Transcript Chapter 20: Pre-Algebra

By Elisabeth, Joanne, and Tim
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Chapter 20 Overview: New Skills
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Pre-skills and Pre-Assessment
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Example Lesson: Intro to Variables and Tables
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Chapter 20 Application Items
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Using the coordinate system to visually represent
values of variables (example lesson to come)
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Finding a missing value in a ratio equation and
other ratio-related problem solving
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Using prime numbers to reduce fractions
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Simplifying expressions that involve exponents
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Operating on integers (using rules to work with
negative numbers, see next slide)
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Rules for combining integers (Format 20.3)
1. If the signs of the numbers are the same, you add
2. If the signs of the numbers are different, you
subtract
3. When you subtract, you start with the number that
is further from zero on the number line and
subtract the other two
4. The sign in the answer is always the sign of the
number that was further from zero
(Another option is to use a number line to
teach the actual concepts involved)
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The traditional four rules for multiplying:
1.
2.
3.
4.
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Positive x Positive = Positive
Negative x Negative = Positive
Negative x Positive = Negative
Positive x Negative = Negative
A one-rule alternative is described (p.451)
Revise any statements that uses its underlined
word incorrectly.
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The specific unit measured by a ruler is how
much the numbers stand for (for example
inches and centimeters are examples of
specific units)
The type of unit measured by a tape measure
is distance
Use the number line for these ones.
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How far is 4 from -3?
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Draw a vertical axis at 0
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Label the x axis and y axis on your drawing
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Plot a point at (3,5)
When a problem involves an unknown
amount, you can use a symbol like x (which
is called a variable) to stand for the unknown
value.
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In algebra, a function is a way to relate two
variables to each other
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The ratio of a dirt bike ramp's height to its
width might be 1 to 2
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2, 3, 5, 7, and 11 are prime numbers
Also think about including some of these on
your pre-assessment:
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Solving problems involving proportion or
percent, multiplying two powers of the same
base number, reducing fractions, and
operating on negative numbers
1
2
3
Variable
Number Line
Coordinate Plane
• Units
• Horizontal
• Vertical
• Table
• Axis (1-dim.)
• Axes (2-dim.)
• Scale
• Coordinate
• Ray vs. Line
• Ordered Pair
• Operations
• Function
• Axes (2-dim.)
3) What is a
Coordinate
Plane?
• Horizontal
• Axis (1-dim.)
• Scale
• Ray vs. Line
• Operations
• Vertical
• Coordinate
• Ordered Pair
• Function
2) How do
you use a
Number
Line?
1) What is a
Variable?
• Unit
• Table
Lesson 1: Pages 443 to 445
Introduction to VARIABLES
1.
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This is the opener
lesson for a unit that
culminates in the use
of Graphical and
Algebraic models
Students first need a
good understanding
of what variables are
and how to represent
them numerically (in
a table)
2.
3.
Given a single variable, students will generate and
consider reasonable values
Given a table with values of one variable, students
will plot points on a number line
Given two related variables, students will generate
and consider reasonable pairings of values and use a
table to display their pairs
Small group (4 or fewer) activity
Make lists of values for units
Present next ideas to large group
Decide on symbols to use for units
Make a table of values for a unit
Mark values on a number line
Individual practice
Turn lists into tables, mark values
Present next ideas to large group
Add a column to a table
Generate reasonable pairs
Use statements to check pairs
Small group practice
Add columns, generate pairs, check, save tables for later
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Hand out envelopes
◦ In these envelopes are cards, each labeled with the
name of a measurable aspect of a person’s life
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Group task (3 min):
◦ each person take one of the cards
◦ on scratch paper, list 3 reasonable values for the
aspect labeled on your card (your own current value
for that aspect is just one that you can use – think
of some for other people too)
◦ Switch cards and do it again until you have listed 3
values for each of the four aspects
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The four cards are examples of aspects about
a person that can have different values.
◦ What should we call things like this that can vary so
much in their value?
◦ Choose a fitting one-letter symbol for each of the
four aspects we looked at
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In algebra, when you have a bunch of values for a
variable, there is more than one way to display those
values
◦ You can display it using a table. The symbol for that
variable goes at the top of the column. Values for the
variable go (in ascending order) below it.
◦ You can also display the values for a variable using a
number line. Here, the number line (or axis) itself is labeled
with the symbol for the variable and dots are made along it
where the values are located.
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Individual task:
◦ Display the numbers from your four lists using the four
tables and four number lines on the handout
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We’ve seen how to display the values for a
variable when it is alone
Variables can also be related to each other
Which of the four aspects make good pairs?
