Section One (Two Credits)
The Placevalue Big Idea • How do numbers work?
The hardest part of a journey is to begin. Enjoy the journey!
Our modern civilisation uses a way of generating numbers used for counting based on the number 10 or the number 2. The 10 system is called decimal and the 2 system is called binary. In the past, sometimes other civilisations used a different number such as 20 or 5.
The decimal system appears to have developed because there are 10 digits (fingers and thumbs) on our two hands. Curiously 20, 5 and 4 (the gaps in fingers) have also been used.
The binary system is used extensively in computer technology and is based upon the idea of “on" and "off” in an electrical switch or “1” and “0”. Base 2 or binary is a fairly recent use.
Write a paragraph or two or discuss with someone “How numbers work?”. How are they made up?; What does larger mean?; What is the smallest number?; What are decimals?; and "How are numbers combined and compared"? "What do you know about binary numbers?" That sort of thing.
A visual mind map or diagram would be a good way to record all these thoughts.
Test your understanding question
In the number 12,345 is there a 2?
(c) There are many 2’s
(d) All the above
(e) Who cares!
Share your answer with a few other people.
The Primary Cause of Concern and Anxiety
Place value understanding, or lack of it, is the single cause of most issues students and many teachers have in being confident when working with mathematics and progressing to more complex ways of thinking and working. The only thing to do here is "get your head around it!" So, spend time checking below.
Core to this understanding is realising that the position or place of a numeral in a multi-digit numbers changes its value; hence the term “place value”.
In the number 12,345 above there is a 10,000, a 2000, a 300, a 40 and a 5. There is no 2 although that is not absolutely correct! We could rewrite 12,345 as 12,343 + 2 or see 12,345 as 6,172 lots of 2’s and a 1. Answer (d) is looking pretty good!
Writing 12,345 as an expanded number we get
12,345 = 1x10,000 + 2x1000 + 3x100 + 4x10 + 5x1
Now the terms 1000’s column, 100’s column and 10’s column and units or 1’s column start to make sense.
We can make huge numbers using this system by just adding a new group that is ten times bigger on the left. The number 812,345 is just 800,000 bigger that 12,345.
There is no largest number! We can always add 1 to the biggest number we can think of so we say there is an infinite number of counting numbers.
There is a really big number that has a name thanks to the Charlie Brown comic strip. This is also where a popular internet search engine established its name.
A googol is 1 followed by 100 zeros. There is an even bigger number that has a name called googolplex. If you tried writing out a googolplex with 0’s and a 1 using long rolls and ink you would discover there is not enough room in the universe to stuff the paper needed let alone finding the ink you need and the time to write it all out. A pointless task!
Write out a googol. Find out what a googolplex is.
Test your understanding
On a line with 0 at one end and a million on the other, mark where you think 1000 appears.
Using position to solve addition, subtract, multiplication and division calculations.
Basic facts are instant recall knowledge such as 3 + 4 = 7 and 8 + 5 = 13. There are also multiplication basic facts such as 7 x 8 = 56. Traditionally the facts for addition and multiplication are memorised to the instant recall level for numbers 1 to 12. An even better approach is normalise instant recall of the family of basic facts and include subtract and division as well.
Memorise your basic facts! Check up below.
The Tens and Powers and being Multiplicative.
A little superscript is put above and right of a number to show how many times it is multiplied by itself. Ponder that little statement and make sure you can explain what is meant.
Hence 23 = 2 x 2 x 2 = 8
Write 3x3x3x3x3 in power form.
The Powers of 10 build in size very quickly.
10, 10x10, 10x10x10, 10x10x10x10 or 10,100,1000,10000 and so on. These are the numbers that are used as the base for our counting system. The are written however from biggest to smallest or 1 (units).
Hence 10,000, 1000, 100, 10, 1 and the number 12,345 is as explained above. Putting these in columns uses the idea of position and should be used in calculations to help keep track of what is happening.
Test yourself with the numbers in this place value table. There are two extra features in there that need explaining.
What are the extra features in the table that need explaining?
The hawk-eye reader will notice a zero and a one written as “0” and “Units”.
The zero is a place-holder. It holds the space and shifts the bigger numbers to where they are meant to be. For example 50020 would look like 52 without the zeros and the value of the numbers is considerably different. The “zero” is a placeholder!
What does the comma do?
The Units column is a bit trickier to explain...
The numbers we use for counting have deep historical reference mainly from Arabic but also Indian sources and these go back many thousands of years.
