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Making Mathematics
Meaningful
Marty Romero
UCLA Graduate School
of Education
[email protected]
2011 UCLA Curtis
Center
Mathematics and
Teaching Conference
March 5, 2011
What is
“It is
only within the last thirty
Mathematics?
years or so that a definition of
mathematics emerged on which most
mathematicians now agree: mathematics
is the science of patterns. What a
mathematician does is examine abstract
patterns – numerical patterns, patterns
of shape, patterns of motion, patterns
of behavior, voting patterns in a
population, patterns of repeating
chance events, and so on. Those
patterns can be either real or
imagined, visual or mental, static or
dynamic, qualitative or quantitative,
purely utilitarian or of little more
than recreational interest, They can
arise from the world around us, from
the depths of space or time, or from
our inner workings of the human mind.”
(Delvin – The Language of Mathematics Pg. 3 )
What is
“employers
want young people who can
Mathematics?
use ‘statistics’ and three-dimensional
geometry, systems thinking, and
estimation skills. Even more important,
they need the disposition to think
through problems that blend
quantitative work with verbal, visual
and mechanical information; and the
ability to deal with situations when
something goes wrong.”(Boaler – What’s
Math Got to Do with It? Pg 6 )
“State business leaders consistently
report that they need a pipeline of
prospective entry-level employees who
can read, write, solve problems,
communicate with others, think
critically, and be responsible for
their work (Tulchin & Muehlenkamp,
2007; de Cos, Chan, & Salling, 2009).
Other organizations echo these
concerns. The Partnership for 21st
Century Skills has highlighted a wide
range of high-level skills that are
important in the fastest growing job
sectors. These skills include critical
thinking and problem solving, excellent
y  2( x  2
x
2
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int[log2 ( x 1)]
S
2
2
4
2
4
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10
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2
4
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)
Algebra 2
POW – The Birthday Cake
Determining the maximum /minimum number of pieces in
which it is possible to divide a circle (birthday cake) for a given
number of cuts is called the circle cutting or pancake cutting
problem. Use the table below to help organize your work to
come up with your answers.
1 Slice
2 Slices
3 Slices
4 Slices
5 Slices
Diagram(s)
Diagram(s)
Diagram(s)
Diagram(s)
Diagram(s)
Minimum
Regions: ___
Minimum
Regions: ___
Minimum
Regions: ___
Minimum
Regions: ___
Minimum
Regions: ___
Maximum
Regions:___
Maximum
Regions:___
Maximum
Regions:___
Maximum
Regions:___
Maximum
Regions:___
Algebra 2
Slices
POW – The Birthday Cake
Min
Max
1
2
3
4
5
n
Extra Serving:
The problem of dividing a circle by lines can
also be generalized to dividing a plane by
circles. As can be seen above, the maximal
numbers of regions obtained from n = 1, 2, 3,
circles are given by 2, 4, 8. What is the
maximum number of regions a plane can be
divided by 4 circles. Show the drawing and
explain the numerical pattern.
Common Core
College/Career
Readiness
Expect students to:
•make inferences
•interpret results
•analyze conflicting explanations of phenomena
•support arguments with evidence
•solve complex problems that have no obvious answer
•reach conclusions
•offer explanations
•conduct research
•engage in the give-and-take of ideas
•think deeply about what they are being taught
(National Research Council, 2002).
Project: Follow Me
Consecutive Numbers
Which natural numbers can be expressed as the sum of two or more consecutive natural numbers?
Use your calculator and your equation solving skills to find the solutions.
n
Sum
n
1
21
2
22
3
1+2
23
4
24
5
25
6
26
7
27
8
28
9
29
10
30
11
31
12
32
13
33
14
34
15
35
16
36
17
37
18
38
19
39
20
40
Sum
10+11, 6+7+8,1+2+3+4+5+6
Project: Follow Me
Consecutive Numbers – Page 2
Describe in a complete sentences three patterns that you found after finishing the
table.
