Early level 3 plan (term 1)

Planning notes
This plan is a starting point for planning a mathematics and statistics teaching programme for a term. The resources listed cover about 50% of your teaching time. Further resources need to be added to meet the specific learning needs of your class, to support your local curriculum and to provide sufficient teaching for a full term.
Resource logo
Level Three
Integrated
Units of Work
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.
Resource logo
Level Three
Number and Algebra
This unit requires students to work with the number of ones, tens, hundreds and thousands in four-digit whole numbers.
  • Recognise the importance of zero as a place holder in whole numbers.
  • Recognise and apply understanding of the base ten system repeated naming pattern of thousands, hundreds, tens and ones.
  • Appreciate that tens are nested in hundreds and thousands and use that knowledge to rename whole numbers...
Resource logo
Level Three
Geometry and Measurement
Units of Work
In this unit we use scale drawings to locate ourselves, and other people and objects, in the classroom and in the local community. We also introduce the compass as an instrument that can tell us in which direction we are facing.
  • Draw and interpret simple scale maps.
  • Use maps or plans to propose actions.
  • Understand the use of a compass to specify and find directions.
Resource logo
Level Three
Number and Algebra
Units of Work
This unit explores situations that involve addition and subtraction of whole numbers. Students are expected to choose among a range of strategies, based on their understanding of place value.
  • Choose appropriately among strategies such as using standard place value, using tidy numbers with compensation, and equal adjustments to solve addition and subtraction problems with whole numbers.
  • Choose correct operations to solve joining, separating, and difference problems with whole numbers...
Resource logo
Level Three
Integrated
Units of Work
This unit introduces students to combinations as a type of situation that multiplicative thinking can be applied to.
  • Use tree diagrams and two-way tables to find all the combinations for a simple pairing situation.
  • Use multiplication to count all the possible combinations (and permutations).
  • Use models of all the possible outcomes, and experimental results, to compare the chances of different outcomes.
Source URL: https://nzmaths.co.nz/user/75803/planning-space/early-level-3-plan-term-1

In this year

Purpose

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.

Specific Learning Outcomes
  • 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.
Description of Mathematics

Rates and ratios both involve the comparison of two numbers. A rate is a comparison of two numbers with different units, whereas a ratio compares two numbers with the same unit.

There are many examples of rates in our everyday lives: speed (kilometres per hour), remunerative pay (dollars paid per hour of work), pulse (heartbeats per minute), or price (cost per unit bought).

The simplest rates are called unit rates and are expressed using one unit of the first measure for every ‘so many’ units of the other measure. For example, one breath every 6 seconds is a unit rate. When working with unit rates we multiply or divide both measures by the same amount to ensure the rate is constant. For example, to find out how long it will be before four breaths elapse multiply both measures by four, 4 x 6 = 24 seconds.

Much of the mathematics in this unit involves the use of rates, particularly rates over time.

Opportunities for Adaptation and Differentiation

The learning opportunities in this unit can be differentiated by varying the scaffolding provided or altering the difficulty of the tasks to make the learning opportunities accessible to a range of learners. For example

  • pairing the students so that they can be supported as they work on tasks
  • working directly with students who need support when working with rates, exploring approaches which involve repeated addition
  • changing the numbers used in the problems so that students are working with whole numbers only.

 

This unit draws on facts about the students and their preferences and experiences. It includes investigating the number of times students blink in a year, how much sleep they get, the food they eat, and their favourite pastimes and travel locations. The activities can be adapted to ensure students’ experiences and interests are included. For example:

  • include foods that you know are freely available within the community and use costs from local shops when calculating
  • include activities that are currently popular with students in when talking about favourite pastimes
  • use locations that are familiar with students when talking about travel within New Zealand. Use a map of the local region and some local places of interest or significance rather than a national map, if this is more appropriate.
Activity

Prior Experience

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.

Session 1

  1. 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. 
     
  2. Explain to students that 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.
     
  3. 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.
     
  4. Show Slide 1 of the PowerPointwhich 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. 
     
  5. Move onto Slide 2 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.
     
  6. Slide 3 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-hand strategy allows for her birth length and her growth spurt from zero to two years. 84cm is the average height for a two-year-old girl. The difference of 56 cm needs to be divided by 8 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.
     

