Number Sense and Operations
(For Grade 7)
By
Bryan Anderson- Cass Lake-Bena Middle School
([email protected])
Jerry Bellefeuille- Frazee High School
([email protected])
1
Overview:
This compilation of lessons is designed to be implemented once a week. Class time was
estimated to be approximately 45 minutes. Some materials (ex: 20 round objects used in “What
is Pi?” ) are not supplied.
These lessons cover all of the Minnesota Standards in Mathematics for 7th grade. Specific
standards are presented at the beginning of each lesson.
Lesson A: Do You Measure Up? 3-4 days
Students will learn the history of length measurements and while creating their own unit of
measurement, they will discover the pitfalls of having to many types of units, the precision of
different units and how units are converted. This unit is best used as an introduction to
proportions and fraction multiplication.
Lesson B: Are You Being Rational? 3 days
Students will prove the premise that all rational numbers can be written as a ratio of integers.
They will determine strategies for creating equivalent fractions, writing proper fraction form
and learn how to convert decimals to fractions for both terminating and repeating decimals.
Calculator rounding and truncating will also be discussed.
Lesson C: My Calculator Was Wrong! 1-2 days
Students will discuss why different calculators give different answers, as well as rounding and
truncating issues. They will determine how to look at a fraction and decide whether or not it
will have a terminating decimal. Reducing fractions and equivalent fractions will also be
addressed.
Lesson D: What is Pi? 2 days
Students will discover where the magical number of Pi originated. They will measure various
circular objects and try to discover what the relationship of Pi is. Students will be introduced to
the fact that Pi is irrational. Rounding and estimating are also discussed.
Lesson E: Heather’s Garden Problem 1 day
Students are asked to determine how much it will cost to create a circular garden. Students will
investigate the importance of rounding and its effect on solutions. Will rounding have more of
an impact on smaller figures than larger?
Lesson F: Heather’s Improved Design 2-3 days
Students will compute the area and perimeter of an irregular shape. They will create a table
dependant on how many pieces of a circular figure is used. Students will then analyze their
table and create a formula on how the area and perimeter is affected. They will be asked to
generalize this for any fractional part of a whole.
2
Lesson G: A Remainder of One 1 day
A story will be read to the students involving remainders. Students will investigate divisibility
rules and try to define some in their own words. Students will then be asked to describe the
patterns in the story.
Lesson H: Soldier Bug Todd 1 day
An extension of “A Remainder of One”, students are posed a problem where if a number is
divided by 2,3,4,5 or 6 the remainder is exactly 1 but the remainder of 7 is 0. Students will then
find a rule to create a set of all solutions to the problem.
Lesson I: The Division Algorithm 1-2 days
Building on the Soldier Bug story and problem, students will be introduced to the Euclidean
method and the Division Algorithm to find the greatest common divisor. Students will do a
worksheet where they can explore either type and then present their ideas to the class. The
teacher will then tie these methods to the factor list and factor tree methods.
Lesson J: Where Do We Go From Here? 2-3 days
Students will dwell into the concept of negative being the opposite in direction. The absolute
value is the length a number is from zero. It is a nice introduction to the coordinate system.
3
Lesson _A_ Do You Measure Up?
Objectives:
- Students will learn the appropriateness of unit of measure.
- Students will discover how make measurements more precise.
- Students will demonstrate how convert various units of measurement.
- Students will develop an understanding of proportionality.
Standards:
- 7.1.2.1 – Add, subtract, multiply, and divide positive and negative rational numbers.
- 7.1.2.5 – Use proportional reasoning to solve problems involving ratios in various
contexts.
Launch:
The Egyptians were the civilization recorded to have a standard unit of length.
It was called a cubit. A cubit was determined to be the length from the elbow to the tip
of an extended finger.
The Greeks used the width of 16 fingers and called it a foot.
The Romans accepted the Greeks standard of a foot and broke it into 12 unicae which
we now refer to as an inch.
King Henry I created the unit referred to as a yard. And the US standardized these
measurements
Explore:
1. Have students break into groups of 2 – 4 so there is an even number of groups.
2. Discuss the pros and cons the Egyptians may have encountered when developing and
utilizing their cubit.
