Mathematics is about relationships and thinking

Mathematics Lessons for Remote LearningStrand: Number and Algebra Target: Y7, 8, 9, 10 –NZC Level 3/4Topic: Algebra – Vital Skills and Knowledge

Starter – Odd One Out3 bab a2b

The idea here is to select a number that you think is the odd one out.

Odd one is ______________. Reason ______________________________________

Just for a bit of extension see if you can think of a reason why each number could be the odd one out.

LearningTHE DEEP UNDERSTANDING OF ALGEBRA

- Being enabled to notice, find and record patterns and relationships, use visual interpretations, to make generalisations and make sense of the world in solving problems.

Muhammad ibn Musa al-Khwarizmi was a 9th-century Muslim mathematician and astronomer. He is known as the "father of algebra", a word derived from the title of his book, Kitab al-Jabr.

Algebra can be thought of as the generalisation of number. Whatever numbers ‘do’ algebra must ‘do’ as well, to be consistent. This is a really useful thing to remember if stuck. Try with numbers and do the same with letters.

For example which of the 4 operations can commute? Does a + b = b + a? A number example 3 + 4 = 4 + 3 shows this to be true and if addition is seen as “joining” then we also see that this property is always true for addition.

• Task 1 – Which of the other three operations subtraction, multiplication and division are commutative like addition?

Variable – At the heart of algebra is a variable. Letters or words or symbols can all be a variable. Examples are the

letter a or x, A or X, alpha or GST, or . The letters a and b above are variables representing any number.

A problem as an example. Worked Problem 1

I am thinking of two numbers whose product is 16 and sum is 10. What are they?

I could just dream up the answer by guess and check but this would not work for a larger 10 digit number! Algebra opens up the possiblity of solving every problem that looks like this.

I will let one number be a and the other number b. These are my two chosen variables.

I read the problem again and write a x b = 16 (product), and a + b = 10 (sum). I transferred the written real world problem into a mathematical world. I like thinking in the maths world. It is logical and tidy. I make connections in this world to things I know and problems I have solved. There, is a good reason for doing practice and homework!

How do I solve ab = 16 and a + b = 10? [Note a x b = ab = a.b, these are just conventions.]

Guess and check is good for the small numbers and limited possibilities as in this problem.I guess 5 and then see it has to be a factor of 16. Try 4. 4x4=16 but 4+4 is not 10. How about 2 and 8? Yes, that works.

A Table will work if there are only a few possibilities. Here we see the solution pops out quickly.

Factors of 16 Sum

1 x 16 = 16 17

2 x 8 = 16 10 TRUE!

4 x 4 = 16 8

Strategic, Creative, Critical and Logical Thinking are Key to Mathematics

Mathematics is about relationships and thinking

From ab = 16, logic says b = 16/a and a = 16/b. This is one skills of moving variables around.Put numbers in to check this. 2x8 =16, so 8 = 16/2 and 2 =16/8. A good reason for knowing tables of facts!

Task 2• Algebra allows us to generalise and be able to figure out harder problems like

Solve “I am thinking of two numbers that multiply to make 481 and add to make 50.”

Visual Problems are fun and are less daunting than masses of letters.

This little gem is from https://brilliant.org/daily-problems/ and simply asks for the value to the ‘?’.The problem is more interesting if the value of each symbol is requested. In algebra the symbol, green circle for example, represents a number and has the same value in each place that it is used.

Task 3• Find out the values of each symbol and the question mark in this picture.

Algebra has three main features- Skills or manipulation- Generalisation or proof- Formula or substitution

Skills - There are many algebra books that develop skills and this is where most school mathematics classes begin and unfortunately end. It is important to know how to expand brackets, factorise and simplify expresssions but these are of no use until a problem is encountered.

Generalisation – The statement “An odd number has the formula 2n + 1 where n can be a Whole Number.” This connects to Number Knowledge and W = [1,2,3,4,5,...}.

Task 4• List the numbers that are in these sets

- N, the Natural or Counting Numbers- I, the Integers- W, the Whole numbers- M3, the Multiples of 3

Formula - These are used widely in society and the workplace and represent some serious thinking that someone has done to solve a particular problem. A rectangle measuring a long and b wide has an area of a times b so Area of Rectangle = length x width = ab is a formula. Formula often simplfy a complex situation to make it easier to use and understand. The number of cases of COVID worldwide is summarised by the formula Number = 100,000 x days since April 1st, unfortunately.

Task 5Find ten more formulae in use.

A Starter Algebra Course by Jo Boaler.https://www.youcubed.org/algebra/

Warmups from Brilliant.orghttps://brilliant.org/algebra/

Task 6 - Make sense of this cryptic statement.

“The difference between the square numbers is very odd!”

Strategic, Creative, Critical and Logical Thinking are Key to Mathematics

Mathematics is about relationships and thinking

JournallingToday I learned ________________________________________________________________

CommentsMake any comment you feel like making here.

Math Language: List all the math words you can find in this document and write what you think it means beside the word. Eg subtraction means to take away or to find the difference. Keeping a list of these words is a very good idea.

AnswersStarter – Odd One Out

3 bab a2b

The idea here is to select a number that you think is the odd one out.

Odd one is _____3_________. Reason ________it is a closed statement_______

Just for a bit of extension see if you can think of a reason why each number could be the odd one out.

b is a single variable and represents anything, ab represents a product, a2b represents the volume of a square prism that measures axa and is b long. A closed statement is 3+1 = 4 and can be true or false. 3 +1 = 5 is an example of a false closed statement. An open statement contains a variabel like x+1 = 4. This could be true or false depending on x.