  7. Provide your students with their own copy of Copymaster One. Ask them to make their own estimates first before solving the problem with a partner. Allow access to calculators as the focus is on conceptual thinking about multiplication rather than calculation strategies. 
     

  8. 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?
       

  9. After a suitable period bring the class together to discuss their answers. The most important issue is how to tell if a situation is multiplicative or not. A situation is multiplicative if a rate exists, that is, a constant change in one measure for every constant change in another measure. Heart beats are a good example, a pulse rate of 70 beats per minute is a rate. To estimate the number of heart beats per year this rate must be assumed to be constant, despite knowing that pulse rate increases with exercise and decreases with sleep.

Session 2

In this session students explore healthy living, consider their current habits, and compare those habits to experts’ recommendations.

Ask students: 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?
 

Sleep

  1. 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.

     
  2. 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.
     
  3. 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.


     
  4. 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?
     
  5. 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.
     
  6. 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
     

Food

  1. 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 or 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.

     
  2. 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.
     
  3. 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.
     
  4. 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)
       
  5. After an appropriate time share some of the answers. Focus on a few food groups.
    If you eat two pieces of fruit everyday how much do you consume in one year? 2 x 365 = 730 pieces of fruit.
    How many kilograms is that?
     
  6. You may bring along some fruits and weigh them to work how many of each fruit weigh 1 kilogram, or you might prefer to look up the average number of pieces in a kilogram online.
     
  7. You might also ask students to work out the daily or annual cost of some foods. For example, if 5 bananas weigh 1 kilogram and bananas cost $2.50 per kilogram, then 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 per person. That’s expensive if you have a large family!
     

Screen Time and Exercise

  1. 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.
     
  2. Invite ways to display the data. You might use dot plots to show the frequencies for each amount of exercise.

Session 3

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.

  1. Tell students: 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?
     
  2. 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.
     
  3. 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?
       
  4. 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.
       
  5. 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.
     
  6. Ask the class: How can we work out the amount of time this year that we will spend on school subjects like reading, writing and mathematics?
     
  7. Model the calculations together using a calculator and being clear about what each number is representing. For example:


     
  8. 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.

Session 4

In this session students consider where in New Zealand they might travel this 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 until you've seen the country.” In this task students find out how long it will take to drive from their location to given attractions.

  1. Provide students with a road map of the North or South Island of New Zealand without a scale, showing the main state highways and the tourist spots listed below. You may find it convenient to make these maps by printing images from Google maps onto A3 paper. Students will need to work in pairs and have access to string, a ruler and a calculator.
     
  2. Give students the following challenge:
    • For North Island based students:
      If you know that the distance from Hamilton to Auckland is 125 km and the driving time is about 1 hour 36 minutes, then 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.
      For South Island based students:
    • If you know that the distance from Christchurch to Timaru is 165 km and the driving time is about 2 hours 12 minutes, then 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.
       
  3. Encourage students to cut a piece of string that is the same length as the reference distance (Hamilton to Auckland or Christchurch to Timaru), cut strings that are the same lengths as the distances from home to the tourist spots, then use the relationship between the two lengths to estimate the distances and drive times. For example, estimating the distance and drive time from Christchurch to Hanmer Springs:

    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.
    If the blue string represents 2 hours 12 minutes, 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 30 = 90 minutes is a good estimate.
     
  4. Watch students as they work on the tasks.
    Do they think proportionally about the distance or time they are estimating and the reference distance and time?
     
  5. Once students have estimated the distances and drive times, they should check their answers online.
    ​​​​​
  6. 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 from their home town.

Session Five

In this session students use a bank of problems set at Level 3 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.

Attachments

Building a multiplicative view of whole number place value

Purpose

This unit requires students to work with the number of ones, tens, hundreds and thousands in four-digit whole numbers.

