Number Sense Activities and Lessons
Final Project for Math 6061
Bemidji State University
Summer 2012
Completed by:
Joan Carter, Math Instructor
Minneapolis Community and Technical College
[email protected]
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Executive Summary
These units, activities and lessons are designed to be used in Math 0060 taught at Minneapolis
Community and Technical College. This course prepares students for Algebra and Technical
Programs. These lessons are also applicable to Minnesota Middle & High School. All of the
lessons in this unit make use of Number and Operation ideas: representing numbers,
relationship of numbers and number systems; understand the meaning of operations and how
they relate to one another; compute fluently and make reasonable estimates.
The activities in these lessons were purposefully chosen to be a concrete as possible. Although
students seem to understand positive and negative numbers as a concept, they do not fully
understand using them in operations (addition, subtraction, multiplication and division). Often
textbooks have them memorize algorithms. This strategy does not work for many students.
And this becomes the barrier to solving equations and moving on in mathematics.
These lessons will be used as supplemental material to the current course curriculum. The
students are given weekly quizzes to assess what they have learned. What this material adds is
the approach to discussing the concepts and why they are learning. These lessons are concrete
representations of Math concepts that many students struggle in the past to learn. The proper
use of language of Math and the meaning of these words is also emphasized.
This project was created while doing coursework at the Bemidji State University in Bemidji,
Minnesota during the summer institute of 2012 in pursuit of a master degree in mathematics
with emphasis on education.
Math 0060 Outline of Major Content Areas Covered in this course: Whole Numbers and
decimal review; Graphs (bar, line, circle); Statistics (mean, median, mode); Metric and English
measurement; Geometry basics; Variables and expressions; Simple linear equations and word
problems; Ratios and percents.
Learning Outcomes
A. Read and write large numbers, estimate sums, differences, products and quotients of whole
numbers.
B. Read and write decimal numbers, express money amounts, compare and round decimals,
arithmetic operations with decimals.
C. Solve application problems with whole numbers and decimals.
D. Use metric units for measuring, convert between metric units using dimensional analysis,
choose appropriate metric units, differentiate between Fahrenheit and Celsius temperatures,
and solve application problems using metric units.
E. Use English units for measuring, convert between English units using dimensional analysis,
choose appropriate English units and solve application problems using English units.
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F. Draw right angles, intersecting, parallel and perpendicular lines, calculate angles in a
triangle, understand formulas for rectangles, squares, triangles, circles, rectangular solids,
and cylinders, and use the Pythagorean Theorem.
G. Use the order of operations, simplify variable expressions, solve simple linear equations
and algebra word problems.
H. Reduce fractions using prime factorization, perform arithmetic operations with fractions,
convert between fractions and decimals, compare fractions, and solve application problems
with fractions.
I. Use ratios and proportions, solve percent problems using proportions, solve percent
problems using percent equations, and use a four-function calculator.
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Lesson Plan Table of Contents
Divisibility Tests
Pages 5-18
Least Common Multiples
Pages 19-24
Measuring Temperature
Pages 25-26
Adding Integers using a Number Line
Pages 27-28
Adding Integers using Chips
Pages 29-31
Subtracting Integers using Chips
Pages 32-33
Multiplying Integers
Pages 34-35
Dividing Integers
Pages 36-40
Distributing and Factoring Using Area
Pages 41-46
Bouncing Tennis Balls
Pages 47-50
Pre-Test/Post-Test
Pages 51-56
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Divisibility Tests
Adapted from: NCTM – Investigations Using Base-Ten Block
Diagrams for Divisibility Tests by Marvin E Harrell and Dawn R
Slavens from Teaching Children Mathematics, February 2009
This lesson will fulfill the class requirement of number sense and preparation for
arithmetic operations with fractions and algebra. The students will develop strategies
for using and understanding divisibility tests. Benefits learning why the tests work can
be transferred to understanding number sense related to large numbers and place value,
simplifying fractions, learning concepts of greatest common factor and least common
multiple, and providing a context to practice mental math strategies.

Launch: Define Divisibility - A number ( ) is divisible by another number ( )
if the
divides evenly into the . That is
has no remainder. How do we know if
a number is divisible by 2? (Even number). Why does this test work? What about for 5
or 10? What about for 4? What about 3?

Explore: Have colored pencils available. Pass out the Activity Sheet 2:
Discovering Divisibility - Divisibility and the Number 5. Have students work through
this exercise and share the results. (Share 1) What is the divisibility test for 5? Can we
relate these results to 10? Next let’s look at 4. Pass out Pass out the Activity Sheet 3:
Discovering Divisibility - Divisibility and the Number 4. What is a divisibility test for
4? Can you use this knowledge to find the smallest four digit number that is divisible
by 4 (1000)? What about the largest four digit number divisible by 4 (9996=9900+96)?
(Share 2) Pass out the Activity Sheet 4: Discovering Divisibility - Divisibility and the
Number 8. This one is different than what we have just done. (Share 3). Pass out the
Activity Sheet 5: Discovering Divisibility - Divisibility and the Number 9. Be sure to
emphasize that students are not to combine the unshaded squares as they did in the
previous (8s) activity. Have students work through this exercise and share. (Share 4)
Pass out the Activity Sheet 6: Discovering Divisibility - Divisibility and the Number 3.
Have students work through this exercise and share. (Share 5) Pass out the Activity
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Sheet 7: Discovering Divisibility - Divisibility and the Number 6. Have students work
through this exercise and share. (Share 6)

