Creating YOUR 21 st Century Classroom One Task At A Time. Sierra Sands, Jan 2015

Transcription

1 Creating YOUR 21 st Century Classroom One Task At A Time Sierra Sands, Jan 2015

2 Practices & Claims Quiz Sierra Sands USD The 8 Standards of Mathematical Practice (SMP) are: 1 P 5 T 2 R 6 P 3 C 7 S 4 M 8 P The 4 Smarter Balance (SBAC) Claims are: 1 C & P 2 P S 3 C R 4 M & D A

3 MPJ s Ultimate Math Lessons 147 STUDENT HANDOUT You have been asked to ship a large broom to your scary Aunt Matilda in Transylvania. The only problem is that the broom is 8 feet long. (Why does she need an eight foot broom, anyway?!?) The post office will ship any box as long as the length and the width (the two shortest sides) do not add up to more than 108 inches. The third side can be any length. After talking to the people at the postal annex, you find out that there are four different boxes that might be large enough to work. Their measurements are listed below: #1: 4 2 x 5 x 5 10 #2: 3 4 x 5 10 x 3 4 #3: 2 6 x 5 10 x 5 10 #4: 7 6 x 10 x 10 Before determining which box is best suited to ship Auntie M s broom, prove the conjecture below. Once you are confident that the conjecture is true, use it to show which boxes will be big enough to hold the broom: Box #1 Box #2 Box #3 Box #4 Of the boxes that are big enough to hold the broom, which is small enough to ship? Why? Conjecture: Given a rectangular box with dimensions of x, y & z, the square of the length of the diagonal of the box will be equal to the sum of the squares of the dimensions of the box. In other words, d 2 = x 2 + y 2 + z 2. Prove the conjecture: Verify the conjecture: Use two different instances to support your proof. (Choose two sets of three numbers and show that the formula is true.) Instance #1 Instance #2

4 Teaching Students to THINK, COMMUNICATE, COLLABORATE & CREATE 4 Claims: Concepts & Procedures, Problem Solving, Communicate Reasoning, Modeling & Data Analysis Questioning (Dig Deeper/Reach Higher) Access for All (Low Floor/High Ceiling) 8 Practices ECI 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Target (Clear & Congruent) Engagement (Most students, most of the time) Feedback (specific & timely) Monitor & Adjust Suggestions for Lesson Improvement? Rigor: Fluency, Deep Understanding, Application, Dual Intensity

5 C. Shore Name Farmer s Fence Farmer John has 18 feet of fence to enclose his sheep corral. He needs to make a rectangular corral, and is curious about the area that it will yield. 1. Draw two instances of a rectangular corral that has a perimeter of 18 feet. Length Width Area Complete the table for the width of the pen given the length. l w = A(l)= 3. Write an equation to generalize the relationship between the length, l, the width, w, and the perimeter, Below is another equation describing the relationship between length and width. Show how this equation can be transformed into the equation that you wrote for # 3. 2l + 2w = Graph your equation for w in terms of l. 6. Discuss the meaning of the a) slope of the line b) and the intercepts.

6 Farmer s Fence (cont d) 7. Complete the table for the area of the pen given the length. 8. Write an equation to generalize the equation of the area, A(l),of the rectangular pen with length l. 9. Graph your equation. 10. a) Find the coordinates of the roots, and label them on your graph. b) What do the roots of the equation say about Farmer John s Pen? c) Why can we not draw the rectangles that are represented by these roots? 11. a) Find the coordinates of the vertex, and label it on your graph. b) What does the vertex of the equation say about Framer John s Pen? c) Draw the rectangle that is represented by the vertex. Extension: Does the vertex form of your equation support your findings for the vertex?

7 Farmer s Fence (cont d) 12. a) Use your area equation to find the length that will yield an area of 14. b) Does this agree with the data in the table? Does this agree with your graph? c) Draw the rectangles that represent your soultion. 13. a) Use your graph to estimate the length that will yield an area of 10 square feet. b) Use your equation to figure the length. c) Draw the rectangles that represent your soultion. (cont d)

8 Farmer s Fence (cont d) 14. a) Using your equation, substitute the maximum value for the area and solve for the length. b) Draw the rectangle that represents your soultion. Why is there only one figure for this value? How does your graph support the idea that there will be only one solution? 15. We know that 25 square feet is not a possible area for this rectangle. a) How does the graph show this fact? b) What should we anticipate for a result when we substitute 25 in for A in our equation? Do so and see if you predict is correct.

