This unit is designed to engage your class for at least the first week of the school year. It provides you, their teacher, with opportunities for you to learn about their current level of achievement.
- Use multiplication to solve rate problems, especially about units of time.
- Use and calculate with metric units for mass and standard units of time.
- Use scales to estimate distances and times.
The most significant piece of mathematics in this unit is about rates. In multiplication a rate is always involved. In simple situations that will be a unit rate.
For example, one breath every 6 seconds is a unit rate since it is expressed using one unit of the first measure for every ‘so many’ units of the other measure (in this case breaths and seconds). To find out how long it will be before four breaths elapse multiply both measures by four, 4 x 6 = 24 seconds.
Since rates involve multiplicative relationships the operations of multiplication and division are needed. Both measures need to be multiplied or divided by the same factor or the rate is changed. Sometime measures need to be converted into different units.
Consider the problem, “If someone blinks ten times per minute, how many times do they blink each day?”
Since there are sixty minutes in one hour the person blinks 60 x 10 = 600 times per hour.
They are probably awake about 16 hours per day, so they blink 16 x 600 = 9 600 times per day.
Specific Teaching Points
Features of student work to look for are included in the session descriptions.
It is expected that students will present a range of prior experience of working with numbers, geometric shape, measurement, and data. Students are expected to be able to apply place value to add and subtract two digit whole numbers.
Talk to your students about the purpose of the unit which is to find out some information about them, so you can help them with their mathematics. In the first session students explore body facts and learn when it appropriate or not appropriate to use multiplication.
In this unit we are going to work out some interesting facts about what will happen to us this year. Here is an interesting fact. You will probably blink about ten times per minute.
Discuss why humans blink. There are two main reasons; to keep our eyes moist and to protect them from light, dust and other potential damage. It is hard to gather data on blinking because people tend to alter their blinking patterns if they are being measured. You could gather data from your students about how often they blink but it is not necessary for this lesson.
Show Slide One of the PowerPoint, which shows Toby trying to work out how many times he blinks in one year. Invite ideas about how Toby can create a reliable estimate. Students might suggest that Toby will not blink when he is asleep. If he sleeps for nine hours, then 15 x 600 = 9 000 blinks are a good estimate for one day. Multiplying 9 000 by 365 (days in one year) gives an estimate of 3 285 000.
Move onto Slide Two which shows Lena who is estimating how much she will grow in height this year. She thinks that portioning her height into ten equal parts gives an estimate of her height gain each year. Do your students recognise that her thinking is flawed for two main reasons?
- She was not zero centimetres tall when she was born.
- Growth rates vary at different ages, children grow very fast in their first two years and early adolescents also grow fast.
Lena is also unlikely to be an exact whole number of years old. For example, she might be 9 years, 7 months old.
Slide Three shows three strategies that Lena might use to estimate her gain in height this year. Discuss the strategies linking them back to the flaws in her original thinking.
The left-hand strategy is division of her total height by her age in years. The middle strategy allows for her height at birth (50 cm) by subtracting that amount from her height at ten years. The right strategy allows for her birth length and her growth spurt from zero to two years. 84cm is average height for a two-year-old girl. The difference of 56 cm needs to be divided by eight since that represents growth over eight years not ten. The right-hand method is the most accurate and 56 ÷ 8 = 7cm is a good estimate of yearly growth in height.
Provide your students with their own copy of Copymaster One. Ask them to find their own estimates first before sharing with a partner. Allow access to calculators as the focus is on conceptual thinking about multiplication rather than calculation strategies.
Wander the room and look at how students approach the problems:
- Do they recognise situations when no change will occur during the year? (body temperature and number of teeth and bones)
- Do they apply the constant rate as the multiplier in multiplicative situations?
- Do they have a sense of the size of numbers, including decimals, and of the units of measure?
After a suitable period bring the class together to discuss their estimates. The most important issue is how to tell if a situation is multiplicative or not. A rate should exist, that is a constant change in one measure for every constant change in another measure. Heart beats are a good example. For a pulse rate of 70 beats per minute is a rate. To estimate number of heart beats per year this rate must be assumed to be constant, despite knowledge that pulse rate increases with exercise and decreases with sleep.
In this session students explore healthy living, consider their current habits, and compare those habits to experts’ recommendations.
What do you need to do this year to stay healthy and learn best?
Students are likely to have heard good health messages previously. They should suggest ideas like:
- eat good food
- exercise everyday
- avoid too much television or computer time
- get plenty of sleep
- restrict the amount of sugar they consume.
You might discuss the ideas further.
- What is good food?
- How much exercise and sleep should they get?
- What foods contain lots of sugar?
Gather some data about your students. Begin with sleep. Ask students to work out how many hours and minutes sleep they got on the previous night. A linear representation may help some students to make the calculation. Demonstrate how to work out the amount of sleep for different start and finish times. Here is a calculation for bedtime at 8:20pm and rising at 7:15am.
