This is a level 3 number strand activity from the Figure It Out series. It relates to Stage 6 of the Number Framework.
A PDF of the student activity is included.
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use tables and rules to describe a linear number pattern
FIO, Level 4, Algebra, Book Three, Stacking Up, pages 4-5
In this activity, the students must first work out the height that each additional container, hat, CD, chair, or desk contributes to the height of the respective stack. Explanations for each object’s height are outlined in the Answers. The table below shows short cuts for working out the height of stacks with different numbers of containers. Students might make a table like this one to help them figure out how the rule works for stacks with any number of containers.
A rule for the height of a stack with any number of containers is the height of the first container, 29 centimetres, plus 9 centimetres for each of the other containers in the stack. So for a stack with x containers, the height, y, is y = 29 + (x – 1) x 9. This
can be expressed as y = 29 + 9(x – 1) or, more simply, as y = 29 + 9x – 9 or 20 + 9x. This is also y = 9x + 20.Students who grasp this symbolic algebra may like to
use this rule to check the values in the table above. A table using y = 9x + 20 is shown here.
The table below shows short cuts for working out the height of stacks with different numbers of hats in question 2. As with the containers in question 1, students may find that a table like the one below will help them figure out how the rule works for stacks with any number of hats.
A rule for the height of a stack with any number of hats is the height of the top hat, 18 centimetres, plus 7 centimetres for each of the other hats in the stack. This can be expressed symbolically as y = 18 + (x – 1) x 7 or y = 18 + 7(x – 1). This can be simplified to y = 18 + 7x – 7 or y = 7x + 11. The following table shows different algebraic expressions for the height, y, of stacks with x hats.
Students completing such a table of values successfully will see that symbolic algebra provides a simple way to figure out the height of any stack of hats.
In question 3, a short cut for the height of 25 CDs is given in the Answers as 25 x 2 + 7 = 57 centimetres. The height, y, of a rack with x CDs is then y = x x 2 + 7 or y = 2x + 7.
Students may approach question 3b in several ways. The most straightforward is to reason as follows: The rack is 97 centimetres tall, but we know that the frame takes up 7 centimetres of this, so the CDs take up 90 centimetres. Each CD requires 2 centimetres, so there must be 90 ÷ 2 = 45 CDs.
Students who are comfortable with algebraic manipulation might be interested to see how we can adapt the rule found above (y = 2x + 7) to solve question 3b. In this question, we are told that y = 97 and are asked to find x, the number of CDs. Algebraically, we are told that 97 = 2x + 7. We can then reason as follows to find x: 90 = 2x (subtracting 7 from each side), so 45 = x (dividing both sides by 2), which is
the answer we found above.
In question 4, the students will need to make their own measurements. Students who have shown skill in working with algebraic symbols should be encouraged to devise and then test their own algebraic expressions for the height of stacks with any number of chairs or desks.
Answers to Activity
1. a. 17 cm. The stack of 4 containers measures 41 cm, and the stack of 2 containers measures 25 cm. So the 2 extra containers add 16 cm (41 – 25), which is 8 cm for each new container. So 1 container measures 25 – 8 = 17 cm.
b. 89 cm
c. A rule for the height of a stack with any number of containers is: the height of the first container, 17 cm, plus 8 cm for each of the other containers in the stack.
2. a. 12 cm. Each hat below the top hat adds 6 cm to the height of the stack. The hat at the top of the 3-hat stack is 24 – 2 x 6 = 12 cm.
b. The height of a stack with 20 hats would be 19 x 6 + 12 = 126 cm. A rule for the height of a stack with any number of hats is: the height of the top hat, 12 cm, plus 6 cm for each of the other hats in the stack.
3. a. 57 cm. The extra 5 CDs need 10 cm, so each additional CD in a rack occupies a height of 10 ÷ 5 = 2 cm. The height of a rack with 10 CDs is 27 cm. The 10 CDs need 20 cm, so an extra 7 cm is used for the frame of the rack. The height
of a rack with 25 CDs must therefore be 25 x 2 + 7 = 57 cm.
b. 45. (97 – 7 = 90. 90 ÷ 2 = 45. So 7 + 45 x 2 = 97.)
4. Answers will vary depending on the measurements.