Building Mathematical Thinkers Number (8)

Essential Standard/Objectives: Understand structure of the Real number system Time: 60 minutes Materials: record sheet

Description: Our number system is based on the number ten. Have a discussion with the class to explore the answers to these questions. How many individual digits are needed in our base ten system? Why? Write the digits and explore the concept of place value and grouping in our system.

Problem: Class discussion---Suppose our number system was based on 5. How many digits would we need? What are these digits? Explore writing two digit numbers in base 5. Discuss the grouping concept and place value in this new system. What would the numeral 100 mean in base 5? Have students add in base 5. Determine how these examples were completed. 134 + 31 = 220 241+113=404

Suggestions: Have students work with a partner to complete more addition examples. Let students write about how they solved each example. You could also explore subtraction as an extension. Building Mathematical Thinkers Algebra (8)

Essential Standard/Objectives: Solve equations Title: Problem solving Time: 45 minutes Materials: record sheet

Description: Students should use a variety of strategies to complete the following word problems.

Problem: A—A barrel full of flour weighs 40 pounds. The same barrel filled with nails weighs 64 pounds. If the nails weigh three times as much as the flour, how much does the empty barrel weigh? B—Two containers each contain the same amount of acid. If 7 grams of the first container are poured into the second container, then the second container has twice as much as the first container. How many grams does the second container now have?

Suggestions: As students solve these problems many with want to write algebraic equations. Encourage students to draw illustrations to verify their answers.

Assessment: Students could create their own word problems and share them with their classmates. Building Mathematical Thinkers Measurement (8)

Essential Standards/Objectives: Apply scale factor in contextual problems Title: Pose A Problem Time: 40 minutes Materials: record sheet

Description: This activity requires students to identify and provide the missing information that will allow a standard problem to be solved. Students are to decide what information is missing and explain their reasoning. They should write a complete sentence that gives the needed information to solve each problem effectively.

Problems: 1) The area of a rectangle is 18 square centimeters. Find the perimeter of the rectangle.

2) The base of a triangle is 28 centimeters. Find the area of the triangle.

3) The perimeter of a square is equal to the perimeter of a rectangle with a length of 8 inches. Find the measure of the side of the square.

4) A rectangular solid has a volume of 60 cubic inches. What is the height of the rectangular solid?

5) Eric’s mother wants to carpet a rectangular floor 26 feet long. How many square yards of carpet does she need?

6) Michael drew a picture of two concentric circles. The smaller one had a radius of 4 inches. What is the area of the region between the two circles?

Suggestions: After students provide the needed information, they should solve each problem and explain their solutions. Compare student’s problems and discuss responses. Building Mathematical Thinkers Problem Solving (8)

Essential Standard/Objectives: Solve linear relationships to solve problems Title: Possibilities Time: 30 minutes Materials: record sheet

Descriptions: These problems explore combinations. Students should practice organizing and displaying their solutions using an organized format. Students may approach these problems using different strategies. It is necessary for students to be able justify and explain their strategy.

Problems: 1) On a recent safari, a group of people and elephants contained 100 knees and 100 trunks. If each person took 3 trunks, how many people and elephants went on the safari?

2) Jack knows that there are 18 animals in the barnyard. Some are chickens and the rest are cows. He counted 50 legs in all. How many animals are chickens and how many are cows?

3) Two multipedes are cantering through the shopping mall. They count up their feet and find that three dozen socks will be just enough to keep all their feet warm. If one of them has eight more feet than the other, how many feet does the multipede have?

4) The owner of the Atlantis Bicycle Shop was quite mad. When he took inventory, instead of counting the number of bicycles and tricycles in the store, he counted the number of pedals and the number of wheels. He had counted 153 wheels and 136 pedals. How many bicycles and tricycles did he have?

5) A carpenter only builds stools and tables. Each stool has three legs and each table has four legs. One day he used exactly 70 legs. What could he have built? How many different possibilities are there? How do you know you have found all the possibilities?

Suggestions: Students must have opportunities to explore different problems. All of these problems have common connections. Have students discuss how their strategies may be similar when completing these kinds of problems. What holds true and are there specific strategies that may work for all of the problems. Building Mathematical Thinkers Geometry (8)

Essential Standard/Objectives: Solve multi-step problems involving combinations of operations Title: Three Verses One Time: 30 minutes Materials: record sheet

Description: This activity allows students to focus on the attributes of specific math topics. Students are to provide examples that satisfy a certain condition and one example that does not fit. This activity requires that students understand the vocabulary, and the problems are open ended so that students will give different solutions that can be shared with the entire class. For each problem, provide three examples that satisfy the condition and one that does not.

CONDITION Three One 1) Write four angle measurements: three that are obtuse and one that is not

2) Write four pairs of angle measures: three that are complementary and one that is not

3) Write four types of triangles: three that represent the classification of triangles by sides and one that does not

4) Draw four figures: three that have a perimeter of 12 units and one that does not

5) Draw four figures three that have an area of 20 square units and one that does not

Suggestions: Have students share their examples and discuss with the class. Students may want to write justifications for their solutions. Building Mathematical Thinkers Geometry (8)

Essential Standard/Objective: Apply geometric properties to solve problems Title: One Picture is Worth a Thousand Words Time: 30 minutes Materials: record sheet

Description: Students are to show their understanding by drawing a picture to represent the given situation. It is important for students to be able to construct representations when solving word problems. Be sure that students label diagrams appropriately. Draw a diagram or picture to represent the situation described in each problem. You should label your diagram with data contained in the problem. Do not attempt to solve the problem. Focus on looking at the concepts and relationships described in the problem. 1) Triangle ABC is an isosceles triangle. The measure of the vertex angle B is forty degrees. What is the measure of the exterior angle C?

2) Can an altitude of a triangle fall outside the triangle?

3) The vertices of an isosceles triangle are (2, 0), (10, 0), and (6, 6).what is the area of the triangle?

4) A 25-foot ladder is placed against a wall. It forms an angle of 30 degrees with the wall. How far up the wall does the ladder reach?

5) Each side of an equilateral triangle is 8 inches long. A small equilateral triangle with a side length of 1 inch is cut from each corner of the original triangle. What is the perimeter of the resulting figure?

6) Two isosceles triangles share a common base but do not share any area. The sides of one triangle measure 5 inches, 5 inches, and 3 inches; the sides of the second triangle measure 10 inches, 10 inches, and 3 inches. What is the perimeter of the figure formed by drawing the two triangles?

Suggestions: Students could solve the problems after checking their diagrams. Have students discuss how the diagrams helped their understanding and made solving easier.