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Find the Four digit number 1 On separate sticky notes write the digits 0 – 9. A special four-digit number has the following traits: All the digits are different The digit in the thousands place is 3 times the digit in the tens place. The number is odd. The sum of the digits is 27. What is the special four-digit number? Handout 3 NCTM (2006). Mathematics Teaching in the Middle School, Volume 11, number 6.

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2 EXPRESSIONS for 10: – 2 2 X 5 30 ÷ 3 1 x ÷ 3 20 ÷ EQUATIONS equaling 10: 5 + 5=10 12 – 2=10 2 x 5=10 30 ÷ 3=10 1 x 6 + 4= ÷ 3=10 20 ÷ 4 + 5=10 Write expressions representing 10. BALANCED EQUATIONS: = 12 – 2 2 x 5 = 30 ÷ 3 1 x = ÷ 3 20 ÷ = Write equations equaling 10. Use the above to write balanced equations.

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3 1 X ÷ 3 = Use number and operation tiles to build Balanced Equation Puzzles = x3+ 0 = 2 ( 6=1+60 Separate the number tiles from the operation tiles. Draw 6 numbers and 2 operations. Put down an equation or draw until an equation is possible = 7 SLE….NO , 2.7.3, 3.7.2, 2.8.1, 2.8.4, ÷ ÷ 1 1 Draw to replace the tiles played. If no tiles are played draw a tile from either pile and advance to the next person. ) Handout 4

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ALGEBRA…. Decisions, Decisions, Decisions Dear Bill, Today is my 55th birthday. I have decided to give away some of my money each year to my relatives. You may choose one of the following options: Option 1: $100 dollars now, $90 next year, then $80 the year after, and so on. Option 2: $10 dollars now, $20 next year, then $30 the year after, and so on. Option 3: $1 now, $2 next year, $4 the year after and so on doubling each year. You will only receive money until I retire. Write to me soon and tell me how you want your money. With Love, Aunt Judy 7 Handout 5

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8 Decisions, Decisions, Decisions Dear Bill, Today is my 55 th birthday. I have decided to give away some of my money each year to my relatives. You may choose one of the following options: Option 1: $100 dollars now, $90 next year, then $80 the year after, and so on. Option 2: $10 dollars now, $20 next year, then $30 the year after, and so on. Option 3: $1 now, $2 next year, $4 the year after and so on doubling each year. You will only receive money until I retire. Write to me soon and tell me how you want your money. With Love, Aunt Judy 1. Bill starts to figure out how much he will receive from the three options. If Judy retires at age 59 which option do you think Bill should choose? Justify your answer. 2. Bill thinks that his Aunt Judy is likely to wait until she is 65 years old to retire. What is his best choice option for her retirement at age 65? Justify your answer 3. Since Bill is not sure what age Aunt Judy will retire, how do you think Bill should reply to Aunt Judys letter? Explain clearly which option you think Bill should choose. Show your reasoning. Name: _________________________________

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BULGING BACKPACKS Students will estimate the weight of his/her backpack, record this number Everyone estimates the weights of three chosen students Weigh the chosen backpacks Graph the actual weight vs. estimate 9 Handout 8 Lessons For Algebraic Thinking, Grades 6-8, Marilyn Burns, Math Solutions What should a person know by looking at the points and lines you added later to your scatter plot? What are the main things you learned from this activity? Some questions to ponder…

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10 Calculator Instructions for TI 84 To clear data in all lists: Press 2 nd MEM. Choose ClrAllLists. Press Enter. Press Enter. You will see DONE Press 2 nd MEM. Choose 7 – Reset Choose 2 – Default Choose 2 - Reset. Enter your data in lists: Press STAT. Press Enter. Enter the estimates in L 1. Press ENTER after each number. Enter actual weights in L 2. Press ENTER after each number.

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11 To create a scatter plot: Press 2 nd Y= (STAT PLOT). Press ENTER for Plot 1. Press ENTER for On. Arrow down to Type: Press ENTER for Scatter Plot. Arrow down to Xlist: Press 2 nd STAT and choose L 1. Arrow down to Ylist: Press 2 nd STAT and choose L 2. Arrow down to Mark: Press ENTER for the box. Press WINDOW. Set the appropriate window for this problem. Press TRACE and use the right and left arrow keys.

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12 To create two box-and-whisker plots: Press 2 nd Y= (STAT PLOT). Press ENTER for Plot 1. Press ENTER for On. Arrow down to Type: Press ENTER on the box and whisker icon. Arrow down to Xlist: Press 2 nd STAT and choose L 1. Press ENTER. Set Freq to 1. Press WINDOW. Set the appropriate window for this problem. Press TRACE and use the right and left arrow keys to see the values for each quartile. Repeat steps 1-7 choose Plot 2 and L 2.