9 C. Shore Name Atrium Design #1 The famous hotel builder, Art Katekt, has been hired to design the atrium of a new hotel. Each room of the hotel opens onto a walkway overlooking the rectangular atrium. The design must include a protective brass railing around the perimeter. Since Art has been given a budget and the cost of brass in fairly high, he has been restricted to using only 640 feet of railing around each floor. You work for Art Katekt and need to help him figure the problem below. #1 Determine the dimensions of the railing so that the guests on each floor could enjoy the maximum scenic view of the atrium below. a) Write an equation to generalize the relationship c) Write an equation for the area, A(w), of the between the length, l, and the width, w. rectangle in terms of the length, w. b) Graph equation (a) d) Graph equation (c) Dimensions for Maximum Area: Maximum Area:

10 C. Shore Name Atrium Design #2 The famous hotel builder, Art Katekt, has been hired to design the atrium of a new hotel. Each room of the hotel opens onto a walkway overlooking the rectangular atrium. The design must include a protective brass railing around the perimeter. Since Art has been given a budget and the cost of brass in fairly high, he has been restricted to using only 640 feet of railing around each floor. You work for Art Katekt and need to help him figure the problem below. #2 One side of the building will be a glass wall. With only three sides of the atrium needing the 640 feet of brass railing, what would be the dimensions of the greatest scenic view in this situation with a glass wall serving as one of the lengths of the atrium? glass wall a) Write an equation to generalize the relationship c) Write an equation for the area, A(w), of the between the length, l, and the width, w. rectangle in terms of the length, w. b) Graph equation (a) d) Graph equation (c) Dimensions for Maximum Area: Maximum Area:

11 C. Shore Name Atrium Design #3 The famous hotel builder, Art Katekt, has been hired to design the atrium of a new hotel. Each room of the hotel opens onto a walkway overlooking the rectangular atrium. The design must include a protective brass railing around the perimeter. Since Art has been given a budget and the cost of brass in fairly high, he has been restricted to using only 640 feet of railing around each floor. You work for Art Katekt and need to help him figure the problem below. #3 Glass elevators will travel along one side of the atrium, so that guests may enjoy the view. The elevators would be 40 feet wide along one of the lengths of the atrium, with the 640 feet of brass railing extending around the remaining edges. What dimensions would create the maximum scenic view of the atrium? glass elevators a) Write an equation to generalize the relationship c) Write an equation for the area, A(w), of the between the length, l, and the width, w. rectangle in terms of the length, w. b) Graph equation (a) d) Graph equation (c) Dimensions for Maximum Area: Maximum Area:

12 Teaching Students to THINK, COMMUNICATE, COLLABORATE & CREATE 4 Claims: Concepts & Procedures, Problem Solving, Communicate Reasoning, Modeling & Data Analysis Questioning (Dig Deeper/Reach Higher) Access for All (Low Floor/High Ceiling) 8 Practices ECI 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Target (Clear & Congruent) Engagement (Most students, most of the time) Feedback (specific & timely) Monitor & Adjust Suggestions for Lesson Improvement? Rigor: Fluency, Deep Understanding, Application, Dual Intensity

13 The Math Projects Journal Page 2 STUDENT HANDOUT Cool Shoes: Linear You own and operate NewCoolShoes.com, an online shoe store. Many people want to order shoes for friends and relatives, but do not know their shoe size. Since it is easier to estimate a person s height than shoe size, you want the customer to be able to enter a person s height and calculate the appropriate shoe size (approximate). You must have either a graph or equation in order to do this. So, your task here is to create both, using sample data from your class. BOYS GIRLS MATH-123

14 The Math Projects Journal Page 3 STUDENT HANDOUT Cool Shoes: linear (continued) 1. Fill the charts with data from your class. Record each person s height and shoe size. 2. Plot data points from the charts. Use one color or symbol (+) for boys and a different one for girls (*). 3. Do you notice any relationship between people's height and shoe size? What kind of correlation is it? 4. Draw an approximate line of best fit for each set of data (one for boys, one for girls). 5. For each line, calculate the rate of change (slope). BOYS: There is a change of sizes for every inches of height, or sizes per every one inch. GIRLS: There is a change of sizes for every inches of height, or sizes per every one inch. 6. a) Calculate the y-intercept of each line. BOYS: GIRLS: b) What do these intercepts imply? Do they match your graph? 7. Write the equations of each line. BOYS: GIRLS: 8. For each set of data, find a height that does NOT appear in the chart. For instance, if no girl in the class is exactly 68" tall, then choose 68 inches for the girls. Use your equation and your chosen value for height to find the corresponding shoe size at that height. Do your solutions match the graphs? BOYS: Height = GIRLS: Height = Shoe Size = Shoe Size = 9. For each set of data, find a shoe size that does NOT appear in the chart. For instance, if no boy in the class has a shoe size of 13.5, then choose 13.5 for the boys. Use your equation and your chosen value for shoe size to find the corresponding height. Do your solutions match the graphs? BOYS: Height = GIRLS: Height = Shoe Size = Shoe Size = MATH-123

15 Cricket Facts Only male crickets chirp. Why do they chirp? Crickets chirp primarily at night. Why? [Answers below] Male crickets rub their wings, not their legs, to chirp. How do crickets make sounds with their wings? [Answers below] Crickets will not chirp if the temperature is below 40 degrees Fahrenheit ( F) or above 100 degrees Fahrenheit ( F). Why? [Answers below]

16 Cricket Chirp Rates Look at the table of data that represents the chirping of a snowy tree cricket in two different conditions. In the first section of the table, a male cricket was recorded in a room that had a warm temperature. The same cricket was recorded in a much cooler room, as shown in the second section of the table. Each of the cricket recordings shown here lasted 20 seconds.