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. 20

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