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Some questions to ponder… What should a person know by looking at the points and lines you added later to your scatter plot? What are the main things you learned from this activity? 13

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14 BULGING BACKPACKS Overview: Students will estimate and find the actual weight for each of their backpacks. The class constructs a scatter plot to display some of the data; then each student creates a personal scatter plot. This lesson gives students experience plotting points on a Cartesian plane and interpreting those points in terms of a real-world situation. The function y=x is used to describe the meaning for points above and below this line. With numerous similar experiences, students develop an intuitive understanding of function and correlation. Prerequisite Concepts and Vocabulary: Finding explicit and recursive rules for patterns Graphing points Scaling axes for specific data Understanding the vocabulary function, scatter plot, coordinate, variable, and axis Time: This lesson takes approximately 90 minutes. Materials: Bathroom scales Student backpacks Large sheet of graph paper for class scatter plot 1 sheet of graph paper per student Graphing Calculator (optional)

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15 BULGING BACKPACKS Lesson Plan: Discuss plan with the class Each student makes a table and records his or her own data (Table included). Students make estimates for only three backpacks, then find their actual weight, to help students get some general idea of how much backpacks weigh. Students use the bathroom scale to weigh each of the backpacks to the nearest half-pound. For the class data have each student record their estimate and actual weight on a table on the board. Label the x-axis actual weight and the y-axis estimates.(This allows us to talk about the points above the line y=x being overestimated and the points below being underestimated as well as anyone on the line as being perfect) Next assign the individual scatter plots for their estimates of all backpacks. The next three can be done as a class on the class scatter plot or assigned for each individual to write about. Draw the function y=x+2 and discuss what that tells us. Draw the function y=x-3 and discuss what that tells us. Draw the function y= 7 and discuss what that tells us. Extension Using the graphing calculator, create a box and whisker plot for estimates in Plot 1 and another box and whisker plot in Plot 2. Display both plots in the same window. Discuss the differences in the range of the estimates and the actual weights. Check for outliers and compare the statistics for estimates and actual weights using the trace feature of the calculator. Adapted from: Lawrence, Ann, and Charlie Hennessy. Lessons for Algebraic Thinking, Grades 6-8. Math Solutions Publications, 2002.

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16 NameActual WeightEstimate Name: _____________________________ BULGING BACKPACKS…STUDENT WORKSHEET

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How much does each shape weigh? Explain. From Teaching Student Centered Mathematics P Most weight is the cylinder; least weight is the cube 2 spheres = 1 cylinder….and 3 cubes = 1 sphere….therefore 6 cubes = 1 cylinder 1 cube = 2 cylinders …so… 1 cube + 1 cylinder = 3 cylinders 3 cylinders = 6 spheres…therefore 1 cylinder = 2 spheres Handout 9 What will balance two spheres? Explain.

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From Teaching Student Centered Mathematics P cylinder + 1 cube = 13 2 spheres + (1 cylinder + 1 cube) = 21 Use substitution: 2 spheres + 13 = 21, 2 spheres = 8, 1 spheres = 4. 1 cube + 2 spheres = 14, 1 cube + 8 = 14, 1 cube = 6. 1 cylinder + 6 = 13, 1 cylinder = 7. 2 cubes + 1 sphere = 8 and 3 spheres = 12 therefore 1 sphere = 4 2 cubes + 4 = 8, 2 cubes =4, 1 cube = 2 Equation 1: 1 sphere + 1 cylinder = 7 Equation 2: 1 sphere + 1 cube = 6 Since 7 is one more than 6 and the spheres are the same, then 1 cylinder = 1 cube + 1 Equation 3: 1 cylinder + 1 cube = 6 Substitute: (1 cube + 1) + 1 cube = 6, 2 cubes + 1 = 9, 2 cubes =8, Therefore 1 cube = 4 Equation 2 Substitute: 1 sphere + 4 = 6, 1 sphere = 2 Equation 1 Substitute: cylinder = 7, 1 cylinder = 5

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How much does each shape weigh? Explain. From Teaching Student Centered Mathematics P Handout 9 What will balance two spheres? Explain. Name: _______________________________________ Student Worksheet

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From Teaching Student Centered Mathematics P

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Arkansas Benchmark Questions 7th grade Handout 10 8th grade 2007

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7 th grade Open Response x + 2x + x=1802.3x + 2x + x=180 6x = x = 30 3.Angle x = 30 0 Angle 2x = 2(30) = 60 0 Angle 3x = 3(30) = 90 0

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23 Pam and Greg are each building a pyramid of tiles. The number of tiles needed is represented by the rule, where n is the number of levels in the pyramid. The pattern for the pyramid is shown below. 8th grade Open Response 1.Draw the next pattern in the sequence. 2.How manytiles would be in a 10-level pyramid? Show your work. 3. Greg has 4-inchtiles, and Pam has 2-inchtiles. They are each going to build a 24-inch tall pyramid. Greg predicts he will need half as many tiles as Pam since his tiles are twice as large. Compare the pyramids to see why Greg is incorrect. n(n+1) 2 55 See handout 10

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