Activities One to Three

In these activities, the students have to work with a lot of data as they find differences between distances, find differences between times, find fractions, calculate average speeds in kilometres per hour, and calculate percentages.
The students need to be aware that although both the data about time and the data about distances are presented in the same decimal format, they actually represent different things. The distance data is in standard decimal format, that is, the figure to the right of the decimal point represents tenths. But the time data is not in this same format. Here, the figures to the right of the decimal place represent the number of
minutes, that is, sixtieths. So when the students want to find the distance travelled in stage 4, they can calculate 16.2 – 11.9. But in finding out how long this stage took to cycle, they will get the wrong answer if they calculate 11.09 – 10.59. Instead, they will have to work out that the difference in time between 10.59 and 11.00 is 1 minute, and then there are another 9 minutes to get to 11.09, a total of 10 minutes altogether. You might like to ask the students to compare this way of recording time with the method used on page 12 of the students’ book.
Several questions in these activities ask the students to find an average speed in kilometres per hour. This rate shows how far someone would travel in 1 hour if their speed was evened out over the whole hour.
For Activity Four, question 3a, the students need to use the information from earlier questions in this activity and from Activity Two. From their earlier answers, they will quickly work out that the group A riders were 13 minutes faster over the same distance than the group C riders.
One way of working out the percentage for question 3b is to think of it in terms of how much faster group A were than group C. Using the average speeds, 24 km/h ÷ 18 km/h = 1.3, so group A were 1/3 or 331/3% faster than group C.
 

Answers to Activities

Activity One

1 a-b

2. 3 hrs
3. a. 80 min. (1 hr, 20 min.)
b. 4/9
4. a. 24 km/h
b. Stage 6
c. Answers will vary. Possible answers could include the impact of tiredness, hills, gravel roads, corners, and compulsory stops.
Activity Two
1. 24.6 km/h, which rounds to 25 km/h
2. a. 6.5 km
b. 22.9 km/h, which rounds to 23 km/h
3. 23.8 km/h, which rounds to 24 km/h
Activity Three
1. 11.47 a.m.
2. a. 23 min.
b. 21.39 km/h, which rounds to 21.4 km/h (1 d.p.)
Activity Four
1. a. 10.31 a.m.
b. 18.57 km/h, which rounds to 19 km/h
2. 17.64 km/h, which rounds to 18 km/h
3. a. Group A (6 km/h faster)
b. 331/3% faster, which rounds to 33% (to the nearest whole number)
4. Answers will vary. Group A ride further than the other two groups (this could be considered an advantage or a disadvantage). Group C were 13 minutes slower than group A over the same route, so biking at the end of the camp may have been
more tiring than biking at the beginning.

Rates are difficult to model with materials and require students to have good, sound multiplicative strategies. The idea of average (mean) is established in the transition from advanced additive to advanced multiplicative.
Question 1, which asks students to calculate metres per second, involves the idea of rate, that is, a fraction used to compare 2 quantities of different measures. To answer this question, the students need to divide 80 metres in turn by each of the 6 waka ama times. This part of the question can easily be done on a calculator, especially if 80 is put into memory and recalled for each calculation.
If the calculation results in a decimal number that includes hundredths or thousandths, you may need to help the students to use the procedure for rounding to 1 decimal place:
i. go up to the next figure if the result in the hundredths column is 5 or greater (for example, 180 ÷ 45 = 1.7777, which rounds to 1.8);
ii. leave the same figure if the result in the hundredths column is 4 or less (for example, 80 ÷ 49 = 1.632653, which rounds to 1.6).
An interesting extension to question 1 would be to work out:

• the mean time taken by the combined 6 waka ama to do the 80 metre sprint (by adding the 6 paddling times and then dividing by 6; rounded to 1 decimal place, this is 51.2 seconds).

• the mean speed of the 6 waka ama taken together (80 ÷ 51.2 = 1.6 metres per second, rounded to 1 decimal place).

