TEACHER NOTE: if you are using the INVESTIGATIONS series, see
MATHEMATICAL THINKING AT GRADE 5 – Investigation 1 and 2.
I.
Grade Level/Unit Number:
Grade 5, Unit 1, Activity Set 1
II.
Unit Title:
Numbers, Operations and Algebra
III.
Unit Length:
2 weeks
IV.
Indicators Addressed:
5-2.7 Generate strategies to find the greatest common factor and the least
common multiple of two whole numbers. C3
5-2.9 Apply divisibility rules for 3, 6, and 9. C3
V.
Materials Needed: See Lessons
Final Project: Students will create game questions for Prime/composite,
GCF/LCM. Each question and answer should be written on a separate index
card. Combine the cards and create student groups to play the game.
Key Learning:
Divisibility, prime and composite numbers, least common multiples and
greatest common factors are all critical components of everything else that
will be covered this year.
1
New Essential Mathematics Vocabulary:
Composite number
Prime number
Students in third grade identified numbers as odd or even and in 4th
grade they found factors of a given number up to 50. These are
prerequisites for students at the fifth grade level who are expected to
identify prime and composite numbers.
Students need experiences that will help to enhance their
development of number sense and in examining the properties of numbers.
Students have the notion that all odd numbers are prime. The notion that all
odd numbers are not prime needs to be emphasized. The misconception can
be addressed and the concept introduced through the suggested lesson.
Suggest that this kind of special number is one that can be useful for
doing mental arithmetic and for working with fractions.
The following activities are from: Prime and Composite Numbers – 5th
Grade – CEEMS document
2
3
•
•
•
Determine whether a given two-digit whole number is prime or
composite with the aid of a corresponding pictorial array.
Identify a one-digit number that is prime.
Determine how many factors a prime number can have.
Lessons that Promote Understanding Include:
Materials Needed:
• Hundreds chart
• Chart markers
Distribute a copy of the hundreds chart. Worksheet 1 Ask students
to put a square around the 1. Next, circle the number 2 and cross out the
multiples of 2. Then circle the 3 and proceed to cross out the multiples of
3. The 4 is already crossed out, so move on to the 5. Circle the 5 and cross
out all multiples of 5. Continue using this method until all the numbers are
4
either circled or crossed out. Closely monitor students. Ask the students to
determine what the circled numbers have in common and how the circled
numbers are different from the crossed out numbers. When it is determined
that all the circled numbers have itself and one as factors, the terms
“prime” and “composite” can be introduced.
Ask students to explain the characteristics of prime numbers and
composite numbers through their journal writing; for example, ask them to
define prime and composite numbers and provide examples to further
illustrate their understanding of this concept. They should also explain in
elementary terms why the number “2” is the only even prime and why 1 is
neither prime nor composite. They should explain by writing why all odd
numbers are not prime and illustrate this concept by drawing a Venn diagram.
Using 2 circles that do not intersect, label one circle for prime numbers and
the other for composite numbers. They can use the numbers 1-100 for this
activity. Instruct them to put any number they found as neither prime nor
composite outside the Venn.
For additional practice of identifying prime and composite numbers,
use the interactive sites listed:
http://www.aaamath.com/fra63a-primecomp.html
http://illuminations.nctm.org/mathlets/factor/index.html
(Note: Remember to check the sites prior to student use as they do change
periodically.)
Top
1. Divisibility Rules: students in grade 5 are required to know divisibility
rules for 3, 6, and 9, but 2, 5, and 10 should also be reviewed.
POWERPOINT - divisibility rules ppt
Worksheet 2 *
(divisibility notes)
*SEE ATTACHMENT worksheet 3 and Key (worksheet with missing
numbers and key)
Top
2. Prime and composite numbers are hard concepts for students to grasp.
One is a special number- neither prime nor composite. This discussion will
help to direct the explanation of all prime and composite numbers.
Composite numbers- numbers with more than 2 factors
Prime numbers- numbers with ONLY 2 factors. (No more, no less)
Reminder: ONE only has one factor.
Ways to Check for Divisibility
5
Factors are the numbers that can be multiplied together to get the
desired product.
EX: factor x factor = product
EX: 54: 1,2,3,6,9,18,27,54 composite
EX: 41: 1,41 prime
http://www.toonuniversity.com/flash.asp?err=499&engine=14: and here is a
cool site for games with prime factoring.
http://www.purplemath.com/modules/factnumb.htm : and one more site that
walks you through this type of factoring. Well done!
Top
3. GREATEST COMMON FACTOR (GCF)
Tell students this skill will benefit them as they begin to work with
reducing fractions!
GCF: the largest factor that is common to both numbers.
LEAST COMMON MULTIPLE (LCM): the smallest common multiple between
two numbers.
Prime Factoring Flip Chart / OR / PowerPoint If you have Promethean on
your computer, simply drag this icon into “My Flipcharts” and then it will
open. If not, you can use the PowerPoint.
PowerPoint
Least Common Multiple Activity* worksheet 4
(hundreds chart w/ bingo chips) - purchase of items can be made at
http://www.alliedbingo.com/home/ab2/smartlist_65/tubs_of_chips.html
* ASSESSMENT – worksheet 5 A , B
(prime factorization, GCF, LCM test and key)
Worksheet 2
back
Divisibility Notes
Rule for 2= must be an even number
Rule for 3= the sum of the digits is divisible by 3
example: 138 1+3+8=12 12 is divisible by 3
Rule for 4= the last 2 digits form a number that is divisible by 4.
Example: 2,324 24 is divisible by 4
6
Rule for 5= Rule for 6= number is divisible by 3 and 2
Example: 2,622 The number is even so it is divisible by 2
2+6+2+2=12 and 12 is divisible by 3
It is divisible by 2 and 3, so it is divisible by 6 also
Rule for 9= the sum of the digits is divisible by 9
Example: 567 5+6+7=18
18 is divisible by 9
Rule for 10= number ends in 0
7
Worksheet 3
back
Name: ___________________________
DIVISIBILITY RULES:
** Make each number divisible by both 3 AND 9.
1.
6___
2.
46___
3.
53___,27___
4.
3,___21
5.
7,___ ___ 5
6.
9___, 4___3
**Make each number divisible by 6.
7.
7 ___
8.
53___
9.
8___,4___2
*** Make this number divisible by 3,6, and 9.
10. 7___4,___2___
Worksheet 3 KEY
Name: _______KEY__
DIVISIBILITY RULES:
** Make each number divisible by both 3 AND 9.
1.
6___ (3)
NOTE, order within ( ) does NOT matter.
2.
46___ (8)
3.
53___,27___ (1,0) (3,7) (1,9) (4,6) (5,5)
4.
3,___21
(3)
5.
7,___ ___ 5 (0,6) (1,5) (2,4) (3,3) (7,8) (9,6)
6.
9___, 4___3 (2,0) (1,1) (2,9) (3,8) (4,7) (5,6)
**Make each number divisible by 6.
7.
7 ___ (2) (8)
8.
53___ (4)
9.
8___,4___2
NOTE: LOTS OF POSSIBILITIES… NUMBER COMBINATIONS
*** Make this number divisible by
3, 6, and
9.SUMS OF EITHER:
SHOULD
HAVE
1, 4, 7, 10, 13, OR 16
10. 7___4,___2___
***Last digit must be EVEN.
The SUM of the 3 missing digits must equal
EITHER 5 or 24.
8
Worksheet 5
back
Explain what GCF is, and how it will help you with fractions.
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
1. Explain what LCM is, and how it will help you with fractions.
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
2. Classify these numbers as Prime or Composite. (there is a tricky one!!!)
1. 5 _____________
2. 30_____________
3. 14_____________
4. 9_____________
5. 57_____________
3. Compare these fractions. (><=)
¼ ____ ½
2/3___ 3/5
4/10___2/5
4. Use PRIME FACTORIZATION to find GCF and LCM.
1. 12: 30
2. 56: 60
3. 90: 35
9
1. Explain what GCF is, and how it will help you with fractions.
_______GCF is used to help reduce fractions to simplest form.
2. Explain what LCM is, and how it will help you with fractions.
_______LCM is used to find common denominators to enable us to
add and subtract fractions.
3. Classify these numbers as Prime or Composite. (there is a tricky one!!!)
a. 5 ____prime_________
b. 30________composite_____
c. 14_____composite________
d. 9______composite_______
e. 57____composite_________
4. Use PRIME FACTORIZATION to find GCF and LCM.
a. 12: 30
GCF: 6
LCM: 60
b. 56: 60
GCF: 4
LCM: 840
c. 90: 35
GCF: 5
LCM: 630
Worksheet 5B key
10
rksheet 4
back
11
Back
Worksheet 1
12
I.
Grade Level/Unit Number: Grade 5/Unit 1, Activity Set 2
II.
Unit Title:
Numbers, Operations and Algebra
III.
Unit Length:
1 week
IV.
Indicators Addressed:
5-3.4 Identify applications of commutative, associative, and distributive
properties with whole numbers. B1
V.
Materials Needed - see each section
Final Project
TEACHER NOTE: When teaching order operations, be certain your students
know that multiplication and division are to be read in order from left to
right. Then, addition and subtraction are read from left to right.
Order of Operations: PEMDAS
Do all multiplication and
division from left to right, not
multiplication then division!!!!
Do all addition and subtraction
from left to right, not addition
then subtraction!!!!
Please (parentheses)
Excuse (exponents)
My (multiplication)
Dear (division)
Aunt (addition)
Sally (subtraction)
13
This is a neat mnemonic device for students to use to memorize the order of
operations.
EX: 2x3+4-2÷2=
6 + 4- 1=9
Parentheses
Exponents
Multiply or Divide
Add or Subtract
Please
Excuse
My Dear
Aunt Sally
•
Do the operations within each level from left-to-right.
•
Write the answer directly below the operation sign.
•
Do not 're-use' any numbers.
•
POWERPOINT
Order of
Operations
By: Lisa Foti
EDT 210 Wednesday 9–10:50
If you have Promethean software, simply drag and drop (or copy and paste)
these two icons into “My Flipcharts” and they will open.
math_order_operations.flp
NumberTheory_OrderofOperations.flp
**** SEE ATTACHMENT 1A worksheet 1 & B worksheet 2**** (worksheet
and key)
Snake
Duration: 30-45 minutes
An AskERIC Lesson Plan
14
Description: Snake is a fun and highly interactive math game for practicing
basic math skills which can be used by substitute teachers, to fill unplanned
time, and as a treat at the end of the day.
Goals: Increase proficiency in basic addition, subtraction, multiplication, and
division facts.