3. Have each group develop their own unit(s) of length.
4. Have them measure the length of various items in the room (i.e. height of desk, length
of a book, height of the teacher) as well as the length of the room itself.
5. Have students discuss the accuracy of their measurements, and what if anything needs to
be done to modify their measuring device.
6. Have groups pair up and compare the measurements of their items. Have them discuss
the relationship between their numbers.
7. Have them develop a method for converting from one type of measurement to another.
8. Have students calculate the perimeter of the room and the difference between the length
and width of the room.
Share:
1. After Explore item 2 have the class share the pros and cons by first listing them on the
board then as a class discussion. Determine where problems could have arouse and
possible solutions.
2. After Explore item 4 have students share how accurate their measurements were and
how they handle it when it came down to a partial unit.
3. Have students share their strategies for converting measurements.
4
Summarize:
1. Emphasize the need to have a standard unit of measure.
2. Smaller units tend to be more precise.
3. Show a physical method (wrapping method) to convert measures.
- Take a piece of string that is the length of the larger unit and wrap it around the smaller
unit of measurement. (i.e. take a piece of string that is a meter long and wrap it around a
yardstick. It will go around the length once and have about 3 inches left over. So 1
meter is about 1 yard and 3 inches or 1 m = 1 1/12 yds.)
4. Show a mathematical method (proportions) for converting between units.
1 foot / 12 inches = x feet / 18 inches
Multiply the numerator of the 1st fraction to the denominator of the second and set it
equal to the product of the denominator of the 1st fraction and the numerator of the
second.
1(18) = 12(x)
Solve for x.
18 = 12x divide both sides by 12
X= 1.5 or 3/2
So 18 inches equals 1.5 or 3/2 feet.
Educator Notes:
a. The information in the Launch was gathered at
www.coe.uh.edu/archive/science/science_lessons/scienceles3/length/length.html
b. This particular lesson may be broken up into 3 parts:
i. The discussion of problems with ancient units of measure.
ii. The development and use of student rulers.
iii. The development of strategies for converting units of length.
5
Lesson _B_ Are You Being Rational?
Objectives:
- Students will be able to demonstrate how rational numbers can be represented as a ratio
of 2 integers.
- Students will learn how to develop and test conjectures.
- Students will understand how calculators truncate or round some numbers.
- Students will create common ratios
Standards
- 7.1.1.1 – Know that every rational number can be written as the ratio of 2 integers or as
a terminating or repeating decimal.
- 7.1.1.2 – Understand that division of 2 integers will always result in a rational number.
Use this information to interpret the decimal result on a calculator.
- 7.1.1.5 – Recognize and generate equivalent representations of positive and negative
rational numbers, including equivalent fractions.
- 7.1.2.3 – Understand that calculators and other computing technologies often truncate or
round numbers.
Launch:
I hate the word “always”. My wife says that I always leave the seat up in the bathroom.
Then I have to find at least one time where I didn’t leave the seat up to prove her wrong.
Have any of you had a parent or friend say to you that you always did/do something?
Mr. Jacobson said that any rational number can always be written as a fraction of 2
integers. I told him how much I hate the word always and bet him that we could find at
least one.
Explore:
1. Have students break into groups of 2-3.
2. Have them write several numbers of various forms and discuss why they feel their
number cannot be written as a fraction with integers. (go to Share 1)
a. Have them write fractions that don’t have integers in the numerator and
denominator. Can these fractions be changed?
b. For integers, are there division problems that give you an equivalent value?
3. Have the students write down several fractions and their decimal equivalents and give a
visual representation for both. (Students may use calculators)
a. When do the values terminate?
b. When do values repeat?
c. Do you think the calculator is giving you an exact answer or is it rounding it off?
