations/attributes of objects.
Different number systems for different contexts: finite and infinite; discrete and continuous

Video

counting is important;
numbers surround us;
various applications of how important number is to careers and everyday life;

Gets into concepts

digits
counting
number systems
Body tallying in Aboriginal cultures
number sense

Then presents the year level descriptions for secondary. Let's focus on year 8 as a summary

Integers and positive rational numbers
Extend understanding of computation with 4 operations
- apply efficiently using patterns for establishing rules for multiplication and division
- apply commutative and associative laws for regrouping, partitioning, place value, patterning, multiplication or division
extend exponent laws to numerical calculations
convert between fraction/decimal forms - locate on a number line
Recognise irrational numbers: $sqrt(2)$, $pi$ and terminating and recurring decimals (e.g. $1/3$)
establish and apply the exponent laws and notation with positive integers and non-zero powers
use mathematical modelling to solve practical problems involving rational numbers and percentages
- financial contexts;
- formulate problems
- interpret and communicate solutions
- review appropriateness

Every day mathematics#

Number Guide#

Learning about number includes

digits and numerals;
different number systems, structures, processes

Fundamentals#

Number types#

Types	Description
Natural numbers	The set = {0,1,2,3 ...} or = {1,2,3 ...} depending on whether counting is started at 0 or 1. The elements of $N$ are also called the counting numbers, used to count the number of elements in finite sets.
Whole numbers	The natural numbers including zero, for example, 0, 34 or 1,789.
integer	An element of the infinite set of numbers = {...-3,-2,-1,0,1,2,3 ...}.
rational number	Can be expressed as a fraction of two integers where the denominator cannot be 0. It is an element of the infinite set of numbers $Q = \frac{m}{n}$ where $m$ and $n$ are integers and $n \neq 0$. It may be expressed in decimal form, e.g. $\frac{1}{8} = 0.125$ and $\frac{4}{9} = 0.444...$. Note: the recurring of the 4 indicates a pattern, that's why this is a rational number.
Irrational numbers	Subset of real numbers that have an infinite, non-repeating string of decimals. The non-repeating means that the fraction has to be non-repeating. For a non-repeating recurrence there are mathematical tricks to find a fraction that gives that answer exactly
Real numbers	Collective name for all the rational and irrational numbers

Exponential notation#

Concept	Definition
Exponential or index numbers	A number written in the form $a^n$ where $a$ is the base and $n$ is the index or exponent.
Scientific notation	A number written in the form $a \times 10^n$ where $1 \leq a < 10$ and $n$ is an integer.

Place value and scientific notation#

The decimal system is a positional system using base 10. Large and small numbers can be represented using scientific notation.

Name	Number	Scientific notation
Thousandths	0.001	1 x 10^-3
hundreds thousands	300 000	3 x 10⁵

Recurring and terminating decimals#

Concept	Definition
Recurring decimal	A decimal number with a pattern of repeating digits e.g. $\frac{1}{3} = 0.3333.... = 0.\dot{3}$ or $\frac{1}{7} = 0.142857142857... = 0.\overline{142857}$
Terminating decimal	A decimal number with a finite number of digits e.g. $\frac{1}{2} = 0.5$ or $\frac{1}{4} = 0.25$

Rounding#

The rounding rule is to round up if the digit to be rounded is 5 or more, and round down if the digit to be rounded is 4 or less.

Operating with numbers#

The order of operations typically given using BODMAS or BEDMAS (brackets, orders/exponents, division, multiplication, addition, subtraction). For example

\[\dfrac{2 + (2+2)^2}{2} - 2 \times 2 = 5\]

Additive thinking#

Ability to think and solve problems using repetition (additive) and cumulative strategies. Representing this through models and annotations.

Properties of addition#

Property	Definition
Identity	The sum of any number and zero is the number itself.
Commutative	The order of the numbers does not affect the result.
Associative	The grouping of the numbers does not affect the result.
Inverse	The sum of a number and its negative is zero.