Achievement Objectives
NA3-4: Know how many tenths, tens, hundreds, and thousands are in whole numbers.
Specific Learning Outcomes
  • Recognise the importance of zero as a place holder in whole numbers.
  • Recognise and apply understanding of the base ten system repeated naming pattern of thousands, hundreds, tens and ones.
  • Appreciate that tens are nested in hundreds and thousands and use that knowledge to rename whole numbers.
  • Rename whole numbers to make the addition and subtraction of collections of tens or hundreds easier.
Description of Mathematics

Understanding place value is crucial if students are to develop the estimation and calculation skills necessary to become numerate adults. Our number system is very sophisticated though it may not look it. Numerals are all around us in the environment but the meaning of digits and the quantities they represent are challenging to understand. Our number system is based on groupings of ten. Ten ones form one ten, ten tens form one hundred, ten hundreds form one thousand, etc. The system continues, giving us the capacity represent very large quantities. The place values, one, ten, one hundred, one thousand, etc., are powers of ten. That means that the place immediately to the left of a given place represents units that are tens time more than the given place, e.g. thousands are ten times greater than hundreds.

Ten digits, 0, 1, 2, 3, 4, 5, 6, 7, 8, and 9 are used to represent all the numbers we could ever need. We don’t need a new number to represent ten because we think of it as one group of ten. Similarly, when we add one to 999, we write 1000 and do not need a separate symbol for one thousand. The position of the 1 in 1000 tells us about the value it represents. Zero has two uses in the number system, as the number for ‘none of something’, e.g. 6 + 0 = 6, and as a place holder, e.g. 7040. Place holder means it occupies a place, or several places, so the reader knows the values represented by the other digits. In 7040 zero is acting as a place holder in the hundreds and tens places.

Place value means that both the position of a digit as well as the value of that digit indicate what quantity it represents. In the number 2753 the position of the 7 is in the hundreds column which means that it represents seven hundred. Two is in the thousands column which means that it represents 2 units of one thousand, called 2000.

Understanding the nested nature of place value becomes very important as students learn to operate on whole numbers and extend their knowledge to decimals. Nested means that the places are connected, e.g. within hundreds there are tens, within ones there are tenths. Renaming a number flexibly is an important application of nested place value.

In particular it is vital that students understand that when ten ones are combined they form a unit of ten, when ten tens are combined they form a unit of one hundred, and when ten hundreds are combined they form a unit of one thousand. For example, the answer to 2610 + 4390 is 7 thousands since 610 and 390 combine to form another thousand. Similarly, when a unit of one thousand is ‘decomposed’ into ten hundreds, the number looks different but still represents the same quantity. For example, 4200 can be viewed as 4 thousands, and 2 hundreds, or 3 thousands and 12 hundreds, or 2 thousands and 22 hundreds, etc. Decomposing is used in subtraction problems such as 7200 – 4800 = □ where it is helpful to view 7200 as 6 thousands and 12 hundreds.

At Level 3 students need to develop a multiplicative view of place value that includes understanding the relative size of quantities represented by different numbers. A nested view of 230 as 23 tens allows multiplicative connection between 23 and 230. 230 is ten times larger than 23, and 23 is ten times smaller than 230. Such knowledge can be expressed with equations, 23 x 10 = 230, 10 x 23 = 230, 230 ÷ 10 = 23. Multiplication and division basic facts can be leveraged for harder calculations, 4 x 3 = 12 so 4 x 30 = 120 (ten times more). 30 x 4 = 120 as well. 12 ÷ 3 = 4 so 120 ÷ 30 = 4.

Opportunities for Adaptation and Differentiation

The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to differentiate include:

  • Providing physical materials so that students can anticipate actions and justify their solutions. Place Value People is a representation that is accessible, allows for combining and partitioning units, and can be used with whole numbers up to 4 digits.
  • Connecting symbols with quantities, especially as the quantities are combined (addition), separated (subtraction). Support students to generalise properties that always occur, e.g. ten units of a place value combine to make one unit of next highest place value.
  • Using important mathematical vocabulary to discuss concepts, in particular words for the places (ones, tens, hundreds, etc.) and words associated with operations, e.g. addends and sum for addition.
  • Encouraging students to collaborate in small groups and to share, and justify, their ideas.
  • Using technology, especially calculators, in predictive, pattern-based ways, e.g. What will happen to 234 if you multiply it by ten?..divide it by ten?
  • Altering the complexity of the whole numbers and operations that are used. You might refer to the Level 2 place value units for some students.

The context used for this unit is people which should be engaging for all students. You might want to relate problems to people throughout the world or in regions of Aotearoa to provide more real-life settings. Encourage students to be creative by accepting a variety of strategies, and asking students to create their own problems for each other to solve.