The complication with adding hours and minutes is that measurement of time is based on sixty, not ten. If sixty minutes amass then an extra hour is created. In the case above this does not occur. Adding the minutes gives 40 + 15 = 55 and adding the hours gives 3 + 7 = 10. The total time is 10 hours 55 minutes.
After students have calculated their sleep time you might graph the data using a stem and leaf plot. Asking students to record their time on Post-its makes creating the graph on a whiteboard or large sheet of paper easy. Cut each post-it into hour and minute parts.
Look up the recommended hours of sleep for kiwi children online. For nine and ten-year-old children the usual amount is 9-11 hours per night.
What fraction of our class is getting enough sleep on weeknights?
Ask students to work out the time they should go to sleep and wake up to meet the requirement of 9-11 hour per night.
You might calculate how many hours sleep each student in the class should get this year. Based on ten hours per night the calculation is:
365 x 10 = 3 650 hours per year 3 650 ÷ 24 = 152 days per year
Next consider what a balanced approach to healthy eating looks like. Guidelines can be found online at https://www.healthed.govt.nz/ which is New Zealand Government sponsored though it is easy to locate other resources. Search ‘Healthy eating for children 2-12 years’ and you will find recommended daily intakes. Copymaster Two contains the recommended number of servings per day for each food group.
To see what a daily food intake might look like you could measure out the given amounts and collect the parts in one place. This activity will give you some indication of your students’ familiarity with standard metric units such as grams and with measurement devices, such as kitchen scales.
Ask your students to record what food they ate yesterday and to match the items from each meal and snack to the food groups. Encourage them to estimate the quantities of each food they consumed and compare what they ate with the recommended portions.
Then pose this problem:
- Imagine you ate according to the foods and amounts on the Copymaster. Work out how much of each type of food you will need to buy for a whole year.
Let the students work in small groups to establish the amounts of each food they will need. Allow them to use calculators. Look to see:
- Do your students use multiplication correctly to establish the amounts?
- Do they show a sense of the size of metric measures? (For example, 1 kilogram equals 1000 grams)
After an appropriate time share some of the answers. Focus on a few food groups.
- If you eat two fruits everyday how much do you consume in one year. 2 x 365 = 730 pieces of fruit.
- How many kilograms is that?
You may bring along some fruits and weigh them to work how many of each fruit weigh 1 kilogram. You might also work out the cost. 730 bananas weigh about 730 ÷ 5 = 146 kilograms and cost about 146 x 2.5 = $365. To eat two bananas costs $1.00 per day.
Consuming 100 grams of meat per day totals 365 x 100 = 36 500 grams per year. That is 36.5 kilograms. At a cost of at least $12.00 per kilogram that totals at least 36.5 x 12 = $438 per year. That’s an expensive food for a large family.
Screen Time and Exercise
Experts recommend that students spend no more than two hours per day on screen time (television, gaming and computers) and spend at least one hour on exercise. Gather data from your students about how much screen time and exercise per day they remember having during the previous weekend.
Invite ways to display the data. You might use dot plots to show the frequencies for each amount of exercise.
There may be a negative relationship between screen time and exercise time though using scatterplots to find that relationship is quite sophisticated at this level. It is important that students understand that each dot represents the data from one student.
In this session students build on the work on amount of sleep and exercise to establish how much time they will spend on other activities. The last two sessions have mostly been about things we have to do. We can’t control our heart or breathing much. In this lesson let’s explore things we like to do. Scheduling in experiences that make us feel good rewards us for doing the things that are more like work. What are some of the things you like to do?
Invite ideas from students about their favourite pastimes. Sport and cultural pursuits like dancing are likely to be popular though many students will mention playing on digital devices. ‘Hanging out’ with friends is also likely to be popular but ask students to explain what they do with their friends, e.g. talk, shop, view. Give each student two cards and ask them to write down their two favourite pastimes. Put all the cards in the middle of the floor and collectively sort the data. If all the cards are the same size, then a bar chart can be created as you sort.
Compare the data for your class with that from the 2017 census at school. In that survey students answered this question:
"Which sport or activity do you most enjoy participating in?"
- In what ways do you think the activities students chose for this question will be different to our choices?
Students might note that the question orientates the respondent towards sport rather than other activities. Therefore, the activities suggested by your students are likely to be more diverse. The data video provided with this unit shows how you can access a dataset from CensusatSchool and sort the data using software that is available on the site (INZite). Look at the bar graph in the video.
- How is the data different to the data from your class?
- Is that what you expected? Why?
- What might ‘other’ mean in the bar chart?
You might ask students to gather and analyse their own sample if technology is available. Ask students to write some statements about favourite pastimes of their classmates and of New Zealand students at years 5 and 6.
Next, ask your students to solve this problem:
- How much time will be you spend this year doing you two favourite activities?
- What fraction of all the time you have this year are your favourite activities?
Let students use calculators as the focus is conceptual. Ask students to record their calculations and show clearly what the numbers they use relate to. Observe as your students work:
Do they create an appropriate calculation to work out the time they spend on each activity?
