Download Number Concepts Mathematics

Number
Concepts
Mathematics
Grade 5
Justin Lundin
[email protected]
Table of Contents
Lesson 1- Comparing Fractions
Lesson 2- Adding and Subtracting using Arrays
Lesson 3- Multiplying Fractions using Rectangular Array Models
Lesson 4- Sense about Numbers
Lesson 5- Investigating the Commutative, & Associative Properties
Lesson 6- Finding Prime #’s through the “Sieve of Eratosthenes”
Lesson 7- Creating Factor Trees
Lesson 8- Set Definition Approach to finding the GCD/GCF
Lesson 9- Using the Prime Factoring Method to find the GCD/GCF
Lesson 10- Set Definition Approach to finding the Least common Multiple
Lesson Plan 1- Comparing Fractions
Standard(s): MCA 5.1.3.2 Model Fractions using a variety of representations.
5.1.3.3 Estimate sums and differences of fractions.
Objectives: Students will:
 represent fractions by modeling them as parts of a rectangle.
 Compare fractions by drawing them out as rectangular arrays.
Launch: As students, “Who would have more cake, the teacher who has 3/4 or the
teacher who has 5/6? When students give an answer, have them give some for of
rational. Then, see if the class can come with a consensus.
Explore: Draw two identical rectangles on the board and call on a couple students to
come up and represent 3/4 in one rectangle with the lines running vertically and represent
5/6 in the other with the lines running horizontally. Check with the class to see whether
or not they feel these two models have helped them see what is the larger amount of cake.
Discuss how representing the models may help compare when they have like
denominators but often when the denominators are different it is much more difficult to
see the difference.
Show the class how to compare the two fractions by drawing the horizontal and vertical
lines in both rectangles. Show how 3/4 has become 9/12 now and 5/6 has become 10/12.
Now students can draw conclusions about the fractions. 9/12<10/12 so 3/4<5/6,
10/12>9/12 so 5/6>3/4.
Share: Give the students the fractions 2/3 and 3/5. In small groups have them first
represent the fractions. Then have them compare and discuss their results. Have them do
the same with 1/2 and 3/7.
To finish the lesson give 3/4 and 2/3 to the groups. Have them go up to the board and
show how they would compare and represent the fractions.
Summarize: Discuss with the class the results on the board and how modeling the
comparison of fractions helps us see which is really bigger and which is smaller.
Evaluation: Show how you might use rectangular arrays to represent 3/5 and 2/4. Then,
show how a person can compare them by using the rectangular array method.
Lesson Plan 2- Adding and Subtracting using Arrays
Standard(s): MCA 5.1.3.2 Model Addition and Subtraction of Fractions.
5.1.3.1 Add and subtract fractions using efficient and generalizable
procedures, including standard algorithms.
Objectives: Students will:
•
use rectangular arrays to represent how to add and subtract fractions with
common and uncommon denominators.
Launch: Suppose Trevor has 2/3 of a box of cookies and Macy has ¼ a box of cookies.
How many cookies do they have all together? Give the class some thing time to try and
come up with ways to solve the problem. After some time, ask the students if they can
think of any ways to represent this problem with pictures?
Explore: Discuss with the class the ideas they have generated. Discuss examples such as
(Number lines, Pie charts, and Rectangular Arrays.) On the board have on student draw a
box with horizontal lines that represent 2/3 and have another student draw a box of
similar size with vertical lines that represents ¼. Then, demonstrate to the class how to
divide the boxes up into equal parts that still represents 2/3 and ¼. Then, draw a third
box for the answer that is divided up just like the first two. Add up the total number of
shaded parts in the first two boxes and shade that many in the third box. Discuss with the
students how that represents how many cookies Trevor and Macy have together. Go
through a few examples together until it seems they have the hang of it. Next ask the
class to try and represent how much more Trevor has than Macy. Discuss their results
and methods. Go through a few subtraction problems until they have the hang of it.
Share: Have the students come up with their own fraction addition and subtraction
problems. Then, have them exchange their problems with a partner and solve their
partner’s problems through the rectangular array method.
Summarize: Bring the class back together and review how drawing pictures that
represent the fractions can help solve fraction problems.
Extension:
Evaluation: Draw a rectangular array to solve the following problems;
1/2 + 1/3 =
5/8 + 1/5=
2/3 – 1/4 =
3/5 - 3/8 =
Lesson Plan 3- Multiplying Fractions using Rectangular Array Models
Standard(s): MCA 6.1.3.1 Multiply fractions using efficient and generalizable
procedures, including standard algorithms.