Objectives: Students will use mental math to compute answers to arithmetic
problems. Students will add columns of numbers to arrive at a total score.
Materials:
pair of 6-sided dice (for older grades use 10-, 12-, or 20-sided dice)
calculator
paper
pencil
Procedure:
Each student writes the word "SNAKE" in large letters at the top of a piece
of paper, making a column under each letter. All students stand by their
desks, and the teacher rolls the dice and announces the two numbers.
Depending on the grade level or the ability of the students, the teacher can
have students add, subtract, multiply, or divide the two numbers. The
students will do so quietly in their heads and then enter their answers in the
first column "S." (The teacher should record each answer on a piece of paper
as well.) The students continue to record their answers in the "S" column
until they choose to sit down and play it safe or until the round ends. When a
student has chosen to sit down, he/she can no longer collect points and must
wait until the next round to stand up and rejoin the game. A round (or
column) ends when one of the following occurs:
1. All the students have chosen to sit down.
2. The teacher has rolled a 1 on one of the dice. In this case, all the
students who are still standing will lose all their points for that column only.
Their total for that column will go to 0.
3. The teacher has rolled "snake eyes." In this case, all the students who are
standing will lose all their points in each completed column and in the current
column. Their score will now be 0.
After a round ends, all the students may stand up again and begin
collecting points for the next column. After all five rounds have been played,
students will add up all of the columns to determine their total score.
Students with the highest overall score win. Students can double check
their calculations with a calculator. The teacher's answer sheet can also be
used to verify students' scores. Be certain to render it appropriate for the
grade level of the students and offer a reward when feasible.
15
One difficulty of the game is trying to keep students from calling out
the answers. If students are silent, then all children can practice their
mental math facts. Also, when students have their scores, they tend to rush
up front to show the teacher. So make a rule that students must remain in
their seats and raise their hands when ready to share scores. Display a
child's total score on the board and ask if anyone has a higher score; repeat
this until the top two or three scores are identified.
Assessment: Observe students' participation throughout the game. Collect
students' sheets and compare their answers with the teacher's answer
sheet.
Properties of Whole Numbers:
Commutative Property: Works only for addition and multiplication. Numbers
can be placed in any order and still have the same sum or product.
Ex: 4+5=9 or 5+4=9
6x3=18 or 3x6=18
Title - The Commutative Property
By - Jennifer Dalke
Lesson 3: The Commutative Property, Using Arrays
Materials
Cans
flashcards
grid paper
small stickers
small bags of buttons
Anticipatory Set
Referring to an array of cans (4 x 6) stacked on the table at the front of
the room, ask students if they have ever seen something similar to this.
Where? Explain that this is called an ‘array’: an arrangement of items in a
number of equal-sized rows. Ask students where else they have seen arrays
of items. Challenge them to identify arrays that are in the classroom. (Be
certain there are a number of possibilities before beginning the lesson.)
Finally, point out a blank bulletin board and tell them they will be making
their own bulletin board with the work they do today.
Guided Practice
1. Call students’ attention to the array of cans at the front of the room,
again. Ask, How many rows across there are of the cans? How many cans in
each row? I will write "4 rows of 6 cans" on the board. How many cans in all?
Write " = 24" next to the existing sentence. Is there another way to write
this? Write "4 x 6 = 24."
2. With students in pairs, distribute bags of small buttons to each group
explaining that we are going to practice making arrays with the buttons.
3. Use a set of flashcards (making sure that the "0" facts are taken out) to
16
pick multiplication problems at random. Write the problem in large print on
the board, and ask them how an array could be made to show this problem.
Use buttons on an overhead to show them.
4. Continue to pick cards at random, asking pairs of students to make arrays
with their buttons. Call on volunteers to demonstrate what they have done
at their desks, using the buttons on the overhead.
5. Ask if any pair of students has done the problems in a different way. In a
perfect situation, the students will discover the Commutative Property on
their own. If not, turn the page on the overhead so that it is sideways. Ask
if this is the same problem. Why or why not? Discuss how this problem is
written and why it is the same.
6. Repeat this activity, with each group showing that the two arrays yield
the same answer. Again, students come up to the overhead to show their
work.
7. Next, collect the buttons and separate the pairs of students.
8. Finally, use an overhead to show how arrays can be drawn on a grid.
Practice doing this together, pointing out that a 2 x 3 array yields covers
the same space as a 3 x 2 array. Label each array appropriately.
Individual Practice
1. Distribute grid paper.
2. Mix up the flashcards, hold them face down and ask students to choose
one. Then give them the corresponding card as well; for example, if a
student picks 8 x 4, give him/her 4 x 8.
3. Tell them to outline, color, and cut out each array on the grid paper.
3. As they work, distribute one piece of construction paper to each
studentso they can glue each array onto the construction paper. (Show them
an example.)
4. Check to see if any students have labeled their arrays, and point this out
to the other students. If no one has done so, explain its importance.
5. Finally, distribute small stickers to be placed in each box of their arrays.
Closure:
When all of students are finished, point to the bulletin board and explain
that this is where we will be hanging up our arrays under the title, "Array
for Multiplication!" (Kind of like ‘hooray!’). Students will come up one by one.
Each student should show his/her array to the class, and explain what
multiplication sentences that each one illustrates. Hang it up on the board.
17
Assessment:
• Observe the students for active participation.
• Check for completion of the array projects with 100% accuracy. If a
child is not doing his/hers correctly, pair him/her with someone who
finishes early so they can help the student understand the concept.
This is the “Very Large Array”
The Very Large Array, one of the world's premier astronomical radio observatories, consists
of 27 radio antennas in a Y-shaped configuration on the Plains of San Agustin fifty miles
west of Socorro, New Mexico. Each antenna is 25 meters (82 feet) in diameter. The data
from the antennas is combined electronically to give the resolution of an antenna 36km (22
miles) across, with the sensitivity of a dish 130 meters (422 feet) in diameter. For more
information, see our overview of the VLA. The array is currently in the A configuration.
Just an interesting note for your kids!
Associative Property: Also works for addition and subtraction only.***
Includes parentheses.
Ex: (9+2)+1=12 or 9+(2+1)= 12
2x(9x3)=54 or (2x9)x3= 54
Distributive Property: Used with multiplication and addition or subtraction.
Ex: 5(9+2)
2(11-3)
(5x9) + (5x2)
(2x11)- (2x3)
45+10=55
22-6=16
Identity Property:
a+0=a
(for addition)
a*1 = a (for multiplication)
POWERPOINT
by D. Fisher
Worksheet on properties worksheet 3
Answer key worksheet 4
18
back
Parentheses
Exponents
Multiply or Divide
Add or Subtract
Please
Excuse
My Dear
Aunt Sally
1). 7 + (1 × 5) × 6 =
2). 7 - (3 + 3) =
3). 0 + 0 + (3 + 4) =
4). (4 - 2) × 1 + 2 =
5). 4 + 1 × 4 × 1 - (2 × 1) =
6). 7 × 7 - 3 - (7 - 5) =
7). 2 × 0 + 7 + 4 + 2 × 5 =
8). 3 × 2 - (5 × 1) =
Worksheet 1
19
back
Parentheses
Exponents
Multiply or
Divide
Add or
Subtract
Please
1). 7 + (1 × 5) × 6 =
7+
30
37
Excuse
My Dear
7+
5
Aunt Sally
x6
2). 7 - (3 + 3) =
76
1
3). 0 + 0 + (3 + 4) =
0+ 0+
7
7
4). (4 - 2) × 1 + 2 =
2
x1 +2
2+2
4
20
5). 4 + 1 × 4 × 1 - (2 × 1) =
4 + 1x 4 x 1 - 2 = (continued above)
4+
4
-2
8
-2
6
6). 7 × 7 - 3 - (7 - 5) =
7 x 7 -3 2
49 -3 2
46 -2
44
7). 2 × 0 + 7 + 4 + 2 × 5 =
0 +7 +4 +2x 5
0 +7 +4 +10
7
+4 +10
11+ 10
21
8). 3 × 2 - (5 × 1) =
3x25
6 5
1
Worksheet 2
21
PROPERTIES
Name:__________________
Match the words in the first column to the best available answer in the
second column.
back
_____ 5x(3+6)=(5x3)+(5x6)
1)Associative Property of
Multiplication(AX)
_____ 0/4=0
2)Associative Property of Addition(A+)
_____ 4x3=3x4
3)Commutative Property of Addition(C+)
_____ (3x5)x2=3x(5x2)
4)Identity Property of 0(ID-0)
_____ (6+8)+(4+6)=(4+6)+(6+8)
5)Multiplication and Division Property of
0(x-0)
_____ (3+5)+5=3+(5+5)
6)Commutative Property of Addition(CA)
_____ 14/1=14
7)Multiplication and Division Property of
0(X-0)
_____ 2x8=8x2
8)Associative Property of
Multiplication(AX)
_____ 0+6=6
9)Distributive Property(Distrib)
_____ (6x25)x4=6x(25x4)
10)Multiplication and Division Property of
0(X-0)
_____ 8x0=8
11)Distributive Property(Distrib)
_____ (8x3)-(6x3)=3x(8-6)
12)Associative Property of
Multiplication(AX)
22
_____ 18x1=18
13)Associative Property of
Multiplication(AX)
_____ 16-16=0
14)Identity Property of 1(ID-1)
_____ 18+12=12+18
15)Identity Property of 0(ID-0)
_____ (5x10)x10=5x(10x10)
16)Commutative Property of
Multiplication(CX)
_____ 3x(2+5)=(3x2)+(3x5)
17)Distributive Property(Distrib)
_____ (4+12)+18=4+(12+18)
18)Identity Property of 1(ID-1)
_____ 21x0=0
19)Associative Property of Addition(A+)
_____ (18x10)x10=18x(10x10)
20)Commutative Property of
Multiplication(CX)
Worksheet 3
23
Answer Key: (Worksheet 3)
back
9
11
17 - 5x(3+6)=(5x3)+(5x6)
5 - 0/4=0
16
20 - 4x3=3x4
1
8
12
13 - (3x5)x2=3x(5x2)
6 - (6+8)+(4+6)=(4+6)+(6+8)
2
19 - (3+5)+5=3+(5+5)
14
18 - 14/1=14
16
20 - 2x8=8x2
4
15 - 0+6=6
1
8
12
13 - (6x25)x4=6x(25x4)
7
10 - 8x0=8
9
11
17 - (8x3)-(6x3)=3x(8-6)
14
18 - 18x1=18
4
24
15 - 16-16=0
3 - 18+12=12+18
1
8
12
13 - (5x10)x10=5x(10x10)
9
11
17 - 3x(2+5)=(3x2)+(3x5)
2
19 - (4+12)+18=4+(12+18)
7
10 - 21x0=0
1
8
12
13 - (18x10)x10=18x(10x10)
25
Worksheet
Ask Dr. Math: FAQ
Glossary of Properties
Dr. Math FAQ || Classic Problems || Formulas || Search Dr. Math || Dr.
