Transcript BLOCK20-Bx - Math GR. 6-8

It is not enough to
know the skills. It is
important to know
how to use these skills
to solve real-world
problems.
Problem solving
touches every aspect
of our lives.
Problem solving is an integral part of all
mathematics learning. In everyday life and in
the workplace, being able to solve problems
can lead to great advantages.
c
a
b
In the real-world
would you be asked
to find the length
of c or …….
Pleasantville
Amanda needs to go to Holiday City. Since she
is in a hurry she needs to take the shortest
route possible.
What route should she take and how many
miles will she save?
32 miles
Merry Town
48 miles
Holiday City
What steps should we take when solving
a word problem?
1. Understand the problem
2. Devise a plan
3. Carry out the plan.
4. Look back
UNDERSTAND THE PROBLEM
Ask yourself….
•What am I asked to find or show?
•What type of answer do I expect?
•What units will be used in the answer?
•Can I give an estimate?
•What information is given?
•If there enough or too little information
given?
•Can I restate the problem in your own words?
DIVISE A PLAN
The plan is usually called a strategy
Problem-solving strategies include:
•Act is out
•Make a drawing or diagram
•Look for a pattern
•Construct a table
•Identify all possibilities
•Guess and check
•Work backward
•Write an open sentence
•Solve a simpler or similar problem
•Change your point of view/logical reasoning
CARRY OUT THE PLAN
Now solve the problem.
The original strategy may
need to be modified, not
every problem will be
solved in the first attempt.
LOOK BACK
This is simply checking all steps and
calculations. Do not assume the problem
is complete once a solution has been
found. Instead, examine the problem to
ensure that the solution makes sense.
What strategy would you use to solve the following problem?
George has written a number pattern that begins with 1, 3, 6, 10, 15. If
he continues this pattern, what are the next four numbers in his
pattern?
What do you need to
find?
How can you solve the problem?
SOLVE:
You need to find 4 numbers after
15 that fit the pattern.
You can find a pattern.
Look at the numbers in the pattern.
3 = 1 + 2 (starting number is 1, add 2 to make 3)
6 = 3 + 3 (starting number is 3, add 3 to make 6)
10 = 6 + 4 (starting number is 6, add 4 to make 10)
15 = 10 + 5 (starting number is 10, add 5 to make 15)
Using a
pattern made
this easy
Following this pattern…
Starting with 15, add 6 to make 21.
Starting with 21, you add 7 to make 28
Starting with 28, you add 8 to make 36.
Starting with 36, you add 9 to make 45
So the next four numbers would be
21, 28, 36, and 45
What strategy would you use to solve this problem?
Karen has 3 green chips, 4 blue chips and 1 red chip in her
bag of chips. What fractional part of the bag of chips is
green?
What do you need to find?
You need to find how many chips in all .
Then you need to find what part of this
total is green.
How can you solve this?
Try drawing a picture.
SOLVE:
Draw 8 chips.
G
G
G
B
3 OUT OF THE 8 CHIPS ARE GREEN
B
3
8
B
B
R
What strategy would you use to solve this problem?
Steven walked from Coral Springs to Margate. It took 1 hour 25 minutes to walk
from Coral Springs to Pompano Beach. Then it took 25 minutes to walk from
Pompano Beach to Margate. He arrived in Margate at 2:45 P.M. At what time did
he leave Coral Springs?
What do you need to find?
How can you solve the problem?
You need to find what the time was when
Steven left Coral Springs.
You can work backwards from the time
Steven reached Margate.
SOLVE:
Start at 2:45. This is the time Steven reached Margate.
Subtract 25 minutes. This is the time it took to get from Pompano Beach to Margate.
The time is 2:00 P.M.
Subtract: 1 hour 25 minutes. This is the time to get from Coral Springs to Pompano
Beach.
Steven left Coral Springs at 12:55 P.M.
Another helpful tool
when organizing your
thoughts and solving
problems is to use
graphic organizers.
There are many different
types you can use.
Types of Graphic Organizers
• Hierarchical diagramming
• Sequence charts
• Compare and contrast charts
A Simple Hierarchical Graphic Organizer
A Simple Hierarchical Graphic Organizer example
Geometry
Algebra
MATH
Calculus
Trigonometry
Compare and Contrast
Category
Illustration/Example
What is it?
Properties/Attributes
Subcategory
Irregular set
What are some
examples?