4. Have them discuss the differences in how the fraction looks and how its equivalent
decimal looks. Have them find strategies to change the decimal to a fraction. (go to
Share 2)
6
Share:
1. Have one person from each group write a possible number that cannot be written as a
fraction with integers and have the class justify whether or not it works. (go to
Summarize 1)
2. Have the students discuss strategies for changing decimals to fractions for both terminal
and repeating decimals. (go to Summarize 2 and 3)
3. Have students discuss how the fraction and decimal forms compare.
Summarize:
1. Demonstrate various methods for changing rational numbers into fractions of integers,
thus reinforcing the definition of a rational number.
a. Integers can be written as the integer over one.
b. When you have a decimal in a fraction, multiply both the numerator and
denominator by the same power of 10.
Note that these also produce equivalent fractions
2. Demonstrate how to change a terminating decimal to a fraction.
a. Remove the decimal point and place it in the numerator.
b. The denominator is the place value of the last digit on the right.
( 3.26 becomes 326 over 100 or 326/100) Note that this fraction is not reduced but it
does fulfill the premise that all rational numbers can be written as a fraction of 2
integers. (Once again you can use this opportunity to discuss equivalent fractions as
you reduce the fraction.)
3. Demonstrate how to change repeating decimals to fractions.
a. Set the decimal equal to n.
b. Make a new equation by multiplying both sides of your existing equation by 10
to a power of the number of values being repeated.
c. Subtract the 2 equations.
d. Divide by the coefficient of n.
e. Reduce the fraction as necessary.
Example
Step a
Step b
Given .54545454…..
.54545454…. = n
2 values are repeated so multiply both sides by 10 ^ 2 OR 100
100 * .545454…. = 100 * n becomes 54.545454…. = 100n
Step c
100n = 54.545454…
- n = .545454…
99n= 54
Step d
99n = 54
99
99
Step e
so n = 54
99
Divide both numerator and denominator by 9, so n= 6
11
7
Lesson _C_ My Calculator Was Wrong!
Objective:
- Students will learn how to develop and test conjectures.
- Students will be able to determine if a ratio will result in a terminating or repeating
decimal.
- Students will understand how calculators truncate or round some numbers.
Standards:
- 7.1.1.1 – Know that every rational number can be written as the ratio of 2 integers or as
a terminating or repeating decimal.
- 7.1.1.2 – Understand that division of 2 integers will always result in a rational number.
Use this information to interpret the decimal result on a calculator.
- 7.1.2.3 – Understand that calculators and other computing technologies often truncate or
round numbers.
Launch:
My calculator lied to me. I know that 2/3 is point 6 repeated, but my calculator showed
.666667 which is 666667 millionths. So whose right? If the calculator is wrong, how often is
it wrong?
Explore:
1. Have students break into groups of 2-3.
2. Have them discuss why the calculator may be wrong. (Go to Share 1)
3. Have them fill out the Repeat or Terminate worksheets.
a. Take the row value divided by the column value.
b. Fill in the fraction and T if it terminates or R if it repeats.
4. Have them discuss whether or not patterns exist.
5. Is there a way to determine whether or not a fraction will yield a terminating decimal?
If so, what is it?
6. Did the negative numbers have an impact on whether or not you got a repeating
decimal?
Share:
1. Have each group write down one reason why the calculator may be incorrect?
2. Have students demonstrate patterns.
3. Have students discuss strategies for determining whether or not a fraction will
terminate.
Summary:
1. Calculators would be continuously running if they were given a repeating decimal.
2. At a certain point, digits will seem insignificant.
3. Point out that if a fraction is reduced and the denominator had only factors of 10, then it
terminated.
4. Negative numbers have no effect on whether or not a fraction results in a terminating
decimal.
8
Repeat or Terminate?
Names: _______________________________________
a/b
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
9
19
20
Repeat or Terminate?
Names: _______________________________________
a/b
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
7
8
9
10
10
8
9
10
Lesson D: What is Pi? (2 class periods)
Objectives:
Students will discover the relationship between the circumference and diameter, Pi.
Students will learn that Calculators round irrational or repeating decimals.
Students will understand that how rounding will affect the outcome.
Minnesota Standards Covered:
Know that every rational number can be written as the ratio of
two integers or as a terminating or repeating decimal. Recognize
Read, write,
7.1.1.1 that π is not rational, but that it can be approximated by rational
represent and
compare positive
numbers such as 22 and 3.14.