Math Home
Operation - Identity - Associative - Commutative - Distributive - Closure Inverse - Equality
An operation works to change numbers. (The word operate comes from Latin
operari, "to work.") There are six operations in arithmetic that "work on"
numbers: addition, subtraction, multiplication, division, raising to powers, and
taking roots.
A binary operation requires two numbers. Addition is a binary operation,
because "5 +" doesn't mean anything by itself. Multiplication is another
binary operation.
From the Web:
Binary Operation, Eric Weisstein's World of Mathematics
Notes on Binary Operations, Peter Williams
Back to Top
Identity
An identity is a special kind of number. When you use an operation to
combine an identity with another number, that number stays the same. Zero
is called the additive identity, because adding zero to a number will not
change it: the number stays the same.
0 + a = a = a + 0.
Since any number multiplied by one remains constant, the multiplicative
identity is 1.
1 * a = a = a * 1.
26
From the Math Forum:
Additive Identity and Other Properties
Basic Real Number Properties
Identifying Algebraic Properties
Number Properties
Properties
Laws of Arithmetic
Definition of a field.
From the Web:
Field Axioms and Identity Element, Eric Weisstein's World of
Mathematics
Identities, Peter Williams
Back to Top
Associative Property
An operation is associative if you can group numbers in any way without
changing the answer. It doesn't matter how you combine them, the answer
will always be the same. Addition and multiplication are both associative.
Here are some addition examples:
1 + (2 + 3) = (1 + 2) + 3
1 + (5) = (3) + 3
6 = 6.
(-1 + 66) + 14 = -1 + (66 + 14)
(65) + 14 = -1 + (80)
79 = 79.
More generally,
a + (b + c) = (a + b) + c.
Here is a multiplication example:
2 * (4 * 3) = (2 * 4) * 3
2 * (12) = (8) * 3
24 = 24.
In general terms, that's
a * (b * c) = (a * b) * c.
From the Math Forum:
Associative Property
Associative, Distributive Properties
Basic Real Number Properties
Properties of Algebra
27
Meanings of Properties
Number Properties
Properties
Algebraic Systems
Is a "new" operation commutative or associative?
Identifying Algebraic Properties
Laws of Arithmetic
Definition of a field.
From the Web:
The Associative Property, "Ask Lois Terms"
Associative, Eric Weisstein's World of Mathematics
Associative Operations, Peter Williams
Back to Top
Commutative Property
An operation is commutative if you can change the order of the numbers
involved without changing the result. Addition and multiplication are both
commutative. Subtraction is not commutative: 2 - 1 is not equal to 1 - 2.
Here are some examples of the commutative properties of addition and
multiplication:
88 + 65 = 65 + 88
153 = 153.
12 * 13 = 13 * 12
156 = 156.
More generally,
a + b = b + a, and
a * b = b * a.
From the Math Forum:
The Commutative Property Around Us
Basic Real Number Properties
Properties of Algebra
Additive Identity and Other Properties
Meanings of Properties
Properties and Postulates
Algebraic Systems
Is a "new" operation commutative or associative?
Properties of Real Numbers
Identifying Algebraic Properties
28
Number Properties
Properties
Laws of Arithmetic
Definition of a field.
From the Web:
The Commutative Property, "Ask Lois Terms"
Commutative, Eric Weisstein's World of Mathematics
Commutative Operations, Peter Williams
Back to Top
Distributive Property
When you distribute something, you give pieces of it to many different
people. One example of distributing objects is handing out papers in class. In
math, people usually talk about the distributive property of one operation
over another.
The most common distributive property is the distribution of multiplication
over addition. It says that when a number is multiplied by the sum of two
other numbers, the first number can be handed out or distributed to both
of those two numbers and multiplied by each of them separately. Here's the
distributive property in symbols:
a * (b + c) = a * b + a * c.
Here's an example:
5 * (2 + 8) = 5 * 2 + 5 * 8
5 * (10) = 10 + 40
50 = 50.
Not all operations are distributive. For instance, you cannot distribute
division over addition. Let's try an example:
14 / (5 + 2)
= 14 / (7)
= 2,
but
14/5 + 14/2
= 2.8 + 7
= 9.8.
Clearly, 2 is not equal to 9.8.
From the Math Forum:
Distributive Property
What is the Distributive Property?
29
Distributive Property
Associative, Distributive Properties
Properties of Algebra
Meanings of Properties
Laws of Arithmetic
Definition of a field.
From the Web:
The Distributive Property, "Ask Lois Terms"
Distributive, Eric Weisstein's World of Mathematics
Distributive Operations, Peter Williams
Back to Top
Inverse
The inverse of something is that thing turned inside out or upside down.
The inverse of an operation undoes the operation: division undoes
multiplication.
A number's additive inverse is another number that you can add to the
original number to get the additive identity. For example, the additive
inverse of 67 is -67, because 67 + -67 = 0, the additive identity.
Similarly, if the product of two numbers is the multiplicative identity, the
numbers are multiplicative inverses. Since 6 * 1/6 = 1 (the multiplicative
identity), the multiplicative inverse of 6 is 1/6.
Zero does not have a multiplicative inverse, since no matter what you
multiply it by, the answer is always 0, not 1.
From the Math Forum:
Basic Real Number Properties
Laws of Arithmetic
Definition of a field.
From the Web:
Multiplicative Inverse and Field Axioms, Eric Weisstein's World of
Mathematics
Inverse, Peter Williams
Back to Top
30
I.
Grade Level/Unit Number:
Grade 5/Unit 1, Activity Set 3
II.
Unit Title:
Numbers, Operations and Algebra
III.
Unit Length:
3 weeks
IV.
Indicators Addressed:
5-3.1 Represent numeric, algebraic, and geometric patterns in words,
symbols, algebraic expressions, and algebraic equations. B2
5-3.2 Analyze patterns and functions with words, tables, and graphs. B4
5-3.3 Match tables, graphs, expressions, equations, and verbal descriptions
of the same problem situation. C2
5-3.5 Analyze situations that show change over time. B4
V.
Materials Needed: see each lesson
Final Project: Worksheet 7
Key Learning: How are patterns and graphs related?
Note to Teacher: Emphasize that patterns are everywhere in life and come
in form of words, numbers, etc. Discuss the various types of graphs and
their purposes. Recognizing, creating and extending patterns are
fundamental problem solving techniques. Therefore, any work done with
patterns and functions should be accomplished through a problem solving
mode. The emphasis in fifth grade is on analyzing and using patterns to solve
problems, not recognizing basic repeating or growing patterns.
Fifth grade students are expected to analyze, describe, extend, and
use numeric patterns (arithmetic sequences and function rules) and
geometric patterns (triangular numbers, perfect squares) to solve problems.
Such problem solving opportunities should specifically include the
relationship among distance, speed, and time. A triangular number is a
natural number such that the shape of an equilateral triangle can be formed
by that number of points. Every triangular number can be written as the sum
1 + 2 + 3 + ... + n for some natural number n. The sequence of triangular
numbers (sequence A000217 in OEIS) for n = 1, 2, 3... is:
1, 3, 6, 10, 15, 21, 28, 36, 45, 55, ...
31
1
In fifth grade students
should build on previous experiences;
3
charting and graphing change over
time lays an important foundation
for later work on linear (straight
6
line on a graph) and nonlinear (not a
straight line on a graph)
relationships.
10
Fourth grade students
primarily focus on using charts and
graphs, whereas at fifth grade they
are required to actually create the
15
charts and graphs to show changes
th
over time. It is at the 5 grade
level that the concept of finding
missing elements in numerical or non-numerical forms is introduced. This is
also the first time that students are required to use function rules to make
generalizations.
In mathematics education today, there is a growing awareness that
children need experiences with problem-solving activities and the use of
calculators should be introduced and applied at every level. All children are
highly motivated by the use of the calculator.
Use these questions to help prompt them in their search:
1. Do any numbers repeat?
2. Can you predict the next row of numbers?
3. What is the rule or “function” for the 2nd row of numbers? The 3rd
row?
4. What sequences do you find?
5. If the pattern continued, what would the numbers be in the 12th
row?
Geometric Patterns ppt
Geometric Patterns
By Monica Yuskaitis
32
Building Function Generators!
Your students will be creating function generators to create their own
patterns! In order to do this, have students bring in a sturdy shoe box.
Each student will need 2 strips of card stock. Follow the diagram below to
complete their function generator. Let students have fun with this and
decorate their own!
Front
Back
Cut a slit on the top front and bottom black
of the student’s shoebox. Then, with the
two pieces of cardstock, create a tunnel
going from each opening. Secure each side
with tape. Students will be able to use
index cards cut in half to use in their
function generators.
Once students have completed their function generators, group students
into pairs. The first student will write a number on one side of the index
card and send it through the function generator to his/her partner. The
partner will then get the index card and write a number to add to this
pattern, and send it back to his partner through his function generator.
This should take place a couple of more times. The partners should pick up
on what type of pattern they are trying to make without saying a word. For
instance, if the first number was 2, and then 5, one may think we’re going to
add 3, while another thinks the rule is x2 +1. It will go back to the original
player to decide which way he will go. Students could create 3 or 4 patterns
on their own before going onto individual work.
Number Sequences.
TLI: To recognise and extend number
sequences formed by counting from any
number in steps of constant size, extending
beyond zero when counting back.
Numbers Sequences ppt
Worksheet 1a & b
33
Pairs of Patterns: Alike and Different
Give students two patterns that are related in some way. To indicate
that they understand the patterns, they should first extend each one by at
least five more numbers. Next, they should write an explanation of how the
two patterns are alike and how the two patterns are different. Here are
some possible pairs to use:
2,4,6,8,10,….. and 3,5,7,9,11,….
2,4,6,8,10,…. and 2,4,8,16,32,…..
3,4,3,4,3,4,….and 7, 9, 7, 9, 7, 9,…
5,10,15,20,20,…and 2,7,12,17,22,…
10,20,30,40,50,…and 50,100,150,200,250,…
The challenge in these patterns is not only to find and extend the
pattern; but, to have students write how they determined what the pattern
was.
