Download Scope and Sequence * Block #6, Standard 1: Integers, Prime and

Scope and Sequence – Block #6,
Standard 1: Integers, Rules of
Divisibility, Prime and Composite, and
Common Multiples/Factors.
1st - Integers: Students have built a number sense using Whole Numbers,
Fractions and Decimals in previous grades. Now we introduce Integers which
include negative numbers . Integers are then incorporated into student’s
understanding of numbers and operations with numbers.
•Students must learn to identify, read,
and locate integers on a number line.
•Zero is Neutral and separates the
positive numbers from the negative
numbers.
•Positive to the Right.
•Negative to the Left.
•Issue: Many students have trouble with
remembering left and right and get anxiety
when you orally teach the line in those terms.
Student’s can also be shown a vertical
number line, which is often used to
describe real- world situations and is
easier for some kids to understand first
before a linear number line.
Future: Becoming proficient at using recognizing negative numbers on a time line in 5th grade will
help students understand how to begin graphing on a Cartesian plane in 6th grade.
Issues: Other models may include unifix blocks
for kids who need greater hands on. This
would be used similar to chips but may feel
more comfortable to students who want it that
concept to take a linear shape too.
Zero
Students need to understand that integers have opposites.
Using chips can also help show opposites
and aid in using negatives within addition
and subtraction operations.
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-1 and 1, -2 and 2, -3 and 3… are
opposites. Meaning they are equal
distance from zero (equivalency?).
As you travel to the left the numbers
decrease in value.
As you travel to the right the
numbers increase in value.
Opposites make Additive Inverses.
This means that [a number + its
opposite = zero (3 + -3 = 0)]
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Blue is Positive
Red is Negative
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Issue: In [3 + -2]. + is the operation and
– is the value of 2. A common issue is
that kids forget about the value.
Places where integers can be used in students’ environment.
$$$$$$
Kids usually relate well to concepts of
money.
Kids are typically interested in science
and nature.
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You can relate negative numbers to
money by telling kids: “if you owe
your friend $2.00 and then you
borrow $3.00 more, how much do
you then owe your friend? [-2 + -3]
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You can relate negative numbers to
depths in the ocean.
You can relate negative numbers to
degrees below zero.
2nd – Rules of Divisibility:
Students have likely mastered there multiplication facts
and skip counting in third and fourth grade and will use these skills to learn and use
the rules of divisibility for 2, 3, 5, 6, 9, and10. However, let students develop as many
as possible.
For the Future: Students will use rules of
divisibility to check their work as they
encounter harder long division problems.
This will help with Prime factorization and
GCF/LCM.
They can also use these rules to engage in
more complex mental math - quickly
grouping items in everyday life.
Issues: Students misuse these rules by
picking the wrong rule. They will also
complete the task for the divisibility rule
and believe they’ve completed the actual
division(careful with low students).
Developing Rules of Divisibility
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2 – Students have learned skip counting
by 2 beginning in 2nd grade and have a
strong understanding that multiples of
2 always end in even numbers.
3 – Divisibility by 3 seems complicated
when reading the rule and looking at
big numbers, [231, 2+3+1= 6, 6/3=2,
231 is divisible]. but kids just need to
look at the pattern:
3,6,9,12,15,18,21,24… if you add up
digits they are divisible by 3.
5 – Skip counting by 5 also happens in
2nd grade and students will notice
pattern ending in 0 or 5.
6 – Students can begin to see when
they do prime factorization how this
rule works. If you can divide by 2 and 3
then the number is obviously divisible
by 6.
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9 – divisibility by 9 is also something
students will likely memorize.
10 – Students have also learned to skip
count by tens in second grade and
mastered quick times tables in 3rd
grade.
3rd - Strategies for finding Prime, Composite, or Neither in numbers up to 50.
Future: Students must begin work with finding Prime Factorizations in order to
become proficient with Common Denominators and Greatest Common Factors.
•Students need to begin by working with
• Once students play with the concrete
the smallest primes (2,3,5,7,11) and
model, and recognize patterns. Students
composites(4,6,8,10,12)to build an
can also use multiplication /division facts
understanding of what Prime and
and divisibility rules to find out if larger
composite means
numbers are prime or composite (ex. Any
•One way to do this is with a rectangular
# divisible by 2, 3, and 5…. are not prime
rearrangement model. If a number can
numbers).