Required Resource Materials
  • Calculators,
  • Place Value People (made from Copymaster One)
  • Scissors
  • Stapler
  • Digit cards or ten-sided dice.
  • Copymasters One, Two, Three, Four, Five, and Six
  • PowerPoints One, Two, and Three
Activity

Session 1

This unit builds on the units at level 2 that explore place value of whole numbers to 1000:

You may want to revisit those units or, at least use some of the independent tasks. It also expected that students will learn a good repertoire of basic multiplication facts either prior to the unit or in the course of it.

Introducing Place Value People as a model

  1. Begin the lesson using PowerPoint One. The first three slides show numbers of people icons. All three collections contain 120 icons. The purpose of the class task is to establish that grouping objects helps us to count the number of objects efficiently and that tens-based grouping is the normal convention (though it was not always historically).
    What is an efficient way to count the number of people?
  2. Slide One: There are five blocks of 4 x 6 = 24 people. Students may approach the task additively, e.g. 6 + 6 = 12, 12 + 12 = 24 to find the number in each block. 24 + 24 = 48, etc. to establish the total. A multiplicative approach might be 4 x 6 = 24, 5 x 20 = 100, 5 x 4 = 12 so there are 120 people.
  3. Slide Two: There are three blocks of four tens. The use of place value units makes it much easier to count the 120 people. Look for students to apply multiplicative structure, such as, 4 x 10 = 40, 3 x 40 = 120. Discuss how 3 x 4 = 12 is used to find 3 x 40 = 120.
  4. Slide Three: The people icons are distributed randomly. Give the students a short time to count the icons since allowing them to continue is non-productive. The point is that counting in ones is time consuming and error prone. The animation creates collections of ten icons that might be used to estimate the total number.
  5. Show the students Copymaster One (place value people). Discuss that the 100 sheet is composed of ten rows of ten people or four groupings of 5 x 5 = 25 people. Slide Four of PowerPoint One illustrates how the number 365 can be made efficiently using the Copymaster. Slide Five poses some interesting questions about 365.
    • Question One: Within each 100 are ten tens. 300 is comprised of 30 tens. Since 60 is six tens there are 36 tens altogether. Some students may say that the other five is half of a ten or point five. That is correct. The answers of 36 tens and 36.5 tens are correct.
    • Question Two: In each ten there are two fives. An efficient way is to double the number of tens to get the number of fives, i.e. 2 x 36 = 72 (add the final five to get 73) or 2 x 36.5 = 73.
    • Question Three: To make 365 into 400, five are needed to make 370 and another 3 tens to make 400. 400 plus 100 equals 500. In total 100 + 30 + 5 needs to be added.

What number am I?

Copymaster Two contains “What number am I?” challenges for the students to solve. The clues involve place value understanding and students are expected to use the Place Value People model to solve the problems if they need to. Look to see if your students:

  • Use a systematic approach to eliminating possible numbers as they work through the clues
  • Show a sense of quantity by making numbers with the Place Value People model
  • Demonstrate nested place value knowledge, e.g. 74 ten equal 740.

Wish-upon-a-digit

A digital form of this game is called Wishball. It can be found at this link.

Students will need copies of Copymaster One (Place Value People) and Copymaster Three (Scoresheets) to play. They will also need a way to randomly generate digits. This could be digit cards to draw from, a 10-sided dice, or an online random number generator. 

The goal of the game is to use place value number operations to get from the starting number to the target number in as few turns as possible.

To set up the game:

  1. Select three digits randomly to form a starting number and mark this on the number line at the top of their recording sheet.
  2. Select three digits randomly to form a target number and mark this on the number line as well.

For each turn:

  1. Select a digit randomly.
  2. Choose whether the digit represents ones, tens or hundreds (for example, a 3 could be worth 3, 30 or 300)
  3. Choose whether to add or subtract this number from the current running total and record this on the recording sheet.

When it will finish the game you have one opportunity to choose the digit you want (for example, if you were up to 467 with a target of 667, you can choose to have a 2, and add 200 to finish the game).

Session 2

In the following sessions students are expected to apply multiplication and division to place value. They learn how many tens and hundreds are nested in whole numbers to four digits, and effect of multiplying or dividing a whole number by ten.