One, two, three, four, five, six, seven, eight, nine. Five was once “fife” for example. The way they are written as numerals has a similar history. We use numbers to order and calculate. Hence 1st, 2nd and third is ordination. Telephone “numbers” are coded groups and are words because it makes no sense to add or multiply them! Calculation or numeration is 2x3 = 6.
The Arabs invented “zero” because they were traders and that meant money had to be accounted for accurately and the idea of zero was established. Our numbers in commmon use became 0,1,2,3,4,5,6,7,8,9 and then we start to use them all over again with a different value and position or place-value. Negative number ideas or debt probably arose then as well but these only became established in late Middle Ages in Europe and England.
The next number becomes 1 ten and 0 ones.
Hence 10,11,12,13,14,15,16 etc and the number system is up and running.
Bilbo in The Hobbit uses ...eight, nine, tenty, tenty one, tenty two, tenty three ... and in te reo the numbers become teko (10), teko ma tahi, teko ma rua and so on. This is 10, 10 and 1, 10 and 2 and is a much more sensible way of counting compared to the illogical “-teens” that English uses.
The pattern 10000, 1000, 100, 10, 1 is generated by dividing by 10. 1000/10 is 100 and so on. The 1 is the result of 10/10. This idea also opens up the decimal system as an extension.
I think this explanation is worthy of pondering deeply because when we write this pattern as powers we see a nice little insight. 104, 103, 102, 101, 100 and we establish by looking at the pattern in the power that 1 = 100. Nice. In truth (anything)0 = 0 and curiously 00 can be also.
An unwritten Axiom or truth in mathematics is “One can be anything I choose one to be.” This become a very powerful idea when we refer to 1 group of numbers or things.
Another unwritten Axiom is “When combining or comparing in mathematics we do so using the same size units or ones”. This is the reason the spaces on a ruler are equal distances apart and we have units of measures such as kg and litres.
Using the position idea we can take numbers apart.
12345 is the same as
when they are all added up.
Now there is enough information to state what the purpose of using number reasoning is when applied to problem solving.
“Number reasoning is about having the ability to take numbers apart, strategise, operate on them and reconstruct the resulting answer in order to solve problems.”
Onwards and Upwards!
Write the number eleven thousand eleven hundred and eleven using numbers.
Before hints are given there is another problem that exposes how important and deeply connected place-value ideas are to solving problems.
Write down a four digit number and then using the same digits make another four digit number. Now find the difference between the two numbers.
The new number is curiously predictable.
• the digits sum or add up to 9 if you keep adding.
• the number is divisible exactly by 9
• the number is a multiple of 9
Example to work through line by line.
1234 and reordered is 4321
The difference is 4321 – 1234 = 3087
• And 3+0+8+7 = 18 and 1+8 = 9
• 3087 divided by 9 = 343 as any calculator will show.
• 9 x 343 = 3087
Check your chosen number has these same properties.
A nice question to ask now is “Why does this happen?” Answering this question is doing mathematics which is the purpose of this course. It might be surprising that any number bigger than 9 has these properties and it can be proven.
Time to Review Progress
There are many connections, new words, and ideas presented so far.
Take 5 minutes and review recording as many new words and ideas, connections and questions you may have. List any questions you have.
Place value is very much a multiplicative idea or one that uses groups. The core group used is 10 in the number system. Then we generate groups of ten. This means we must introduce groups and sets of numbers and objects from the moment a child breathes air if we are to normalise this thinking across homes and classrooms. New Zealand is not a country yet where society has normalised the expectation that everyone will enjoy mathematics.
Some extra resources to take a look at:
Now to use the place value ideas to build operations on numbers and measures.
Section One (Two Credits)
The Placevalue Big Idea • How do numbers work?
1. Write 54,321 in words in at least five different ways.
2. What is the value of 44?
3. Write 1,000,000,000 in the form 10x.
4. Explain the term “place value” when writing numbers.
5. Write the number twelve thousand twelve hundred and twelve as a single number.
6. Bilbo Bobbins would say 21 is twoty – one. What are the numbers he would say after (a) 32 (b) 70 and (c) 99?
7. What does it mean to say “0” is a place holder?
8. Complete the multiplication basic fact below.
Give yourself a grade!
9. Write any other comments you can think of or questions you might have.
10. Search and record the WWW for a site that helps explains Place Value used in Mathematics.
11. Bonus question. How many times did you email Jim? [One mark for each time.]
To gain 2 Credits for this section screenshot or photograph your answers or otherwise email a copy to firstname.lastname@example.org for registering(first time), checking and getting feedback. This is intended to be a painless process and all questions are accepted. Once you have been awarded the 2 credits a fee of $10 will be requested.
Well done, Section One over! On
to Section 2! Go back to the Introduction to find the