Share: 1) Ask students to share their answers. Discuss what they found. How
does the shading relate to the quotient and remainder when you divided by 5 (or 4 or
3)? How can you relate what you found for groups of 5 to groups of 10? 2) Have
students work through this exercise and share the results. Share the results for
divisibility by 4. 3) For AS 4 (Number 8) Students should see that a unit has no groups
of 8, a long (10s) has 1 group of 8 with 2 unshaded, a flat (100s) has 12 groups of 8
with 4 unshaded, 1000 is divisible by 8 and any larger block can be thought of as
multiples of 1000. So you need only determine if the at the last three digits are divisible
by 8. 4) As students work through the questions they are guided to the critical discovery
that the total number of unshaded squares for each type of base-ten block corresponds
to the digits in each of the place values within the number that is being investigated.
That leads to the realization that if the sum of the digits is divisible by 9 the original
number is too. 5) As students work through the questions they realize that when
shading groups of 3 the results are the same as it was with 9s, that the total number of
unshaded squares for each type of base-ten block corresponds to the digits in each of
the place values within the number that is being investigated. That leads to the
realization that if the sum of the digits is divisible by 3 the original number is too. 6) As
students work through this exercise they realize that a number is divisible by 6 is it is
divisible by both 2 and by 3. And now we can see that if a number is divisible by both 2
and by 5 it is also divisible by 10.

Summarize: Divisibility Tests for 2, 3, 4, 5, 8, 9 and 10
Divisibility by 2: a whole number is divisible by 2 if it is an even number.
Divisibility by 3: a whole number is divisible by 3 if the sum of its digits is divisible by 3.
Divisibility by 4: a whole number is divisible by 4 if the number represented by the two digits
in the tens and ones place is divisible by 4.
Divisibility by 5: a whole number is divisible by 5 if its ones place digit is a 0 or 1.
Divisibility by 8: a whole number is divisible by 8 if the number represented by the three digits
in the hundreds, tens and ones place is divisible by 8.
Divisibility by 9: a whole number is divisible by 9 if the sum of its digits is divisible by 9.
Divisibility by 10: a whole number is divisible by 10 if its ones place digit is a 0.
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Least Common Multiples
This lesson will fulfill the class requirement of number sense and preparation for
arithmetic operations with fractions and algebra. The students will develop strategies
for finding the Least Common Multiples of two or more numbers.

Launch: A polygon is a closed form, straight-sided figure. Some common
polygons are Triangle which is a 3-gon; a Square which is a 4-gon; a Pentagon which is
a 5-gon; a Hexagon which is a 6-gon; a Heptagon which is a 7-gon; and an Octagon
which is an 8-gon. Today we will look at a new twist on polygons.

Explore: Pass out the worksheet and the shapes needed for the activity (see
Launch)/ Demonstrate using a square and triangle as shown in the worksheet. Line up
the dots. One shape is the fixed or stationary shape (in example this is the square). And
one shape will be rotated (in the example this is the triangle). The square stays fixed
and we rotate the triangle around the square. Be sure to rotate and not slide the triangle.
With each turn we count. We rotate until the dots line up again. We write down the
answer in the proper box on the table (Fixed shape: Square and Turning Shape:
Triangle). Carry on filling in the table and see if you can start to predict the numbers
based on the number of sides of the two polygons. Continue the activity checking to see
if your prediction is correct. Can you find any exceptions to your rule?

Share: Ask students to share their answers. Discuss what they found. Did
anyone have a rule that turned out to be incorrect? What do these numbers represent?
What do you notice about 6-gon and 7-gon together? Is this different from 7-gon and 6gon? What do you notice about 8-gon and 6-gon? What do you notice about 4-gon and
8-gon? How about 7-gon and 7-gon?

Summarize: Define Least Common Multiple (LCM) of two given numbers is
the smallest whole number that is a multiple of both of the given numbers. Ex. LCM of
6 and 7:
Multiples of 6:
6, 12, 18, 24, 30, 36, 42, 48, …
Multiples of 7:
7, 14, 21, 28, 35, 42, 49, 56 …
42 is the LCM of 6 and 7.
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A New Twist on Polygons
Start with a Fixed Square and a Turning Triangle.
Line up the dots for the starting position. Rotate the Turning Shape only.
One turn rotates the triangle around the vertex of the square. Note the new
position of dot.
How many turns are needed to make the dots line up again?
Enter this number on the table. The square is the fixed shape so we need the
column for the square and the turning shape is a triangle so we want to fill in the
row for triangle. Carry on filling in the table and see if you can predict the number
based on the number of sides of the two polygons. Continue the activity checking
to see if your prediction is correct.
Table of Number of Turns:
Triangle
Turning
Shape
Square
Fixed Shape
Pentagon Hexagon Heptagon Octagon
Triangle
Square
Pentagon
Hexagon
Heptagon
Octagon
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Least Common Multiples Using Prime Factorization
This lesson will fulfill the class requirement of number sense and preparation for
arithmetic operations with fractions and algebra. The students will develop strategies
for finding the Least Common Multiples of two or more numbers.

