The Volcanoes Erupt

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Purpose

This is a level 4 number activity from the Figure It Out series. It relates to Stage 7 of the Number Framework. 

A PDF of the student activity is included.

Achievement Objectives
NA4-2: Understand addition and subtraction of fractions, decimals, and integers.
Student Activity

 

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Specific Learning Outcomes

solve addition and subtraction problems involving integers

 

Required Resource Materials

FIO, Level 3-4, Number, Book 3, The Volcancoes Erupt, pages 20-21

Activity

Although negative numbers are introduced at level 4 of Mathematics in the New Zealand Curriculum, they are readily understood by students at the advanced additive stage or beyond of the Number Framework.
Volcanoes provide a meaningful context in New Zealand for mathematical investigations. The eruption times listed in this activity also serve to remind us that, in geological time, some of these eruptions have occurred very recently and, realistically, there is every likelihood that further eruptions will occur.
Before the students begin this activity, you may need to discuss with them the abbreviations BC and AD. (BC means before Christ and AD is the abbreviation of the Latin phrase Anno Domini, which means in the year of the Lord. Another term, used most commonly by non-Christians, is BCE, which means before the common era, indicating dates before the Christian era.)
The students may be interested to find out that there is, in fact, no year 0. Although Amrit uses 0 to divide BC and AD on his number line as a way of simplifying the mathematics involved, the sequence is actually ... 2 BC, 1 BC, 1 AD, 2 AD ... There is no zero in Roman numerals, which were used in the sixth century when the system we use to count years was invented.
The students also need to understand that years BC are effectively negative numbers. Thus, to find the difference between 100 BC and 200 AD, it is necessary to work out the difference between –100 and + 200, which is 300 years.
A number line is a useful model to use for working out and understanding the difference between years BC and years AD or between negative and positive integers. On a number line, the difference between 100 BC and 200 AD would be:

number line.
A good context for helping the students to understand positive and negative numbers is owing money. Being in debt means having a negative amount of money, regardless of whether it is a $100,000 mortgage on a house or borrowing $5 from an older brother or sister. A student who borrows $5 and spends it has to earn $5 before they get back to a nil balance. (In a mortgage, of course, there is also the issue of interest charged.)
One approach to the questions in this activity would be to draw on a chart or the whiteboard a number line from –10 to 10. The students could use the number line to explore ways of finding the difference between a number on the positive side and a number on the negative side. Eventually, they may be able to see that finding the difference requires adding the two amounts. For example, the difference between –5 and 4 requires adding 5 to get from –5 to 0 and then another 4 to get to 4, that is, a total of 9. Mathematically, they may come to realise that 4 – –5 = 4 + 5, which is 9.
Ask “What is the difference between 4 and –5?”
To calculate the answers to the various questions in this activity, the students need to employ a range of strategies. Questions 1 and 2 both refer to volcanoes that have erupted since the year 0. The students could solve this using a missing addends strategy (1886 + = 1914), which can be easily solved using an empty number line:

number line,
The answers to questions 3 and 4 can be worked out in similar ways. For example, the Rotokawau eruption was in 1500 BC and the Tùhua eruption was in 4300 BC so, counting back from 1500 BC, 1 500 plus 500 years takes it to 2000 BC and another 2 300 years takes it to 4300 BC, a total of 2 800 years. This could be shown on an empty number line like this:

number line.
The trickier calculations are the ones that cross the year 0. In these cases, the strategies described earlier can still be used. For example, in question 5a, the first Taranaki eruption was in 1200 BC and the second in 1740 AD, so the students might say that it takes 1 200 years to reach the year 0 and another 1 740 to reach 1740 AD, a total of 2 940 years. This could be shown on an empty number line as:

number line.
To answer questions 7a and c, encourage the students to think about what they will have to subtract from the years given in order to arrive at year 0. They may decide that subtracting 2 000 years would be about right, and this is quite reasonable. The next step, subtracting 2 000 from 10 700, would result in 8 700, which is approximately the time of the Maungarei eruption. Using approximation is perfectly valid in these circumstances because the eruption dates themselves are best approximations based on geological evidence. In the case of question 7b, the students can simply subtract 1 800 from 2 000. This results in the year 200 AD, which is roughly the time of the last Taupo eruption.


Investigation

The students will need to do some book or online research to find the answers in this investigation.
A suitable book resource is:
Hicks, G. and Campbell, H., eds (1998). Awesome Forces: The Natural Hazards That Threaten New Zealand. Wellington: Te Papa Press.
A website that provides relevant information on New Zealand volcanoes is:
http://www.gns.cri.nz/Home/Learning/Science-Topics/Volcanoes/New-Zealand-Volcanoes


Answers to Activity

1. 28 years
2. 440 years
3. a. 1 300 years
b. 4 000 years
4. a. 2 800 years
b. 7 200 years
5. a. 2 940 years
b. 4 495 years
c. 1 736 years
6. a. Haroharo
b. Rotokawau
7. a. Maungarei
b. Taupo
c. Taranaki
Investigation
Results will vary. They may include the eruptions that eventually formed the Lyttelton, Akaroa, and Otago harbours, some of the mountains in the western Waikato,
some of the cones in the vicinity of Auckland City, and Mount Edgecombe, Mount Ngāuruhoe, Mount Tongariro, and White Island.

 

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Level Four