Using Integrated Tasks in
“Helping Teachers Develop as School-based Leaders”

Mathematical Agenda: Responding to Critical Ideas (Connecting, Relearning, and Integrating Content Areas)

Tasks – Integrating two or more mathematical strands
· Should the tasks be at the ELEM level or simply for teachers?
· Should SOME of the tasks be at the ELEM level?
· Tasks that clearly are integrated or those that may appear to be from a particular strand, but can be approached from more than one mathematical vantage point?

Tasks – Integrating two or more mathematical strands

• Most Square Task – Number, geometry, measurement (A new housing subdivision offers lots of three different sizes: 185 feet by 245 feet, 75 feet by 114 feet, and 455 feet by 508 feet. If you were to view these lots from above, which would appear most square? Which would appear least square? EXPLAIN your answers.)
• Radius Squares – Measurement, geometry (How many "radius squares" are there in a circle? Use the materials from this page and the next to see how many squares with dimension the same as the radius of the circle it will take to completely fill the circle.)
• Marcy’s Dots – Number, algebra, geometry
• How Many Rectangles – Number, geometry (For each number 1 to 30, how many different rectangles are there with that area (e.g., for a rectangle with area 12, there are 1 × 12, 2 × 6, 3 × 4, 4 × 3, 6 × 2, and 12 × 1 or 6 different rectangles).
• Divide that Paper – Measurement, geometry (In one class, imagine a student wants to divide an 8½ inch by 11 inch sheet of paper into three columns of equal width. The student is ready to measure off lengths of 2 5/6 inches, but the teacher says, "Let me show you a carpenter's trick." He places a 12-inch ruler at an angle on the page so that the 0-inch and 12-inch marks on the ruler are on the left-and right-hand edges, respectively, and makes marks at the 4-and 8-inch points on the ruler. He then repeats the procedure, with the ruler farther down the page. Drawing lines through the 4-inch marks and the 8-inch marks divides the page neatly into three equal parts. The teacher then says, "Carpenters use this trick to divide boards into thirds.” Why does this work? Can you use a similar approach to divide paper into fourths? Fifths?)
• Celsius-Fahrenheit Task – Number, algebra (Is there a temperature where both the Celsius and Fahrenheit scales report the same number? If so, what is it? If not, argue why you do not believe there is a common temperature. Note: The freezing point of water is 0º C and 32º F. The boiling point of water is 100º C and 212º F.)
• Where Would a Meteor Hit? – Measurement, Data Analysis, Probability (Setting: If a meteor was to come hurtling from outer space and strike the planet Earth, where would it hit? More specifically, how likely would it be to strike land versus water? The following simulation provides an opportunity for you to consider this question in an interesting manner.)
• Rolling Triangles – Geometry, measurement (Setting: When the lengths of the sides of a triangle are rolled randomly from dice, what kinds of triangles are formed — right, acute, obtuse, equilateral, isosceles, scalene, or even no triangle? Roll your set of three dice 20 times and using geostrips, tell what kind of triangle was formed. Then make any tentative assertion you feel comfortable in making about the probability of various triangles.
• CD Throw – Data analysis, measurement, probability (Setting: The following activity takes place in a classroom with one-foot tiles on the floor. Using a CD, you will toss it on the floor, keeping track of the times it hits and does not hit a line where two (or more) tiles join. Before Collecting Data: What do you think will happen? That is, given the size of the floor tiles and the size of the CD, how often do you think the CD will hit a line? Could you calculate the theoretical probability to compare to the experimental probability that you will calculate below?
• The Pentagon Challenge: The Sum of the Interior Angles – Measurement, geometry (In Ms. Potter’s mathematics class, students are working with angles. Just last week the class discovered that the sum of the interior angles of a triangle is 180. They are fairly sure this is true because they used various approaches (measuring each angle with an angle ruler, tearing the angles and putting them together to make a line, using Sketchpad, etc.). Yesterday, Ms. Potter issued a new challenge: make a conjecture for the sum of the interior angles of a pentagon. Students tried various methods. Three groups used similar approaches (partitioning the pentagon into triangles), but made different conjectures. Analyze the modified student work below and be ready to answer the questions that follow.
• Area of a Blob – Measurement, data analysis (To help students develop a richer view of area, I suggest starting with an irregular polygon, a "blob," or anything where a quick and easy formula for area does not exist. Task: Estimate the area of the blob provided. Explain your thinking and process.
• Fraction Division – Number, geometry (Use the area below to draw a pictorial representation of 3 ÷ b. Briefly explain what the various parts of the drawing represent. Write a context (story problem) that would be modeled by 3 ÷ b. Then connect the result of the division to the context.)
• Division with Remainder – Number, real context (Write three different story problems that would be solved by dividing 51 by 4 and for which the answers listed below would be correct. You should have realistic problems. The real-world situation should dictate which of the three answers is appropriate for each problem as you would do it in the world. Do not use arbitrary statements such as “round up.” a. 12¾; b. 13; c. 12)
• Fraction Sense – Number, real context (A researcher concluded from the literature stating that 1/3 of all females have been abused as children and 1/6 of all males have been abused as children then ½ of all adults have been abused as children. How would you respond to this person?)
• What Fraction Is Blue? – Number, geometry (The following “ribbon” was made with pattern blocks by students in a 5th-6th grade class. The teacher asked, “what fraction of the ribbon is blue?” Following the picture are responses by children, with their reasoning stripped away. Given just the response, what do you think the child was thinking? (From “Communicating about Fractions with Pattern Blocks” by Janet Caldwell, Teaching Children Mathematics, November 1995, pp. 156-161.)
• Skeleton Tower – Geometry, number, algebra (A. How many cubes are needed to build this tower? B. How many cubes are needed to build a tower like this, but 12 cubes high? C. Explain how you worked out your answer to part (B). D. How would you calculate the number of cubes needed for a tower n cubes high?)

