Wednesday, November 11, 2009

Cycling Digits



Cycling Digits from "Thinking Mathematically" page 165

I have in mind a number which when you remove the units digit and place it at the front, gives the same result as multiplying the original number by 2. Am I telling the truth?


Choose {a, b, c, d, e} each elements of the set (0, 1, 2, 3, 4, 5, 6, 7, 8, 9), forming numbers of the type with:

one digit that is a = 2a | Implies a = 0 the only solution

two digit that is ba = 2(ab) or | converting to units

10b + a = 2(ab) = 20a + 2b |simplify
8b = 19a | Must belong to the set above,
| if a =8 then b =19 and it can not

ba = ab + 9(b-a) | Here is the solution and it is not two

| times the reverse number. False.

| See solution on left above. The reverse

| of ba is ab plus 9 times difference of b

|and a, therefor not equal to twice ab.

b=a gives the two digit Palindromes 11, 22, ... | This does not give two times original

|number! False.

Three digit numbers 2(cba) = acb | converting to units

200c +20b + 2a = 100a + 10c + b | simplifying

190c + 19b = 98a which is 19(10c+b)=98a | integer solution c=9, b=8 and a=19 False.

cba = acb + 9( combination of 21 for #s 1 apart?) |work on solution as they are not equal

cba = acb + n (189= 21*9) | the RHS is acb plus or minus some

cba = acb + n (189= 21*9) |multiple of 9, sign depending order of -

|Again not equal to 2acb.

100c+ 10b+ a = 100a + 10c+ b | simplify

90c + 9b = 99a which is 10c + b = 11a | a=b=c set of (0, 1, ..., 8, 9) three digit

|Palindromes 111, 222, ..., 888, 999

Perhaps ab = n(ba) | Something to try!


This problem has potential for more investigation when time permits or with your help.

Sorry about the image quality tha is why I left the type, but the alignment is ???


Saturday, November 7, 2009

Some things to consider in choosing or assessing a mathematics textbook

The textbook we have chosen is Applied Mathematics 11.

How heavy is it? This book seems to be a reasonable weight for a grade 11 student.

How does it feel? The hard cover seems durable enough to last several school years.

How would it be to carry it from home to class? Less than paper size so should fit well into back pack.

How durable is it (covers, pages, binding, etc.)? Well constructed with thick enough to be tougher than normal sheet paper.

How many years of use will it have? It should last five or more years.

Is it good value for the money? Provided it will last five years then the content and durability will make it a good value.

What does it smell like? I thought that this was a strange question until I handled other texts and found that Addison Wesley Math 11 and 12 smelled obnoxious. I do not know if this was from the ink or if it could be an allergen to students.

What do the pages feel like? Very nice in the Applied math book.

Is it pleasing or unpleasing as an object? I think it is an acceptable book.

How have people altered this object? All books are subject to their user abusing them.

Has the school rebound it? NO.

Have kids written or drawn in it? Yes many students degrade their textbooks and they should pay a fee for that.

What year was it published, where and by which company? Pearson Education Centre, 2000.

Is this a special edition of the book? Yes it is for Applied Math as taught in Western Canada.

Who wrote it? Do you know any of the writers? Math teachers, educators and consultants. No writers known to this reviewer. Publisher –– Claire Burnette

Author – Lot’s of different people

Where are they from and what are their credentials? Any bias? Canadian writers. Unknown.

Is there a preface or introduction? What does it say? Not a must read!

What does the table of contents tell you? Clearly written and useful.

The index? Well spaced, easy to read

Does it have a glossary, answer key, data tables, supplementary problems, enrichment material? Well spaced, easy to read. Colorful with many diagrams. Good data tables. Anything else of note?

Look at the overall design of the book. Does it use colour to help guide users to certain features? Is there much white space, or are the pages full of type? Are there photos, diagrams, pictures, graphs, border decorations? How do these affect your impression/kids’ impression of the book?

Supplementary – How to use calculator and other tools (micrometer) in back. Exercises are replaced by tutorials to make more friendly for students.