Lucy’s Number Trick

  1. Play the first four slides of PowerPoint Two to introduce Lucy’s look for a pattern in the 11 x tables with two-digit whole numbers. Let students talk in pairs as each answer comes up.
    Is there a pattern?
    What is the pattern?
  2. While showing slides 5-7, invite your students to predict the answer to other 11 x facts. Prediction encourages them to conjecture on possible rules. Access to calculators will support students to check their predictions. After slide 7, discuss how the answer might be predicted. In general, for any two-digit number, 10a + b (written as ab), the answer has a ones digit of b. The tens digit is a + b and the hundreds digit is a. If a + b is less than ten the pattern is easier to spot. For example, 11 x 45 has an answer of 5 ones, 4 + 5 = 9 tens, and 4 hundreds (495).
  3. If a + b is greater than ten then some renaming occurs. Take 11 x 57 for example. The answer has 7 ones, 5 + 7 = 12 tens, and 5 hundreds. Since 12 tens are 120 another hundred is created. Therefore, the answer becomes 6 hundreds, 2 tens, and 7 ones (627).
  4. Noticing the pattern is important. The next step is to structure the pattern, that is, find out why it behaves the way it does. Challenge the students to use Place Value People (Copymaster One) to model some examples of the 11 x pattern. Slides 8 and 9 of PowerPoint Two illustrate 11 x 27. You might use the slides initially or allow students the opportunity to represent calculations themselves first. Essentially the pattern works because of the distributive property. Multiplication by eleven is the sum of multiplying by ten and one.
    Symbolically this property can be written as 11 x 27 = (10 x 27) + (1 x 27) = 270 + 27. The vertical algorithm on Slide Nine better illustrates how the digits in the second factor contribute hundreds, tens and ones to the product.
  5. What happens if the two digits in the second factor add to a number greater than nine?
  6. Students might recognise that if the sum of a + b > 9 (for any number 10a + b) there is more renaming required as tens are renamed to form hundreds. Slide Ten shows the calculation of 11 x 79 which is a good example.
  7. Challenge the students to practise their skills in calculating 11 x any 2-digit whole number. You might have a quiz to see how quickly they can use the digit pattern to find products. You might express the method as an algorithm (step by step procedure), such as:
  8. If time permits set related extension problems:
    1. Develop an algorithm for multiplying a three-digit whole number by 11.
    2. Develop an algorithm for multiplying a two-digit number by 101.
    3. Develop an algorithm for multiplying a two-digit whole number by 111.
    The solutions are:
    1. For any three-digit number, 100a + 10b + c (written as abc) the product when it is multiplied by 11 is a thousands, a + b hundreds, b + c tens, and c ones.
      For example, 345 x 11 = 3 795. Note that renaming is needed when a + b and/or b + c equal ten or more.
    2. For any two digit number, 10a + b (written as ab) the product is a thousands, b hundreds, a tens and b ones. No renaming will be needed.
      For example, 68 x 101 = 6 868.
    3. For any two-digit number, 10a + b (written as ab) the product when it is multiplied by 111 is a thousands, a + b hundreds, a + b tens, and b ones. Note the renaming needed if a + b > 9. For example 35 x 111 = 3 885.

Session 3

In this session students investigate how many tens are in a whole number to four places.