Launch: Review the results of the New Twist on Polygons. Define Least
Common Multiple (LCM) of two given numbers is the smallest whole number that is a
multiple of both of the given numbers. Ex. LCM of 6 and 7: 42 is the LCM of 6 and 7.
Let’s see if we can find another way to find LCMs.

Explore: Define: Prime Number is a whole number greater than 1 that has only
two factors, 1 and itself. Composite Number is a whole number greater than 1 that is
not a prime. In other words a composite is the product of 1 and at least two prime
factors. (Note: the Whole Numbers 0 & 1 are neither prime nor composite). Reminder factors are the numbers we multiply together to get a product. What are the factors of
12? We know that
different factorizations of 12 are
. So the four
. Prime Factorization is
the process where given a product, we find all the prime number factors that when
multiplied back together give us the original product for 12 the prime factors are
. Let find the prime factorization for some of the numbers we found there.
What are the Prime Factors of 6? What are the Prime Factors of 7?
Use the Venn Diagram: Factors of 7 in one side and factors of 6 in the other.
Notice there are no numbers in the intersection. Find the Prime Factors and use the
Venn Diagram to chart the following pairs of numbers: 8 & 6, 4 & 8 and 7 & 7.
Compare what you find with the results of the number of turns in a New Twist on
Polygons. Using the Venn Diagram can you see another way to find Least Common
Multiples?

Share: Ask students to share their answers. Discuss what they found. Do we
see a rule here? Try the rule for 32 and 48 (answer: 96). And again for 32, 48 and 30
(480).
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
Summarize: To find the Least Common Multiple (LCM) of given numbers
Find the Prime Factorization of each number and plot the numbers in a Venn Diagram.
Multiply all the factors together to find the LCM of the given numbers.
Finding LCM by Using Division by Primes: Find the LCM of 32, 48 and 30.
2
32
48
30
2
16
24
15
2
8
12
15 *
2
4
6
15 *
2
2
3
15 *
3
1
3*
15 *
5
1*
1
5
1
1
1
Process: Find the smallest Prime Factor of any of the numbers. Divide by the Prime. If
not divisible by the given prime * bring down to the next level unchanged. Continue
until all 1s. The LCM is the Product of the Primes. LCM of 32, 48 and 30 is
.
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Prime Factorization: Venn Diagrams
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Prime Factorization: Venn Diagrams
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Adapted from: Unit 1.3, Accentuate the Negative - Integers –
Connected Mathematics Dale Seymour Publications, Menlo Park,
CA, 1998
This lesson will be taught in a 45 minute session. It will fulfill the class requirement of
number sense and preparation for algebra.

Launch: Normally we think of a number line as horizontal. Show number line.
We call the numbers to the right of zero positive integers. Those to the left are negative
integers. Negatives are opposites of positives. A thermometer is a vertical number line.
Here in Minnesota (sometimes) the temperature falls below zero. These are Negative
Temperatures.

Explore: Pass out Problem Sheet. Let students work. Walk around and
observe, answer and help. Pair people as needed.

Share: Ask students to share their answers. It is always helpful for the students
to have to explain to others. Ask: If the thermometer was horizontal instead of vertical,
where would the positive temperatures be in relation to zero? How about the negative
temperatures? Are temperatures integers? Is there more than one unit of measure for
temperature? Can you name them?
Now suppose the temperature is +5º. Where is this located on a vertical number line?
Now suppose the temperature is -5º. Where is this located on a vertical number line?
Can you find two other numbers that are the same distance from zero? Numbers that are
the same distance from zero but on different sides of zero are called opposites. These
pairs of numbers are also called zero pairs.

Summarize: Review the definitions of integers, positive numbers, negative
numbers, opposites, zero pairs. No need to remind them that it is cold in Minnesota in
the wintertime.
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Measuring Temperature Problem Sheet
Section A: Arrange the following temperatures in order from lowest to highest.
1. -8º, 4º, 12º, -2º, 0º, -15º
Which is hottest? Which is coldest?
Section B: The temperature reading on a thermometer is 5ºF. Show how you got there. Decide
what the new reading will be if the temperature:
1. Rises 10º
2. Falls 2º
3. Falls 10º
4. Rises 7º
Section C: The temperature reading on a thermometer is -5ºF. Show how you got there.
Decide what the new reading will be if the temperature:
1. Falls 3º
2. Rises 3º
3. Falls 10º
4. Rises 10º
Section D: Find the halfway temperature between the two given temperatures. Show how you
got there.
1. 0º and 10º
2. -5º and 15º
3. 5º and -15º
4. 0º and -20º
5. -8º and 8º
5. -6º and -16º
Section E: Decide which of the given numbers is further from -2º. Show how you got there.
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1. -6º and 6º
2. -7º and 3º
4. 2º and -5º
4. -10º and 5º
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Adding Integers using a Number Line
Adapted from: Unit 2.1, Accentuate the Negative - Integers –
Connected Mathematics Dale Seymour Publications, Menlo Park,
CA, 1998
This lesson will be taught in a 45 minute session. It will fulfill the class requirement of
number sense and preparation for algebra.