Tasks (6/10, 6/11)

• Balloon Switching- Number, algebra. Ten people arrive at a party. Each of them have a balloon. If each person trades their balloon with everyone else at a party, how many balloon switches will take place?
• How long 'til a Million?- Number, algebra. You have $1. Each week your amount of money doubles. How long until you have $1 million?
  
• Biggie Size It (Squares)- Algebra, number, geometry. If you double the side length of a square, what happens to the perimeter? To the area? What if you triple the side length? What if you quadruple the side length?
  
• Biggie Size it- Number, algebra, geometry. Given a triangle, what happens to the perimeter and area if you double the side lengths? What if triple the side length? Does the type of triangle matter?
• Double the Fun- Number, geometry. Suppose we want to double the dimensions of this rectangle. Draw your figure using as many different methods as you can. Explain your methods.
• Fraction Cookie Jar- Number, Algebra. You have a jar of cookies. Sid eats 1/2 of them. David eats 1/3 of the remaining cookies. Drew eats 3/4 of the remaining cookies? Ron eats 3 cookies. If there are 2 cookies left how many cookies were originally in the jar? Use a picture and equations to explore the task.
• Multiple Strategies (Addition)-The teacher posed the problem 237+198. Tyrisha solved the problem by adding 200+100+30+90+70+7+8. Rodney solved the problem by adding 237+3+5+10+50+40+100. Jesus solved the problem by adding 235+200. Mary solved the problem by adding 237+100+90+3+5. Solve the task each of the ways. How are they similar? How are they different?
• Multiple Strategies (Subtraction)- The teacher posed the problem 201-98. Simon changed the problem to 203-100. Krystal approached the problem by solving 201-2-96. Bernise changed the problem to 201-1-90-7. Solve the task each of the ways. How are they similar? How are they different?
• What's in Between?- Yasmine claims that in order to find a fraction in between two fractions you simply add the numerators and denominators separately. For example, A fraction between a/b and c/d would be (a+c)/(b+d). Is this true? Does it always work?
  
• Planning Cruises- Number, algebra. A cruise line has 3-day, 4-day, and 7-day cruises. After each cruise, a ship returns for one day and repeats the pattern. If one cruise of each type leaves today, when will all three cruises leave again on the same day?
• Broken Calculator tasks- Number, algebra. There are tons of these..here's one.... Use your calculator to generate the number 1,011 without using a 0 and a 1 in any of the operations.
• Consecutive Odds and Evens- Choose four consecutive odd counting numbers. Take the product of the middle two numbers and subtract the product of the first number and the last number. Try a few samples and formulate a rule. Explain why the rule works.
• McNugget Numbers- McDonald's sells Chicken McNuggets in boxes of 6, 9, or 20. Obviously one could purchase exactly 15 McNuggets by buying a box of 6 and a box of 9. Using only combinations of boxes of 6, 9, and/or 20 McNuggets, can you buy exactly 17 nuggets? 53 nuggets? What is the largest number of nuggets that is IMPOSSIBLE to purchase.
• Hot Air Balloon- A hot air balloon moves up 200 feet during its first minute of flight. Then it will continue to rise each minute thereafter for a distance of 4/5 (or 80%) of the distance traveled the previous minute. If you ignore air pressure, will the balloon fly out into space (in other words, will it rise indefinitely)? Explain why or why not.
• Almost a Square- Determine the dimensions of a rectangle whose dimensions are most square-like.
• Sum of Natural Numbers- Given that 1 = 1, 1 + 2 = 3, 1 + 2 + 3 = 6, and 1 + 2 + 3 + 4 = 10, what is the sum of the numbers from 1 to 100? Explain how you found your answer.