  1. Remind students about their learning from Session Two.
    What is the product of 10 x 26? How do you know?
    What is the product of 10 x 49? How do you know?
    What is the product of 100 x 83? How do you know?
  2. If necessary, use Place Value People to model the 10 x calculations (ten sets of the other factor). Discuss why modelling 100 x 83 might not be efficient and that thinking about pattern is more helpful.
    What is 83 composed (made up) of? (8 tens and 3 ones)
    What is 100 x 3? Why is the product the same as 3 x 100? (Commutative property)
    What is 100 x 8? So, what is the product of 100 x 80? Why? (Do students recognise the effect on the digits in the product of multiplying by ten?)
  3. Ask students to model the number 245 with Place Value People (Copymaster One).
    You need to make teams of ten for sports day. How many teams of ten can you make with 245 people?
  4. Ask students to anticipate the result of making tens then justify why they believe 24 tens exist in 245. Their explanations should include the fact that ten tens are nested in each hundred.
    How might we record what we have found in an equation?
  5. Let students offer possibilities like 24 × 10 = 250 or 240 ÷ 10 = 24. Discuss the meaning of the symbols, especially the division sign. In this case the process is measurement, i.e. How many tens make 240?
  6. Ask: What might we do with the extra five people?
    Some students might suggest that five is half of ten or that there are 24.5 tens in 245.
  7. Ask: How many tens can be made with …609 people? 444 people? 930 people?
  8. Do the students justify their answers by referring to the tens nested within the hundreds? Make 1 000 with Place Value People by attaching ten one hundred sheets together with a single staple.
    How many people are in this stack? (1000 since there are ten hundreds)
  9. Make the number 1 728 using Place Value people. Work out how teams of ten people can be made with that number.
  10. Look to see that students correctly model 1 728 with one thousand stack, seven hundreds sheets, two ten strips and eight singles.
    Do they unpack 100 tens in 100, 70 tens in 700, and 2 tens in 20?
    Do they name and justify that there are 172 tens in 1 728 (or 172.8)?
  11. Ask: How many teams of one hundred people could be made with 1 728 people?
  12. Look to see if your students recognise that ten hundreds exist in 1000, and another 7 hundreds exist in 700, making 17 hundreds altogether. If students give an answer of 17.28, recognise that this a very sophisticated answer since 0.28 means 28 hundredths. Record the operations of finding out how many tens and hundreds are in 1 728 as equations; 1720 ÷ 10 = 172 and 1700 ÷ 100 = 17.
    What patterns do you see in dividing by ten and one hundred?
  13. Provide your students with Copymaster Four that contains a set of related problems to work from. Look for students to develop fluency with renaming whole numbers.
  14. Mark the Copymaster together and discuss the strategies students used to solve the problems.

Session 4

In this session students put the concept of nested place value to work by solving problems. Most of the problems involve addition and subtraction, but a multiplicative view of place value is essential to the development of fluent strategies.

  1. Introduce Slides 1 and 2 of PowerPoint Three. You could use a large set of Place Value People to model the operations of 63 + 8 = 71 and 630 + 80 = 71 or simply play the animations and discuss what happens. The key idea is that students recognise the connection in both operations, that is 630 + 80 is the addition of 63 + 8 in units of ten.
    What is the sum of 6 300 and 800? How do you know? (63 + 8 = 71 in units of 100, so 7 100)
  2. Give your students other examples to calculate, with or without the support of Place Value People, depending on their fluency. Good problems are:450 + 70 =                  
    880 + 60 =                  
    390 + 90 =                  
    940 + 80 =  
  3. Slides Three and Four of PowerPoint Three deal with the same principle applied to subtraction. The examples of 82 – 7 = 75 and 820 -70 = 750 illustrate how a subtraction result with ones can be used to work out a result with units of ten. The principle works with units of any equal size, e.g. 8200 – 700 = 7 500 (units of 100).
  4. Give your students other examples to calculate. Check to see that they can demonstrate what is occurring with the quantities in the calculations, using Place Value People. Good examples might be:
    430 – 60 =                  
    910 – 80 =                  
    540 – 90 =                  
    820 – 70 =      
  5. Provide your students with Copymaster Five to work from in pairs. This worksheet consolidates and extends the concept of renaming a number to make calculations easier. Note that the principle is extended to four-digit whole numbers. Some students may need support, including modelling with Place Value People (Copymaster One).    

Session 5

  1. To consolidate the concept of nested place value, introduce and play the game Cover Cathy Crocodile 4 digits. The game can be made using Copymaster Six. Easier forms of the same game are available in the level two place value units (Cover Cathy Crocodile: 3 digits and  Cover Cathy Crocodile: 2 digits)
  2. The purpose of the game is to develop students’ fluency in renaming whole numbers up to four digits. To assess students’ ability to rename pose this question:
    Using thousands, hundreds, tens and ones, make up at least five names for 4768.
    For example, the first name might be 4 thousands, 7 hundreds, 6 tens, and 8 ones. The second name might start with 47 hundreds…
    Which name would be most useful to solve the problem 4768 + 900 =  ?