Launch: You have been adding numbers for a long time, and you all know
how to find the sum of two or more whole numbers. My question is this: What is
addition really about? What does it mean when you add two numbers? Take answers.
We combine two amounts – we add – to find the total number of something.

Explore: Pass out Number Line worksheet. Can you think of a way to show
addition on the number line? Find a partner and discuss. Pair people as needed. Let
students work. Walk around and observe, answer and help.
Next ask: Now that we have some strategies for adding numbers like 7 and 8, let’s look
at a different set of problems and think about how we can use the number line to help
us add them. Consider -5 + -4. With your partner, use the number line to show the
addition of these numbers. Now try +5+-4; and -5 + 4. Predict first and then use the
number line. Give more paper and other problems as needed.

Share: Ask students to share their answers. It is always helpful for the students
to have to explain to others. One way to use a number line is to start at 0 draw an arrow
above the number line the length of the first number to the left if it is negative to the
right if it is positive. Represent addition of the second number by drawing a second
arrow above the first arrow, starting at the point where the first arrow ended and going
left or right depending if the second integer is positive or negative. What did you get for
-5 + -4? Discuss. Now try +5+-4; and -5 + 4. Predict first and then use the number line.

Summarize: Review the definition addition and using a number line. Positives
to the right and Negatives to the left.
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Adding Integers Using a Number Line
Problem:
Problem:
Problem:
Problem:
Problem:
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Adding Integers using Chips
Adapted from: Unit 2.2, Accentuate the Negative - Integers –
Connected Mathematics Dale Seymour Publications, Menlo Park,
CA, 1998
This lesson will be taught in a 45 minute session. It will fulfill the class requirement of
number sense and preparation for algebra.

Launch: We have used number lines to add integers. Do you remember what
we called the pairs of numbers that were equidistant from zero? (Oh yeah, zero pairs,
what do they add to? Zero). Today we will try something new to represent addition. In
the language of business, sometimes people say ‘we are in the black’ or ‘in the red’.
What do you think that means? Profits are a positive situation and are referred to as “in
the black”. Losses are a negative situation is “in the red”.

Explore: What patterns do you remember from adding with the number line?
What do you know about the sum of two positive numbers? Can you give an example?
What do you know about the sum of two negative numbers? Can you give me an
example? What do you know about the sum of a positive and a negative? Can you give
an example? Is this true for all cases? Let’s explore a different way to model the
addition of integers. As you do this activity, think about what happens when you add
positive and negative integers. Distribute chips and Chip Boards (any paper is fine).
Explain that the chips represent the addends (the two numbers being added). Red chips
are negative and black chips are positive. Give them some chip board examples on doc
camera. Give some written problems. In pairs, one person works the problem with
chips. The other person writes down the problem and solution using proper notation.
Give a variety of problems. Partners should take turns in the different roles.

Share: Ask students to share their answers. Discuss how to reduce the board
when there are two color chips. Zero pairs can be removed. Discuss why. Now that you
have done this activity, discuss what happens when you add positive and negative
integers. Define “Absolute Value”
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
Summarize: When adding two numbers with the same sign the answer (sum)
has the same sign as the addends. When adding two numbers with different sign the
answer (sum) has the same sign as the addend that is furthest from zero. Mention zero
pairs again. And Absolute Value.
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Adding Integers Using Chips – Instructor Reference
Problem 1: Show a Chip Board with 11 black and 14 red chips.
(11 + -14 = -3)
Can you give another combination that would simplify to this same
answer?
Problem 2: Seven black chips are added.
(-3 + 7 = 4)
Three more black chips are added.
(4 + 3 = 7)
Twelve red chips are added.
(7 + -12= -5)
Problem 3: Find two combinations of b & r that simplify to -11
Write a sentence to represent each combination.
Write an equation to represent each combination.
Problem 4: Now can you find these sums:
-
105 + +65 =
+
-
-
-
+
99 + -47 =
90 + -90 =
1050 + -150 =
120 + -225 =
35 + -35 =
Problem 5: In Duluth, Minnesota, the temperature at 6:00 AM on January 1 was -30ºF.
During the next 8 hours, the temperature rose 38 ºF. Then, during the next 12 hours, the
temperature dropped 12 ºF. Finally, in the next 4 hrs, it rose 15 ºF. What was the temperature at
6:00 AM on January 2?
(-30 º + 38 º + -12 º + 15 º = 11 ºF)
For reference if needed:
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Subtracting Integers using Chips
Adapted from: Unit 3.1, Accentuate the Negative - Integers –
Connected Mathematics Dale Seymour Publications, Menlo Park,
CA, 1998
This lesson will be taught in a 45 minute session. It will fulfill the class requirement of
number sense and preparation for algebra.