Objectives: Students will:
 use rectangular arrays to represent how to multiply fractions with like and
unlike denominators.
Launch: Put 3/4 and 2/3 on the board. Ask student to use their prior knowledge of
fractions and rectangular arrays to try and come up with a way to represent 3/4*2/3. Give
them time to work out their thoughts and discuss their ideas with their classmates.
Explore: Have a class discussion and give the students a chance to explain their thoughts
and ideas of how to represent multiplication of fractions. Then demonstrate for them
how to show the multiplication of fractions using rectangular arrays. Work through
2/5*1/3 with the class.
Share: Put the students into small groups and have them work through 1/2*2/3, 1/4*3/5,
4/7*2/3, and 3/4*1/2. After the groups have had time to work out the problems. Give
each group the chance to go up to t he board and show the class how to represent one of
the multiplication problems using the rectangular array model.
Summarize: Review with the class the rectangular array model for multiplication and
have them come up with some reason why it may be useful to understand this method.
Evaluation: Using the rectangular array model, show how one might solve this
multiplication problem. 2/5*3/4=
Lesson Plan 4- Sense about Numbers
Standard(s): MCA 5.1.1.2 Consider the context in which a problem is situated to select
the most reasonable solutions and use the context to interpret the solution appropriately.
Objectives: Students will:
• be able to use context clues to place the numbers in the appropriate places.
• be able to use mathematical strategies and number sense to place appropriate
numbers into context.
Launch: Using the Number Sense handout from the Number Concepts class, at the
beginning of Social Studies class, put the paragraph about Thomas Jefferson under the
document camera. Ask the class, “What might we be able to figure out about T.
Jefferson by using our math skills?” Let students lead the discussion and share ways that
they were able to make deductions based of their knowledge in math. Use this as an
example of where a person might use math outside of math class.
Explore: During the following math class, look at and discuss many other
developmentally appropriate number sense activities from the packet.
Share: When you feel the students are ready, have them come up with their own number
sense paragraph problem using many numbers. When students are ready, have them
share their problems with the class.
Summarize: Review with the class the different ways and strategies they came up with to
put the numbers in the right places in the paragraphs.
Evaluation:
Mr. Mo bought a new car for __A__. He had to pay __B__ % sales tax on his
new car. Mr. Mo also had to pay __C__ dollars for a new licenses plate. While waiting
for the paperwork to get done Mr. Mo got hungry and bought a cheesy, cheeseburger for
__D__ and a bottle from the drink machine that cost him another __E__ dollars. By the
time Mr. Mo drove away with his new car he was shocked, he had just spent __F__
dollars at the car lot.
23,000
1.75
7.50
3.9
340
23349.25
Lesson Plan 5- Investigating the Commutative, & Associative Properties
Standard(s): MCA 5.2.2.1 Apply the commutative and associative properties and order of
operations to solve problems involving whole numbers.
Objectives: Students will:
• learn what the characteristics are of the commutative and associative
properties.
• analyze tables of numbers to find out if they are commutative & associative.
Launch: Have the class share what their prior background and knowledge is about the
commutative and associative properties. Explain to them that for example the
commutative property of multiplication is a * b = b * a. The associative property for
multiplication is (a * b) * c = a * (b * c). Have the class come up with some examples
that would prove these properties to be true. Then, see if they can come up with
examples of the two properties for addition.
Explore: Pass out the Investigating the Commutative & Associative Properties
worksheet. Investigate the first problem with the class to determine whether or not the
table follows the Commutative and Associative properties. While discussing the table
introduce the concept of closure to the students. Go through the second problem the
same way.
Share: Have the students get into small groups and discuss/analyze the third and fourth
charts. After they have been given some time to come up with reasonable solutions, have
the groups discuss their results with the class and explain why they have reached their
conclusion.
Summarize: Review with the class the to properties they worked on in this lesson, the
commutative property and the associative property. See if they can give examples of
each in addition and multiplication.
Evaluation:
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Lesson Plan 6- Finding Prime #’s through the “Sieve of Eratosthenes”
Standard(s): MCA (Can’t find anything on primes in the 4-7th grade benchmarks) Aagh!
Objectives: Students will:
 learn the characteristics of prime numbers.
 learn strategies for finding prime numbers.
 learn ways prime numbers are related to each other.
Launch: Put a list of numbers on the board. (1, 3 9, 7, 8, 2.) Ask the class to think about
these questions; Which numbers are prime? What makes them prime? How do we know
they are prime? How often does a prime number occur in counting number?