7
and negative
rational numbers,
expressed as
Compare positive and negative rational numbers expressed in
integers, fractions
various forms using the symbols <, >, ≤, ≥.
7.1.1.4
and decimals.
For example: − 1 < −0.36 .
2
Number &
Operation
Add, subtract, multiply and divide positive and negative
rational numbers that are integers, fractions and terminating
decimals; use efficient and generalizable procedures,
7.1.2.1 including standard algorithms; raise positive rational numbers
to whole-number exponents.
Calculate with
2
positive and
For example: 34 × ( 1 ) = 81 .
2
4
negative rational
numbers, and
Understand that calculators and other computing technologies
rational numbers
often truncate or round numbers.
with whole number
7.1.2.3
exponents, to solve
For example: A decimal that repeats or terminates after a large number of digits
real-world and
is truncated or rounded.
mathematical
problems.
Solve problems in various contexts involving calculations with
positive and negative rational numbers and positive integer
7.1.2.4
exponents, including computing simple and compound
interest.
Bold highlighting indicates partial standards met.
elsewhere in this unit.
No highlighting means the standard is met
11
Launch:
I called my mother the other day and she asked me how my day was. I told her that I really liked Pi, I
could use it for a variety of things. At this point I realized the other side of the phone line got dead
silent. I asked, “What is wrong mom?” She replied, “How would you use pie for anything else but
eating?” That is when I realized that I might have to have a talk with mom about what Pi is.
So what is Pi? (Get various responses from students, hopefully circles/diameters/radii will come up.
Depending on how the discussion goes, you might opt to not tell them anything else, or opt to let them
know that it is a relationship between the circumference and diameter- but do not tell them what the
relationship is.)
To get a better understanding of Pi, the students will do an activity trying to find out what relationship
it is. Using various circular objects and a tape measure, students will record the circumference and
diameter of each.
Explore:
Discovering Pi
Materials Needed:
20 various sized circles
Tape measures
Paper or grid sheet and pencils
Calculators
Set Up:
Place one circle and tape measure per station.
Have students make some sort of grid for recording
their data.
Activity:
Students will measure the diameter and circumference of each circle.
They will then record it on their grid. Students will have to determine
what measure system they will use and how accurate their
measurements will be. After students collect all their data, they will
have to determine what mathematical operation gives them Pi, thus
finding the relationship hinted to earlier.
Share:
On a grid either on the board or overhead, have students record their findings for circumference,
diameter and Pi. What strategies did students use to find Pi? How do you know your answer is
correct? Why do you think you didn’t get an exact value for Pi? What things in your activity could
you have changed to get a more precise value?
Summarize:
Pi is the relationship between the circumference and the diameter.
Pi is an irrational number, it neither terminates nor repeats.
Calculators often truncate repeating or irrational numbers.
Accuracy affects outcome.
12
Lesson E: Heather’s Garden Problem (1 class period)
Objective:
Students will learn what effect estimations will have on problems.
Students will learn what degree of estimation is appropriate for the problem.
Students will review circles, circumference, area, diameter and Pi.
Minnesota Standards Covered:
Know that every rational number can be written as the ratio of
two integers or as a terminating or repeating decimal. Recognize
Read, write,
7.1.1.1 that π is not rational, but that it can be approximated by rational
represent and
compare positive
numbers such as 22 and 3.14.
7
and negative
rational numbers,
expressed as
Compare positive and negative rational numbers expressed in
integers, fractions
various forms using the symbols <, >, ≤, ≥.
7.1.1.4
and decimals.
For example: − 1 < −0.36 .
2
Number &
Operation
Add, subtract, multiply and divide positive and negative
rational numbers that are integers, fractions and terminating
decimals; use efficient and generalizable procedures,
7.1.2.1 including standard algorithms; raise positive rational
numbers to whole-number exponents.
Calculate with
2
positive and
For example: 34 × ( 1 ) = 81 .
2
4
negative rational
numbers, and
Understand that calculators and other computing technologies
rational numbers
often truncate or round numbers.
with whole number
7.1.2.3
exponents, to solve
For example: A decimal that repeats or terminates after a large number of digits
real-world and
is truncated or rounded.
mathematical
problems.