Task 1 – Which of the other three operations subtraction, multiplication and division are commutative like addition?Multiplication is commutative. The order of the numbers does not change the result like addition. 3x4=4x3Division and Subtraction are not always commutative. 4 – 3 ≠ 3 – 4 and 6/2 ≠ 2/6. It can sometimes be true, 2-2 = 2-2 and 6/6 = 6/6 but these are exceptions.

Task 2Algebra allows us to generalise and be able to figure out harder problems like

“I am thinking of two numbers that multiply to make 481 and add to make 50.”

Guess and check is quite hard in this case. Building a table likewise so we are forced to use other solutions.

The picture shows a screen shot of a graphing calculator. The two equations show the problem restated in the maths world and the little aqua blue square shows the solution of 37,15 or where these two equations intersect. The other intersection is the same solution but 15,37 or the other way around.

A visual solution like this is a modern way of solving algebra problems and every student should be learn how to use a graphing calculator. It is a fast way to get an answer and allows more time to figure out the question and ponder the answer.

This graphing calculator is from https://www.pacifict.com/ . The App store has many and it is more a problem of finding one that suits you and is easy to use.

More formally and traditionally the two equations x + y = 50 and xy = 481 can be solved by substituting 50 – y for x in the other equation to get (50 – y)y = 481. This can be expanded to 50y – y2 = 481 and rearranged to y2 - 50y + 481 = 0. This can then be solved using the quadratic formula to reveal the two solutions for y and the re-substituted to find the corresponding x values. Here is a complete solution https://www.calculatorsoup.com/calculators/algebra/quadratic-formula-calculator.php . What would you rather use?

Strategic, Creative, Critical and Logical Thinking are Key to Mathematics

Mathematics is about relationships and thinking

Task 3Find out the values of each symbol and the question mark in this picture.

Thsre are several ways to solve this problem and like all good problems it should be solved in several ways becuase new insights are gained. Learning!

Way 1. A good insight is that the totals across must be the same as the totals down. There is a common 19 so 15 + 13 = 14 + ? Hence quite easily ? = 14.

Way 2. Adding the two left columns gives 2xGreen +2x Blue + 2x Red = 28 which is twice the first row. ? = 14.

Way 3. Now that we know ? = 14 the first two rows show that Yellow = Red +5. Using this in the third row we see Red + Red + (Red +5) = 14, or 3x Red = 9, or Red = 3.Using this information in Column 1 means 2xGreen +3 = 15 or Green = 6.Using the same information in Column 2 means 2x Blue +3 = 13 or Blue = 5.

Check using any convenient row or column. Substituting these into row 1 for example shows 6 + 5 + 3 = 14

There are more ways of solving this problem but the methods are all paths similar to Way 3.

Task 4List the numbers that are in these sets

- N, the Natural or Counting Numbers = {1,2,3,...} and is an infinite set.- I, the Integers = { ...,-3,-2,-1,0,1,2,3,...}, doubly infinite!- W, the Whole numbers = {0,1,2,3,...} infinite again. - M3, the Multiples of 3 = {3,6,9,12,15,...], infinite.

Notice there is zero difference between N and W! That is a maths joke or a ‘moke”.

Task 5 There are hundreds of formula in common use. A famous one is Einstein’s Equation E = mc2

Here c is the speed of light or 3x108m/s and m is mass in kilograms. Mass is a very dense form of Energy!

Task 6 - Make sense of this cryptic statement.

“The difference between the square numbers is very odd!”

The square numbers are 1x1, 2x2, 3x3, 4x4 etc or 12, 22, 32, 42, ...using the power or exponent notation. Below is a picture showing these. Look at the pink dots and notice they are the odd numbers. 1,3,5,7,9, ... or as I prefer to see them 2x0+1, 2x1+1, 2x2+1, 2x3+1, 2,4+1, ... which tries to expose the two sides added on and an extra one in the corner. The statement above should now make sense!

This problem opens the cans of wonder called polygonal numbers https://en.wikipedia.org/wiki/Polygonal_number

Here are the triangular numbers (because they form triangles) and are made up by summing the Natural Numbers

Strategic, Creative, Critical and Logical Thinking are Key to Mathematics

Mathematics is about relationships and thinking

1, 1+2, 1+ 2+ 3, 1+2+3+4, ... to make 1,3,6,10, ... and in general (last)x(next)/2 = n(n+1)/2 or ½ n(n+1). Hence the 4th triangular number is 4x5/2=10. The triangular numbers turn up in many places and is one that must be studied. The formula is actually (the middle number)x(number of numbers) = (n+1)/2 x n . You might need to ponder that.

FeedbackStudents and teachers are welcome to email jimhogan2@icloud.com with comments. This was a lesson that could be given to a NZC Level 2, 3, 4, 5 student for some placevalue learning and revision. Students should select a set time each day and perhaps using the timer on a cell phone set 45 minutes or so to learn and practice mathematics. Keep trying on problems and expect to struggle. Persevering and struggling are great competencies to develop. You can learn more about these from https://www.youcubed.org/resource/growth-mindset/. We have a great math website in Nzwith a special resource called e-AKO https://nzmaths.co.nz/information-about-e-ako-pld-360 .

Strategic, Creative, Critical and Logical Thinking are Key to Mathematics