6
be rearranged from a single column
rectangle (1 by__ ) into a two column
•Students can cross off numbers on a 10
rectangle ( 2 by __ ), then it is divisible
x 10 multiplication chart pretty quickly in
by 2 and thus is not prime. 2 is divisible
order to isolate the primes.
by itself and 1 only and is the first prime.
4
3
2
Next, students can begin to find Prime Factorizations of common numbers, using
simple division by 2, 3, 5, and 7 (the first four primes).
Students should know that this is referred to as the “Fundamental Theorem of
Arithmetic”.
Prime factorization of 72 is 3 squared x 2
Prime factorization of 60 is [2x2x3x5].
cubed [3x3x2x2x2].
Prime factorization of 28 id [2x2x7].
Primes to 50: 2,3,5,7,11,13,17,19,23,29,31,37,41,43,and 47d47
4th – Greatest Common Factor (GCF) and Least Common Multiple (LCM).
LCM: 5th and 6th graders are asked to add
and subtract fractions with different
denominators.
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LCM should first be explored by listing all
the multiples of easy numbers such as 4
and 6.
4: 4, 8, 12, 16, 20, 24
6: 6, 12, 18, 24
12 is the smallest/least.
The prime factorization of these two is 4:
2x2 and 6: 2x3. Eventually transition into
the understanding that multiplying the
prime factors of the largest power in each
number will give you the LCM. 2x2x3
Issue: The rule/prime factorization
method for LCM will not easily be
understood by students. This will require
lots of scaffolding and noting patterns.
Building on 4th grade concepts of equivalent
fractions… GCF is used to reduce fractions
into their simplest forms.
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GCF should again be explored using simple
numbers.
First get the Prime Factorization of 4 and 6
4: 2 x 2 and 6: 2 x 3
Take the biggest/greatest factor which
they both have (i.e. greatest common
factor). The GCF is 2.
So to reduce a fraction like 4/6 divide both
the numerator and denominator by the
GCF 2. Thus 4/6 = 2/3.
Block #6, Standard 2:
Order of Operations, Commutative,
Associative and Distributive
Properties
1st - Order of Operations
2nd - Commutative, Associative, and Distributive Properties.
PEMDAS – Please Excuse My Dear Aunt
Sally
• P, Parentheses ()
• E, Exponents ^• M&D, Multiplication & Division
(left to right)
• A&S, Addition & Subtraction
(left to right)
Ex.
3(2+3) = 3(5) = 15
(3x2)+ 3 = 6+3 = 9
15
9
The Commutative, Associative, and
Distributive Properties
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These Properties are covered using
manipulatives in 3rd Grade.
5th graders need to become more
familiar with using these properties
in the abstract form.
Commutative Property
3x2 = 2x3 = 6 OR 3+2 = 2+3 = 5
Associative Property:
3+(2+4) = (3+2)+4 = 9
Distributive Property:
3(4+2) = 3(4) + 3(2) = 18
Block #6, Standard 3:
PLOTTING INTEGERS
Plotting Integers
With the introduction of Integers our x, y graph we travel in the negative direction on
each axis, giving us four quadrants/the Cartesian plane.
Ordered Pairs (x,y) = (-3,2) & (3,2)
•Throughout the lower grades, students
learn to identify whole numbers on a
linear graph.
•In 4th grade students begin to plot points
in the first quadrant where whole
numbers /positive x and y exist.
In 5th grade:
• Students graph in all four quadrants.
• Students begin to investigate with
finding distances between points.
In 6th grade students will become more
adept at working on with the
Cartesian plane as they find
transformations of polygons.
Block #6, Standard 5:
PROBABILITY
Probability
• Probability is discussed in 3rd grade. It
also show how probability is inclusive
(between 0 -1, ¼ + ¼ + ¼ + ¼ = 1)
•In 3rd and 5th they impossible, likely,
unlikely, impossible are all used
•Probability is revisited in 5th grade with
new notation is 1:6 (x:y) (1 out of 6)
Issues: If students role the die 6 times
they will almost certainly NOT role
each number once. This can be
confusing to them.
It is important that students
understand that with each
INDIVIDUAL role of the die the
chances of getting any of the
numbers is equal at 1:6.