Solve problems in various contexts involving calculations with
positive and negative rational numbers and positive integer
7.1.2.4
exponents, including computing simple and compound
interest.
Bold highlighting indicates partial standards met.
elsewhere in this unit.
No highlighting means the standard is met
13
Launch:
My wife wants me to make our house look nice. She has great plans on landscaping, home
improvement, fencing; you know all the hard manual labor types of jobs. One of the big focal points
to our new house will be a huge circular-shaped flower garden (I’m going to call it a landscaping
project for the house, it sounds more macho). When I went to T&K Outdoors to talk to them about
this project, I found out that edging blocks for the garden were going to cost __________, and fill for
garden was __________. Heather has great design ideas so I would like some help in estimating the
cost of this “improvement”.
Explore:
Students need to have knowledge of Circumference, Pi, Diameter, Radius and Area
Students will work in groups of 2-3.
Garden Problem:
Hand out fraction circles worksheet (use 1/6th circle template) for students. Tell the
students that this is the Flower Garden template, but that they do not need to make any cuts
yet. Ask students what type of vocabulary they associate with circles. Student replies should
converge to Circumference, Pi, Radius, Diameter and Area. Tell the students that the
proposed Radius of the garden is 6 feet and have students find diameter, circumference and
area of the garden.
Ask the students what Pi is, most will give the number 3.14. Remind students that at earlier
grades, Pi was 3, now they use 3.14. Have students check the effect on the circumference and
area of the garden using their old value of 3. After students have had some time with their
calculations, tell the students that Pi is irrational, and show the students Pi to the 8th decimal
place. Ask students what effect using 4, 6 or even 8 decimal places will have on our
circumference and area. Then have students create a table and calculate values to test their
hypothesis.
Pi = 3.14159 26535
Share:
Have students record their values on a overhead or table on the board. Have students talk about the
change in circumference and area and how that will affect the flower garden. Is it important to be
accurate or is the approximation of 3.14 acceptable? Have students relate this to projects where
circles will be smaller or vastly larger (ex: cutting tile to allow for pipes/drains to circular sports
arenas). Is it important to be accurate with Pi? What situations will approximations be acceptable?
Summarize:
Estimating does cause error.
Choose appropriate estimations for a given problem.
Pi is an irrational number with infinite decimal places.
14
Heather’s Flower Garden Template
15
Lesson F: Heather’s Improved Design (2-3 class periods)
Objective:
Students will compute the area and perimeter of partial circles.
Students will create tables and generate relationships for shapes using a different number of fractional
pieces.
Minnesota Standards Covered:
Read, write,
represent and
compare positive
and negative
rational numbers,
expressed as
integers, fractions
and decimals.
Know that every rational number can be written as the ratio of
two integers or as a terminating or repeating decimal. Recognize
7.1.1.1 that π is not rational, but that it can be approximated by rational
numbers such as 22 and 3.14.
7
Understand that division of two integers will always result in a
rational number. Use this information to interpret the decimal
result of a division problem when using a calculator.
gives 4.16666667 on a calculator. This answer is not exact.
7.1.1.2 For example: 125
30
The exact answer can be expressed as 4 1 , which is the same as 4.16 . The
6
calculator expression does not guarantee that the 6 is repeated, but that
possibility should be anticipated.
Compare positive and negative rational numbers expressed in
7.1.1.4 various forms using the symbols <, >, ≤, ≥.
For example: − 1 < −0.36 .
2
Number &
Operation
Add, subtract, multiply and divide positive and negative
rational numbers that are integers, fractions and terminating
decimals; use efficient and generalizable procedures,
7.1.2.1 including standard algorithms; raise positive rational
numbers to whole-number exponents.
For example: 3 × ( ) = .
2
4
Calculate with
positive and
Understand that calculators and other computing technologies
negative rational
7.1.2.3 often truncate or round numbers.
numbers, and
For example: A decimal that repeats or terminates after a large number of digits
rational numbers
is truncated or rounded.
with whole number
exponents, to solve
Solve problems in various contexts involving calculations with
real-world and
positive and negative rational numbers and positive integer
7.1.2.4
mathematical
exponents, including computing simple and compound
problems.
interest.