What is it like?
Compare and Contrast - example
Numbers
Illustration/Example
What is it?
6, 17, 25, 100
-3, -8, -4000
Properties/Attributes
Positive Integers
Whole
Numbers
0
Negative Integers
Zero
Fractions
What are some
examples?
What is it like?
Venn Diagram
Venn Diagram - example
Prime Numbers
5
7
11
13
2
3
Even Numbers
Multiples of 3
4
8
6
10
6
9
15
21
Multiple Meanings
Multiple Meanings – example
Right
Equiangular
3 sides
3 sides
3 angles
3 angles
1 angle = 90°
3 angles = 60°
TRIANGLES
Acute
Obtuse
3 sides
3 sides
3 angles
3 angles
3 angles < 90°
1 angle > 90°
Series of Definitions
Word = Category
=
+ Attribute
+
Definitions: ______________________
________________________________
________________________________
Series of Definitions – example
Word = Category
Square
=
+ Attribute
Quadrilateral
+
4 equal sides &
4 equal angles (90°)
Definition: A four-sided figure with four
equal sides and four right angles.
Four-Square Graphic Organizer
1. Word:
4. Definition
2. Example:
3. Non-example:
Four-Square Graphic Organizer – example
1. Word: semicircle
4. Definition
A semicircle is half of a
circle.
2. Example:
3. Non-example:
Matching Activity
• Divide into groups
• Match the problem sets with the
appropriate graphic organizer
Matching Activity
• Which graphic organizer would be most
suitable for showing these relationships?
• Why did you choose this type?
• Are there alternative choices?
Problem Set 1
Parallelogram
Square
Polygon
Irregular polygon
Isosceles Trapezoid
Rhombus
Quadrilateral
Kite
Trapezoid
Rectangle
Problem Set 2
Counting Numbers: 1, 2, 3, 4, 5, 6, . . .
Whole Numbers: 0, 1, 2, 3, 4, . . .
Integers: . . . -3, -2, -1, 0, 1, 2, 3, 4. . .
Rationals: 0, …1/10, …1/5, …1/4, ... 33, …1/2, …1
Reals: all numbers
Irrationals: π, non-repeating decimal
Problem Set 3
Addition
a+b
a plus b
sum of a and b
Multiplication
a times b
axb
a(b)
ab
Subtraction
a–b
a minus b
a less b
Division
a/b
a divided by b
a÷b
Problem Set 4
Use the following words to organize into
categories and subcategories of
Mathematics:
NUMBERS, OPERATIONS, Postulates, RULE,
Triangles, GEOMETRIC FIGURES, SYMBOLS,
corollaries, squares, rational, prime, Integers,
addition, hexagon, irrational, {1, 2, 3…},
multiplication, composite, m || n, whole,
quadrilateral, subtraction, division.
Possible Solution to PS #1
POLYGON
Parallelogram:
has 2 pairs of
parallel sides
Square, rectangle,
rhombus
Trapezoid,
isosceles trapezoid
Quadrilateral
Trapezoid: has 1
set of parallel
sides
Kite: has 0 sets of
parallel sides
Kite Kite
Irregular: 4 sides
w/irregular shape
Possible Solution to PS #2
REAL NUMBERS
Possible Solution PS #3
Addition
Subtraction
____a + b____
____a - b_____
___a plus b___
__a minus b___
Sum of a and b
___a less b____
Multiplication
___a times b___
____a x b_____
_____a(b)_____
_____ab______
Operations
Division
____a / b_____
_a divided by b_
_____a  b_____
Possible Solution to PS #4
Mathematics
Numbers
Rational
Prime
Integer
Operations
Addition
Subtraction
Rules
Symbols
Postulate
m║n
Corollary
√4
Geometric
Figures
Triangle
Hexagon
Multiplication
Irrational
Whole
Composite
{1,2,3…}
Division
Quadrilateral
Mike, Juliana, Diane, and
Dakota are entered in a 4person relay race. In how
many orders can they run the
relay, if Mike must run list? List
them.
Mrs. Stevens earns $18.00 an hour
at her job. She had $171.00 after
paying $9.00 for subway fare. Find
how many hours Mrs. Stevens
worked.
Try solving this problem by working
backwards.
Use the work backwards strategy to
solve this problem.
A number is multiplied by -3. Then 6
is subtracted from the product. After
adding -7, the result is -25. What is
the number?