Technomaths

One of my favourite Maths lessons is about the Fibonacci series, using the YouTube video “Nature by Numbers“. Firstly, I ask students to add 0 and 1 and then the two previous numbers to make a sequence. 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89 etc. All students are capable of this, even if some use a calculator.
I then show students the video and we discuss all the examples from nature that show the Fibonacci sequence. We search for examples on Flickr – sunflowers, pine cones, succulents and nautilus shells. Although some people claim this is spooky, the pragmatist in me believes that nature exhibits this sequence as it is the most efficient way to pack objects (leaves, petals, seeds, she’ll compartments) into a small space.
If we divide each number in the sequence by the number before, the answer approaches what is called “phi” or the golden ratio, approximately equal to 1.618. Now we can discuss how this ratio was used by ancient Greeks, Euclid and perhaps even Leonardo da Vinci in the “Vitruvian Man”. Many architects, artists, photographers and others believe that this ratio represents the perfect proportion to render the subject most beautiful.

Phi and the Golden Ratio in Art

Golden Ratio in Art and Architecture

Fibonacci Numbers in Nature

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Learning Intention: Students will understand how to collect data using a tally and create a frequency table and bar graph using the data. They will understand how to convert fractions to decimals and percentages. They will create a pie chart using this data by converting 100% to 360 degrees.

Success Criteria: Each student will produce a poster that includes a frequency table (including fractions, decimals and percentages), bar graph and pie chart of their chosen data, collected from the Year 7 Maths Survey.

1. First collect your data in tally form.
2. Add each category and find the total.
3. Represent each category as a fraction.
4. Convert to a decimal (2/25 = 16/100)
5. Convert to a percentage 16/100 = 16%
6. Create a bar graph using this data
7. Remember to add SALT to your graph – Scale, Axes, Labels, Title
8. Turn your bar chart into a pie chart (multiply percentage by 3.6 because 100% = 360 degrees)
9. Make sure you have a key to interpret your data.
10. Add a beautiful title and colour to present your poster.
11. Go to Create-A-Graph and use your data to check that your graphs are correct.
12. Print out the computer generated graphs to add to your poster.

Learning Intention: Students will understand how to calculate and simplify ratios using body measurements as an example.

Success Criteria: Students will be able to correctly estimate the ratio of various body proportions, using average class data.

Do you think that “Barbie” dolls portray a realistic image of a human figure? What about “Bratz” dolls and Manga figures? What is it about these figures that make them so appealing to children? We are going to investigate the proportions of “Barbie” dolls and other figures to compare their ratios with the average year 7 student from Hawkesdale college. We now have the data that gives us some average measurements of height, head circumference, arm and leg length and chest, waist and hip measurements of the average student in our class.

• As you can see from the figure above, “Barbie” has a head to height ratio of 4 to 28cm or 1:7.
• Her leg to height ratio is 14cm to 28cm or 1:2.
• Her neck to height ratio is 2cm to 28cm or 1:14.
• Her head circumference is larger than her waist circumference.
• Her foot length to leg length is 2cm to 14cm, so, 1:7.

Calculate the same ratios for the average Year 7 student, using our data. What can you conclude?

Bill Genereux has written a terrific post about the proportions of superheroes, from a book that teaches people how to draw comic book heroes. Malyn Mawby has also written a great post about using Da Vinci’s “Vitruvian Man” to learn about ratio and proportion.

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Learning Intention: Students will understand the meaning of mean, median, mode and range. They will understand how to calculate an average from given data and be able to work out the range of that data. They will then use the data to explore and compare ratios.

Success Criteria: Students will correctly calculate the average and range of the given data and use the data to make some comparisons of ratio.

Statistical values, such as mean, median, mode and range, help us to determine trends in the data. They are used in a variety of ways in business, education and government organisations. For example,  a company might want to find out about which customers spend the most money in their shops, so they can target their advertising better. Governments might need to find out about the age groups in their community, so they can distribute funding to health care, aged care, kindergartens, schools, universities, hospitals and other services.