4
1
2
81
Use proportional reasoning to solve problems involving ratios in
various contexts.
7.1.2.5
For example: A recipe calls for milk, flour and sugar in a ratio of 4:6:3 (this is
how recipes are often given in large institutions, such as hospitals). How much
flour and milk would be needed with 1 cup of sugar?
Bold highlighting indicates partial standards met.
elsewhere in this unit.
No highlighting means the standard is met
16
Launch:
Heather decided that she didn’t want a normal, round flowerbed. She wanted a design that was more
interesting and would spark conversations all over the neighborhood. This morning, she showed me
her new plan for the garden, and it looked like this:
What will my new cost for edging and fill be?
Explore:
Students need to have knowledge of Circumference, Pi, Diameter, Radius and Area
Students will have all the materials they need for this activity from previous circle garden problem.
Students will work in groups of 2-3.
Garden Problem 2:
Have students cut out the fraction circles that were handed out from previous circle
garden problem. Students will then model the new design on their desks using their
newly created pie pieces. Students should be devising strategies on how they will figure out
the new cost. Each group should have an estimate for the new project cost.
Share:
Have various groups share their solution and explain how they came to the total they did. Encourage
students to use the model on the board or bring up their model to clarify their answer.
Explore:
What would happen if Heather decided to only use 4 of the pieces? What if she only
wanted 3? Have students return to their groups and construct a chart depicting the
circumference, area and cost of edging and fill for a garden using all 6, only 5,4,3,2 or 1
piece of the circle. How would these tables change if we used different parts of the
whole?
Share:
Students will be allowed to present their data by filling in a chart on the overhead. What form of Pi
did students use for their answers? Was this acceptable or was a more accurate verison more
appropriate? Could students repeat this process if Heather wanted a garden using 9ths of a circle?
Sometimes you also use the terms perimeter with circular shapes. What types of patterns were found
when you eliminated one piece of the circle each time? What other types of real life applications can
students link to this type of problem?
Summarize:
Students will organize a set of data.
Students will see a pattern in that data.
Students will create a formula based on that pattern.
Extension:
Students should be able to devise a general formula to find area and perimeter for any fractional part
of a whole. Have them create a formula and make another table with at least 3 different fractional
parts (5ths, 7ths, 9ths for example)
17
Lesson G: A Remainder of One (1 class period)
Objective:
Students will discover and learn divisibility rules for the numbers 1-10.
Minnesota Standards Covered:
Add, subtract, multiply and divide positive and negative
rational numbers that are integers, fractions and terminating
decimals; use efficient and generalizable procedures, including
7.1.2.1 standard algorithms; raise positive rational numbers to wholenumber exponents.
Number
Operation
Calculate with
positive and
negative rational
numbers, and
2
For example: 34 × ( 1 ) = 81 .
& rational numbers
2
4
with whole number
exponents, to solve
real-world and
Solve problems in various contexts involving calculations with
mathematical
7.1.2.4 positive and negative rational numbers and positive integer
problems.
exponents, including computing simple and compound interest.
Bold highlighting indicates a partial standard met.
elsewhere in this unit.
No highlighting means the standard is met
Launch:
Read the book, “A Remainder of One” by Elinor Pinczes. Ask students what concept the book was
written after.
Explore:
Students will work in groups of 2-3.
Remainder Problem:
Have students write the numbers 1-10 in a column on their paper. Tell the students that they
need to find a divisibility rule for each one on the page.
Share:
Have students record their divisibility rules on the board, and discuss how they formed their rule. Ask
the class if there were any other ways to get the same rule or if someone else had a different rule.
Have students make a reference sheet with the divisibility rules that they like and understand. Review
the student’s rules for divisibility, and discuss any rules that they may have missed or didn’t
understand.
18
Summarize:
Students will have a list of divisibility rules (not all may appear as follows)
2 => even numbers
3 => if the sum of the digits is divisible by 3, the original number is
4 => if the last 2 digits of the number is divisible by 4, the original number is
5 => if the number ends in 0 or 5
6 => if the number is divisible by both 2 and 3
7 => take the last digit, double it, and subtract it from the rest of the number;
if the answer is divisible by 7 (including 0), then the number is also.