1.
2.
3.
Which numbers cannot be written as a consecutive sum? What is special about these
numbers?
Write 95 as the sum of 5 consecutive natural numbers. Write 64 as the sum of 4
consecutive numbers.
In algebra, the sum of any two consecutive numbers is x + (x+1)= 2x +1. Complete
the table below to express the sum of different lengths of consecutive numbers.
Sum of Consecutive
Numbers
Expression
Result
2 Numbers
x + (x+1)
2x+1
3 Numbers
x + (x+1) + (x+2)
4 Numbers
5 Numbers
6 Numbers
10 Numbers
(Try to use a pattern
to find the result)
Follow Me- Homework Consecutive Integers Word Problems
Solve each problem below by a guess and check method and then by writing and solving an equation.
Show your work in the table
Word
Problem
Equation
Simplified Equation
Solution
Find two
consecutive
integers whose
sum is 45.
x + (x+1) = 45
2x+1 = 45
The numbers
are 22 and 23
Find two
consecutive
integers whose
sum is 99.
Find three
consecutive
integers whose
sum is 99.
Find three
consecutive
integers whose
sum is 207.
Find two
consecutive odd
integers whose
sum is 92.
Find two
consecutive even
integers whose
sum is 54.
Find three
consecutive odd
integers whose
sum is 369.
x + (x+2) = 92
Follow Me- Practice Quiz
Solve each problem below by a guess and check method and then by writing and solving an equation.
Show your work in the table
Word
Problem
Equation
Simplified Equation
Solution
Find two
consecutive
integers whose
sum is 45.
x + (x+1) = 45
2x+1 = 45
2x + 1 – 1 = 45 – 1
2x = 44
X = 22
The numbers
are 22 and 23
x + (x+2) = 92
2x + 2 = 92
2x + 2 – 2 = 92 – 2
2x = 90
X = 45
The numbers
are 45 and 47
Find two
consecutive
integers whose
sum is 199.
Find three
consecutive
integers whose
sum is 123.
Find three
consecutive
integers whose
sum is 93.
Find two
consecutive odd
integers whose
sum is 92.
Find two
consecutive even
integers whose
sum is 154.
Find three
consecutive odd
integers whose
sum is 306.
Follow Me
- Quiz
Solve each problem below by a guess and check method and then by writing and solving an equation.
Show your work in the table
Word
Problem
Find two
consecutive
integers whose
sum is 123.
Find three
consecutive
integers whose
sum is 333.
Find three
consecutive
integers whose
sum is 138.
Find two
consecutive even
integers whose
sum is 514.
Find three
consecutive odd
integers whose
sum is 156.
Equation
Simplified Equation
Solution
Algebra 2
Problem Solving - Word Problems
Use a guess and check table to write an equation that could be
used to solve the following word problems.
Tom and Jerry together have $111. If Tom has $17 more than Jerry,
how much does each person have?
TOM ($)
JERRY ($)
MONEY TOGETHER
The sum of three consecutive integers is 753. What are the
integers?
st
1 #
2nd
#
3rd
#
Sum
ANSWER
Answer
753
Algebra 2
Problem Solving - Word Problems
Use a guess and check table to write an equation that could be used to
solve the following word problems.
1.
A 150-centimeter board is cut into two pieces. One piece is 24 centimeters
longer than the other. How long is each piece?
2.
Admission to the school dance was $3 in advance and $4 at the door. There
were 30 more tickets sold at the door than in advance, and ticket sales totalled
$1590. How many of each kind of ticket were sold?
3.
Find three consecutive odd integers such that the sum of the smallest and seven
times the largest is 68.
4.
Maya has three times as many dimes as nickels. She has four more quarters than
nickels. If the value of the coins is $9.40, how many nickels does Maya have?
5.
Gloria is twice as old as her brother Jaime. Jaime is two years older than
Susanna. The sum of the ages of the three children is 26. Find the age of each
child.