Launch: You know how to find the difference numbers. My question is this:
What is subtraction really about? What does it mean when you add two numbers? Take
answers. How is this different than addition? Subtraction is taking something away and
that is the opposite of addition. When else did we talk about opposites? We subtract to
find the difference in two numbers.

Explore: Model subtraction using a Chip Board. What happens when what we
want to take away the opposite of what is on the board (15 - -8)? Add zero pairs as
needed to have 8 reds to take away. Distribute chips and Chip Boards (any paper is
fine). Remind them that the red chips are negative and black chips are positive. Give
them some chip board examples on doc camera. Give some written problems. In pairs,
one person works the problem with chips. The other person writes down the problem
and solution using proper notation. Give a variety of problems.

Share: Ask students to share their answers. Discuss zero pairs and how to
reduce the board when there are two color chips. Zero pairs can be removed. Discuss
why. Now that you have done this activity, how does subtraction compare to when you
add positive and negative integers? Show some examples and discuss.

Summarize: Subtraction means take away. It is the opposite of addition.
Sometimes we need to add zero pairs to be able to take the correct amount away.
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Subtracting Integers Using Chips – Instructor Reference
Problem 1: Use a Chip Board to show 14 - 11 = 3
Can you give another combination that would simplify to this same
answer?
Problem 2: Start with 10 chips. Seven black chips are subtracted. (10 - 7 = 3)
Three more black chips are subtracted.
(3 - 2 = 1)
Five more black chips are subtracted.
(add zero pairs)
(1 - 5 = -4)
Problem 3: Start with 10 chips. Seven red chips are subtracted.
(10 - -7 = 17)
Three more red chips are subtracted.
(17 - -3 = 20)
Five black chips are subtracted.
(20 - 5 = 15)
Problem 4: How does subtraction relate to addition? Find
1050 - 150 =
+
-
-
120 - 225
+
35 - 35 =
1050 + -150 =
120 + -225 =
+
35 + -35 =
Problem 5: Use the number line to look at 7- -5 and -7-5. What does the solution tell you?
(how far the numbers are apart.)
Problem 5: Use the number line to look at 5 - 12 and -5 + 12. What does the solution tell you?
(how far the numbers are apart.)
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Multiplying Integers
Adapted from: Investigation 4, Accentuate the Negative Integers – Connected Mathematics Dale Seymour Publications,
Menlo Park, CA, 1998
This lesson will fulfill the class requirement of number sense and preparation for
algebra. The students will develop strategies for multiplying and dividing integers;
recognize and use the relationship of multiplication and division as inverse operations
and solve problems. In this lesson we will look at multiplying two positive integers and
a positive and a negative integer.

Launch: Let’s think about temperature. We’ll say that when temperature rises
we will use a positive symbol. When temperature decreases we will use a negative
symbol. Let’s start by looking at a 3º increase over several hours. Define Product.

Explore: Together we will look at a table with the temperature increasing 3º
over several hours. What patterns do you see? How would you find the next
temperature? (Say it, write it in words, and write it in symbols). This is called a
recursive equation. Is there a way we could get directly to the 5th iteration? This is
called an explicit equation. How are adding and multiplying related? Now do the same
for 3º decrease (-3º). Now find the temperature after 10 hours, assuming the pattern
continues. Now work together to find the answers to the rest of the questions. Make
sure you discuss your answers with a partner.

Share: Ask students to share their answers. Help them analyze how each pair
of addition and multiplication sentences are related. What does 3x2=6 mean? (that you
have 3 groups of 2). What does the operation of multiplication mean? (Could be
repeated addition). Use consistent language. So what is the expression 8 x 4 mean? (8
groups of 4 = 32). Now write 8 x -4 on the board. What does this mean? (8 groups of -4
= -32). Is this different than -4 x 8? (No) Is that always true? Yes.

Summarize: Multiplication is repeated addition.
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Multiplying Integers
1: Suppose the temperature changed by an average of +3º per hour for a 10 hour period.
Number of hours
Total Temperature
1
2
3
4
5
+3º
change
Write a sentence that tells how you find the next temperature.
Write an expression that tells how you find the next temperature.
Write a multiplication sentence that represents the total change in temperature for the
entire 10 hour period.
2: Suppose the temperature changed by an average of -3º per hour for a 10 hour period.
Number of hours
Total Temperature
1
2
3
4
5
-3º
change
Write a sentence that tells how you find the next temperature.
Write an expression that tells how you find the next temperature.
Write a multiplication sentence that represents the total change in temperature for the
entire 10 hour period.
3: Make up a situation about temperatures that can be expressed as 4 -10
4: Find each product.
-4
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-5
4
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Dividing Integers
Adapted from: Investigation 4, Accentuate the Negative Integers – Connected Mathematics Dale Seymour Publications,
Menlo Park, CA, 1998
This lesson will be taught in a 45 minute session. It will fulfill the class requirement of
number sense and preparation for algebra. The students will develop strategies for
multiplying and dividing integers; recognize and use the relationship of multiplication
and division as inverse operations and solve problems. In this lesson the students
develop rules for dividing integers based on what they know about multiplication.