Answer key – Yes, only answers

What fonts and type size are used? Do these help or hinder readers?

Are there sidebars along the edges of the page? If so, what purpose do they serve?

Do graphic features contribute to a clear, readable, interesting design?

As a graphic design – Well done, colorful, white space, nice spacing between the lines making it easy to read. Lots of pictures and photos. (Could it be too colorful?). Good font size. Nothing in borders except a solid line down the left when showing examples.

Are all topics in the IRPs covered? Yes, this can meet the needs of the IRPs.

Any extra topics included? Yes some enrichment topics.

Do you like the approach taken? Very practical which goes with the Applied approach.

Is there a logical sequence to the chapters and sections? Yes, tries to cover everyday life topics. Structure of the chapters are reasonable. However it does not have a whole lot on how to approach it. Good number of examples, tutorials, and a section called investigation. Explanations in Reference section in the back.

How is each chapter/unit structured? How are new topics introduced? Are adequate examples, activities, problems and summaries provided?

Are important points highlighted graphically (boxed, indented, highlighted in color, etc.) for easy reference?

Important points highlighted in red. The text book is organized such as you the teacher with the students work together through the text and if you don’t understand something it tells you to refer to the references in the back.

Does the textbook embody the goals of the mathematics curriculum? More specifically:

Does it promote the NCTM principles and standards? Yes

Does it support the BC IRPs three principles of learning? Yes

Does it include the PLOs mandated in the provincial curriculum?

Does it support principles of citizenship and human rights implicit in the BC curricula? (Note that this may be tacit rather than overt.) It covers the PLO topics.

Text as Curriculum – Multicultural. Pictures of smiling kids all over the place of all types of ethnicity.

Is this textbook designed as a resource for teachers and/or students? Good for both beginning and experienced teachers.

Is it designed for student self-directed study?

Does it support classroom interactions? Individual/group work?

Open-ended problem solving? Multi-modal inquiry? IT links?

How do you think it would work for a new teacher? for an experienced teacher? What would be most and least helpful for each of these? Is there a useful teachers’ guide?

Does every student need a copy of the textbook, or would it be enough for the teacher to have one copy? Every student in the classroom would need one.

Text Book in a Classroom context – Student self directed study in the form of investigation. Encourages classroom interaction and both individual and group work. Open ended giving outside of the classroom work for fun.

Thorough teaching guide to supplement the text.


If you were asked to make a choice between several recommended textbooks, how would you decide? Content and level of questions to fit the course objectives.

What is the price of this textbook? Unknown. How would price factor into your decision? Has to fit into a budget that is continually squeezed.

Do you think it is a good idea to choose one textbook series for a school? for a district? for the province? Yes, that way universal exams or standards could be assessed.

What advantages and disadvantages are there to this kind of uniformity? Teachers can always add extras as long as the core material is taught.

Could you teach math without class sets of textbooks? If so, how would you do it?

Yes, but it would take more preparation time to bring it up to current, internet and web standards with interactive components.

Our groups recommendation is that it could be used as a text for the Applied Math 11 course.

The only question that we would like to know is the cost to see if it fits within budgets?

Divisibility by ZERO

If I but divide one whole
The larger my bottom
A smaller piece feeds me!
The smaller my bottom
Greater numbers of pieces eye see
My slice is so thin
I cannot see
At none I know not that number they be

This exercise may be a good one, but, I found it took much longer to do than I estimated. With time limited, the quality of the poem may be less, however the thinking should be more.

Before the exercise of writing begins I think that one should go over some numerical patterns with the group of students to see that there understanding of what is happening is going in the right direction.

1/2 to 1/3 to 1/4 to ... 1/n where n gets very large then the result will become ___________ .

and 1/(1/2) = _____ 1/(1/3) = ______ 1/(1/4) = ____ and .... 1/n = _________

when n goes to zero.

Any exercise that gets students to think and use there knowledge to understand a process to arrive at an answer to a problem is a useful exercise.

A Practicum Experience

I was just ready to start teaching my first Math 10 class on the subject of slopes of a line, when a young lady put up her hand.

