Books in Boxes

Make sure there is a set of maths books, dictionaries, or similar and a plastic box (the dimensions given in the activity are those of a standard “cube” crate) in the classroom. The students can then explore ways of stacking the books. They will be able to see and feel what 20 kilograms is like.
If there is a crate and a set of books similar in size to the box and atlases in question 1a, it will be easy for the students to experiment until they find the most economical way of stacking the books. If these are not available, they should be able to sketch the different possibilities and use a calculator to help them work out the dimensions of the various stacks.

When working out mass in questions 1b and 1c, the students need to remember to take the mass of the plastic box (2.3 kilograms) into account.
Question 1c asks the students to work out how many atlases could fit in the box without exceeding the 20 kilogram limit. They should estimate the answer before they calculate it. Question 1d is a 3-part problem:

• calculate the volume of the atlases
• calculate the volume of the box
• calculate the percentage of the volume of the box that is occupied by the atlases.

Encourage your students to explain how they got the percentage. Many push the percentage key without having any idea about what the calculator is doing. (It may be a good idea to tell them not to use the percentage key at all.)
Question 2 repeats the same questions but uses books with different dimensions and mass. This time, the books will fill the crate without going over the 20 kilogram limit. Your students can think about why this is and should conclude that some books are heavier than others (that is, they have a greater density), which they knew all along.
Question 3 involves Figure It Out books. The school office may have a set of digital scales for measuring the mass of parcels, in which case, the students will be able to find the mass of a single book. If this is not possible, they can put a stack of 10 or 20 books on whatever scales are available, find the total mass, then divide by 10 or 20 to get a fairly accurate mass for 1 book. A similar principle can be used to find the thickness of 1 book: find the thickness of a stack of books and then divide by the number of books in the stack.
In 3c, the answer turns out to be a percentage that exceeds 100. Let the students work out what this means.
 

Cross-curricular links
 

Health and Physical Education
The principal puts a weight limit on the boxes because he doesn’t want the students or staff to strain their backs when lifting them. Your students should investigate whether this is a suitable limit.
This activity could lead into a discussion concerning the importance of looking after one’s back, an investigation of the incidence or causes of back injury, and/or research into safe lifting techniques. The ACC has relevant information in pamphlet form. Findings could be worked into a poster or a computer presentation. The
students could find out about the use of devices such as trolleys and hand trucks for shifting large loads and the hoists used in hospitals for lifting patients into bed. They could also find out how piano shifters manage.
Often there are only 2 people to shift a heavy piano up and down flights of steps.
Achievement Objective

•  access and use information to make and action safe choices in a range of contexts (Personal Health and Physical Development, level 4)

Answers to Activity

1. a. The maximum is probably 23 books (18 stacked on edge with 2 more fitted down the side and 3 flat on top).
b. 36.6 kg. (23 x 1.49 + 2.3 kg)
c. 11 books. (20 – 2.3 = 17.7 kg available for books. 17.7 ÷ 1.49 = 11.9, which needs to be rounded down to 11, because 12 books would exceed the limit on mass.)
d. (11 x 29 x 22 x 1.8) ÷ (33 x 33 x 28) = 0.41 or 41% (rounded to the nearest whole number)
2. a. The maximum is probably 26 books. (2 rows of 8 books stacked on their long edge, 1 row on top of the other, 1 row of 8 books stacked on their bottom edge, and 2 books lying flat in the space on top)
b. 15.6 kg. (26 x 0.510 + 2.3 kg)
c. All 26 will fit because they do not exceed the 20 kg limit on mass.
d. (26 x 19.5 x 13 x 3.8) ÷ (33 x 33 x 28) = 0.82 or 82% (rounded to the nearest whole number)
3. a. 150 books.
(Each weighs 118 g. 17.7 ÷ 0.118 = 150)
b. Answers will vary but should be in the range 32 500–33 000 cm^{3.
(One book is 29.7 x 21 cm, and a stack of 10 books is about 3.5 cm high. Based on these dimensions, the volume of 150 books will be 15 x 29.7 x 21 x 3.5 = 32 744 cm3 [rounded to the nearest whole number].)
c. Answers will vary, but based on the dimensions in b, the answer would be 107%.
(15 x 29.7 x 21 x 3.5) ÷ (33 x 33 x 28) = 1.07 or 107% [rounded to the nearest whole number]. As 100% means the box is totally full, 107% means that the volume of 150 books exceeds the volume of the box.}