After solving several pairs of related number patterns, students can
make up their own pairs of patterns and challenge other students to discover
how they are alike and different.
Once students have created their own patterns, have them walk
around the room to music and when the music stops, turn the person closest
to them. The students should compare their patterns and try to find
similarities.
Activities from Elementary and Middle School Mathematic - Teaching
Developmentally, 4th Edition. By John A. Van De Walle.
Number Theory Worksheet 2
Patio Tiling
Materials Needed:
• 30 white tiles (for each student or pair of students)
• 30 brown tiles (for each student or pair of students)
Each student or pair of students will need 30 white tiles and 30 brown
tiles. Present the following task:
Dusty Lee is designing square patios. Each patio has a square garden
area in the center. Dusty uses brown tiles to represent the soil of the
garden. Around each garden, he designs a border of white tiles.
Ask students to build three patios using brown and white tiles to show
the garden and the border. Record the number of white and brown tiles for
each patio in a table. Continue the table for the next two squares. Ask
34
students to describe the patterns they see and identify by naming the rule
for the number of tiles on the patio and the number on the border. Ask
them to test their rule with another example of a patio. The students
should use the chart to name the number of brown tiles in the seventh patio.
(There are 49 brown tiles in the seventh patio.)
Extensions:
Find the number of brown (white) ties in the fifteenth patio: in the
thirtieth patio; in the hundredth patio. (Patio 15 has 225 brown tiles and 64
white tiles; patio 30 has 900 brown tiles and 124 white tiles; patio 100 and
10,000 brown tiles and 404 white tiles.)
Patio
Number
1
2
3
4
5
6
Number of
Brown Tiles
1
4
9
16
25
36
Number of
White Tiles
8
12
16
20
24
28
Adapted from Navigating through Algebra in Grades 3-5.
Total Number of
Brown and White
Tiles
9
16
25
36
49
64
35
Zach’s Walk-a-Thon
After completing the following lesson, students will see the
correlation between the longer the time the greater the distance. It is
important that they see that one (say distance) is dependant on the other
(time/speed).
Example:
Zach recently participated in the school walk-a-thon. He walked on
average 4 miles per hour. Using the data in the table shown below, have the
students create a line graph. Make sure they label the y and x axis and
provide a title for the graph (this should be related to their work on naming
coordinate pairs). Ask them to estimate how far Zach will walk in 8 hours.
Time (hour)
1
2
3
4
Distance
4
8
12
16
(miles)
If time were x and distance were y, what would be the function rule?
How could we write an expression to represent this?
What could be the general rule for this pattern?
Is the change in this pattern predictable or non-predictable? Explain.
From the graph, were there any changes in speed?
For additional activities on distance, speed, and time visit the following
website:
http://www.shodor.org/interactivate/lessons/ft1.html
A Calculator Based Ranger (CBR) is a great interactive tool that allows
students to become actively engaged in exploring the relationship among
time, distance, and speed. (This was adapted for CEEMM materials)
Line graph activity Worksheet 4
Heart-Pumping Math!
Give each student a copy of the handout below. Instruct students on
how to measure their heartbeats. (Find their heartbeat on their neck or
wrist after each exercise, and count how many beats in 10 seconds. They
will then use the number of beat per 10 seconds and multiply it by 6 to find
HPM…Heartbeats per minute). When all exercises are completed and data
recorded, students will create a line graph to display results and make
generalizations based on their graph.
Heart Pumping Math Worksheet 3
36
Title - Big Mac & Slammin' Sammy, The Home Run Race, 99!!
This is a great on-going graph lesson. It can be simple or complex. This
graph compares, and charts the daily home run race of Mark McGwire and
Sammy Sosa (could be used with any players, but it's been exciting lately
with these two). Use 4 columns for both players a total of eight. First
column, the date they hit it, second, a cut out baseball that has the number
they hit that day, third, the distance it went, and the fourth column what
field it went to. It is fun for the students to come in and write the date, and
glue up the baseball, write in the distance and what field. Daily information
is available at cnnsi.com.
Ask students to create the question they want answered at the end of
the race.
For example:
1. Who will hit the most?
2. What field do they hit most often?
3. Who will hit the farthest?
4. What is the total distance each one hit, and together? Is it higher than
Mt. Everest?
Bar graph Worksheet 5
Review for Final Assessment
To review the skills covered in this activity set, use this game. Turn
the lights down so students will get the real effect of who wants to be a
millionaire!
Who Wants to be a Millionaire?
Final Assessment
Test Worksheet 6
Other resources for this unit to use at your discretion:
Dinner Time Promethean Flipchart (If you have Promethean software,
simply drag and drop this icon into My Flipcharts and it should work fine.)
time_dinner.flp
Patterns Promethean Flipchart (If you have Promethean software, simply
drag and drop this icon into My Flipcharts and it should work fine.)
Patterns.flp
37
Number sequences
Complete the number sequences by filling in the spaces below. State
whether the sequences are ascending or descending order.
Ascending
or descending?
1.
2, 9,
16,
___, ___, ___, ___,
___________
2.
20,
31,
42,
___, ___, ___, ___,
___________
3.
133, 130, 127, ___, ___, ___, ___,
___________
4.
11,
17,
23,
___, ___, ___, ___,
___________
5.
9,
0,
-9,
___, ___, ___, ___,
___________
6.
26,
16,
6,
___, ___, ___, ___,
___________
7.
-5,
0,
5,
___, ___, ___, ___,
___________
8.
2.5, 3.0, 3.5, ___, ___, ___, ___,
___________
Now have a go at these more tricky sequences!
9.
2,
4,
8,
16,
___, ___, ___,
___________
10. 6,
18,
54,
162, ___, ___, ___,
___________
Think very carefully now. Can you write down the short hand rule for each
of the above sequences? The first one has been done for you. NN = new
number; LN = last number
2.
______________
1.
NN = LN + 7
3.
_______________
4.
______________
5.
_______________
6.
______________
7.
_______________
8.
______________
9.
_______________
10.
______________
Worksheet1a
38
(Answer Key 1a) Number sequences
Complete the number sequences by filling in the spaces below. State
whether the sequences are ascending or descending order.
Ascending
or descending?
1.
2, 9,
16,
23,
30,
37,
44,
Ascending
2.
20,
31,
42,
53,
64,
75,
86,
Ascending
3.
133, 130, 127, 124, 121, 118, 115,
descending
4.
11,
17,
23,
29,
35,
41,
47,
Ascending
5.
9,
0,
-9,
-18, -27, -35, -42,
descending
6.
26, 16,
6,
-4,
-14, -24, -34,
descending
7.
-5, 0,
5,
10,
15,
20, 25,
Ascending
8.
2.5, 3.0, 3.5, 4.0, 4.5, 5.0, 5.5,
Ascending
Now have a go at these more tricky sequences!
9.
2,
4,
8,
16,
32,
64,
128,
Ascending
10. 6,
18,
54, 162, 486, 1458, 4374,
Ascending
Think very carefully now. Can you write down the short hand rule for each
of the above sequences? The first one has been done for you.
1.
NN = LN + 7
2.
NN = LN + 11
3.
NN = LN -3
4.
NN = LN + 6
5.
NN = LN -9
6.
NN = LN - 10
7.
NN = LN + 5
8.
NN = LN + 0.5
9.
NN = LN x 2
10.
NN = LN x 3
39
Number sequences
Complete the number sequences by filling in the spaces below. State
whether they are ascending or descending order.
Ascending
or descending?
1.
13,
25,
37,
___, ___, ___, ___,
___________
2.
20,
31,
42,
___, ___, ___, ___,
___________
3.
26,
23,
20,
___, ___, ___, ___,
___________
4.
11,
17,
23,
___, ___, ___, ___,
___________
5.
-16, -9,
-2,
___, ___, ___, ___,
___________
6.
133, 130, 127, ___, ___, ___, ___,
___________
7.
-25, -20, -15, ___, ___, ___, ___,
___________
8.
9,
0,
-9,
___, ___, ___, ___,
___________
9.
91,
102, 113, ___, ___, ___, ___,
___________
10. 97,
64,
31,
___, ___, ___, ___,
___________
Think very carefully now. Can you write down the short hand rule for each
of the above sequences? The first one has been done for you.
1.
NN = LN + 12
2.
______________
3.
_______________
4.
______________
5.
_______________
6.
______________
7.
_______________
8.
______________
9.
_______________
10.
______________
Worksheet 1b
40
(Answer Key 1b)Number Sequences
Complete the number sequences by filling in the spaces below. State
whether they are ascending or descending order.
Ascending
or descending?
1.
13,
25,
37,
49,
61,
73,
85,
Ascending
2.
20,
31,
42,
53,
64,
75,
86,
Ascending
3.
26,
23,
20,
17,
14,
11,
8,
Descending
4.
11,
17,
23,
29,
35,
41,
47,
Ascending
5.
-16, -9,
-2,
5,
12,
19,
26,
Ascending
6.
133, 130, 127, 124, 121, 118, 115,
Descending
7.
-25, -20, -15, -10, -5,
0,
5,
Ascending
8.
9,
0,
-9,
-18, -27, -36, -45,
Descending
9.
91,
102, 113, 124, 135, 146, 157,
Ascending
10. 97,
64,
31,
-2,
-35, -68, -101,
Descending
Think very carefully now. Can you write down the short hand rule for each
of the above sequences? The first one has been done for you. NN= new
Number; LN = last number
1.
NN = LN + 12
2.
NN = LN + 11
3.
NN = LN - 3
4.
NN = LN + 6
5.
NN = LN + 7
6.
NN = LN - 3
7.
_______________
8.
______________
9.
_______________
10.
______________
Worksheet 1b
41
Worksheet 2
back
Date
___________________
Name
_____________________________
Fill in the missing multiple.
1.
3.
5.
7.
9.
11.
13.
15.
17.
19.
21.
23.
25.
27.
29.
Number Theory
(Answer ID # 0708428)
24, 30, 36, 42, ______
12, 14, 16, ______, 20
45, 54, 63, 72, ______
______, 28, 32, 36, 40
14, 21, 28, ______, 42
______, 20, 25, 30, 35
56, 64, 72, ______, 88
38, ______, 76, 95, 114
70, 84, 98, 112, ______
49, 56, ______, 70, 77
6, 9, 12, ______, 18
______, 56, 64, 72, 80
______, 88, 99, 110, 121
90, 105, 120, 135, ______
6, ______, 10, 12, 14
2.
4.
6.
8.
10.
12.
14.
16.
18.
20.
22.
24.
26.
28.
30.