Block #7, Standard 1
Represent Fractions, Decimals,
Percents, and Fractional
Equivalencies, Comparing Fractions
by finding Common Denominators
and Converting Mixed Numbers and
Improper Fractions
Models of Fractions
Number-line Model of Fractions
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In 3rd Grade students must represent
halves, thirds, fourths, sixths, and
eighths.
In 4th Grade students also represent
fifths and tenths.
In 5th Grade students also represent
improper fractions and mixed
numbers.
Mixed number
11/2
Area Model for 1 ½
Set Model for 1 ½
Models of Fractions
Geo-board: Using a Geo-board is concrete
and students can help kids build different
fractions. Below students can easily build
3/6. Students can easily build an
understanding of equivalent fractions, it’s
easy to see that 3/6 = 1/2 = 12/24
This is a representational model which is
scaffolding students to working with an
abstract model.
Issues: Kids have problems
understanding fractions, but the candy
bar scenario seems to usually be helpful.
You get 1/2 of a candy bar. 1/2 means
dividing the candy into ___(2)___
shares, and keeping __(1)__ share.
You’ll want to give them LOTS of
repetition with these types of
exercises.
Decimals and Percents on a number-line
and Decimal/ Fraction Equivalency
Issues: equivalencies can be hard for kids so this is a good model to use.
This model is a cool way to take area models and have
students link it to a number-line. Students will be able
to link the visual representation of which fractions are
biggest and how we order fractions on a number-line.
0%
30%
100%
Comparing Fractions by finding
Common Denominators
The Concrete Model:
The Pictorial model:
We saw this models used in 4th Grade.
Let students discover various multiples
smaller fractions which are the same
height as lesser multiples of other larger
fractions.
This scaffolds toward the
rules/algorithm for adding fractions with
different denominators.
1/2 + 3/8 = ? Look how many 8ths are in
1/2. There are 4, so we now have 4/8 +
3/8 = 7/8
Converting Mixed Numbers and
Improper Fractions
Pictorial Models:
Improper
Number-Line Model:
Represent Commonly Used Fractions as
Decimals and Percents
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1/2 - $1.00/2 is 50 cents = 50%
1/4 - $1.00/4 is 25 cents = 25%
1/5 – Stays from our typical monetary
intervals, but is a good way to transition
students to thinking more about the
concept of percents.
There are 100 pennies in $1.00 and if
you cut it into 5 parts =
100pennies/5parts = 20pennies = 20%
$1.00/100parts = 100pennies/100 parts
= 1 penny = 1%.
So if you divide any number by 100
parts you get 1% of that number.
So, 220/100 = 2.2 which is 1% of 220.
Now if you need to find 10% or 15%
you can multiply by 10 or 15.
Issues: Students often have problems
understanding percents as being
hundredths.
Help students connect cutting/dividing a
whole into 100 parts with the
hundredths place on a place value chart.
Equivalent Fractions and Recognition
of Simplest Form
Fraction strips/pictorial model with
equivalencies highlighted.
Transition to abstract model, with
special attention paid to the simplest
form.
Simplest form of equivalent fractions is
found by dividing the denominator by
the numerator. Generally reducing
fractions into their simplest form is done
by dividing both the numerator and the
denominator by their GCF.
Rename Whole Numbers as Different
Fractions
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1 = 1/1, 2/2, 3/3, 4/4, 5/5, 6/6, 7/7,………….
2 = 2/1, 4/2, 6/3, 8/4, 10/5, 12/6, ……………
Etcetera, etcetera, etcetera…………………..
Students can count up
parts of fraction strips
to see that there are 2- 1/2s,
5 – 1/5s,
Or…. 8 – 1/8s which is 8/8s = 1.
This activity can help students
prove this concept to
themselves.
Block #7, Standard 2:
Analyzing Patterns and Making
Predictions
Patterns
• Input/output chart based on knowledge of exponents. Students must also
learn how to write algebraic expressions: x^2 = y
Input
x
2
4
6
8
10
Out
-put
y
4
16
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• Based on the following pattern what will the 20th letter be?
a,b,c,d,a,b,c,d,a,b,c,d,a,b
• Kids can see that every 4th letter is d and they know that 20/4 = 5 with no
remainder. Thus the 20th letter will be d in the 5th repeated sequence.
• Or kids can draw lines as place holders out to the twentieth spot and just
fill in the pattern.