For example, a student who learns a musical instrument might include 1 hour for a lesson and 3 hours practice each week. They might allow for not practising for a few weeks in the holidays. So, their calculation might be 48 x 4 = 192 hours in the year.
Do they correctly calculate the amount of time they have in the year using an appropriate unit and conversions between units?
Since activities usually take hours, using hours as the unit makes sense, but some students may choose minutes for more accuracy. Watch to see if students allow for time being based on sixty, not ten or 100. Some students might remove sleep time as that is not available for activity.
Do they record the amount of time as a fraction and are they open to the possibility of simplifying the fraction?
For example, 365 x 24 = 8760 hours available. Allowing for 9 hours of sleep per day the hours reduce to 365 x 15 = 5 475. This is a small fraction, 3.5%. Students might try to use percentages or decimals to get a sense of the proportion of time they spend on leisure activities.
After a suitable time gather the class and share some findings. Students might notice that people dedicated to an activity, particularly elite sports or arts, spend a lot of time perfecting it. Some sports scientists believe 10 000 hours of practice are needed to hone skills to a professional level and the time may be double that for musicians. Students might also notice that the time spent on enjoyable activities is relatively small compared to other tasks.
How can we work out the amount of time this year that we will spend on school subjects like reading, writing and mathematics?
Model the calculations together using a calculator and being clear about what each number is representing. For example:
Allow for variations that arise from the ideas of students, such as “We read at home as well,” or “Is researching online still reading?” Express the amounts of time as fractions and compare them to those found for enjoyable activities.
In this session students consider where they might travel this year.
Kiwis are renowned travellers. In 2016 New Zealanders made 2.9 million trips overseas.
How many people do we have? (4.5 million)
What is 2.9 out of 4.5 as a fraction?
You might use a linear or decimat model to show the fraction.
Students should note that the fraction is more than one half. In fact, it is very close to two thirds. That means two overseas trips for every three New Zealanders. Of course, some kiwis make many trips per year.
Sometimes kiwis do not appreciate their own country. In the mid-1980’s an advertising campaign was launched to encourage New Zealanders to visit their own country before going overseas. You can search for the video using “Don’t leave home to you see the country.” In their first task students find out how long it will take to drive from their location to given attractions. Provide students with a road map of New Zealand showing the main state highways (these can be found online if hard copies are not available. Students are expected to use their personal knowledge of travel times and distances, combined with simple scales, to estimate times and distances. They will need to work in pairs and have access to string, a ruler and a calculator.
The challenge is as follows:
For North Island based students:
Work out the distance and driving time from your home to each of these tourist spots:
Paihia, Hot Water Beach, Raglan, Waihau Bay, Waitomo, Rotorua, Taupo, New Plymouth, Napier, Wellington
To help you here is a factoid: The distance from Hamilton to Auckland is 125 km and the driving time is about 1 hour 36 minutes. The road is mainly straight.
For South Island based students:
Work out the distance and driving time from your home to each of these tourist spots:
Picton, Nelson, Kaikoura, Hanmer Springs, Franz Joseph Glacier, Lake Tekapo, Queenstown, Moeraki, Bluff.
To help you here is a factoid: The distance from Christchurch to Timaru is 165 km and the driving time is about 2 hour 12 minutes. The road is mainly straight.
Watch students as they work on the tasks.
Do they think proportionally about the distance or time they are finding and the reference distance and time?
For example, if they live in Palmerston North students should notice that the distance by road to Rotorua is about two and one half times that from Hamilton to Auckland. An estimate of 2.5 x 125 = 312.5 km is very accurate. The road is windy in some places, but 2.5 x 96 = 240 minutes or 4 hours is also very accurate.
A distance time calculator is available on the New Zealand Automobile Association site as well as other sites. Students can check their answers.
After a suitable time gather the glass to discuss learning issues. A likely complexity was how to deal with parts (fractions of the reference distance and time). You might illustrate this issue with the distance from Christchurch to Hanmer Springs. Imagine two different pieces of string like this:
When the pieces are straightened and aligned they will look like this:
If the blue string represents 165km, what does the red string represent?
Students might use halving to find that the red string is a bit more than three quarters of the blue string. So, 3 x 41 = 123 km is a good estimate.
For the final part of the lesson ask students to find two different New Zealand destinations they would like to visit this year. For each destination ask them to calculate the distance and car travel time.
In this session students use a bank of problems set at Level Three of the mathematics curriculum to establish some learning goals for the year.
Copymaster Three contains a set of problems for students to solve independently. Do not allow the students to use calculators so you get a window into their mental and written strategies. Encourage them to record as much information as they can about how they solved the problems. They should write on the copy as an achievement record. You might use this same set of problems at the end of the year to demonstrate growth.
After completing the problems provide the students with the answers so they can check their attempts. Ask students to use the problems to identify areas of mathematics they think they are good at and those they need to improve at. They might record mathematics goals for the year in their exercise book.