22, 33, 44, ______, 66
39, 52, 65, ______, 91
112, 128, 144, 160, ______
40, ______, 60, 70, 80
105, 120, 135, ______, 165
______, 108, 126, 144, 162
72, ______, 96, 108, 120
68, 85, 102, 119, ______
______, 12, 15, 18, 21
______, 30, 40, 50, 60
______, 18, 24, 30, 36
51, ______, 85, 102, 119
16, ______, 24, 28, 32
26, 39, 52, 65, ______
60, 72, 84, 96, ______
Back
42
Heart-pumping Math!
Directions: You are going to measure your heart rate with each exercise.
Do each exercise for 30 seconds each. You will collect your HPM (Heart rate
per minute) after each exercise. Once you have completed all exercises,
place your data on a line graph. Line graphs show change over time, so it
would be the most appropriate graph to use. Worksheet 3
Exercise:
# of Heartbeats per minute
Resting
Walking in place
Jumping jacks
Jogging in place
Running in place
Walking in place
Resting
Create your line graph in the space below.
back
43
Worksheet 4
Date
___________________
Name
_____________________________
Line Graphs
(Answer ID # 0729729)
Make a line graph using the data in the table.
1.
Plant's Height
Plant's Height
End of
Week
Height
(cm)
#1
3
#2
8
#3
10
#4
12
#5
14
a. At the end of which week(s)
was the plant no more than 12
centimeters tall?
b. Between which two weeks
was there the greatest
increase in height?
c. How much did the plant
grow from the end of
week #3 to the end of
week #4?
44
2.
Attendance
Attendance
Game
Attendance
Game #1
300
Game #2
500
Game #3
150
Game #4
475
a. Which games(s) did not
have an attendance of at least
500 people?
b. How many more people
came to game #4 than to game
#3?
c. Which games(s) had an
attendance of more than 300
people?
back
Worksheet 5
45
Name
_____________________________
Date
___________________
Bar Graphs
(Answer ID # 0535332)
Complete.
1.
Points Scored
a. Which player(s) scored more than 6
points?
b. Who scored the most points?
c. What are the total points for all five
teams?
2.
Number of pushups done in physical a. How many more pushups did Connor
education class
do than Michael?
b. How many pushups did Taylor do?
c. How many fewer pushups did Kaitlyn
do than Rebecca?
3.
Picking Apples from Trees
a. How many fewer apples were picked
from #5 than from #2?
46
b. How many more apples were picked
from #3 than from #5?
c. There were twenty more apples
picked from tree #3 than from tree
____.
back
47
Worksheet 6
Patterns and Functions Test
Name________________________
Date:______________________
1. Stylists at a hair salon charge $26 for each haircut. If they gave 63
haircuts, how much money did they collect, not including tips?
A $89
B $504
C $1,538
D $1,638
2. Which point is located at (6, 0)?
F
Point P
G
Point Q
H
Point R
J
Point S
3. The numbers 1, 4, 9, and 16 are called square numbers. You can
use the figures below to determine the next three square numbers.
1
4
A
B
C
D
32,
24,
30,
25,
9
64,
35,
40,
36,
128
48
50
49
48
4. The numbers 1, 3, 6, and 10 are called triangular numbers. You
Use the figures below to determine the next three triangular numbers.
1
A
B
C
D
3
6
13, 17, 22
15, 21, 28
17, 25, 34
20, 40, 80
5. Look at the pattern below:
101 = 10
102 = 100
103 = 1,000
What will 106 = ?
A 10,000
B 100,000
C 1,000,000
D 10,000,000
6. Find the pattern and choose the next three numbers:
1 2 3 4 5
6
, , ,
,
,
, __, __, __
2 4 6 8 10 12
7
8
9
A
,
,
13 14 15
8
10 12
B
,
,
14 16 18
7
8
9
C
,
,
14 16 18
7
8
9
D
,
,
13 15 17
49
7. Find the pattern and choose the next three numbers:
2
2
2
4 , 5, 9 , 10, 14 , 15, __, __, __
3
3
3
A
18
B
19
C
D
2
2
, 19, 20
3
3
2
2
, 20, 24
3
3
2
2
16 , 17 , 18
3
3
2
2
2
19 , 20 , 24
3
3
3
8. What number is missing in this table?
In
Out
A
B
C
D
1
3
2.4 5.4
5
7
?
9.4
5.5
7.2
7.4
8.4
9. Which of the following would be the correct rule to find the output?
A
B
C
D
In
1.
2.
3.
4.1
Out
2.
5
7.
8.2
take half of the input
add 2 to the input
double the input
multiply the input by 3
50
10. Look at this table to determine the rule:
1
2
3
4
5
A
B
C
D
(
(
(
(
3
5
7
9
11
x 2) + 1 =

x
) − 1 =
+ 1) x 2 =
− 2) x 3 =
back
51
Stained Glass Window
Worksheet 7
Here is a Cargill Project. It incorporates many standards as well as
calculations. If you wish to complicate this a bit, you could change money to
include decimals. To do this, you will need pattern blocks for kids to create
their original design. You may leave it at that, or you may have the
trace/draw what they have created onto paper and color as they wish.
Geometry Project
Name:
__________________________________
You have been asked to design a stain glassed window for a new library. The
design that is the most attractive, unique, follows the guideline given, and
meets the budget will be selected to be placed above the entry to the
building. Please be creative and help us use this window to attract patrons.
Guidelines
1. Design must have a pattern
2. Design must have symmetry. Rotational symmetry would be awesome.
3. Design should be multi-colored and multi-shaped.
4. Price of the window cannot be more than $10,000
5. Design of the window must be submitted with a total number of each
piece needed and a total cost for each piece and then a final cost of
the window.
Prices for stain glass pieces.
Hexagon- $ 250.00
Large Rhombus- $50.00
Parallelogram- $100.00
Triangle- $25.00
Square- $50.00
Small Rhombus- $25.00
While you are designing your window, be sure to keep up with the cost. If
you go over budget we will not be able to use your design!
Helpful Hints:
1. Lay out small parts of the design and then trace the pattern blocks.
Then continue with another section.
2. Keep tally marks as you work, so you won’t have to go back and count.
3. Work from the center out.
52
Name of Piece
Tally of pieces
used
Total amount of
pieces used
Cost of
each piece
Total Cost
Hexagon
Parallelogram
Large Rhombus
Triangle
Square
Small Rhombus
Final Project Cost
__________________________________________________
Back
53
I.
Grade Level/Unit Number: Grade 5: Unit 1 Activity Set 4
II.
Unit Title:
Numbers and Operations
III.
Unit Length:
3 weeks
IV.
Indicators Addressed:
5-2.2
5-2.3
5-2.4
5-3.3
V.
Apply an algorithm to divide whole numbers fluently.
Understand the relationship among the divisor, dividend,
and quotient.
Compare whole numbers, decimals, and fractions by using
the symbols <, >, and =.
Match tables, graphs, expressions, equations, and or verbal
descriptions of the same problem situation.
C3
B2
C2
Materials Needed: see each lesson
Final Project: Division Project
Preparing to Teach:
Activity 1 back
Students need to be able to show how to compare numbers with objects as
well as using numbers. Any unit of measure is useful here such as counters,
money, mass (using a scale), length (using a ruler) or liquid measurement (using
a variety of graduated cylinders) making this a practical skill.
PowerPoint
Worksheet Worksheet 1
Answer Key Worksheet 2
Comparing
Numbers
TEACHER’S NOTES: For 5-2.2, 5-2.3, 5-2.4
Division is the process of determining how
many times one number is contained in
another number. When numbers are
divided, the result is the quotient and a
A division sign (÷)
A division sign (√)
A horizontal line with the dividend above
the line and the divisor below the line. (—)
54
remainder. The remainder is what remains
after division. The number divided by
another number is called the dividend; the
number divided into the dividend is called
the divisor. Division is indicated by any of
the following:
Thus, the relationship between the
dividend, divisor, and quotient is as shown:
Unlike multiplication, the division process
is neither associative nor commutative.
The commutative law for multiplication
permitted reversing the order of the
factors without changing the product. In
division the dividend and divisor cannot be
reversed. Use the equation form:
For example, the quotient of 1 8 6 is not
the same as the quotient of 6 1 8 . 1 8
divided by 6 equals 3 : 6 divided by 1 8
equals 0 . 3 3 .
The associative law for multiplication
permitted multiplication of factors in any
order. In division, this is not allowed.
Or a slanting line a/b meaning a divided by
b
a÷b ≠ b÷a
55
When dividing two numbers, the divisor
and dividend are lined up horizontally with
the divisor to the left of the dividend.
Division starts from the left of the
dividend and the quotient is written on a
line above the dividend.
Starting from the left of the dividend,
the divisor is divided into the first digit
or set of digits it divides into. In this
case, 5 is divided into 34; the result is 6,
which is placed above the 4.
This result (6) is then multiplied by
the divisor, and the product is subtracted
from the set of digits in the dividend
first selected. 6 x 5 equals 30; 30
subtracted from 34 equals 4.
The next digit to the right in the dividend
is then brought down, and the divisor is
divided into this number. In this case, the
7 is brought down, and 5 is divided into
47; the result is 9, which is placed above
the 7.
Again, this result is multiplied by the
divisor, and the product is subtracted
from the last number used for division. 9
x 5 equals 45; 45 subtracted from 47
equals 2. This process is repeated until all
of the digits in the dividend have been
brought down. In this case, there are no
more digits in the dividend. The result of
the last subtraction is the remainder. The
number placed above the dividend is the
quotient. In this case, 347 divided by
5 yields a quotient of 69 with a remainder
of 2.
Example 1:
Divide 347 by 5
Solution:
Example 2: Divide 738 by 83
Solution:
Example 3: Divide 6409 by 28
Solution:
56
*****Division can be verified by
multiplying the quotient by the divisor and
adding the remainder. The result should
be the dividend. Using Example 3, multiply
228 by 28 to check the quotient.
**** Division of whole numbers can easily
be taught using hands-on methods.
***** BE SURE TO STRESS THE
RELATIONSHIP BETWEEN DIVISION
AND MULTIPLICATION BY HAVING
STUDENTS CHECK THEIR PROBLEMS
BY USING MULTPLICATION.
The Doorbell Rang Activity 2 back
Materials/resources
"The Doorbell Rang" by Pat Hutchins
• The Doorbell Rang sheet
• PowerPoint presentation
• At least 12 cookies (can be real chocolate chip cookies or paper ones)
• Bell
•
Technology resources
You will need a video projector in order to show the PowerPoint presentation
on division.
Pre-activities
Students need to know their multiplication facts.