In question 2a, the students need to think logically to work out that 6 strokes is 1/3 of 18 strokes, so the time for 6 strokes must be 1/3 of 21 seconds, namely, 7 seconds. A double number line is a useful strategy here:

For question 2b, the students can use their answer from 1a: 6 strokes (1 set) takes 7 seconds. They can then use their knowledge of basic multiplication facts to answer 7 x = 49. The thinking that the students do for this question is relevant to question 2c (see the Answers).
Question 4, like question 1, focuses on rate, but this time calculating metres per minute. To work out the average (mean) speed in metres per minute of the paddling times, the students need to recognise that 12 kilometres is 12 000 metres. Then they can use the same approach as in question 1, namely, dividing the 12 000 by the time in minutes taken by each waka ama and rounding where necessary to 1 decimal place.
Possible extensions to question 4 are similar to those outlined above for question 1:

• calculating the mean time of the 6 waka ama over the 12 kilometre distance (121.7 minutes)

• calculating the mean speed of the 6 waka ama (98.6 metres per minute).

Answers to Activity

1.

2. a. 7 s
b. 7 sets
c. 6 sets. 48 ÷ 7 = 6, plus 5 of the next set of 6. They crossed the line before they
completed the last stroke of the 7th set.
3. a. 6. (72 ÷ 12)
b. 11. (132 ÷ 12)
c. 18 (with 2 strokes on the last side because 218 ÷ 12 is 18 and 2 remainder)

4.

Rate problems such as those in this activity are difficult to model with concrete materials, so students who attempt this activity should be strongly multiplicative and at least at the advanced multiplicative stage of the Number Framework.
This activity provides basic scientific information on distance, time, and speed and requires students to use this information in their calculations. The use of ratio tables can enhance the students’ understanding of the relationship between time, distance, and speed. The formulae that they need to use are: distance = speed x time, speed = distance ÷ time, or time = distance ÷ speed.
In question 1a: distance = 5 metres per minute x 28 minutes = 140 metres
In 1b: time = 62 metres ÷ 5 metres per minute = 12.4 minutes (or 12 minutes 24 seconds)
In question 1c, the speed is converted from metres per minute to metres per hour by multiplying by 60 (because there are 60 minutes in 1 hour): 2 metres per minute x 60 = 120 metres per hour.
Question 2 is ideal for using ratio tables (or double number lines):

So, in 2a: 15 metres + 30 metres = 45 metres, so 1 hour + 2 hours = 3 hours.
In question 3a, the students need to remember that 10 minutes is 1/6 of an hour because the speed is in kilometres per hour.
Distance = speed x time
= 8 kilometres per hour x 10 ÷ 60 hours (or 8 ÷ 6)
= 11/3 kilometres
In 3b: time = distance ÷ speed
= 0.1 kilometres ÷ 8 kilometres per hour
= 0.0125 hours
= 0.0125 x 60 minutes
= 0.75 minutes x 60 seconds
= 45 seconds

 

Investigation

The investigation could be linked with the investigation on page 14 of the students’ book. See also the comments on investigations in the notes for page 11.

Answers to Activity

1. a. About 140 m
b. About 12.4 min. (12 min 24 s)
c. About 120 m/h
2. a. About 3 hrs
b. About 3.75 m
c. About 40 min. (The path measures about 20 cm, which is 10 m according to the scale. 10 m is 2/3 of 15 m, and 2/3 of 60 min is 40 min.)
3. a. About 11/3 km
b. About 0.75 min. (45 s)
Investigation
Answers will vary.

This activity includes questions on length, time, and rates. It builds on the previous page in that it uses number sense and number calculations to make calculations involving rate. Students who are advanced additive could answer the rate questions using strategies such as repeated addition. In this case, such strategies will work, but they will be inefficient. Students who are advanced multiplicative and beyond should be using multiplication strategies for most of the questions.
Most students will assume for question 1 that the paddlers will try and avoid all the taniwha. Each bend, whether paddling or dragging, is assumed to be 500 metres, and the total journey is 55 kilometres. The total time travelled = number of 500 metres walked x 15 minutes + kilometres paddled x 12 minutes.
If Wiremu, Tāmoko, and Ngāhuia drag the waka overland for 10 bends, they will walk 10 x 500 metres (5 kilometres). It takes 15 minutes to drag the waka over each 500 metres: 10 x 15 minutes = 2 hours 30 minutes. This leaves 50 kilometres (55 – 5 = 50) to paddle.
Total time = 2 hours 30 minutes + 50 x 12 minutes
= 2 hours 30 minutes + 10 hours
= 12 hours 30 minutes
In question 2, the students take into account the effect of the 5 friendly taniwha, but they can use similar reasoning to that for question 1.
You could use the information on the page to explore various scenarios of walking versus paddling. Encourage the students to justify their answers with mathematical information. For example, “We made them walk 5 bends. That’s 21/2 kilometres, so that would take 1 hour 15 minutes. That leaves 52.5 kilometres to paddle at the rate of 12 minutes per kilometre, so that would take 10 hours 30 minutes. So the total time would be 11 hours and 45 minutes.”
You could give the students a blank version of this table to fill in:

The table allows the students to see the relationships between the times and distances. It can also be used to develop an algebraic formula that describes the relationship between the distance walked and the time taken and the distance paddled and the time taken (using kilometres and minutes).
For example, when comparing the distance walked and drag time: “What do you multiply 1 by to get 30? 2 by to get 60? 2.5 by to get 75?” (The students should be able to identify the factor of 30.)
When comparing the distance paddled and the paddle time, ask: “What do you multiply 55 by to get 660?” (12) “Does this factor work for all the relationships?” This can be quickly tested with a calculator by multiplying all the distances paddled by 12. (This will establish the relationship if the students are struggling with the mental calculations.)

Total time in minutes = 30 x distance walked (kilometres) + 12 x distance paddled (kilometres).

Answers to Activity

1. 12 hrs 30 min, assuming they avoid all 10 taniwha. (If they dragged the waka for about 500 m round each bend where there is a taniwha, that would take them 150 min. They would paddle for 50 km: 50 x 12 min = 600 min. 150 + 600 = 750 min, which is 12 hrs 30 min.)

2. With 5 friendly taniwha letting them paddle past safely, they would only drag the waka around 5 stretches of 500 m (5 x 15 = 75 min) and paddle for 52.5 km (52.5 x 12 = 630). 75 + 630 = 705 min, which is 11 hrs 45 min.

Activities One to Three

In these activities, the students have to work with a lot of data as they find differences between distances, find differences between times, find fractions, calculate average speeds in kilometres per hour, and calculate percentages.
The students need to be aware that although both the data about time and the data about distances are presented in the same decimal format, they actually represent different things. The distance data is in standard decimal format, that is, the figure to the right of the decimal point represents tenths. But the time data is not in this same format. Here, the figures to the right of the decimal place represent the number of
minutes, that is, sixtieths. So when the students want to find the distance travelled in stage 4, they can calculate 16.2 – 11.9. But in finding out how long this stage took to cycle, they will get the wrong answer if they calculate 11.09 – 10.59. Instead, they will have to work out that the difference in time between 10.59 and 11.00 is 1 minute, and then there are another 9 minutes to get to 11.09, a total of 10 minutes altogether. You might like to ask the students to compare this way of recording time with the method used on page 12 of the students’ book.
Several questions in these activities ask the students to find an average speed in kilometres per hour. This rate shows how far someone would travel in 1 hour if their speed was evened out over the whole hour.
For Activity Four, question 3a, the students need to use the information from earlier questions in this activity and from Activity Two. From their earlier answers, they will quickly work out that the group A riders were 13 minutes faster over the same distance than the group C riders.
One way of working out the percentage for question 3b is to think of it in terms of how much faster group A were than group C. Using the average speeds, 24 km/h ÷ 18 km/h = 1.3, so group A were 1/3 or 331/3% faster than group C.
 

Answers to Activities

Activity One

1 a-b

2. 3 hrs
3. a. 80 min. (1 hr, 20 min.)
b. 4/9
4. a. 24 km/h
b. Stage 6
c. Answers will vary. Possible answers could include the impact of tiredness, hills, gravel roads, corners, and compulsory stops.
Activity Two
1. 24.6 km/h, which rounds to 25 km/h
2. a. 6.5 km
b. 22.9 km/h, which rounds to 23 km/h
3. 23.8 km/h, which rounds to 24 km/h
Activity Three
1. 11.47 a.m.
2. a. 23 min.
b. 21.39 km/h, which rounds to 21.4 km/h (1 d.p.)
Activity Four
1. a. 10.31 a.m.
b. 18.57 km/h, which rounds to 19 km/h
2. 17.64 km/h, which rounds to 18 km/h
3. a. Group A (6 km/h faster)
b. 331/3% faster, which rounds to 33% (to the nearest whole number)
4. Answers will vary. Group A ride further than the other two groups (this could be considered an advantage or a disadvantage). Group C were 13 minutes slower than group A over the same route, so biking at the end of the camp may have been
more tiring than biking at the beginning.