8 => if the last 3 digits are divisible by 8, the original number is
9 => if the sum of the digits is divisible by 9, the original number is.
10 => if the number ends in 0
19
Lesson H: Soldier Bug Todd (1 class period)
Objective:
Students will use their divisibility rules to solve a “Soldier Bug Todd” problem.
Minnesota Standards Covered:
Add, subtract, multiply and divide positive and negative
rational numbers that are integers, fractions and terminating
decimals; use efficient and generalizable procedures, including
7.1.2.1 standard algorithms; raise positive rational numbers to wholenumber exponents.
Number
Operation
Calculate with
positive and
negative rational
numbers, and
2
For example: 34 × ( 1 ) = 81 .
& rational numbers
2
4
with whole number
exponents, to solve
real-world and
Solve problems in various contexts involving calculations with
mathematical
7.1.2.4 positive and negative rational numbers and positive integer
problems.
exponents, including computing simple and compound interest.
Bold highlighting indicates a partial standard met.
elsewhere in this unit.
No highlighting means the standard is met
Launch:
My college professor, Solder Bug Todd, had a similar problem to the bug in the story. But his
problem was even worse. He was the lone remainder when his troop divided into groups of 2,3,4,5
and 6. It was only when they grouped in rows of 7 that he finally wasn’t alone. How many bugs were
in Todd’s troop?
Explore:
Students will work in groups of 2.
Soldier Bug Todd Problem:
Students will answer the soldier bug Todd problem.
Share:
Have each group come to the front and give their answer to the problem. Ask each group to share
how they thought about the problem and what they tried that didn’t work. If students find a correct
answer, ask if they can explain how to get other numbers that will also work on the problem. Talk
about correct strategies to get multiple answers after finding one.
Summarize:
Review correct process for solving “Soldier Bug Todd”
Multiples of the correct answer will generate a set of solutions for the problem.
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Lesson I: The Division Algorithm (1-2 class periods)
Objective:
Students learn the Division Algorithm.
Students will learn an alternate way to find the Greatest Common Divisor.
Minnesota Standards Covered:
Add, subtract, multiply and divide positive and negative
rational numbers that are integers, fractions and terminating
decimals; use efficient and generalizable procedures, including
7.1.2.1 standard algorithms; raise positive rational numbers to wholenumber exponents.
Number
Operation
Calculate with
positive and
negative rational
numbers, and
2
For example: 34 × ( 1 ) = 81 .
& rational numbers
2
4
with whole number
exponents, to solve
real-world and
Solve problems in various contexts involving calculations with
mathematical
7.1.2.4 positive and negative rational numbers and positive integer
problems.
exponents, including computing simple and compound interest.
Bold highlighting indicates a partial standard met.
elsewhere in this unit.
No highlighting means the standard is met
Launch:
The soldier bug story and problem was about division. How do we divide? Can someone show me
on the board how to divide 25 by 6? How far do we have to go to get our answer? We will explore
how these methods relate math that we already know.
Explore: (1 class period)
Show students the Euclidean Method:
70/40 = 40/40 + 30/40
40/30 = 30/30 + 10/30
30/10 = 3
Since 10 is our last divisor, it is the GCD of 70 and 40.
- give students 2 problems to practice the Euclidean Method and GCD
After students feel comfortable with Euclid’s Method, show them the Division Algorithm.
A/B =>
A = B*Q + R; 0<R<B
70 = 40*1 + 30
40 = 30*1 + 10
30 = 10*3 + 0
Since our last nonzero remainder is 10, that is also the GCD
- give students 2 problems to practice the Division Algorithm and GCD
Have students break into groups of 2-3 and work on a worksheet containing 5 problems.
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Share:
Have each group come to the front and give an answer to one of the problems. Ask each group to
share how they thought about the problem and why they decided to use the method they did. Ask
what problems the group had while solving this problem. Have students record in their journal which
method they prefer and why.
Summarize:
Show students Euclid’s Method.