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If we transfer the values for one of the (cause)
variables in a pair to a 2-column table, we
can think about the resulting values for the
other variable in the pair
On your own:
◦ Choose one of your pairs of related variables
◦ Put the values for the causing partner into the first
column of a 2-column table (on back side of sheet)
◦ Put reasonable accompanying values for the other
variable in the second column
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A number line is only able to display values
for one variable
◦ (There is only one axis and it can stand for only one
type of value)
How can we display with dots when there are
two variables involved and we have to show the
value of one based on the value of the other?
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1. Complete the following function table.
x
Function
x times 4
Answer
y
0
1
2
3
4
5
0x4
1x4
2x4
3x4
4x4
5x4
0
4
8
12
16
20
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2. Construct a set of problems for use in Format 20.3 for combining integers.
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Make a poster board titled: Steps to Combining Integers
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Step 1: If number signs (negative, negative) and (positive, positive) are the same: add numbers.
Step 2: If number signs (positive, negative) and (negative, positive) are different: subtract numbers.
Subtract from the number farthest from the zero on number line.
Step 3: Solve the problem.
Step 3: Always, look and take the number sign from the higher number closest on the number line from zero and attach it to your
answer.
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For example: -20 + 8 = ____
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The number signs are different, so subtract 8 from 20. (negative, positive)
Result: -12
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Twenty is the larger number and it has a negative sign in front of it, put a negative sign in front of 12. Final answer is -12
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Another example: 15 – 8 = ____
The number signs are different, so subtract 8 from 15. (positive, negative)
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Fifteen is the larger number and it is positive, so put a plus sign or leave it blank in front of 7. Final answer is 7.
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One more example: 4 + 5 = ____
The number signs are the same, so add them both together.
Five is the larger number and it is positive, so put a plus sign or leave it blank in front of 9. Final answer is 9.
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3. Construct a set of examples and nonexamples for students to use in
detemining if a number is a prime number.
Rule: All prime numbers have only two different factors: 1 and itself,
that’s it.
Non-example 1 x 1 = 1 all the same factors
Example 1 x 2 = 2, 1 and 2 are the 1 and itself factors
Non-example with 4, 1 x 4 = 4, 2 x 2 = 4, 3 different factors, 1, 2, and
4, does not follow rule
Example: 1 x 5 = 5, 1 and 5 are the 1 and itself factors
Non-example with 10, 1 x 10 = 10, 2 x 5 = 10, 4 different factors, 1, 2,
5 and 10 does not follow rule
4. Explain why initial teaching and practice of the concept of absolute value is
best accomplished using the phrase farther from zero rather than absolute
value.
Integers are made up of positive and negative numbers. Conceptually it can be
more challenging, some students may find it difficult to understand negative
numbers, therefore, Stein recommends referring to a number line for visual
representation with the number zero as the reference point for the necessary
steps, such as left of line as negative and right of line as positive. Also for
number sign to final answer by looking for the greater number from the
number zero. Unlike the combining integers strategy, teaching the conventional
notation (teaching absolute value, first), the student needs to understand the
first number with the opposite number and determine correct number sign, for
example, -4 – 5 =, the student would have to understand that conceptually the
problem is
4 – 5 = because the absolute value of -4 is 4. More complicated problems, could
be even more difficult to understand, such as 4 – (-5) = 4 – 5 =, and so forth.
Another concept to grasp, when the problem has a plus sign for computation,
whether it is (positive, positive, or negative, negative) the final answer will
always be positive. 4 + 3 = 7, -4 + (-3) = 7. Furthermore, the absolute value
rules are different for multiplication and division.
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5. Outline the steps in teaching students to find the prime factors of a number.
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Let’s take the numbers 18 and 32. We can also make this a fraction 18/32.
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Work out all the factors.
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18 divided by 2 = 9
9 divided by 3 = 3
Now the prime number rule: two different factors, 1 and itself.
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Take all the prime factors: 2 x 3 x 3 = 18
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32 divided by 2 = 16
16 divided by 2 = 8
8 divided by 2 = 4
4 divided by 2 = 2
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2 x 2 x 2 x 2 x2 = 32
2 x 3 x 3 = 18
3x3=9
2 x 2 x 2 x 2 x 2 = 32 can reduce, 2 x 2 x 2 x 2 = 16
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6. Write a classification problem with fractions to
be solved with ratio table, and show the solution.
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Three-fourth of the fifth grade class are boys.
There are 10 girls in the fifth grade class. There
are a total of 40 fifth grade students. How many
Fraction
Ratio
Quantity
boys are there?
5th grade
3/4
3
30
1/4
1
10
4/4 = 1
4
40
boys
5th grade
girls
Total fifth
graders
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7. Show the solution process to the following
comparison problem with percentages, using the
ratio table shown.
Show the solution process to the following
comparison problem with percentages, using the
ratio table shown.
Ben is 35% taller than his mom. If Ben is 21 inches
taller than his mom, how tall is Ben and how tall is his
mom? (Hint: Ben is being compared to his mom, so
who is equal to one or 100%)
Solution process: 21=.35X.