• The mean (or average) of a set of numbers is calculated by adding all the numbers together and dividing by the number of values. It will always be somewhere between the smallest and largest value. What is the average height of students in Year 7 at Hawkesdale P12 College? Is there a height difference between males and females?
• The mode is the most frequently represented score. A shoe shop will need to order more pairs of popular sizes – 8, 9 and 10 – than smaller or larger sizes, for example.
• The median is the middle value – halfway between the smallest and largest value.
• The range is calculated by subtracting the smallest number from the largest number. What is the range of arm lengths in Year 7?

Fractions as we know them today weren’t used in Europe until the 17th century. However, Egyptians have been using fractions since at least 1800BC, although they never wrote fractions with a numerator greater than one. These are called unit fractions. Fractions with a numerator greater than one were expressed as the sum of unit fractions. Find out more at the History of fractions and Egyptian fractions.

The National Library of Virtual Manipulatives has a range of interactive applets that you can access to learn about fractions:

• Fractions – parts of a whole (easy)
• Fraction Number Line Bars (easy)
• Visualizing fractions (easy)
• Naming fractions (easy)
• Fraction bars
• Equivalent fractions
• Adding fractions (more difficult)
• Comparing fractions (more difficult)
• Multiplication of fractions using rectangles (more difficult)
• Fractions, decimals and percentages (more difficult)

Try at least three of these interactives and write a comment below about what you have learned.

Image is a screenshot from the Cool Math 4 Kids site.

When adding or subtracting fractions the first thing you need to do is make sure that the denominators (bottom numbers) are the same. If all the denominators are the same you can simply add or subtract the numerators (top numbers) and then simplify the answer if required. If the denominators are different, you need to find a common multiple and convert both fractions, so that the denominators are the same. Activities 4 and 5 below show how this is done:

1. Adding Fractions from Cool Maths 4 Kids.

2. Three simple steps to adding fractions from “Maths is Fun”.

3. BBC Bitesize – fractions activities – Choose the fractions activities from BBC Bitesize (Equivalent fractions and ordering and comparing fractions).

4. Adding fractions with different denominators from “Maths Playground”.

5. Adding fractions with different denominators from YouTube – Maths Made Simple Series.

Learning intention: Students will be able to identify and name equivalent fractions (halves, thirds, quarters, fifths and sixths) and describe how they are calculated.

Success criteria: Students will successfully identify equivalent fractions on their fraction walls and name equivalent fractions on a number line.

Maths Playground – Visual fractions (the visuals are good, but the program doesn’t always allow the right answer?)

Maths is Fun – Equivalent fractions

Maths Games – Matching equivalent fractions

Problem solving strategy board from Maths300

Over the last few weeks you have learned about the following concepts:

• numbers less than zero, called negative numbers
• factors
• multiples
• prime numbers (only two factors, one and themselves)
• Composite numbers (any number with more than two factors)
• Square numbers (numbers with an odd number of factors)
• Square root (the symbol over a number that indicates you calculate the number that is multiplied by itself to get the original number)

We played the game “Multo” which helped to consolidate your knowledge of number facts and made you think about which numbers were most frequently called (common multiples and not prime numbers greater than 7).

Remember you can access Mathsmate Skill Builder sheets at their website if you need help with your Mathsmate. You should also be accessing Mathletics to complete three activities each week. I have found that the Google Chrome or Mozilla Firefox browsers seem to access Mathletics from home better than Windows Explorer.

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Learning Intention: Students will understand what Multiples, Factors and Prime Numbers are and why they are useful.

Success criteria: Students will be able to recognise multiples, factors and prime numbers and calculate the Highest Common Factors (HCF) and Lowest Common Multiples (LCM) of given numbers.

• Resources from NRich – Factors and Multiples
• Factors and Multiples Jeopardy Game
• Factoring – Factors and multiples