6.
Ms. Speedi has 40 pets, all canaries and cats. The number of birds is twenty-two
more than twice the number of cats. How many of each animal does she have?
7.
The longest side of a triangle twice the length of the shortest side. The third side
is two centimeters longer than the shortest side and the perimeter of the triangle
is 38 centimeters. Find the lengths of the three sides.
8.
A rectangle has perimeter 48 feet. The length is three times the width. Find the
length and width of the rectangle. (Remember, the perimeter of a rectangle is the
sum of all four sides.)
9.
Francie and Angie are twins. Their little sister, Denise is half as old as they are.
The sum of all three sisters' ages is 45. Find the age of each girl.
10.
When Alex banged on the Coke machine, all the coins came out, but no Coke.
There were nickels, dimes, and quarters. There were three times as many dimes
as nickels, and 40 more quarters than nickels. If the total amount of money was
$17.20, how many coins fell out of the machine?
Algebra 2
Problem Solving - Word Problems
Use a guess and check table to write an equation that could be used
to solve the following word problems.
1. Jessica and Jazylynn are twins. Their little sister, Jezebel is half as old as
they are. The sum of all three sisters' ages is 45. Find the age of each girl.
2. Eddie and Alejandro together have $125. If Eddie has $15 more than
Alejandro, how much does each person have?
3. A 102-centimeter board is cut into two pieces so that one piece is five times
as long as the other. How long is each piece?
4. Admission to the school dance was $3 in advance and $4 at the door.
There were 30 more tickets sold at the door than in advance, and ticket
sales totaled $1590. How many of each kind of ticket were sold?
5. Find three consecutive odd integers such that the sum of the smallest and
seven times the largest is 68.
6. Carla has three times as many dimes as nickels. She has four more
quarters than nickels. If the value of the coins is $9.40, how many nickels
does Carla have?
7. A rectangle has perimeter 48 feet. The length is three times the width. Find
the length and width of the rectangle. (Remember, the perimeter of a rectangle
is the sum of all four sides.)
Match the word problem to the algebraic expression. Put the number of the problem
in the box below the equation.
.05( x)  .10(3x)  .25( x  4)  9.40
x  x  3x  3x  48
x  5x  102
x  7( x  4)  68
xx
x  ( x  15)  125
3x  4( x  30)  1590
x
 45
2
GEOMETRY
POW #1 – The Checkerboard
How many squares are there on an 8x8 checkerboard? And the answer is
not 64.
Oh yeah, the answers is not 65 either.
Wilson High School
Activity Card
Wilson High School wants to boost interest in sports and
school activities. It has decided to sell an activity card that
will allow the holder to enter all sporting events free as well
as get discounts on other school activities. The entire
student body was surveyed and asked the question, “What
is the most you would pay for an activity card?” The results
of the survey are given below. Use the data to determine the
optimal ticket price.
Maximum Price
Total Number Willing to
Pay for Activity Card
50
75
85
105
115
135
150
170
765
620
565
460
405
285
210
115
Algebra
Quadratic Recap – Putting It All Together
Modeling with Quadratics Investigation
x
General Form
Vertex Form
y
Use the table to record
points from the parabola
in the picture.
Factored/Root Form
Wall Ball Investigation:
Find an Equation Algebraically
x (horizontal)
y (vertical)
Polynomials in the real world – Lagrange Polynomials Investigation
The equation to model the top part of the roller coaster is:
F ( x)  y0   0 ( x)  y1   1 ( x)  y2   2 ( x)  y3   3 ( x)
(___, ___)
x0
y0
(___, ___)
x1
y1
(___, ___)
x2
y2
(___, ___)
x3
 0 ( x) 
x  x1 x  x2 x  x3


x0  x1 x0  x2 x0  x3
 1 ( x) 
x  x0 x  x2 x  x3


x1  x0 x1  x2 x1  x3
 2 ( x) 
x  x0 x  x1 x  x3


x2  x0 x2  x1 x2  x3
 3 ( x) 
x  x0 x  x1 x  x2


x3  x0 x3  x1 x3  x2
F ( x)  y0 
F ( x) 
0
( x)  y1 
1
( x)  y2 
2
( x)  y3 
3
( x)
y3
Polynomial Function:
A polynomial function is a function of the form
f(x)=anxn + an-1xn-1 + … + a1x + a0
where an ≠ 0, the exponents are all whole numbers, and the coefficients are
all real numbers.