Read "The Doorbell Rang" once for students so they are familiar with the
story before starting the activity.
Activities
1. Begin by selecting 13 volunteers to "act" as you are reading the story.
Tell each of the students the name of the character that he or she
•
57
will be. One student will be the doorbell ringer. (Instruct the doorbell
ringer to ring the bell each time you read "as the doorbell rings.")
Instruct the actors/actresses to come up front when they hear their
name in the book.
2. Read "The Doorbell Rang" by Pat Hutchins. Have two students start up
at the front. These two students will have to divide the 12 cookies up
between themselves. Each time the doorbell rings, more students
come up. Each time students need to divide the cookies among
themselves. As students divide these cookies up, you can demonstrate
on the board how to model what they are doing. (Ex. Model 12 divided
by 2).
3. You may want to read the book the second time and have different
students "act" this time so that everyone has the opportunity.
4. Discuss with students what they have just done--model some problems
on the board.
5. Have students complete "The Doorbell Rang" sheet. Students will use
this sheet to model division as they are showing how many cookies
each of the children got in the story each time that the doorbell rang.
6. Closure review how to divide by presenting the PowerPoint
presentation--this is interactive.
Assessment
Students will be able to model division problems on the "The Doorbell Rang"
sheets. Worksheet 3
********My favorite way of teaching division of whole numbers is to buy
dried beans from the grocery store (or use bingo chips from activity set 1)
and put them into plastic ware. I give each group of four students a
container and make a list of problems on the board. ***Make sure that they
have enough beans for the dividend. Though this seems simplistic, often
students reach this point and don’t understand what division MEANS.
Example: 20/3
Students would count out 20 beans and divide them into groups of 3- the
beans that are left over are the remainders.
Answer- 6 groups of 3 and 2 left over
= 6R2
This activity will help them see a concrete example of what division is.
Base-ten blocks can also be used to show division.
**** At-home project for students****
58
Assignment- Find items at home that will help you with division. Explain how
you used these items to complete the homework problems.
Homework sheet
One-Digit divisor
Comparing Numbers
Worksheet 4
assessment
Greater Than >
Less Than <
Homework key
Worksheet 7
Worksheet 5
One-Digit divisor
Mental Math
key Worksheet 8
PowerPoint
Morning Activity
puzzle Worksheet
Worksheet 6
11
Ma and Pa Kettle
video
Comparing numbers using the ONES place
Comparing numbers using the TENS place
Comparing numbers using the HUNDREDS place
Division with 2-Digit Divisors
***Using the same method as one-digit division, explain that guess and check
is ok to use when working with larger numbers. Tell students that making an
educated guess is best to eliminate unreasonable guesses
Practice Worksheet 9
Answer key 10
Tables and graphs
Activity 3 back
CIRCLE/PIE GRAPHS
When you have finished this, try the Circle/Pie Graph Quiz. (online quiz) and
Hardcopy Worksheet 12(2 pages)
Circle or pie graphs are particularly good illustrations when considering
how many parts are in a whole. In this table both the number of hours in a
whole day devoted to certain activities is listed as well as the percent of
time for each. The pie chart is then divided much as a baker's pie would be
in slices that represent the proportional amounts of time spent on each
activity. To the right of the pie chart is a legend that tells which color
stands for which category and the percents are near the slice that stands
for that amount of time.
Percent of Hours of a Day Spent on Activities
ACTIVITY
HOURS
PERCENT OF DAY
59
Sleep
6
25
School
6
25
Job
4
17
Entertainment
4
17
Meals
2
8
Homework
2
8
Back Worksheet 1
Name _______________________
Date ___________________
SuperKids Math Worksheet
Horizontal Equalities Equations with numbers between 1 and 1000
84
495
503
357
550
83
735
781
863
910
499
611
298
76
44
550
911
300
456
962
980
527
157
94
860
999
135
145
767
895
60
122
3
410
134
447
14
986
433
773
605
476
219
Back Worksheet 2 Name
_______________
Date ___________________
SuperKids Math Worksheet
Horizontal Equalities Equations with numbers between 1 and
1000
550
84 <
495
503
> 357
735 <
781
863
< 910
44
< 550
911
> 300
980
> 527
157
> 94
298
> 76
456
< 962
> 83
499 <
611
860 <
999
135
< 145
767
122 >
3
410
> 134
447
> 14
> 433
773
> 605
476
> 219
986
< 895
Back
The Doorbell Rang WORKSHEET 3
61
By Pat Hutchins
Number of Cookies for
Each Division Problem
People at the Table
Victoria and Sam
Victoria, Sam, Tom,
Hannah
Victoria, Sam, Tom,
Hannah, Peter and Little
Brother
WORKSHEET 4
Back
Date ___________________
®
SuperKids Math Worksheet
Division with Integer Answers
using divisors between 1 and 10
62
63
Back WORKSHEET 5
Date ___________________
SuperKids® Math Worksheet - Answers
Division with Integer Answers
using divisors between 1 and 10
64
back
WORKSHEET 6
Back WORKSHEET 7
Name _______________________
Date ___________________
SuperKids Math Worksheet
Division with Integer Answers
using divisors between 1 and 10
4 )5356
1 ) 441
4 )2260
6 )8496
5 )2130
65
5 )7125
6 )5832
1 )5752
9 )9117
9 )3627
10 )5790
6 )7782
2 )4106
3 )3450
1 )3657
2 )1676
7 )4858
4 )3760
4 )8268
4 )7252
1 )5939
10 )3180
4 )1948
Back
Worksheet 8
10 )9190
5 )4310
Date ___________________
SuperKids® Math Worksheet - Answers
Division with Integer Answers
using divisors between 1 and 10
1339
4) 5356
441
1) 441
565
4)2260
1416
6)8496
426
5)2130
66
1425
5) 7125
972
6)5832
5752
1) 5752
1013
9) 9117
403
9)3627
579
10)5790
1297
6) 7782
2053
2) 4106
1150
3)3450
3657
1) 3657
838
2)1676
694
7)4858
940
4)3760
2067
4) 8268
1813
4)7252
5939
1) 5939
318
10)3180
487
4)1948
Back Name
_______________________
919
10)9190
862
5)4310
Date ___________________
SuperKids Math Worksheet
Division with Integer Answers
using divisors between 2 and 50
47 ) 611
33 ) 99
40 ) 440
24 ) 24
48 ) 576
30 ) 750
41 ) 369
40 ) 680
37 ) 555
9 ) 171
67
19 ) 323
27 ) 999
30 ) 660
35 ) 175
34 ) 510
15 ) 915
13 ) 949
13 ) 481
19 ) 627
6 ) 576
8 ) 760
10 ) 730
38 ) 950
21 ) 693
31 ) 806
WORKSHEET 9
Key #864
Date ___________________
SuperKids® Math Worksheet - Answers
Division with Integer Answers
using divisors between 2 and 50
13
47) 611
3
33) 99
11
40)440
1
24) 24
12
48) 576
68
25
30) 750
9
41) 369
17
40) 680
15
37) 555
19
9) 171
17
19) 323
37
27) 999
22
30) 660
5
35) 175
15
34) 510
61
15) 915
73
13) 949
37
13) 481
33
19) 627
96
6) 576
95
8) 760
73
10) 730
25
38) 950
33
21) 693
26
31) 806
69
WORKSHEET
10
WORKSHEET 11 back
Back
Circle/Pie Graph Quiz
Check the circle in front of the correct answers to the questions.
70
1. A part of a circle/pie graph that explains the colors that represent each
part or slice of the graph is a
legend
grid
axis
2. These two activities took up half of the time of the day.
Entertainment and school
Meals and school
Sleep and school
3. These two activities took up the least amount of time.
Sleep and school
Meals and homework
Sleep and job
4. Which of these took up one fourth of the day?
Worksheet 12a
Entertainment
Sleep
Homework
5. What percent of the day does homework take up?
2
8
25
6. Which of these takes up the same amount of time as meals and
entertainment together?
Job
School
Homework
WORKSHEET 12
71
Challenge / Thinking Final Project
• To check for understanding of the relationship between divisor,
quotient, and dividend:
Given a divisor and quotient, have students determine the dividend.
Ex: divisor = 8 , quotient = 112 , dividend = ________
• To check for understanding of the inverse relationship between
multiplication and division:
Given a multiplication problem, have students create a corresponding
division problem, vise versa
Ex: 18 X 7 = 126 , create a corresponding division problem.
• To check for strategies to use in division, have students show
alternate methods of solving a given problem. (make sure examples
are conducive to this)Ex:
1818
÷
13000
÷ 20 =
,
450
÷
15 ,
6
Worksheet 13
Back
72
I.
Grade Level/Unit Number: Grade 5, Unit 1, Activity Set 5
http://www.bsisonline.com/Standards/mathematics_support_guide.htm
II.
Unit Title: Comparing Whole Numbers, Adding and Subtracting
Decimals and Fractions
III.
Unit Length: 6 weeks
IV.
Indicators Addressed:
5-2.1
5-2.4
5-2.5
5-2.8
5-3.3
5-3.5
V.
Analyze the magnitude of a digit on the basis of
its place value, using whole numbers and decimal
numbers through thousandths.
Compare whole numbers, decimals and fractions by
using the symbols <, >, and =.
Apply an algorithm to add and subtract decimals
through thousandths
Generate strategies to add and subtract fractions
with like and unlike denominators.
Match tables, graphs, expressions, equations, and
or verbal descriptions of the same problem
situation.
Analyze situations that show change over time.
C3
B2
B4 C3
B3 B4
C2
B4
Materials Needed: see each lesson
Final Project: Using Tangrams, have students identify the fraction, decimal
and percentage value of each of the pieces. (Given that the entire square is
a whole) or
Using pattern blocks, the teacher should assign a “value” to a given piece (
ex: hexagram = 1 ) and have students find the fraction, decimal, and
percentage value of each of the other shapes.
Tangram Pattern worksheet 13
Pattern Block worksheet 14
Activity 1 back
*** Before beginning this set, please ensure that all students have an
understanding of what whole numbers, fractions, and decimals are, and how
they are written.
73
Writing decimals powerpoint
DECIMAL -- READING
AND WRITING
What is a fraction? powerpoint
Math Flash
Fractions I
By Monica Yuskaitis
FIVE AND SEVEN TENTHS
5.7
Before being able to compare fractions, whole numbers, and decimals,
students must be able to see the relationship between all three and be able
to convert.
Activity 2 back
Fractions to Decimals
To convert fractions to decimals students must learn that the numerator is
always divided by the denominator, even when the numerator is smaller than
the denominator. This will be a great way to introduce dividing decimals also.