Adding and subtracting integers#

Provides two models for exploring the properties of positive and negative numbers

Model	Description
Counter model	Two types of counters (positive - blue; negative - red) are used to model addition and subtraction of integers.
Air balloon model	Puffs of air into/out of a ballon and descent/ascent

Adding and subtracting fractions#

only possible if the denominators are the same
where different, the fractions need to be converted to equivalent fractions with the same denominator

Finding equivalent fractions can be aided by common diagrams/models

Diagram	Description
Area model	A rectangle is divided into equal parts. The fraction is represented by the number of parts shaded.
fraction walls
fraction strips	aka tape diagrams
line numbers

Multiplicative thinking#

As developing number sense, using a range of additive strategies, build capacity to think multiplicatively

Properties of multiplication#

Property	Definition
Identity	The product of any number and one is the number itself.
Commutative	The order of the numbers does not affect the result.
Associative	The grouping of the numbers does not affect the result.
distributive	For all real numbers $a$, $b$ and $c$, $a \times (b + c) = a \times b + a \times c$
Inverse	The product of a number and its reciprocal is one. e.g. $2 \times \frac{1}{2} = 1$

Multiplying and dividing integers#

two positive integers results in a positive integer
two negative integers results in a positive integer
one positive and one negative integer results in a negative integer

Multiplying and dividing fractions#

Multiplying proper fractions results in a smaller answer
Reading the multiplication of 2 fractions as "of" can help, but also has some issues
Dividing fractions results in a larger answer

Financial application of percentages

Calculating percentage of quantities done using the rule $ amount = percentage \times quantity $
A fraction can be changed into a percentage $p% = \frac{p}{100}$
interest calculation (uses simple interest, doesn't mention complex)

Proportional comparison

Use of percentages to compare different quantities

Proportional reasoning#

A complex process - it is a process of thinking about relationships in multiplicative terms, and identifying and describing what is being compared
Aided by developing an understanding that quantities have the same relative size or same ratio

Term	Definition
Ratios	Variables that have the same unit of measure
Rate	A ratio between two measurements that have different units (e.g. kmh)

Example - mixing paint#

Given ratios mixing paint to achieve a shade, the ratio is 3 parts white paint for every 2 parts blue pain. If I have 6 litres of blue paint. How much white paint do I need?"

I have 40 litres of pain. How much white and blue paint would have been used if the ratio was followed?"

Proportional reasoning strategies#

e.g.

comparision involving rates - which is faster 100km/hr or 6 m/min?
missing value problems: A box of fruit has 3 apples for every 4 oranges. How many apples are needed if there are 36 oranges?

These can be difficult - then language. Supporting strategies

using materials/diagrams to model the situation
Using newmans-error-analysis to find where students struggle.

Misunderstandings#

aka mathematical-misconceptions arising from

a learner misinterpreting the original intent/idea
used a "helpful" rule without understanding the implications of using in a new context

Common misconceptions in Number

multiplication
Multiplication always results in a larger number
division
Division is commutative like multiplication
You can divide smaller numbers into larger numbers regardless of order e.g. $5 \div 10 = 2$
?? what about division resulting in smaller numbers ??
exponentials
exponents are the same as multiplying
a negative exponent implies the number is negative
fractions
Confusion about place value e.g. 7 hundredths is $0.700$ and not $0.07$
The decimal point separates two whole numbers e.g. $15.67$ should not be read as fifteen point 67.
Not recognising the size of a decimal number e.g. thinking $1.25 > 1.3$
Thinking that multiplying by 10 always adds a zero
thinking that numerator and denominator are whole numbers and thus thinking they can add/subject each part individually
confusion over relationship between fractions and decimals e.g. $\frac{1}{10} = 0.1$ so $\frac{1}{8} = 0.08$
thinking that mixed numbers will always be bigger e.g. $1 \frac{1}{2} > \frac{15}{5}$

Digital technology tools#

Desmos - scientific calculator
Partial product finder - enter a simple multiplication which is represented as a rectangle and then can be interacted with to reveal partial products - ways to calculate partial products
Geogebra - mentioned as having many tools - e.g. ratios
TIMES modules from AMSI

Challenges#

i.e. puzzles, problems, interesting ideas in mathematics etc. Seen as effective ways to engage students and check understanding across strands.

Fermi problems - estimation#

aka order-of-magntiude problem. Challenging to answer exactly. Aim is to break down the problem into smaller steps. For example

How many people in the world are on their mobile phones right now?
How many jelly beans fit a bucket?
Could you fit $1M worth of $1 coins in your classroom?

No right or wrong way.

Pose questions#

What is your initial thought about the problem? Will the answer be a large number or a small number?
How could you begin to work on this problem? What do you know, or what could you find out?
Can you think of a way to simplify or downsize the problem to be something you can manage to get an idea about the problem?