Location, Location

Purpose

In this unit we use scale drawings to locate ourselves, and other people and objects, in the classroom and in the local community. We also introduce the compass as an instrument that can tell us in which direction we are facing.

Achievement Objectives
GM3-5: Use a co-ordinate system or the language of direction and distance to specify locations and describe paths.
Specific Learning Outcomes
  • Draw and interpret simple scale maps.
  • Use maps or plans to propose actions.
  • Understand the use of a compass to specify and find directions.
Description of Mathematics

Scale drawings are used extensively in real life. Plans of buildings are used by builders both to make, and to remodel, buildings. Plans are also used by workmen who need to find appropriate parts of buildings if something has gone wrong. Maps are used by car drivers to find their way around town and between towns. Maps are used by pilots and yachties to enable them to navigate too. Plans and maps are used on a daily basis. Knowing how to use them is one of the necessary skills of life.

The underlying mathematics is quite simple. It is just a matter of using ratios. The scale of the plan or map is the conversion factor that changes the size in real life to the size on paper. Although the lengths change from life to its representation, the shape and angles of real life objects don’t change. This makes it easy for us to follow what the plan is representing.

Things that don’t change are called invariants. Invariants are important objects of study in their own right. Much of mathematics is involved in looking at invariant properties of objects. For instance, the sum of the angles of a triangle is invariant under scaling, as is the shape of a given figure. A square always has four equal sides and four angles that are right angles, no matter how large or how small the square is. In higher mathematics, things called matrices affect direction. An important invariant here for individual matrices is the directions that remain fixed. These have useful ramifications. So dealing with scale drawings in the shape of plans and maps is a first step at looking into a fundamental idea that is to be found in most of mathematics.

Opportunities for Adaptation and Differentiation

The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to support students include:

  • providing students with a pre-drawn outline of the shape of the classroom
  • providing scale examples of furniture for students to trace around
  • encouraging students to work in pairs or small groups.

The context of mapping the classroom should be an engaging one, as all students will be familiar with it. You may wish to allow students to make maps of other buildings or rooms, such as:

  • their own bedroom/house
  • a local sports club, marae, church hall or playground.
Required Resource Materials
  • Street map of local area
  • Compasses (for bearings)
  • Large sheets of paper
  • Metre rulers
  • Connected 3, 1999, Mechanical Harry makes a map.
Activity

Getting Started

We begin the week by working in pairs to draw a scale plan of the classroom and the objects in it. To do this accurately we first need to discuss the use of scale.

  1. Show the students an outline shape of their classroom drawn to a scale of 10cm to 1m. (A sheet of manila paper will be about the right size for classrooms that are approximately 8 metres by 6.5 metres).
    outline of classroom

    What can you tell me about this shape?
    What could it represent?

  2. If I told you that this was a floor plan of a room that is 10 times larger than this piece of paper, what do you think it could be a plan of? (If no one guesses correctly add windows to the plan).
    classroom with windows
  3. Use the scale plan to work out the dimensions of the classroom.
    How wide is our classroom?
    How long?

    Check this by measuring the actual classroom.
  4. As a class add the classroom door(s) to the plan.
    Where will we put the door(s)?
    How large will this be on the plan?
    What part of the door do we need to measure?
    (width only)
  5. Have a student measure the door.
    If the door is 80cm wide (which is approximately the width of most doors) how large will it be on the plan?
  6. Add the doors to the plan by drawing the semi-circular opening space.
    classroom with doors
  7. Have pairs of students prepare a floor plan of the classroom with windows and opening doors. This plan will be used in the following days as the students draw a detailed floor plan of the class as it is now and how they would like it to be in future.

Exploring

Over the next 2 to 3 days the students prepare detailed scale plans of the classroom to include desks and other major features.

  1. Ask and discuss the students' ideas about how they will complete the task. These ideas may include:
    Making a scale model of a desk and then tracing around it for the desks in the room.
    One student measuring the object and the other making the scale model.
    Working with other pairs to share information.
  2. As the students work, ask questions that focus on their understanding and use of scale.
    How did you work out the size of the desk on your plan?
    What remains the same when you make a scale model?
    (shape, angle)
    What changes? (length of sides, area)
    How much smaller is it?
    (Where appropriate, ask the students to quantify the change in length and area, i.e, length of scale model = one-tenth of real object, area of scale = one-hundredth of real object.)
  3. Share and compare completed scale floor plans.
    Are our plans the same?
    Why are there differences?
    What did you find difficult about this task?
    What did you learn while completing it?
    Why do you think that scale plans are useful?
    (for building, for designing and redesigning, for planning, etc.)