Show students the Division Algorithm.
Tie these methods to the Factor List and Factor Tree Method for finding GCD.
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Greatest Common Factor Worksheet
Name:
Date:
1) 15, 18 =
2) 25, 35 =
3) 168, 189 =
4) 63, 21 =
5) 153, 136 =
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Lesson _J_ Where Do We Go From Here?
Objectives:
- Students will demonstrate an understanding of negatives as a form of opposite.
- Students will demonstrate how to plot negative values on a number line and in a coordinate
system.
- Students will recognize absolute value as a relationship of distance.
- Students will be able to compare values on a number line in terms of <, >, ≤, ≥ .
Standards:
- 7.1.1.3 – Locate positive and negative rational numbers on the number line, understand the
concept of opposites, and plot pairs of positive and negative rational numbers on a coordinate
grid.
- 7.1.1.5 – Compare positive and negative rational numbers expressed in various forms using
symbols <, >, ≤, ≥.
- 7.1.2.6 – Demonstrate an understanding of the relationship between the absolute value of a
rational number and distance on a number line.
- 7.1.2.2 – Use real world context and the inverse relationship between addition and subtraction
to explain why the procedures of arithmetic with negative rational numbers make sense.
Launch:
I have gotten increasingly lazy or “More efficient” as I get older. Even writing the words UP
and DOWN or FORWARD and BACK seem tedious. So when my brother wanted to visit me I faxed
him the following directions.
“Dear Wayne, I know you know where the school is at, so when you get in the front doors go:
(0, 9), (-4,2), (0,10), (23, -1) and you’ll be at my door. See Ya Tomorrow.”
He never showed up, He said my directions didn’t make sense. I told him a 7th grader could
understand it.
Explore:
1. Have the students pair up.
2. Have the students discuss the meaning behind the numbers in the letter. What is meant by a
Positive number and what is meant by a Negative number. (Go to Share 1)
3. Have the students make a horizontal line and place a dot in the center of their line and label it
0 (zero), place another dot on the right side of the line and label it 2.
4. Have them decide where they feel the following numbers should go on the line:
a. -2, 1, -1, ½, -1/2, 1.3, -1.3, ¾, -3/4, .2, -.2, 9/5 and -9/5
Justify why they placed the values where they did. ( Go to Share 2)
b. Place the numbers in order from least to greatest and use the symbol < between them
5. On a piece of graph paper, have the students highlight a horizontal line in the center of the
paper, also highlight a vertical line in the center of the paper.
6. Have them graph the coordinate (-1, 3) and place the letter A at the location and justify why
they placed it where they did. Was there another possibility for the location? How do we
avoid confusion? (Go to share 3)
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Share:
1. Have the class discuss the meaning of positive and negative. (Go to Summary 1)
2. Have the class discuss how they determined the location the points.
a. How do the distances compare from 0 to a number and 0 and its opposite?
b. How do the values compare to each other?
c. What would be the effect of multiplying a value by -1 both physically and
arithmetically?
d. Can you ever have a negative length? ( go to Summary 4)
3. Have students share there location on the board and why they chose the location.
4. How do we avoid confusion? (go to Summary 6)
Summary:
1. Bring across the point that negative is an opposite to the same magnitude.
2. Compare various values and note that if -2 < -9/5 then -9/5 > -2 and so on.
3. Multiplying by -1 changes the direction not the magnitude.
4. Note that this distance can be referred to as the Absolute Value of a number. And the symbol
used is | | . Example |-2| = 2 , since the distance from 0 to -2 is 2 units. Similarly |3| = 3 ,
since the distance from 0 to 3 is 3 units.
5. It might be beneficial to do several examples with the class.
a. |4| = ?, |-5| = ?, | -5/6| = ?, |4.54| = ?
b. Reiterate that absolute value does not mean opposite.
6. Note that the location where the lines meet is (0,0) or the Origin.
7. The 1st value in the coordinates refers to right(+) and left (-), and the 2nd value refers to up (+)
and down(-) from the origin.
8. Give them the door as the origin and facing into the classroom as up. Have them determine
the location of their desk by the tiles on the floor.
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