Building Mathematical Thinkers s1

Building Mathematical Thinkers Number (7)

Essential Standard/Objectives: Solve multi-operation first degree problems Time: 30 minutes Title: Hit and Run Materials: student record sheet

Description: Students are given the context of an individual who is trying to remember a license plate after an accident. Students are given a variety of clues to determine the plate number.

Problem: The policeman asks the poor victim-Did you get his license plate? The victim answers YES— His license plate was in two parts—A two digit and a three digit number—The two digit number was a prime and the sum of the two digits was a two-digit prime—The tens digit was larger than the units digit—In the three digit part, the digits were all odd and different —The sum of the three digits was palindromic—The sum of the first and third digit was one- half the sum of the first and second. That’s all I remember states the victim. Can you help decipher the license plate numbers?

Suggestions: Have students develop a strategy using a chart and collecting the appropriate clues for the license number. They may have difficulty knowing where to begin. Have a class discussion to help students organize their thoughts and collect all the data. Students may have solved this problem in different ways. Be sure to have the class share their methods. Building Mathematical Thinkers Problem Solving (7)

Essential Standard: Solve multi-operation equations Title: Whodunit Puzzle Time: 30 minutes Materials: record sheet

Description: A whodunit puzzle involves a group of people or objects or more sets of characteristics associated with the people or objects. Each characteristic can be matched one-to-one with each person or object. Solve these whodunits. A—Finley, Garber, and Harris are a banker, a computer programmer, and a secretary, but not necessarily in that order. Their first names are Alex, Bob, and Cynthia. Finley is neither the banker nor the secretary. Harris is not the secretary, and Alex is not the banker. Cynthia is older than both Garber and Harris. What is the complete name and occupation of each? B—Peggy, Jacob, Lee, and Martha are the architect, composer, dancer and singer. Peggy and her husband invited the composer and his wife to dinner. The dancer said that he enjoyed playing chess with Lee. The singer complimented Martha on her excellent recipe for cabbage soup. What is the occupation of each person and is that person male or female?

Suggestions: Explore a variety of strategies for these problems. Creating a chart is a typical method for these types of problems. Have students discuss solutions and their procedures. Organize of clues and logical thinking is an integral part of these kinds of examples. Building Mathematical Thinkers Problem Solving (7)

Essential Standard/Objectives: Analyze patterns and sequences Title: Relationships-Analogies Time: 40 minutes Materials: handout

Description: The most sophisticated method of making comparisons and classifications is analogical thinking. You are probably familiar with word analogies such as: shoe to foot or glove to hand. Analogical relationships can also be found in numbers and other math content. In exploring such relationships, see how members are alike. Try to find the most exact possible relationship. For example, if you were to try to name the relationship between 3, 15, 45, you could say that all are odd numbers—A more exact statement (because it eliminates so many other odd numbers) would be that all can be divided by 3, or all are multiples of 3.

Problems: Have students complete these analogies. 1) 1,2,3,5 ______

2) 3,19,29,31,2______

3) 6,9,12,15______

4) 7,17,27,37______

5) .5,.33,.015,.16______

6) 2.5,5,7.5,10______

7) 20,16,12,8______

8) 3,21,147,1029______

Suggestions: Be sure that, as a class, students understand that they are looking for the common element in each group. They must be able to describe the relationship for each group. Building Mathematical Thinkers Algebra (7)

Essential Standards/Objectives: Solve multi-operation equations and inequalities Title: Stamp collection and packaging Time: 30 minutes Materials: record sheet

Description: Use your thinking strategies and skills to solve these word problems. Be sure to show all your work and explain your solutions. Problems: 1) I have four 3-cent stamps and three 5-cent stamps. Using one or more of these stamps, how many different amounts of postage can I make?

2) In a stationary store, pencils cost one amount and pens cost a different amount. The total cost of 2 pencils and 3 pens is $.78. The total face value of 3 pencils and 2 pens is $.72. What is the cost of one pencil?

3) A dealer packages marbles in two different box sizes. One size holds 5 marbles and the other size holds 12 marbles. This morning the dealer packaged exactly 99 marbles and used more than 10 boxes. How many boxes of each size were used?

Suggestions: Students may want to create equations for solving these problems. Be sure that they describe their solution strategies and record steps. Have students utilize the equations by drawing pictures to clarify the answers. Building Mathematical Thinker Algebra (7)

Essential Standard/Objectives: Apply distributive property Title: Which one Doesn’t Belong? Time: 30 minutes Materials: record sheet

Description: This activity allows students to concentrate on the critical attributes associated with specific topics in math. They are asked to determine how three of the four given items are related. Students must indicate the characteristic common to the three items. In each row circle the item that does not belong. Record the reason for the answer you selected. A B C D 1) 5x 4x 3y -2x

2) 3(x+4)=3x+7 2(x+5)=2x+10 5(x+1)=5x+5 7(x+4) = 7x + 28

3) y=2x +1 2x-y=4 6x-3y=9 2y=x+3

4) (-2)(-3)(-7)(4) (-7)(-5) -(-4)(5) (-2)(-3)(5)

5) 4a a+a+a+a a(a)(a)(a) (4)a

Suggestions: Students should work in teams to create additional examples. Have students explain their reasoning and responses with their classmates. Building Mathematical Thinkers Data (7)

Essential Standards/Objectives: Analyze data Title: Lock it up Time: 45 minutes Materials: activity sheet (Create from the questions below.)