Yes, I said motioning to her.

We are the dumbest math class in the school!, she said.

I was caught of guard for a moment, and then I responded, "No one is dumb, it is usually a matter of not knowing something and this tells you what you need to review or study so that you do know the material that you missed or forgot.

I started the class out with a question, "What is the slope in words?"

One student said, "that the slope was calculated from the difference of two points, with the rise being the y difference and the run being the x difference."

Another student said, "putting the two together with the rise divided by the run gives us the slope."

I wrote: slope = m = (y2-y1) / (x2-x1) and put the numbers in for the two points (2,5) and (1,3)

(5-3) / (2-1) = 2/1=2

Feeling that the class was doing fairly well I continued, " The slope for a line segment which you took yesterday is the same method with the two points picked from points on the line."

I demonstrated with an interactive drawing program that if you pick two points and I had students give me the values of them, then this is how the slope would be calculated and this is what the line would look like. I gave them the formula that they took the day before for a line segment and showed them that it worked for a line as well.

"Now, I said, that if you take one of the points along with the slope you just calculated and put it into the same relationship and simplify, you get the equation of a line."

I assumed that they knew about additive inverses or the rule a +(-a) =0 to eliminate the constant from one side of the equation to another and I named and showed them how to eliminate the constant from one side by doing an addition to both sides of the equation. I then simplified the equation into the form y = mx + b.

Another girl in the back raised her hand and asked, "what level university math do you teach at UBC?"

My response was that this was grade 8 or 9 level math.

I had a worksheet with a clearly worked out example in detail with what we covered. I went around the room and ask if anyone needed help? I gave them some HW.


For the next class I rewrote the lesson plan three times to make it simpler, with a better worksheet.

The results from the worksheet were poor, not from the understanding of the method but from being unable to add, subtract, deal with a negative number, deal with fractions and multiply by zero.

I was left in a state of disappointment from the classes poor arithmetic skills.

Group microteaching reflection - Logarithms

There was a lot to pack into a short time. I felt the opening was good, but that the rules of logarithms could have been clearer and better presented in their typesetting. The examples were good, but may have been to challenging for many in the audience.

From the comments of our audience we had 'things that went well' were:

'the history part was good'
'defined what a log is and the laws'
'interesting intro to John Napier'
'examples were somewhat engaging'
'relation to calculator informing'
'Good Hook'
'liked that you picked a few rules in the end and reintroduced them'
'Good discussion about problem'
'This is a tough subject to tackle'
'Having notes prewritten'
'Nice history lesson but too much detail'


From the audience, 'These areas need work':

'watch out for hands in pockets'
'the notes given in Word were cluttered'
'ideas presented in a kind of confusing way, very abstract for high school students'
'this is dry'
'don't like being put on the spot'
'participation should be more thorough an activity'
'bit hard to see the board, be careful not to stand in front of it'
'maybe a worksheet with fill in the blanks'
'put the title of the lesson first on the presentation'
'derivation not needed'
'more concrete example'
'keep rules on same page as questions'
'don't erase or change times law for dividing' 'don't just erase part of the board to show another idea. It causes confusion'

Thank you to all the participants that gave constructive criticism. The many good ideas will be incorporated.

Every presentation will be reviewed afterwards and improvements made to make the next presentation better for the audience and the presenters.

Friday, November 6, 2009

Zero

The most interesting number in addition and subtraction-nothing, zero, nil, zilch, and 0.

The opposites attract and annihilate -1 + 1 = 0.

ZERO added to "anything" is still "anything" a + 0 = 0

"Symbol of any kind" + 0 = "Symbol of any kind", 7 - 7 = 0 , sine squared plus cosine squared - 1 = 0.

When you think of zero it is not nothing, but something and its opposite or inverse. Properties of Zero in addition a + 0 = 0 and multiplication a * 0 = a(b - b).

The beginning of time. The start of a journey, what is left over when everyone leaves. No Music, not a sound. Knot or not nothing is something. When we have spent our last dime. On the journey, when the gas tank is empty the beginning and the end?