Launch:
We have discussed what it means to add, subtract, and multiply. We know
addition as the operation of combining and subtraction as the operation of
taking away. We have also discussed how addition and subtraction are related.
We refer to these as opposites, or inverse, operations. We can think of them as
undoing each other. If we add an integer to another integer, we can undo the
addition by subtracting the integer we added from the sum. If we subtract one
integer from another, we can undo the subtraction by adding the integer that we
subtracted to the difference.
We now look at the operation of division. What does it mean to divide two
integers? Give an example of Division and tell how it calls for division, and tell
how your example shows a way to interpret division.
Division is sometimes referred to as “sharing” operation. One interpretation of division
is breaking an amount into groups of the same size. For example: I have 36 cupcakes.
How many should I give to each of my four friends if I want to share them equally? To
answer this question, we need to group the cupcakes into four sets and find how many
are in each set. Write 35 ÷ 4 = 9 on the board.
How do you read this sentence?
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The sentence is read “36 divided by 4 equals 9”. This equation means there are 36
things that are put into groups of 4 of equal size and there will be 9 things in each
group. Or it can be thought about as putting 36 things in groups of 4 and finding that
there are enough to make 9 groups.
How are the operations of multiplication and division related?
In multiplication, groups of the same size are put together to find a total amount. In
division, the total is taken apart or partitioned to make groups of equal size.
Multiplication and division are opposite, or inverse operations; they undo each other.
Multiplication puts equal-size groups together to find the total; division partitions the
total to find the size of the groups or the number of groups of a given size.
If multiplication and division are opposite operations that undo each other, how
could we write a sentence that would undo the sentence
? What
would the new sentence mean?
This multiplication sentence says that the total of 3 groups of 12 each is 36. If we start
with the total and divide by 3, we will be partitioning the total to find the size of each
equal-size group (
. If we divide the total by 12, we will be finding how
many groups of size 12 we begin with
).
If students are making sense of this relationship, explain they will use their
understanding of the relationship between multiplication and division to do the problem
and find patterns that will help them to predict the quotient of the division of two
numbers, including integers. If your class is still confused, work through more
examples of two division sentences that are related to any multiplication sentence. Have
students work in pairs on the problem.

Explore: Division is the opposite, or inverse of multiplication, and for any
multiplication sentence you can write a division sentence that undoes the
multiplication. For example
, you can write two division sentences
. Now work together to find the answers to the rest of the
questions. Make sure you discuss your answers with a partner.
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
Share: Ask students to share their answers. Help them analyze how each pair
of multiplication and division sentences are related. What does -30÷6=-5 mean? (that if
you group -30 into groups of 6 you will have -5 groups). What does the operation of
division mean? (Could be the undo of multiplication). Use consistent language. So what
is the expression 8 x 4 mean? (8 groups of 4 = 32). Now write 8 x -4 on the board. What
does this mean? (8 groups of -4 = -32). Is this different than -4 x 8? (No) Is that always
true? Yes. Now think of

Summarize: Have students share their answers. If there is disagreement about
any of the answers have other students share their solutions and explain why they make
sense.
Division is the opposite of multiplication.
Let’s try to agree on rules for the following:
The quotient when dividing a positive integer by a positive integer will always be
The quotient when dividing a negative integer by a negative integer will always be
The quotient when dividing a positive integer by a negative integer will always be
Is this different from
The quotient when dividing a negative integer by a positive integer will always be
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Dividing Integers
1: Find each product.
a:
c: -4
b:
20
d: -5
4
2: Write two division equations that are equivalent to the products you found in 1.
a:
b:
c:
d:
3: Write the division equation that solves each problem.
a:
b:
c:
d:
4: Write the division or multiplication equation that solves each problem.
a:
b:
c:
d:
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5: Find the quotient
a:
b:
c:
d:
e: What do you notice about these quotients?
f: How can you decide if the quotient of two integers is positive?
Negative? Or zero?
6: Find the quotient:
a:
b:
c:
d:
e:
f:
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Distributing and Factoring Using Area
Adapted from: NCTM – Illuminations - this lesson was created by
Annika Tran. http://illuminations.nctm.org/LessonDetail.aspx?id=L744
This lesson will be taught in a 1 ½ hour session. It will fulfill the class requirement of
order of operations, understanding formulas for area, factoring and distribution, and
preparation for algebra. In this lesson, expressions representing area of a rectangle are
used to enhance understanding of the distributive property. The concept of area of a
rectangle can provide a visual tool for students to factor monomials from expressions.
By introducing area to students as a way to represent multiplication of terms, students
have exposure to another tool for understanding and remembering why the distributive
property works.