Description: Have students record all locker numbers represented in their class Discuss the probability of each student’s being assigned the specific locker number he or she actually received. Focus on possible rationale for the system used to assign lockers to all students in the building. Divide students into small groups and give each group the following activity sheet. Complete questions in the small group and have students report to the class.

Questions: 1) The fifty lockers in the hall at the school have been painted over the summer break in the two school colors. The custodian has been assigned the task of replacing the numbers on the lockers. The lockers will be numbered consecutively from 1 to 50. Help the custodian figure how many of each digit are needed to complete the task of numbering the lockers 1 to 50. Explain how your group went about solving this problem. Describe your method in words.

2) The warehouse calls to say that the digit 8 is temporarily out of stock. What is the probability of being assigned a locker having a missing digit? The warehouse also discovers that they are out of the digit 2. What is the probability that a locker is missing at least one of these digits?

3) You are assigned locker number 46, and your locker combination is 3-17-32. This combination is just one of the many using these three numbers. How many other students might have locks that have combinations having these three numbers but in a different order? Explain your answers.

Suggestions: Have students make a presentation to class explaining their use of probability. Be sure that they organize their data and show appropriate steps. Building Mathematical Thinkers Geometry (7)

Essential Standards/Objectives: Title: How Do You Know That? Materials: record sheet

Description: this activity provides an opportunity for students to use words to express key ideas in math. The focus of this activity is relationships rather than computation. For each question presented in this section, students are asked to write one or more complete sentences to answer each question. Students should use their knowledge of definitions and estimation skills.

Questions: 1) How do you know that a triangle can’t have two obtuse angles?

2) How do you know that the area of square is not doubled when each side is doubled?

3) How do you know that a triangle can’t have sides that are 6, 8, and 14 units long?

4) How do you know that two sides of a triangle can’t be parallel?

5) How do you know that three angles of a triangle can’t measure 50 degrees,60 degrees, and 80 degrees?

Suggestions: Students may to work as a team to complete the questions. Divergent thinking is encouraged. It is important for students to be able to justify their responses and explain their validity. Building Mathematical Thinkers Algebra (7)

Essential Standard/Objectives: Analyze linear relationships Title: Drop and Bounce Time: 45 minutes Materials: record sheet and graph paper

Description: this activity uses a ball that rises to half its previous height each time it bounces. Students will explore the number of times the ball will bounce before it peaks at a certain height. They must provide tables, graphs, and algebraic expressions as part of their solutions. Problem: A ball is dropped from a height of 16 feet. At its first bounce, the ball reaches a peak height of 8 feet. Each successive time that the ball bounces, it reaches a peak height that is half that of the bounce just before. 1) How many times will the ball bounce until it bounces to a peak height of 1 foot? Show how you found your answer.

2) Make a table that shows the peak bounce height of the ball for each number of bounces.

3) Make a graph that shows the relationship between the number of bounces and the peak height of the ball.

4) Write an algebraic expression that shows the relationship between the number of bounces and the peak height of the ball.

Suggestions: Students may want to use this activity as an actual experiment. This would allow for additional practice with measurement and collecting data. A discussion related to functions should be integral part of the lesson. Students should be able to represent an exponential function in a table and as a graph. Building Mathematical Thinkers Data (7)

Essential Standards/Objectives: Compare experimental and theoretical probability Title: Lucky Draw Time: 40 minutes Materials: red and blue colored counters or cubes and large cups

Description: Students will utilize theoretical and experimental probability to determine the profitability of a carnival game. Students will have to analyze and reason about probability and to justify their decision related to the profit of this game.

At the Bank Street School’s carnival, the sixth grade class is planning to run a booth to raise money for their favorite charity. The class created the following game: There will be three large barrels. There are an equal number of red and blue balls buried in sawdust in each barrel. The rules are as follows: One turn allows you to make One Lucky Draw from each barrel for twenty-five cents. If you draw three balls of the same color on one turn, you win one dollar. The chairman of the carnival liked the idea but she wants to make sure that the game is a good money- maker. She has asked the class to prepare a report for the carnival committee. This report must include recommendations and clear explanations. The report must be based on mathematical understandings.