This can be taught easiest by using base ten blocks or hundreds chart to fill
in the fraction and see the decimal value:
Example: 2/10
= 20/100= 1/5
Worksheet 1
Answer key worksheet 1b
74
Fractions to
Decimals
NUMERATOR
DENOMINATOR
75
Every fraction can be expressed as decimal (a number with a decimal point).
One way to convert a fraction to a decimal is to divide the numerator by the
denominator. For example, 1/2 is equal to 1 divided by 2, which is equal to
0.5.
Activity 3 back
Decimals to Fractions
To convert decimals to fractions students must be familiar with place value,
especially tenths and hundredths. 0.45= forty-five hundredths or 45/100
0.4= four tenths or 4/10
****Students can use the GCF to reduce to fractions to their simplest form.
Powerpoint
Worksheet 2
Answer Key Worksheet 2b
Changing Decimal
Though some of these use mixed numbers, you may
Numbers to
Fractions
either skip them or use them as challenge questions
for advanced students.
6/5/2007
Michele Webb
1
Activity 4 back
Simplifying Fractions to Lowest Terms, Equivalent Fractions
A fraction is in simplest form (lowest terms) if the Greatest Common Factor
– GCF – of the numerator and denominator is 1. For example, ½ is in lowest
terms, but 2/4 is not.
Equivalent Fractions:
Equivalent fractions are different fractions that are equal to the same
number and can be simplified and written as the same fraction (for example,
3/6 = 2/4 = 1/2 and 3/9 = 2/9 = 1/3).
To reduce a fraction to lowest terms (also called its simplest form), divide
both the numerator and denominator by the GCD. For example, 2/3 is in
lowest form, but 4/6 is not in lowest form (the GCD of 4 and 6 is 2) and 4/6
can be expressed as 2/3.
76
You can do this because the value of a fraction is not changed if both the
numerator and denominator are multiplied or divided by the same number.
Comparing Whole Numbers
Comparing whole numbers is the easiest comparison. Students simply choose
the number that is the largest.
Powerpoint
Comparing Numbers
Greater Than >
Less Than <
Comparing numbers using the ONES place
Comparing numbers using the TENS place
Comparing numbers using the HUNDREDS place
Comparing fractions to fractions
Though we do not teach multiplication of fractions as such in 5th grade, this
is one simple way for students to compare fractions.
Discuss cross multiplication and how it can be used to compare fractions.
Fractions with like denominators can be compared easily by looking at the
numerator. Fractions with like numerators can be compared by looking at the
denominator. ***** The smallest denominator is actually the largest
fraction*** use base ten blocks or drawings to show why the fraction with
the larger denominator is actually smaller. When comparing, give students
worksheet and crayon to color in their fractions to see which is the largest.
Powerpoint
Worksheet 5
Answer Key Worksheet 5b
Fractions XIV
Multiplication of Fractions
by Monica Yuskaitis
*****Explain that cross multiplication can be used with all types of
fractions.
If mixed numbers are present, compare whole numbers first.
77
Activity 5 back
Comparing fractions to decimals
Student can convert to make both numbers fractions or make both numbers
decimals. If they convert to fractions, they can cross multiply to compare.
Worksheet 6
Answer Key Worksheet 6b
http://www.visualfractions.com/
http://www.homeschoolmath.net/math_resources_3.php Internet
interactive site
Fraction/Decimal Worksheet: Change these fractions to decimal numbers
Back Worksheet 1
78
79
Fraction/Decimal Worksheet Key
Change the following fractions to decimal numbers.
Decimal/Fraction Worksheet
Worksheet 1b Back
Change the following decimals to fractions.
1a.
3.45 =
1b.
0.77 =
1c.
5.26 =
2a.
9.94 =
2b.
0.52 =
2c.
2.24 =
80
3a.
0.72 =
3b.
4.55 =
3c.
0.62 =
4a.
0.88 =
4b.
6.98 =
4c.
1.1 =
5a.
0.5 =
5b.
4.91 =
5c.
9.5 =
6a.
0.49 =
6b.
5.21 =
6c.
8.9 =
7a.
9.33 =
7b.
0.76 =
7c.
0.65 =
8a.
0.7 =
8b.
0.46 =
8c.
4.54 =
9a.
1.6 =
9b.
5.3 =
9c.
4.21 =
10a.
5.57 =
10b.
0.74 =
10c.
4.59 =
Worksheet 2
Back
Answer Key for Decimal/Fraction
Answer fractions are only simplified if denominator is less than or equal to 1000.
81
Worksheet
2b Back
82
Worksheet 5Back
FRACTION COMPARISON
Compare using <,>,=
Color in the fractions with a crayon to compare.
5/7 ____ 3/7
3/10 ____ 3/9
4/21 ____ 4/17
3/16 ____ 5/16
9/13 ____ 9/12
83
FRACTION COMPARISON
Compare using <,>,=
Color in the fractions with a crayon to compare.
5/7 __>__ 3/7
3/10 __<__ 3/9
4/21 __<__ 4/17
3/16 __<__ 5/16
9/13 __<__ 9/12
Worksheet 5b
Back
84
Worksheet 6
BackName_______________________
Date___________________
Frank The Fraction:
..The Decimal
Showdown..
Directions:
Determine if item on the left is Greater Than ( >), Less Than (<), or Equal To
(=) the item on the right. Place the appropriate symbol to make the
statement true.
1.
2.
18
99
_______ 0.33
_______ 0.02
100
3.
100
46
40
4.
68
_______
100
83
_______
100
5.
100
6.
100
60
_______ 0.52
0.67 _______ 0.14
100
7.
8.
75
0.71 _______
0.92 _______ 0.17
100
9.
10.
11
_______ 0.25
0.44 _______ 0.34
100
11.
12.
0.63 _______ 0.1
0.16 _______ 0.56
85
13.
14.
73
22
_______ 0.15
_______
100
15.
30
16.
100
100
35
4
0.06 _______
_______
100
17.
48
100
100
18.
0.53 _______
96
0.51 _______
100
19.
21
7
100
20.
51
0.82 _______
_______ 0.74
100
100
86
Worksheet 6b Back
Date___________________
Name_______________________
Frank The Fraction:
..The Decimal
Showdown..
Directions:
Determine if item on the left is Greater Than ( >), Less Than (<), or Equal To
(=) the item on the right. Place the appropriate symbol to make the
statement true.
1.
2.
18
99
_____>__ 0.02
__<_____ 0.33
100
100
3.
46
40
4.
68
100
83
____<___
___>____
100
5.
100
6.
100
60
____>___ 0.52
0.67 ___>____ 0.14
100
7.
8.
75
0.71 ___<____
0.92 ___>____ 0.17
100
9.
10.
11
___<____ 0.25
0.44 ___>___ 0.34
100
11.
12.
0.63 ___>____ 0.1
0.16 ___<____ 0.56
87
13.
14.
73
22
____>___ 0.15
100
15.
16.
35
0.06 ___<____
4
____>___
100
48
100
100
18.
0.53 ____>___
96
0.51 ___<____
100
19.
100
100
30
17.
21
___>____
7
100
20.
51
0.82 ____>___
____<___ 0.74
100
100
88
Activity Set 6, Lesson 2: Adding and Subtracting Decimals back
Introduce/ Review place value of whole numbers to millions, emphasizing
that the decimal system is based on powers of 10.
Activity: Number Jumble
Materials Needed: Index cards with single digits written on them
Teacher should place 7 single digits on 7 different index cards. Cards
should be given to 7 students. Teacher should ask that they arrange
themselves to create:
1. the largest possible 7 digit number
2. the smallest 7 digit number
3. a number between ___ and ____
Repeat with different digits/different students.
Focusing on the “ones” place, discuss the need to breakdown into pieces or
parts.
(Point out that these also have to be powers (divisions) of 10)
Activity 7: Using grids have students represent values for, tenths,
hundredths, thousandths. Back
CEEMM examples This gives excellent visuals for helping students
understand place value.
Link:
Grids Worksheet 7a, 7b. 7c (Use these for students to represent
given decimal values)
Link: tenths grid
http://www.teachervision.fen.com/tv/printables/scottforesman/Math_4_T
TT_12.pdf
Link: hundredths grid
http://www.teachervision.fen.com/tv/printables/scottforesman/Math_3_T
TT_11.pdf
Link: thousandths grid
http://www.math-drills.com/decimal/blmthousandthsgrid.pdf
89
Additional Resources/Practice:
Computer Activity (online game):
http://www.bbc.co.uk/skillswise/numbers/fractiondecimalpercentage/deci
mals/introductiontodecimals/activity.shtml
This website offers “factsheets” “worksheets” and “activities/games”.
View and select appropriate levels for your students.
http://www.bbc.co.uk/skillswise/numbers/fractiondecimalpercentage/decim
als/introductiontodecimals/factsheet.shtml
Activity 8
back
From CEEMM:
Place Value Chart See Example Worksheet 8
Application lesson: Whole Class
Objective: Students demonstrate the concept of symmetry around the ones
place in the decimal place-value scheme.
(Materials needed: labeled place value chart)
• On the place value chart, write the number 33,333.333 -displaying
each numeral in the proper place on the chart.
• Display the place-value chart.
Point out that the ones place is in the center of the chart.
• Ask: “What is the value of the 3 at the left of the ones place? (3
tens) “What is the value of the 3 at the right of the ones place?” (3
tenths)
• Repeat with the 3’s in the hundreds and the hundredth’s places, the
3’s in the thousands and the thousandths places.
• Discuss that the ones place is the point of symmetry, or balance, and
that the tens place to its immediate left and the tenths place is to
its immediate right, the hundreds place is two places to the left and
the hundredths place two places to the right, and so on.
• Point out that each place-value position has a place value that is ten
times greater than the position to its immediate right and a value
that is one-tenth as much as that of the position to its immediate
left, regardless of where it is in relation to the ones place.
90
•
To check for understanding, pose a new number and allow students
to reflect in the math journal on the place values and their
relationships to each other.
Also, allow students to play the following game in order to reinforce
place value.
Comparison Game: Glencoe (Directions are given for how to prepare for
and play the game)
Activity 9
back
From CEEMM
Estimating to whole numbers before adding and subtracting decimals
allows students to achieve a reasonable sum or difference before the actual
computation. This lessens the difficulties with misaligned decimal points.
Begin by giving a simple problem such as the following: “Mary ran the
race in 5.324 minutes. Taylor completed the race in 6.348 minutes. How
much faster was Mary than Taylor?”
First ask the students to estimate. Then challenge them to find the exact
answer.