Flag fractions#

Use country flags that echo area models that can be used to calculate fractions.

e.g. French flag shows thirds. Columbian, Thai.

Pose question#

What could a flag that is half green and half gold look like?
How many different flags could we make as a class? Could everyone design a half-and-half flag that is different (not different colours but shows the halves in different ways)

Scaffolding knowledge#

Individual example lessons using the CRA model

Ratio and proportional reasoning#

Cooking ratios and similar real world examples of ratios seem a better approach than most of the examples used below. Later give examples of Pharmacists, veterinarians, manufacturers, builders.

Concrete#

Basically get students to use counters etc to engage tarts with the problem of a tennis camps. 24 students. 3 : 5 ratio of beginners to advanced. Also give other examples:

60 students 3:9.
4 students rode bikes, 10 walked - what is the ratio? Can that ratio be simplified.
A gold alloy is 2 parts of gold to 5 parts of silver by mass. What fraction of the allow is gold?
$\frac{1}{3}$ of the class is absent. There are 18 students present. How many students are in the whole class?

Ratio is 1:2 - $frac{1}{3}$ are absent and $\frac{2}{3}$ are present. 18 is $\frac{2}{3}$ of the class. So the whole class is 27. But also using the relationship between ratios and fractions the number absent is $\frac{1}{2} \times 18 = 9 $.

The connection here is that the total of the ratio is a factor of the total (3+5=8, 8*3=24). Allowing representation with counters to be done very easily.

A simple rule that leads to a misconception - perhaps not?

I feel that this simple rule - add the ratios together neatly might lead to misconceptions? But then a ratio always has to be a factor of the total? Otherwise the ratio doesn't work.

Relationship between ratios and fractions

Intro material mentions that ratio can be expressed as fractions and later discussion mentions similar proportional ratios e.g. 3:5 and 6:10. I'm stuck thinking how I'd explain the relationship between ratio and fractions. This resource offers an explanation

Ratio indicates either

the number of times contains the other amount, or
is contained by the other amount.

e.g. a group of squares and triangles in a ratio of 2:3 squares to triangles. Can also be described that there a $\frac{2}{3}$ as many squares as triangles, or $\frac{3}{2}$ as many triangles as squares.

24 students with 3:5 ratio means 9 are beginners and 15 are advanced. $\frac{3}{5} \times 15 = 9$ and $\frac{5}{3} \times 9 = 15$

The idea is to give students multiple examples using manipulatives.

Pose questions

What are ratios?
Why is the order of the numbers in a ratio important?
How can you find equivalent ratios? How do you know you are correct?
How can you use the counters to explain equivalent ratios?
How do you simplify a ratio?
What processes are you using when finding equivalent ratios?

Representational#

Moving from physical materials to diagrams, photos etc. e.g. the bar model. Both to continue the exploration (somewhat) but mainly to start using their growing understanding of ratio to create representations

Pose questions

What are the advantages of using the bar model over individual items such as counters?
What other shapes could be used to represent ratios or proportions?
How could a single picture, such as a sports drink bottle, be used to represent a ratio such as

Check for understanding

Can students create bar graphs using single row grids?
Can students create a graph from the information given?
Can students answer ratio questions from graphs?

Abstract#

During this stage students

use notation for representing ratios, simplifying, and multiplying
- simplification done by dividing by HCF
writing a ratio when component parts have been given in different units
finding the component parts when provided with the total and one part of the ratio
sharing out a given total into 3 or more parts
solving ratio problems with real-world contexts.

Examples given

What is the simplest ratio for the measurements 1.2 m and 60 cm?
Concrete is made by mixing gravel, sand and cement in the ratio by volume. How much sand will be needed to make of concrete?
A teenager does some gardening jobs for pocket money. Mowing lawns is $30 an hour, weeding $15 an hour, trimming $20 an hour. She has decided to save 75% of her weekly earnings and put aside 25% for board and bills. In a week 5 hours are spent on mowing, 8 hours on weeding and 4 hours on trimming. How much money does she save? How much does she set aside for board and bills?

Pose questions

How does ratio show the multiplicative relationship between quantities?
How are equivalent ratios obtained? Can you use mathematical notation?
How are measurements increased or decreased?
How are ratios used in our daily lives?