Explore: Discuss what the students know about prime numbers. Point out that prime
numbers are all numbers that have only two factors, itself and 1. 1 is not a prime number
because it only has one factor. Put a chart on the board with N=16 and have them in 4
columns. Demonstrate to the class how the Sieve of Eratosthenes works. Do one
example with the class where N=20 and there are 5 columns.
Put the class into groups. Assign one group to do a table where N=100 & Col. =10,
another N=100 & col. =8, another group N=100 & Col. =6, and the last group N=100 &
the col. =4. Have the groups construct their Sieve of Eratosthenes on a hug sheet of
paper that can be hung to the wall.
Share: Have each group share their chart and what they found along the way. Look at
each group’s results and try to see if the class can pick out any patterns. Is one particular
number of columns easier than the rest to predict prime numbers? Why? Point out what
“Twin Primes” are. Twin primes differ by one number. i.e.- (2,3) (17,19) (29, 31)
Summarize: Put this list of numbers back on the board. (1, 3 9, 7, 8, 2.) Ask the class to
answer these questions; Which numbers are prime? What makes them prime? How do
we know they are prime? How often does a prime number occur in counting number?
Evaluation: Write out the numbers 1-25. Put them on a chart that has 5 columns. Using
what you have learned from the “Sieve of Eratosthenes” go through your chart and circle
all the prime numbers and cross out the composite (not prime) numbers.
Lesson Plan 7- Creating Factor Trees
Standard(s): MCA 6.1.1.5 Factor whole numbers; express a whole number as a product of
prime factors with exponents.
Objectives: Students will:
 factor numbers down to their prime factorization.
 come up with multiple ways to arrive at the prime factorization of a number.
 express a whole number as a product of prime factors with exponents.
Launch: Put the number 24 on the board. Ask the question, “What are the prime factors
of 24?” Give the students some think and discussion time before looking at the question
and answer in a more depth manor.
Explore: Discuss the class’s thoughts and ideas of prime factors. Have the students pull
from previous knowledge about prime numbers and factors. Every counting number
greater than one is a unique product of prime numbers. Start off by asking what two
factors equal 24? (i.e.- 4*6) Discuss whether or not those numbers are prime or
composite. If they are composite continue to break them down until there are only prime
number being multiplied together. (i.e. 2*2*2*3, or 2^3*3) Discuss how those numbers
might fit the description of the prime factors of 24. Have the class see what they get if
they factor 24 out starting with 3*8, 2*12, 1*24. Discuss the results.
Share: Have the students work with partners and factor out the following numbers 12,
32, 36, 25, and 16. Remind them to write their answer in the form of exponents if
whenever possible. When students have completed these, call on groups to go up to the
board and share what their answer is and how they ended up with the prime factorization.
Summarize: Review the steps for making a factor tree and arriving at the prime
factorization of whole numbers.
Activity: Practice factoring out numbers, comparing them, and coming up with the GCF
& LCF of them at http://www.mathplayground.com/factortrees.html
Evaluation: Using the factor tree method express 42 and 80 as a product of prime factors
with exponents.
Lesson Plan 8- Set Definition Approach to finding the GCD/GCF
Standard(s): MCA 6.1.1.6 Determine GCF of whole numbers. Use common factors to
find equivalent fractions.
Objectives: Students will:
 be able to determine the Greatest Common Divisor or GCF using the “Set
Definition Approach.”
 be able to describe a situation when on has to use the GCD/GCF.
Launch: Discuss with the class what their prior knowledge is on greatest common
factors. Explain to them that a GCF is really a greatest common divisor or GCD. A
GCD is the largest divisor that goes into two numbers that are being compared. Ex.- The
GCD of 12 & 20 would be 4 because it is the largest number that will divide both
numbers. Brainstorm times when GCD will come into play. An example, it is used
when renaming fractions in lowest terms.
Explore: Put the numbers 9 and 24 on the board. Walk the students through the “Set
Definition Approach.” Step 1) List all the divisors of 9, then list them all for 24. Step 2)
Circle all the divisors they have in common and write them down. Step 3) Pick out the
GCD.
Work through some examples with the students. (8, 20) (16, 32) Remind students of the
importance of following the steps and writing everything out.
Share: Once the class has the hang of it partner them up and have them go through some
together. (18, 36) (21, 28) (24, 42) After they have had a chance to work them out have
the students compare their results with other groups. Then, have each group put one of
their problems on the board to discuss with the whole class.