Materials: Base-Ten Blocks
If remediation of 4th grade work in necessary, begin with decimals in
the hundredths. Let students work in pairs to estimate and solve a given
problem. Have base ten blocks available if needed by some students. If some
students desire to use base ten blocks, review that the 100’ flat is one, the
ten strip is .1 and the units block is .01. Let students share their answers
with the class. If any of the students used base ten blocks, let those
students demonstrate their work on the overhead.
Next, move to decimals in the thousandths. Allow students to work in
pairs to solve given problems and share their strategies with the class.
Please remember that 5th grade students should add and subtract decimals
symbolically through thousandths.
Activity:
Materials needed: decimal squares
Using decimal squares have students shade in the given decimal values.
91
Ex: .34 + .4 = _______
or .72 - .34 = _____
This gives an excellent “visual”.
Note: for subtraction, students can
‘strike through’ the squares to see what is left.
Addition & Subtraction of Decimals
by
PowerPoint with Notes/examples
Click here
Doug McFarland
Horizon Academy
Roeland Park, Kansas
www.horizon-academy.com
1) 7 + 15.876 + 0.56
7.000
15.876
0.560
23.436
PowerPoint
addition/Subtraction
Practice
Click Here
Concentration game: http://www.quia.com/cc/291087.html
Websites: http://www.aaamath.com/B/dec312x3.htm
http://www.math.com/school/subject1/lessons/S1U1L4EX.html
Teacher resource: Addition/Subtraction example sheet Worksheet 11
Student worksheets: addition/subtraction
http://www.bbc.co.uk/skillswise/numbers/fractiondecimalpercentage/decim
als/usingdecimals/worksheet.shtml
http://www.math-drills.com/decimal.shtml
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Activity 10 back
Following is an NCTM activity which has students take a written problem,
organize data into a chart, interpret the data, then incorporate a working
knowledge of decimals in order to compute baseball statistics.
Teachers may want to assign this to groups, or work on this as a whole group
activity.
http://illuminations.nctm.org/LessonDetail.aspx?id=L257
***Website that has multiple activities/games that address this Activity
Set:
http://classroom.jc-schools.net/basic/math-decim.html
ASSESSMENT: For a final assessment of this activity set:
Given a set of numbers, students will:
1. Place them in order,
2. Round each to the nearest whole, tenth, and hundredth.
3. Represent a given decimal number(s) on a grid.
4. Translate to word form.
5. Add and subtract selected numbers in the set, estimating first.
Worksheet 7a
Back
93
94
Worksheet 7b
Back
95
Worksheet 7c
Back
96
EXAMPLE:
Place values and place holders
'Decimal' simply means based on ten. Our numbers are organized in a system
based on multiples or sub-multiples of ten.
The position of a digit in a number shows its 'place value':
This system continues after the decimal point. Places to the right of the decimal
point are called decimal places, with tenths, hundredths and thousandths in the
first three decimal places. (When speaking numbers, be careful to make the
difference between 'tens' and 'tenths', for example, clear.)
Notice that a zero 'place holder' was needed in 8.03, to keep the 3 in its correct
position.
Place holders are also needed in each of the numbers below.
You can find Skillswise at http://www.bbc.co.uk/skillswise
This factsheet is BBC Copyri
Worksheet 8
Back
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Worksheet 9
Back
98
N2/L1.5
Addition and subtraction of decimals
When numbers include decimal fractions, it is important to remember that the
decimal point marks the end of the whole numbers. So the rule is line up the
decimal points underneath one another and put a decimal point in place ready for
the answer. You then start from the right.
Example 1
62 + 1.05 + 0.9
Adding extra zeros is useful, and will help you to avoid mistakes.
Remember also to round up or down the decimal number to the nearest whole
number to get an estimate for the final answer. Then check your answer.
In this case the estimate is 62 + 1 + 1 = 64. The actual answer is 63.95.
So you can see that the estimated answer is about the right size.
Worksheet 10
Back
99
Example 2
46.7 - 4.15
The answer is 42.55 and the estimate is 43 (47 - 4), so the estimate is the
right size. Worksheet 11
Back
Activity 11: Adding and Subtracting Fractions back
Essential Question: How do students add and subtract fractions
with like and unlike denominators.
It is expected by the time students reach 5th grade they will be
fluent in adding and subtracting whole numbers. Therefore, the emphasis in
5th grade with regard to addition and subtraction is on fractions (including
decimals). There is a major difference in the way 5th grade students should
deal with fractions compared to the way they should deal with decimals.
In 4th grade students had experience writing equivalent fractions. 5th
grade is the first time they use that knowledge to add and subtract
fractions. When adding and subtracting fractions, students should use only
concrete and pictorial models. 5th grade students are NOT expected to add
and subtract fractions symbolically (numbers only). In addition, students
should have had experiences with mixed, proper and improper fractions. For
more information on 5th grade expectations with regard to mixed, proper,
and improper fractions, see “Fractional Relationships”.
Regarding decimals, on the other hand, they should be able to
estimate, add, and subtract decimals symbolically (numbers only) through
thousandths. That builds on the adding and subtracting through hundredths
that they experienced in 4th grade.
5th graders should use a variety of addition and subtraction strategies
to create and solve problems. They should be given opportunities to estimate
solutions.
Suggested reading: Gator Pie
100
Lessons that Promote Understanding Include:
Materials Needed:
• pattern blocks
• graph paper
• overhead pattern blocks
• overhead projector
• colored pencils
NOTE:
All mathematical problem solving should be given in context. For
the context to be meaningful rather than contrived, make up problems based
on situations from your classroom. Because the following introductory lesson
suggestions are for a varied audience and because the problem should be
based on your classroom experiences, context is not included. However,
please provide context for any problems given to students.
Because this is the first experience students have had adding and
subtracting fractions, they should start with fractions that have common
denominators.
Lesson 1:
Using pattern blocks to add and subtract fractions is a visual way to
promote understanding of the concept. Let the hexagon = 1 whole, the
1
1
1
trapezoid = , the rhombus = , and the triangle = .
2
3
6
•
•
•
•
Allow students some free time to explore with pattern blocks if you
have not already done so.
1 1
1
Ask students to demonstrate 1 whole, , , and
using the pattern
2 3
6
blocks.
1
1
1
1
1
1
Then ask students to add + ,
+ ,
+ , etc.
2
2
3
3
6
6
As students build their models, they should copy them onto graph
paper using colored pencils to keep track of the fractional parts. Also,
require that students label the fractional parts of the model and the
final sum. All answers should be in simple form.
101
•
Students should share their solutions and explain their reasoning.
Provide overhead pattern blocks as they demonstrate to the class.
Work involving subtraction of fractions should be introduced in the same
manner using pattern blocks and beginning with common denominators.
PowerPoint Presentations:
Adding –like denominators
Subtracting- like denominators
Lesson 2:
After students are comfortable adding and subtracting fractions with
common denominators, move to adding and subtracting with unlike
denominators. NOTE: Again, it is important that all problems be given in
context relative to your classroom. Let the hexagon = 1 whole, the
1
1
1
trapezoid = , the rhombus = , and the triangle = .
2
3
6
• Using pattern blocks, let students work in pairs to solve problems
4
1
such as
+
(4 triangles and one rhombus equals one). Problems of
6
3
this type do not require “trading” of pattern blocks and are good
introductions to unlike denominators.
• As students build their models, they should copy them onto graph
paper using colored pencils to keep track of the fractional parts. Also,
require that students label the fractional parts of the model and the
final sum. All answers should be in simple form (students worked with
equivalent fractions in 4th grade.).
• Ask students to share their solutions and explain their reasoning.
Provide overhead pattern blocks as they demonstrate.
Work involving subtraction of fractions with unlike denominators should
be introduced in the same manner using pattern blocks.
After students are comfortable adding and subtracting fractions with unlike
denominators, advanced students may move to adding and subtracting mixed
fractions. Again, use manipulatives and pictorial models and give problems in
the context of your own classroom.
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PowerPoint Presentations: (after clicking on the link, go to edit slide to open
the show)
LCM and LCD
Adding-unlike denominators
Good Web sites to help with remediation are:
http://math.rice.edu/~lanius/fractions/frac4.html
http://www.aaamath.com/fra410a-addfractld.html
http://www.matti.usu.edu/nlvm/nav/frames_asid_106_g_2_t_1.html
http://www.aaamath.com/fra57b-subfractld.html
NOTE: CHECK all Web sites before students begin to work. Sites do
change.
The above lessons are located in the CEEMM document.
Assessment: Adding & Subtracting Fractions Worksheet 12 back
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Assessment Adding & Subtraction Fractions
1. Samantha has a pepperoni pizza to share. Alison ate 1/3, Abby ate ¼ and Samantha ate 1/6.
How much pizza was eaten? How much was left for Amy?
2. Daniel brought meat pizzas to share. He ate 2/6, Alan ate 2/3 and Ian ate ¼ and Hal ate 5/12.
How much pizza was eaten?
3. Keith & Mark each had a bag of cookies. Each bag had the same number of cookies. Keith ate
¾ of his cookies. Mark ate 7/8 of his cookies. Who ate more cookies?
4. Marion swims 1/2 of the length of a swimming pool in the same time that Marcus completes 5/8
of the length. Who swims farther?
5. Frank & Kyle got the same number of cookies at snack time. Frank ate 3/8 of his cookies and
Kyle ate ¾ of his. Who ate fewer cookies?
6. Pat & Cam played the same number of basketball games. Pat won ¼ of his games. Cam won
3/8 of his. Who won more games?
7. If you spend 45 minutes each night on your math homework, what fractional part of one hour is
that?
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8. Hailey exercises for one hour everyday. Today she had to stop after 20 minutes. What
fractional part of her exercise did she finish?
9. Mrs. Smith has 20 students. Four students are sick. Six students are working on the news staff.
What fraction of the class is left with Mrs. Smith?
_____10.
Which problem represents the picture
below?
+
____ 11.
A
1 1
+
4 2
B
2 1
+
8 4
C
1 1
+
8 4
D
2 1
+
4 4
(III.E.1)
Which problem represents the picture
below?
−
A
1 1
−
4 8
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B
1 1
−
2 4
C
1 1
−
2 8
D
1 1
−
2 6
(III.E.1)
____ 12.
Which problem represents the
picture below?
−
A
3 2
−
5 5
B
1
2
−
2 10
C
5 10
−
3
2
D
3
2
−
5 10
Worksheet 12
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106
TANGRAM PATTERN
Back
worksheet 13
107
Worksheet 14
Back
108