Check for understanding

an students understand that equivalent ratios are similar to equivalent fractions?
Can students calculate the ratio of two or more similar quantities?
Can students reduce the ratio to its simplest form?
Can students describe ratios as a comparison between quantities?
Can students increase or decrease quantities while keeping the ratio constant?

Discovering irrational numbers#

A lesson sequence linked to rational/irrational numbers; circles; right-angled triangles. Typically because students first encounter with irrational numbers is $pi$. Though the common origin story is $sqrt(2)$ and the Pythagorean theorem (see for more). And AC9M8N01 explicit mentions $sqrt(2)$ and $pi$.

Personally, I can see a sequence starting with revision of rational numbers (and number sets in general) and Pythagoras and their love of rational numbers. Then leading to some CRA around that, perhaps bringing in $pi$ at some stage. Though $pi$ being a ratio of circumference to diameter that cannot be expressed as a rational number is a related way in.

Where do students see that $pi$ can't be represented by a fraction?

In the following, it's not clear to me where students get the "aha" moment that $pi$ can't be represented by a fraction. Which seems to be the core intent. How would that happen with $sqrt(2)$?

source
full precision calculator - plugging $\frac{22}{7}$ and $pi$ into this will demonstrate

Concrete#

Physical objects are provided. Students choose 5 and use string to measure and record the circumeference and diameter.

Pose questions

For the given objects, which measurement is always larger -- the circumference or diameter?
Approximately how many times larger is the circumference than the diameter? How can you work this out?
Does this relationship hold for really large or really small circles? How could you test this?
How could you describe this relationship in words?

Check for understanding

Can students explain the relationship between the circumference and the diameter?
Do they identify that the ratio of the circumference to the diameter is constant?
Do they recognise this constant as $pi$?
Ask students to explain their thinking. Do they identify the constant ratio of circumference to diameter?

Representational#

Establish use of $pi$ various disciplines. Time to explore the relationship a bit more.

Students produce a scatter plot of the values from the concrete activity. It should be a straight line.

Pose questions

What type of pattern does the data appear to suggest?
What does the graph tell us about the ratio between the circumference and the diameter of a circle?
How could we use this graph to find the circumference of a circle if we only knew its diameter?

From there other activities can be launched to leverage their representational work. e.g. The world's largest rice cake was 3.7m in diameter.

Reflection questions

Were students able to explain the method used to calculate the circumference?
How might students calculate the diameter given only the circumference? Did they make the connection between the circumference and diameter?
Have students test their method by calculating the circumference of other circles using only the diameter. Were they accurate? Ask them to present their findings as fractions. What do they notice?

Abstract#

Explicitly positions $pi$ as the entry point for year 8s and $sqrt(2)$ for year 9. In year 9, this opens up further exploration using geometric constructions. Leading into explorations e.g spiral of Theodorus

For year 8s, the suggestion seems to stop with

Students in Year 8 explore ways to transpose the equation to be able to use this for questions that ask them to find the circumference. For example, to find the circumference of a circle they could use or . They could even explain how the two rules are equivalent.

Reducing your carbon footprint - multiplicative nature of percentage change#

Based on carbon footprint calculators (e.g. Carbon Positive Australia)

Messaging about carbon footprints suggests achieving percentage reductions

Linked to AC9M8N05

Concrete#

Explore the effect of 20% reduction each year for 2 years. Use base 10 blocks for physical work. Link percentages to parts per 100.

Students will

reason about changes 20% each year
identify the maths required
communicate how their solution works
explain how 20% reduction over two years is less than 40% reduction in one year

Pose questions

How did the carbon footprint reduce over the years?
How do you explain this mathematically?
Does this explanation match the blocks?

Check for understanding

How do students calculate a percentage decrease from one quantity to another?
Can students use the four operations and strategies flexibly?

Representational#

Getting students to draw diagrams exploring year-on-year reduction (e.g. 30% over 2 years)

Pose questions

Can you predict what reduction it would be after 5 years?
What did the data show you?
Is that what you expected?

Check for understanding

How are students using the area model to make sense of calculations?
How do students apply their knowledge of percentages?

Abstract#

Using symbols to show the cumulative nature of percentage change.

Start with same questions but with symbols

$80% of original = 80 \div 100 \times original = 0.8 \times original$

The idea is to get them to see that the percentage change is multiplicative.

