A B C

Solutions for Session 10, Part C

See solutions for Problems: C1 | C2 | C3 | C4

 Problem C1 Solution: Using 24 square tiles, you can make four distinct rectangles: 1 by 24; 2 by 12; 3 by 8; and 4 by 6. Even though the lengths and widths differ from rectangle to rectangle, their product is always the same, and equal to the area of the rectangles -- 24. Each of these rectangles represents an array of tiles with particular dimensions. If we count the tiles in each array, we get the area of the rectangle. The rule for finding the area of any rectangle, given its length and width, is length multiplied by width.

a.

This problem prompts students to derive the formula for area of a rectangle. It also delves into their understanding of multiplication by having them explore the relationship between multiplication and determining area.

b.

This problem builds on students' prior experiences with multiplication, area, and patterns. It also lays the groundwork for the kind of pattern recognition and generalization students will use in algebra.

c.

The concept of area and the derivation of its formula were extensively explored in the course. Different arrangements of tiles also draw upon ideas such as conservation of area.

d.

How do you know you've found all the combinations? Given an area and one dimension of a rectangle, find the missing dimension.

e.

Use this pattern approach for deriving the area formula for parallelograms. In this case, it would be helpful to have students work on grid or graph paper so that they could count side length and area. Students could also work on finding other "formulas" that involve multiplication; for example, finding the total number of students when given a class size and number of classes, or finding the number of feet or toes when given a set number of people.

Problem C2

Solution:
The measurements aren't the same because the students may have used different measuring tools and techniques. Also, physically measuring an object is likely to produce some degree of measurement error. The measurement process is, by its nature, never exact. Precision is affected by the measuring tool. The smaller the unit on a measuring tool, the more precise it is.

a.

The measurement content of this lesson is the idea that measurements are approximations and that differences in units affect precision.

b.

The idea of measurement error forms the basis for the study of standard deviation. This problem builds on students' prior experiences with measurement units, measuring tools, and decimals.

c.

This problem relates to one of the big ideas of the course, namely that measurement is an approximation. Also, concepts such as measurement error, precision, and accuracy are evident in this type of problem.

d.

How do you decide at what point a measurement is inaccurate? In other words, how much error is acceptable? How important is measurement precision in different contexts (i.e., building a bridge or cutting a piece of wrapping paper to wrap a box)?

e.

Have students measure time and discuss the accuracy of the measurements. Using stopwatches or wristwatches with a stopwatch function, have students try to record a set length of time. Using another watch or clock to track the time, tell the students to "start" their stopwatches. Fifteen seconds later, say "stop" to have the students stop their watches. Students should then write down the time, as precisely as possible, on their watch (e.g., 00:15:09 or 00:15:13 or 00:14:97). Theoretically, all the students should have recorded the same length of time. Their times, however, will likely vary because measurement is an estimate. Discuss why measurements aren't the same, if they are accurate, and what affects precision.

 Problem C3 Solution: The number of centimeter cubes that will fit into a 4-by-6-by-2 box is 48. The number of 1-by-2-by-2 cubes that will fit into this box is 24. Answers will vary for the type of strategy that can be used to predict how many packages of a particular size will fit into the box. One strategy is to first see how many centimeter cubes are needed to cover the base of the box (4-by-6 rectangle), which is 24. Then, because the height of the box will be 2, multiply the number of cubes in the base by 2 to get 48. Similarly, you can find out how many 1-by-1-by-2 cubes will cover the base of the box. Since the height of the box is the same as one dimension of the 1-by-1-by-2 cube, you can place the cubes vertically to determine how many will cover the base of the larger box, and in the process, entirely fill the space that will be encompassed by the folded box. Answers will vary on what kinds of approaches help determine the number of cubes in a box. One method is to multiply all three dimensions of a box to determine the number of centimeter cubes that will fit in that box

a.

The measurement content of this problem is the determination of the volume of a rectangular prism. The big ideas include understanding volume, cubic units, moving from two dimensions to three dimensions, and developing a formula for finding the volume of a rectangular prism.

b.

This problem builds on students' prior experiences with three-dimensional figures and discovering formulas (as in Problem C1). It also leads to a whole host of packing problems. Filling a box with packages of different dimensions (for example, 1 by 1 by 2) is harder than filling it with unit cubes, because you may not be able to fill the box completely, depending on how you arrange the packages. This can help students develop number sense. Students can also try packing more complicated shapes, such as spheres, into rectangular boxes. Now that they cannot fill the space completely, what is optimal?

c.

The concepts of volume and the derivation of its formula were covered in this course. Understanding the relationship between two-dimensional representations and their three-dimensional counterparts was also a part of this course.

d.

Many students will not "see" the formula with only two examples. This activity may instead lead them to compare volume and surface area of a solid. Most students will need many more exercises just like these (giving dimensions, cutting out a net, predicting volume, and checking volume based on given dimensions and a net, etc.) before they can generalize a formula. As they attempt more examples, it may be helpful for students to keep their data organized in a table. For students who already have a good grasp of volume of a solid, consider the following questions: Why does a 1-by-1-by-2 package completely fill the box? Are there other smaller-sized packages that will completely fill the box? Why or why not?

e.

A certain toy company makes sets of children's blocks. The blocks are 1 in. cubes. The company is looking for a rectangular box that will hold a set of 64 blocks with no leftover space. Design a box for the company. Explain why yours is the best design. Additionally, students could explore the relationship between surface area and volume of a rectangular prism by generating all possible rectangular prisms with a given volume.

Problem C4

Solution:
Depending on the triangles drawn, the measurements of the particular angles will vary (with the exception of the equilateral triangle, whose angles will all measure 60 degrees). The sum of the angles for every triangle is 180 degrees, and this will remain true for any triangle. Testing more triangles will demonstrate this further, but it is not a proof.

a.

The measurement content of this problem is the discovery of the fact that the sum of the interior angles of a triangle is 180 degrees.

b.

This problem builds on students' prior experiences finding patterns, identifying various types of triangles, and measuring angles. It lays the groundwork for using this (sum of interior angles equals 180 degrees) and other attributes of triangles, as well as angle relationships, to determine angle measures without using measuring tools such as protractors. The patterning nature of this problem continues to build a strong generalizing foundation.

c.

The concepts of measurement of angles and the sum of angles in a triangle were discussed in this course.

d.

Students could apply what they now know about the sum of the angles in a triangle to find missing angle measures without measuring. For example, given two angle measures in a triangle, find the measure of the third angle.

e.

Have students use geometry software or pattern blocks to determine the sum of interior angles of other regular polygons, such as squares and hexagons. Students can cover a new shape with triangles and use the fact that the sum of interior angles in a triangle is always 180 degrees to determine the sum of the angles in the new shape. If the students use pattern blocks, they can also find each angle of a regular polygon by placing several of that same type of polygon side by side around a point until the blocks fill up 360 degrees. The students can then divide 360 by the number of polygons around that point to obtain the measure of each angle. For example, it takes three hexagons meeting at a single point to complete 360 degrees. When you divide 360 degrees by 3, you get 120 degrees, which is the measure of each interior angle of a regular hexagon.