House of Cards. Below is a house of cards that is 3 stories high.

Transcription

1 Below is a house of cards that is 3 stories high. The top story is made of 2 cards leaned against each other, with 1 card as its base. The middle story is made of 4 cards (2 sets of 2 cards leaning against each other), with 2 cards as its base. The bottom story is made of 6 cards (3 sets of 2 cards leaning against each other), with 3 cards as its base. Determine the total number of cards needed to build a house of cards that is 5 stories high. Determine the total number of cards needed to build a house of cards that is 10 stories high. Determine the total number of cards needed to build a house of cards that is n stories high. 1 of 13

2 Suggested Grade Span 6 8 Grade(s) in Which Task Was Piloted 7 and 8 Task Below is a house of cards that is 3 stories high. The top story is made of 2 cards leaned against each other, with 1 card as its base. The middle story is made of 4 cards (2 sets of 2 cards leaning against each other), with 2 cards as its base. The bottom story is made of 6 cards (3 sets of 2 cards leaning against each other), with 3 cards as its base. Determine the total number of cards needed to build a house of cards that is 5 stories high. Determine the total number of cards needed to build a house of cards that is 10 stories high. Determine the total number of cards needed to build a house of cards that is n stories high. Alternative Versions of Task More Accessible Version: (Refer to the card image that appears in the original version of the task.) 2 of 13

3 Below is a house of cards that is 3 stories high. The top story is made of 2 cards leaned against each other, with 1 card as its base. The middle story is made of 4 cards (2 sets of 2 cards leaning against each other), with 2 cards as its base. The bottom story is made of 6 cards (3 sets of 2 cards leaning against each other), with 3 cards as its base. Determine the total number of cards needed to build a house of cards that is 4 stories high. Determine the total number of cards needed to build a house of cards that is 5 stories high. Determine the total number of cards needed to build a house of cards that is 10 stories high. Teacher Note: See page 7 of the PDF to print a complete worksheet with graphics. More Challenging Version: The original task, and The cards in a standard sized deck measure 3.5 inches by 2.5 inches. What would the area and volume be of the house of cards that is 5 stories high? What would the area and volume be of the house of cards that is 10 stories high? Teacher Note: See page 8 of the PDF to print a complete worksheet with graphics. NCTM Content Standards and Evidence Algebra Standards for Grades 6 8: Instructional programs from pre-kindergarten through grade 12 should enable all students to Understand patterns, relations and functions. NCTM Evidence: Represent, analyze and generalize a variety of patterns with tables, graphs, words and, when possible, symbolic rules. Exemplars Task-Specific Evidence: This task requires students to analyze patterns for determining the number of cards in each story and to find a pattern for determining the accumulated total. Students then are required to generalize the patterns to solve for N stories. 3 of 13

4 Use mathematical models to represent and understand quantitative relationships. NCTM Evidence: Model and solve contextualized problems using various representations, such as graphs, tables and equations. Exemplars Task-Specific Evidence: This task requires students to analyze patterns for determining the number of cards in each story and to find a pattern for determining the accumulated total. Most students do this by creating tables and using the information in the table to generate equations. Time/Context/Qualifiers/Tip(s) From Piloting Teacher This is a medium- to long-length task, depending on the amount of time it takes each student to discover patterns and rules. Many students enjoy building card towers, so are naturally engaged in solving the task. I made decks of playing cards available to students who needed to build the actual house of cards to visualize the task. Many students asked whether the bottom of the house needed base cards. I told them that they could decide but should communicate their decision in their solutions. To print a copy of the more accessible version of the task with the graphic image, refer to page 7. To print a copy of the more challenging version of the task with the graphic image, refer to page 8. Links This task could link to units on construction, architecture or famous towers. Common Strategies Used to Solve This Task Most students create a chart in which to record data and analyze patterns and then extend the patterns to draw conclusions and generalized solutions. 4 of 13

5 Possible Solutions Story Number Cards in Story Accumulated Total n 3n {3[n(n+1)]}/2 or (1.5nÔ+1.5n) or 3n(n/2+.5) More Accessible Version Solution: Story Number Cards in Story Accumulated Total of 13

6 More Challenging Version Solution: Orginal task solution, and In the 5 story house of cards, the length of each side is: 2.5 inches x 5 = 12.5 inches The area is (12.5Ô x â3) /4 å inchesô The volume is x 3.5 å inchesõ In the 10 story house of cards, the length of each side is: 2.5 inches x 10 = 25 inches The area is (25Ô x â3) /4 å inchesô The volume is x 3.5 å inchesõ Task-Specific Assessment Notes General Notes This task lends itself to students using algebraic notation to describe and generalize patterns. It is, however, unnecessary for rules to be written using correct algebraic notation, or any algebraic notation, for that matter. Many students may find a rule for determining the number of cards in each layer, but finding a rule for determining the accumulated total presents a challenge. Novice The Novice's work will be shown, but it will be unclear as to what it represents. There will be no approach evident, and little or no correct math language will be used. Apprentice The Apprentice will have parts of the solution correct. However, an incorrect solution will be achieved due to a computation error or an error in finding and generalizing the pattern. Some math language will be used. The work will have gaps or unclear parts. Practitioner The Practitioner will achieve a correct solution. Algebraic notation will be used to communicate the solution, and all work will be shown and labeled. The approach will be evident or explained, computation will be accurate, and math representations will be used to communicate the solution. Expert The Expert will create a rule for solving all parts of the problem. All work will be shown and the approach and reasoning will be explained. The Expert will demonstrate how the rule can be generalized to a house with any number of stories. S/he will use algebraic notation with defined variables. 6 of 13

7 More Accessible Version Worksheet Below is a house of cards that is 3 stories high. The top story is made of 2 cards leaned against each other, with 1 card as its base. The middle story is made of 4 cards (2 sets of 2 cards leaning against each other), with 2 cards as its base. The bottom story is made of 6 cards (3 sets of 2 cards leaning against each other), with 3 cards as its base. Determine the total number of cards needed to build a house of cards that is 4 stories high. Determine the total number of cards needed to build a house of cards that is 5 stories high. Determine the total number of cards needed to build a house of cards that is 10 stories high. 7 of 13

8 More Challenging Version Worksheet Below is a house of cards that is 3 stories high. The top story is made of 2 cards leaned against each other, with 1 card as its base. The middle story is made of 4 cards (2 sets of 2 cards leaning against each other), with 2 cards as its base. The bottom story is made of 6 cards (3 sets of 2 cards leaning against each other), with 3 cards as its base. Determine the total number of cards needed to build a house of cards that is 5 stories high. Determine the total number of cards needed to build a house of cards that is 10 stories high. Determine the total number of cards needed to build a house of cards that is n stories high. The cards in a standard sized deck measure 3.5 inches by 2.5 inches. What would the area and volume be of the house of cards that is 5 stories high? What would the area and volume be of the house of cards that is 10 stories high? 8 of 13

9 Novice 9 of 13

10 Apprentice 10 of 13

11 Practitioner 11 of 13

12 Practitioner 12 of 13

13 Expert 13 of 13