$(80 \div 100) \times (80 \div 100) = 0.8 \times 0.8 = 0.64$

Which mathematically becomes $Percentage reduction = (1 - %decrease \div 100)^n$

e.g. - 20% reduction over 2 years = $(1 - 20 \div 100)^2 = (1-.2)^2 = 0.8^2 = 0.64$

There is a cognitive leap that would have to be navigated.

Reflection questions

How do students express the amount of change as a percentage of the original quantity?
How do students calculate a value following a percentage decrease?
Does this match their visuals and materials?
Are they able to explain the process?

Maths investigations#

Provide students with experience real-world applications of mathematical thinking/understanding. Provides a number of contexts

real world applications

Proportions in produce, budgeting, and cost per unit.
careers

Waste management engineer, builder (concrete)

ANd now activity ideas for ratios and differences between rational/irrational numbers and terminating and recurring decimals.

Exploration ratios (year 7)#

AC9M7N08 and AC9M7M06

Orchestra and families of instruments as scenario for ratios. Apply to different contexts - sport, choir etc.

Orchestrate Pose questions
- What are the ratios of instruments in a typical 100-piece orchestra? For example, what is the ratio of violins to clarinets?
- Why do you think maintaining specific ratios may be important?
- Is it possible to create a 50-piece orchestra with the same ratio of instruments as the 100-piece orchestra? If not, explain why not.
- How could the number of instruments in each section be changed to have a smaller orchestra with a similar sound?
- What type of ratios can be directly converted to a fraction?
- What were the mathematical steps involved in solving the ratio problems?
school choir
- How many different parts are in a standard choir? (For example; soprano, alto, tenor and base)
- How many conductors are there for the choir?
- How would the choir's sound change if the ratio of parts were to change? (For example; if the number of sopranos halved and the number of bass singers doubled)
- What ratios create a nice harmony?
- What is the ratio of parts in the school choir? What sound would this choir produce?
- How many people would be required in each part for a 1,000 voice concert?
sports teams
- How many different positions are in your sport? (For example; forwards, centres and backs in football, or leg-side and off-side fielders in cricket)
- How many coaches and umpires/referees are there for a match?
- How would the team's performance or strategy change if the ratio of positions were to change? (For example; if the number of forwards doubled and the number of backs halved)
- What ratios provide a well-rounded team?
- What is the ratio of positions in the school team?
- How many people would be required for each position if you were hoping to create 10 teams?

Reflection questions

How did students solve problems when using the concept of ratio?
Were students able to express the number of each part as part-to-part ratios?
Were students able to express the number of each part as part-to-whole ratios?
Did students recognise that only part-to-whole ratios may be directly converted to fractions?
How did students use their knowledge of fractions to solve ratio problems?

Assessment

How did students solve problems when using the concept of ratio?
Were students able to express the number of each part as part-to-part ratios?
Were students able to express the number of each part as part-to-whole ratios?
Did students recognise that only part-to-whole ratios may be directly converted to fractions?
How did students use their knowledge of fractions to solve ratio problems?

Terminating and recurring decimals (year 8)#

AC9M7N04, AC9M7N05, AC9M8N01, AC9M8N03

Exploring various representations of real numbers (fractions, decimals, percentages, ratios), understand them and move flexibly between them.

Activity#

Start by asking students to convert certain fractions (ninths and their multiples) into decimals - explain what resulted? What was interesting?
- What patterns do you notice?
- What's the answer for $\frac{9}{9}$ what about $frac{10}{9}$
Repeat with (sevenths and multiples)

There are patterns
Ask students to find the fraction for any recurring decimal

For example

\[n = 0.3333... 10n = 3.3333... 10n - n = 3.3333... - 0.3333... 9n = 3 n = \frac{3}{9} = \frac{1}{3}\]
Which is an algebraic argument that proves that $0.99... = 1$

\[n = 0.9999... 10n = 9.9999... 10n - n = 9.9999... - 0.9999... 9n = 9 n = \frac{9}{9} = 1\]

Importance of understanding what recurring decimals means

See this Recurring means infinite, add them all up and you will get to 1 (poor oversummarisation).

Concept	Definition
Exponential or index numbers	A number written in the form \(a^n\) where \(a\) is the base and \(n\) is the index or exponent.
Scientific notation	A number written in the form \(a \times 10^n\) where \(1 \leq a < 10\) and \(n\) is an integer.