an is called the leading coefficient, n is the degree, a0 is the constant term.
23 3
1 2 1321
F ( x)  
x 
x 
x7
1260
630
1260
Mathematical Survivor
Project: Last One Standing
The cast of “Mathematical Survivor” is a collection of n not necessarily distinct real numbers {x1,x2,x3,…,xn},
where n=2. From this collection, we select any two numbers, say xa and xb, delete these from the collection, and
insert the number xaxb+xa+xb into the collection. The process to find the mathematical survivor includes
continuing the selection of two random numbers from the new collection and performing the deletion and
insertion process. Proceed until the collection has a single number left. This number is the survivor.
Part 1
Find the survivor for the set of numbers {2, 4, 6, 8}. You can randomly choose the two numbers to start
with. Once you find the survivor, repeat the process two more times to find the surviving number. Make
sure to alter your choice of numbers to delete.
How many different ways can a survivor be produced when starting with 4 numbers in your
collection.You know there has to be at least 3 different ways since you completed part 1.
EXTRA:
Part 2
Using you inductive reasoning skills, make a conclusion about the survivor from the set of numbers {2, 4,
6, 8}. Choose your own set of 4 numbers and find the survivor. Find the survivor two more times for
your set. What conclusions can you make now?
Part 3
The next obvious question is whether or not a mathematical survivor of a collection of numbers is
predictable at the outset and is it totally independent of the selection process made throughout the process.
The answer to the question is YES. The theorem below explains the results.
If S = {x1, x2,…, xn} is a collection of n not necessarily distinct real numbers, where nù2,then the
mathematical survivor of S is guaranteed to be
(x1+1)(x2+1)…(xn+1)-1
Use the theorem to verify that you found the correct survivor for the sets in Part1 and Part 2.
Part 4
Use the theorem above to find the survivors for the following sets.
{2, 2}
{5,5}
{2, 2, 2}
{2, 2, 2, 2}
{5,5,5}
{5,5,5,5}
{2, 2, 2, 2, 2}
{5,5,5,5,5}
Use the results from above to find the survivor for {c,c,…,c} (n copies), where c is some number .
EXTRA, EXTRA
1 1 1
1
Find the mathematical survivor for the following set: {1, , , ,..., }
2 3 4
n
Marty Romero, Wallis Annenberg High School, Adapted from Math Horizons Magazine, February 2007
What Goes Around Comes Around
Project: Hailstone Numbers
A particularly famous problem in number theory, the hailstone problem, has
fascinated mathematicians for several decades. It has been studied primarily
because it is so simple to state yet apparently intractably hard to solve. This
problem is also known as the 3n+1 problem, the Collatz algorithm, and the
Syracuse problem.
If the number is even, divide by 2, if it is odd, multiply by 3 and add 1.
Number
1
2
3
Steps to 1
1
7
2
11
12
13
Number
4
5
6
7
8
9
10
14
15
16
17
18
19
20
Steps to 1
Answer the following questions
1. What is the pattern for the number of steps for 2, 4, 8, and 16? Predict how
many steps the number 32 will have? How many steps for 128?
2. What is the pattern for any of the numbers that are doubles of each other? For
example, 5 and 10, 7 and 14?
3. Predict how many steps are needed for the numbers 40, 80 and 100. How many
steps does it take 76 to get to 1?