Summarize: Review with the students what a GFD or GCF is and how a person may go
about finding it. Review a practical time when one might need to use their knowledge
about GCD.
Evaluation: 1.) Using the “Set Definition Approach” find the GCD of (24, 36)
and (32, 56). Remember to show all the steps.
2.) When would being able to find the GCD be useful?
Lesson Plan 9- Using the Prime Factoring Method to find the GCD/GCF
Standard(s): MCA 6.1.1.6 Determine GCF of whole numbers. Use common factors to
find equivalent fractions.
Objectives: Students will:
 be able to find the GCD (GCF) by using the “Prime Factorization Method.”
Launch: Brainstorm with the class to see if the students can think of any other ways to
find the GCD. After some discussion let them know they will be learning about the
“Prime Factorization Method.” Have a brief discussion with the class to see if any of
them might be able to think about this name and come up with an idea of what this
method is about.
Explore: Relate this method to the student’s prior knowledge of factor trees. Start with
the numbers (9,24). Step 1) Make a factor tree for the number 9 and a tree for the
number 24. Step 2) Write down 9=3*3 and 24=2*2*2*3. Step 3) Circle prime factors
that the two numbers have in common. For this problem it would be 3. So, the GCD of
(9, 24) is 3. Work another example with the class (70, 180). 70=2*5*7 180=2*2*3*3*5.
They have a 2 and 5 in common. 2*5=10. So, 10 is the GCD of (70, 180).
Share: Partner up the class and have them work out some problems together. (80, 100)
(36, 42) (60, 90) (24, 48) After they have had a chance to work them out have the
students compare their results with other groups. Then, have each group put one of their
problems on the board to discuss with the whole class.
Summarize: Review with the class the steps to finding the GCD using the “Prime
Factorization Method.”
Evaluation: Using the “Prime Factorization Method,” find the GCD of (56, 72) and (48,
96). Remember to show all the steps.
Lesson Plan 10- Set Definition Approach to finding the Least common Multiple
Standard(s): MCA 6.1.1.6 Determine LCM of whole numbers. Use common multiples to
calculate with fractions and find equivalent fractions.
Objectives: Students will:
 be able to compare 2-3 integers and find what common multiples they share.
 be able to find the LCM of 2-3 integers using the “Set Definition Approach.”
 be able to explain situations in where they would use their knowledge about
least common multiples.
Launch: Put the integers 4 and 6 on the board. Explain to the class that for the past
couple days we have been finding the GCD of integers. Today, we will start comparing
numbers by their “Multiples.” Ask the class to try to come up with an idea of what a
multiple is or even what might the multiples of 4 and 6 are.
Explore: Discuss the class’s ideas about multiples. Lead them into the discussion that if
“Divisors” divide integers then “Multiples” multiply integers. Talk about the multiples
of 4 and 6. After making a list of their multiples brainstorm with the class on their
thoughts about what the last multiple of 4 and 6 are. In the end, the class should conclude
that multiples of numbers are infinite, they go on forever. On the board, work through
the “Set Definition Approach” to finding the LCM. ***Stress the importance of writing
everything out.
Step 1) Make a list of multiples for 4 and 6. Ex- 4{4, 8, 12, 16, 20, 24, 28, 32, 36, …}
Explain to the class that we always put … at the end of our list because the multiples go
on and on.
Step 2) List 3 common multiples. Example for 4 and 6- {12, 24, 36…}. Let the students
know that it is important to list three so we can clearly see a pattern. At the end of the
lesson discuss the pattern that all common multiples follow. (Once the first is found, the
next is twice the first, the third is three times the first and so on.)
Step 3) Write down the “Least Common Multiple.” {12}
Do a couple examples with the class. Once again stress the importance of being very
particular about writing everything out in the language of mathematics. {5, 7} {3, 8}
Brainstorm and try to find any areas in math where finding the LCM is useful. Ex- When
adding and subtracting fractions with unlike denominators.
Share: Have the students partner up and find the LCM of {4, 7} {6, 9} {8, 12}. After
they have had a chance to work them out have each group put one of their problems on
the board to discuss with the whole class.
Summarize: Review with the class what multiples, common multiples, and least common
multiples are. Have a student restate where they might use them in life.
Evaluation: 1.) Using the “Set Definition Approach” show how you would find the LCM
of {3, 5} and {7, 9}. Remember to show all the steps.
2.) When would being able to find the LCM be useful?
Investigating the Commutative, & Associative
Properties
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