Launch: Today we are going to look at the distributive property. (Define).
What is area? How do we find area of a rectangle? Ok but what does that have to do
with distribution? That is what we will find out today.

Explore: Pass out Problem Sheet. The first activity sheet can be used to introduce the
concept of the distributive property but this sheet is designed to acquaint students with the area
representation for the distributive property, rather than develops the concept for first exposure
to the rule. The first section introduces students to the idea of writing the area of a rectangle as
an expression of the length × width, even when one or more dimensions may be represented by
a variable. Let students work. Walk around and observe, answer and help. Pair people as
needed. The second page helps students recognize and factor out an integer common factor.
Help students who pause when the common factor is a negative number. Be sure that they
change the sign of the second term. (Example: -2a + 10 = -2(a – 5)). The third page helps
students recognize and factor out a variable common factor or an integer other than the leading
coefficient. Encourage students who pause when the common factor may not seem to follow a
pattern. Students will have to consider both terms when deciding on a common factor. The
fourth page helps students recognize and factor out a common factor that may be an integer, a
variable or a product of both an integer and a variable to find the greatest common factor.
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Students should recognize that when they find a common factor, it may not be the greatest
common factor.

Share: Ask students to share their answers. Ask students to write about the
relationship between the product and sum representation of the area model. Ask
students to write about what they think distributing has to do with the distributive
property.
.