4. Fill in the missing numbers from the number chain. There are two answers.
_____, 15, 46, 23, …
_____, 15, 46, 23, …
5. Of the first 1,000 integers more than 350 have their maximum at 9,232. Find
one of the integers that has a maximum of 9232.
What Goes Around Comes Around
Project: Hailstone Numbers
Hailstone Numbers By Ivars Peterson Muse, February 2003, p. 17.
Nothing could be grayer, more predictable, or less surprising than the endless
sequence of whole numbers. Right? That's why people count to calm down and
count to put themselves to sleep. Whole numbers define booooooooring. Not so
fast. Many mathematicians like playing with numbers, and sometimes they
discover weird patterns that are hard to explain. Here's a mysterious one you can
try on your calculator. Pick any whole number. If it's odd, multiply the number by
3, then add 1. If it's even, divide it by 2. Now, apply the same rules to the answer
that you just obtained. Do this over and over again, applying the rules to each new
answer. For example, suppose you start with 5. The number 5 is odd, so you
multiply it by 3 to get 15, and add 1 to get 16. Because 16 is even, you divide it by
2 to get 8. Then you get 4, then 2, then 1, and so on. The final three numbers
keep repeating. Try it with another number. If you start with 11, you would get 34,
17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, and so on. You eventually end up at
the same set of repeating numbers: 4, 2, 1. Amazing! The numbers generated by
these rules are sometimes called "hailstone numbers" because their values go up
and down wildly—as if, like growing hailstones, they were being tossed around in
stormy air—before crashing to the ground as the repeating string 4, 2,
1.Mathematicians have tried every whole number up to at least a billion times a
billion, and it works every time. Sometimes it takes only a few steps to reach 4, 2,
1; sometimes it takes a huge number of steps to get there. But you get there
every time. Does that mean it would work for any whole number you can think of—
no matter how big? No one knows for sure. Just because it works for every number
we've tried doesn't guarantee that it would work for all numbers. In fact,
mathematicians have spent weeks and weeks trying to prove that there are no
exceptions, but they haven't succeeded yet. Why this number pattern keeps
popping up remains a mystery.
Algebra 2
Lecture Page 1
Objectives:
KNOWLEDGE:
Students will know…
The different representations for logarithms of
different bases.
How to correctly verbalize logarithmic notation using
proper math terminology
What the common logarithm is
The definition of logarithm
How to represent a logarithmic equation in
exponential form
Algebra 2
Lecture Page 1
Objectives:
Do Now: Answer the question and evaluate each numerical expression.
What number multiplied with itself is
equal to 81?
10 1 
1
2
144 
49 
Silent Board Game: Without talking to anyone, determine the numbers that complete the table.
X
16
Y
4
1
4
1
1
2
-3
4
2
1
32
-4
Record the relationship between the two variables . Then plot the points from the table to see the
graphical relationship.
Definition of __________________:
-5
Algebra 2
Lecture Page 2
Example 1: Evaluating Logarithms
Logarithmic
Expression
log 2 16 
How do you say it?
Log base 2 of 16 is …
What question is being
asked?
2 raised to what number
is 16?
2?  16
Answer
24  16  log 2 16  4
log 3 3 
log 10 100 
1
log 4   
4
Quick Note (Common Logarithm):
Guided Practice: Evaluate each logarithmic expression
log 3 81 
log 49 7 
log 0.1 
Example 2: Rewrite each logarithmic equation in exponential form.
Logarithmic Form
log 4 16  2
How do you say it?
Log base 4 of 16 is 2?
Exponential Form
4 2  16
log 1000  3
log 2 0.5  1
Make up your own examples and have a partner rewrite the logarithmic equation in exponential form
Think and Reflect:
When I was in high school, my Algebra 2 teacher Mr. Gunner used to say that a logarithm is just an
exponent. What did he mean by that? Write complete sentences using the appropriate vocabulary to
explain what he meant.