Suggestions: Students may need to be guided as they begin this activity. The students should have some foundation with theoretical and experimental approaches. The teacher may want to model the game with counters and cups. Have students share their findings and develop a conclusion based on students’ results. Building Mathematical Thinkers Number (7)

Essential Standard/Objectives: Apply geometric properties to solve problems Title: Sometimes Time: 30 minutes Materials: record sheet

Description: This activity is designed for students to focus on a specific condition which may be true in some cases but not in others. It is important that students understand that for a statement to be true, it must be true under all situations. Each condition is true in some cases and false in at least one instance. Provide an example of when the condition is true and one that shows when the condition is false. 1) Prime numbers are odd.

True False

2) The greatest common factor of two numbers is less than either number.

True False 3) When you double a number, the result is even.

True False 4) When you raise 3 to a power, the ones digit in the result is a 3 or a 9.

True False 5) The least common multiple of two odd numbers is the product of the two numbers.

True False 6) Numbers have an even number of factors.

True False 7) A whole number is either prime or composite.

True False 8) The cube of a number is greater than the square of a number.

True False

Suggestions: Have students explain their solutions and share with classmates. Building Mathematical Thinkers Algebra (7)

Essential Standards/Objectives; Solve multi-operation equations Title: Arthur’s Dilemma Time: 30 minute Materials: record sheet

Description: These problems are designed to give additional practice with translating equations and solving problems. Before students solve the equations, they should translate each example into equivalent English sentences. Problems: A. Can you help Arthur? He is stumped on this equation: 2N + 3N =85 As a class discuss the meaning of the problem. 1) Can you write the equation using sentences?

B. I’m thinking of a number. Twice the number is added to three times the same number which gives the result of 85. What is the number? 2) Have students create additional problems as a team and share with the class.

3) Let students create story problems that utilize the equations designed by the teams.

Extension: Students who have been exposed to properties of whole numbers and like terms can use inverse steps as they complete the problems. C. Solve the following problems: 5(F +1) – 2F =17 3(G +4) + 3(G-4) = 50 H +H + H =50 4N -1 =30 5Y +13= 5 How can these problems be translated and interpreted using English sentences? Suggestions: These types of problems can be modified to meet the needs of your class and to provide varying discussion depending on the concepts to review or introduce. Have students see the relationships as you complete each example. Building Mathematical Thinkers Problem Solving (7)

Essential Standard/Objectives: Understand use of fractions and decimals in context Title: Fractions Everywhere Time: 45 minutes Materials: record sheet

Description: These word problems are designed to engage students in an understanding of fractions and relationships. Students should be encouraged to use a variety of strategies. Drawing pictures, creating tables, and writing equations are just a few of these strategies.

Problems to explore:

1) David’s youth lasted one -sixth of his life. He grew a beard after one- twelfth more of his life. After one-seventh more of his life, David married. Five years later he had a son. The son lived one-half as long as the father, and David died just four years after his son’s death. How old was David when he died?

2) Daniel cut a board in half. Then he cut one half into three equal pieces. Finally, he cut one of these three pieces into four equal pieces. The four equal pieces were each eight inches long. How many feet long was the original piece?

3) You can cut a pizza into seven pieces with only three straight cuts. What is the greatest number of pieces you can make with five straight cuts?

4) Joe has the same number of sisters and brothers. Each of his sisters has only half as many sisters as brothers. How many brothers and sisters are there?

Suggestions: These problems may be given to a team of students. Each team must complete all of the problems, but one of the problems would be assigned to each team. Their goal would be to report their strategy and solution to the class for the assigned problem. Building Mathematical Thinkers Problem Solving (7)

Essential Standard/Objectives: Understand use of fractions and decimals in context Title: Fractions Everywhere Time: 45 minutes Materials: record sheet

Description: These word problems are designed to engage students in an understanding of fractions and relationships. Students should be encouraged to use a variety of strategies. Drawing pictures, creating tables, and writing equations are just a few of these strategies.

Problems to explore:

5) David’s youth lasted one -sixth of his life. He grew a beard after one- twelfth more of his life. After one-seventh more of his life, David married. Five years later he had a son. The son lived one-half as long as the father, and David died just four years after his son’s death. How old was David when he died?

6) Daniel cut a board in half. Then he cut one half into three equal pieces. Finally, he cut one of these three pieces into four equal pieces. The four equal pieces were each eight inches long. How many feet long was the original piece?

7) You can cut a pizza into seven pieces with only three straight cuts. What is the greatest number of pieces you can make with five straight cuts?

8) Joe has the same number of sisters and brothers. Each of his sisters has only half as many sisters as brothers. How many brothers and sisters are there?

Suggestions: These problems may be given to a team of students. Each team must complete all of the problems, but one of the problems would be assigned to each team. Their goal would be to report their strategy and solution to the class for the assigned problem.