Summarize: Describe the distributive property and how the concept of area of
a rectangle can provide a visual tool for students to factor monomials from expressions.
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Bouncing Tennis Balls
Taken from: NCTM – Navigations Series, Navigating through Algebra.
Grades 6-8 Chapter 2 Analyzing Change in Various Contexts, 2009
Goal
To assess students’—
• ability to collect data and record data in a table;
• ability to make a graph to display data using correct labels, scale, and so on;
• recognition of what varies in an experiment;
• ability to name the independent and dependent variables in a problem.
Materials and Equipment
• a copy of the blackline master “Bouncing Tennis Balls” for each student
• Tennis balls, one for each team of four students
• Access to a clock or watch with a second hand
• Centimeter graph paper, a spreadsheet program, or a graphing calculator
Activity
In teams of four, students bounce a ball to solve this problem:
How many times can each team member bounce and catch a tennis ball in two minutes?
A bounce is defined as dropping the ball from the student’s waist. One student keeps the time
while the second student bounces and catches the ball, the third student counts the bounces,
and the fourth student records the data in a table showing both the number of bounces during
each ten-second interval and the cumulative number of bounces. Each trial consists of a twominute experiment, with the number of bounces recorded after every ten seconds (or twenty
seconds for fewer data points). The timekeeper calls out the time at ten-second intervals. When
the time is called, the counter calls out the number of bounces that occurred during that tensecond interval. The recorder records this count and keeps track of the cumulative number of
bounces. The same process is followed by each student, with the students rotating roles, so that
each student can collect a set of data. All the students must bounce the ball on the same surface
(e.g., tile, carpet, concrete) because differences in the surface could affect the number of
bounces. Once the data have been collected, each student prepares a graph showing the
cumulative bounces over two minutes. This graph can be constructed by hand, by using a
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graphing calculator, or by using a spreadsheet, depending on the students’ experiences and on
what information the teacher wants to gather about what the students know and are able to do.
Discussion
The data from one student’s experiment are recorded in table 2.1.The graph in figure 2.1 was
made using the graphing calculator, and the graph in figure 2.2 was made using a spreadsheet
for the sample data set.
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Students present their results to classmates by showing their graphs. The discussion can
involve what the students found easy and what they found difficult in completing this task.
Students’ discussions can be revealing: Can the students identify what varies in the
experiment? Do they comment on the dependent and independent variables either implicitly, in
their conversations about the graphs, or explicitly, using correct terminology? Do they discuss
whether the points should be connected with a line? The numbers of bounces are discrete data,
so they should not be connected. Decisions about the scale for each of the axes are important;
do the students understand what the graphs would look like if the scales changed? When
directed to sketch lines on their graphs in order to notice trends, do they demonstrate some
sense that the steepness of a line is related to the number of bounces per second? Your
observations related to these and other questions will yield information about what your
students appear to know and are able to do that will guide you in making instructional
decisions. An extension of this activity would be for each student to conduct an experiment
using, for example, concrete floors and then carpeted floors to investigate the effect of
differences in the surfaces.
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Selected Instructional Activities
Highlighted here are some fundamental components of a curriculum that addresses content and
develops students’ understanding by focusing on analyzing change.
Building a Sense of Time and Its Relation to Distance and Speed
Initially students need to become aware of their own understanding of time, change over time,
and the use of new kinds of measure (i.e. rates). Posing such questions as those listed below
focuses their attention on these ideas (adapted from Kleiman et al. 1998).
• How do you measure time? Distance? Speed?
• Give an example of something that might be able to travel at two feet per second.
• What is the difference between traveling at two feet per second and two feet per minute or
two feet per hour?
Students can also explore different activities that test their sense of time. They can do the
following activities in pairs; in each instance they may want to observe if they overestimate or
underestimate the time and try the task again.
• Clap your hands so you clap exactly one clap per second for ten seconds.
• Turn a page in a book at exactly one page every two seconds for twenty seconds.
• Sit still for thirty seconds, letting the timer know by raising your hand when you think thirty
seconds has passed.
• Walk at the speed of one foot per second for fifteen seconds.
• Walk the length of your classroom in exactly ten seconds. At what speed were you traveling?
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PRE-TEST/POST-TEST Name: __________________________ Date: _____________
1. Give the place value of the digit 8 in the number 684,159.
A) thousands
B) ten thousands
C) hundred thousands
D) millions
2. Convert to standard form: four hundred twenty-eight thousand.
A) 428,000
B) 4,028
C) 42,800
D) 4,002,800
3. Is 80 divisible by 5?
A) yes
B) no
4. Add: 108 + 13 + 9.
A) 130
B) 120
C) 125
D) 140
5. Add. 5760 + 10,355
A) 15,015
B) 16,015
C) 15,115
D) 16,115
6. An employee earning $45,000 a year had her yearly salary increased by $2500. What is
her new salary?
A) $70,000
B) $45,250
C) $47,500
D) $47,000
7. Subtract.
A) 29
B) 39
C) 30
D) 20
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8. A farmer wishes to place 3572 eggs in containers holding 12 eggs each. How many
containers will be filled completely, and how many eggs will be left over?
A) 296 containers will be filled completely with 8 eggs left over.
B) 296 containers will be filled completely with 6 eggs left over.
C) 297 containers will be filled completely with 8 eggs left over.
D) 297 containers will be filled completely with 6 eggs left over.
9. Kelly will need 66 paving stones to construct a pathway in the garden. If she has 26
paving stones in the shed, how many more will she need to purchase?
A) 37 paving stones
B) 38 paving stones
C) 39 paving stones
D) 40 paving stones
10. Find the Least Common Multiple of 6 and 8
A) 48
B) 64
C) 24
D) 6
11. Which property is illustrated by the following statement?
18  1 = 18
A) Commutative property of multiplication
B) Multiplication property of zero
C) Multiplication property of one
12. Which property is illustrated by the following statement?
(17 · 3) ·4 = 17 · (3 · 4)
A) Commutative property of multiplication
B) Associative property of multiplication
C) Distributive property of multiplication over addition
13. An ounce of a certain dark chocolate bar contains 17 grams of fat. If Angela eats 3
ounces of the chocolate a day for 5 days, how many grams of fat will she ingest?
A) 25 grams
B) 255 grams
C) 265 grams
D) 100 grams
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14. The math department has $300 to spend on a set of calculators. If the calculators cost
$32 each, how many calculators can the department purchase, and how much money
will be left over?
A) 8 calculators; $2 left over
B) 8 calculators; $12 left over
C) 9 calculators; $2 left over
D) 9 calculators; $12 left over
15. Find the Least Common Multiple of 21 and 14
A) 14
B) 21
C) 42
D) 294
16. Is the number 8,702,124 divisible by 3?
A) Yes
B) No
17. Write the fraction as a division problem and simplify, if possible.
0
8
A) 0 ÷ 8; 0
B) 0 ÷ 8; undefined
C) 8 ÷ 0; 0
D) 8 ÷ 0; undefined
18. A teacher has 34 students in her class. Can she distribute a package of 70 candies
evenly to her students?
A) yes
B) no
19. The number 43 is ____________ .
A) composite
B) prime
C) neither prime nor composite
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20. The number 0 is _____________.
A) prime
B) composite
C) neither prime nor composite
21. List all the factors of 45.
A) 1, 3, 5, 9, 45
B) 1, 3, 5, 9, 15
C) 1, 3, 5, 9, 15, 45
D) 1, 3, 5, 6, 15, 45
22. Plot the numbers on the number line.
2, –4
A)
B)
C)
D)
23. Place the correct symbol between the two numbers.
8
–8
A) <
B) >
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Use the following to answer questions 24-25:
Refer to the number line to add the integers
24. Add. –3 + 2
A) –1
B) 1
C) 5
D) –5
25. Add. –1 + (–4)
A) –3
B) 3
C) –5
D) 5
26. Find the Least Common Multiple of 20, 42 and 35
A) 35
B) 420
C) 29,400
D) 42
27. At 6:00 a.m. the temperature was –6°F. By noon, the temperature had risen by 11°F.
What was the temperature at noon?
A) –5°F
B) 5°F
C) –17°F
D) 17°F
28. Find the range of temperatures.
–9°, –1°, 4°, 6°, –2°, –10°, 12°
A) 1°
B) 0°
C) 22°
D) –22°
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PRE-TEST/POST-TEST Answer Key
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
B
A
A
A
D
C
A
C
D
C
C
B
B
D
C
A
A
B
B
C
C
B
B
A
C
B
B
C
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