Next we give the students an opportunity to redesign the classroom floor plan with the aim of selecting one plan as the basis for a real reorganisation.

  1. Discuss:
    What do you think is good about the way our classroom is arranged?
    What would you like to change?
  2. I would like to rearrange the classroom and I would like your ideas. We are going to work in pairs to create a new floor plan. We will then select one and rearrange our classroom on Friday.
    What do we need to remember when we do this?
  3. Brainstorm and list considerations. The classroom exits must remain clear; there must be room to move between the desks, etc.
  4. Discuss use of cardboard scale models of the furniture so that they can be rearranged in the process of deciding on a final arrangement of the furniture.
  5. Display completed floor plans.
  6. Discuss the plans, getting the students to ask questions of the "architects" of the plans. For example:
    Why have you put the teacher’s desk there?
    Is there room to walk to the book corner if you are seated at X?
    How would we get out quickly if the fire alarm goes?
    Can everyone see the whiteboard?
  7. Number the plans and get students to vote for the plan that they would like.
  8. Rearrange the room according to the best plan.

Reflecting

In the final part of our Location, Location unit, we consider a map of the local region and use a compass to put a bearing on our classroom plan.

  1. Display and discuss a map of the local region.
    Can you locate our school?
    Your home?
    The local shop?
    The park?
    The swimming pool?
    What do the symbols on the map mean?
  2. Hopefully one of the symbols is the compass bearing for North. If not display a map that does have a compass bearing. Work with the class to add a compass bearing to the classroom floor plan.

Addition and subtraction with whole numbers

Purpose

This unit explores situations that involve addition and subtraction of whole numbers. Students are expected to choose among a range of strategies, based on their understanding of place value.

Achievement Objectives
NA3-1: Use a range of additive and simple multiplicative strategies with whole numbers, fractions, decimals, and percentages.
NA3-4: Know how many tenths, tens, hundreds, and thousands are in whole numbers.
Specific Learning Outcomes
  • Choose appropriately among strategies such as using standard place value, using tidy numbers with compensation, and equal adjustments to solve addition and subtraction problems with whole numbers.
  • Choose correct operations to solve joining, separating, and difference problems with whole numbers.
  • Record solutions using equations, algorithms, and calculator inputs.
Description of Mathematics

There are three types of situation to which addition and subtraction are applied.

  1. Joining parts to form a whole
    Example: You have 35 toy cars and are given 47 more. How many cars do you have?
    Equation: 35 + 47 = 82
  2. Separating parts from a whole
    Example: You have 82 toy cars and give 47 away. How many cars do you have?
    Equation: 82 - 47 = 35
  3. Comparing parts or comparing whole.
    Example: You have 82 toy cars, and your friend has 35 toy cars. How many more cars do you have than her?
    Equation: 35 + □ = 82 or 82 – 35 = □.

The role of the unknown changes the difficulty of the problem and the operation needed to solve it.

  1. Joining change unknown
    Example: You have 35 toy cars and get some more cars. Now you have 82 cars. How many cars did you get?
    Equation: 35 + □ = 82
  2. Separating start unknown
    Example: You have some toy cars and give 47 cars to your friend. Now you have 35 cars. How many cars did you have to start with?
    Equation: □ - 47 = 82
  3. Comparing start unknown
    Example: You have 82 toy cars.  That is 47 more cars than your friend. How many cars does your friend have?
    Equation: □ + 47 = 82

Notice that the same equation can represent a range of different situations.

Opportunities for Adaptation and Differentiation

The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to differentiate include:

  • Provide place value materials so that students can anticipate actions and justify their solutions. Use materials like place value blocks, bundled iceblock sticks, and beaNZ in canisters that are proportional, e.g. ten ones comprise the tens unit. Physical representation diminishes the degree of abstraction required and helps students see what is occurring with the quantities that numbers represent.
  • Connect symbols and mathematical vocabulary, especially the symbols for addition and subtraction (+, -) and for equality (=). Explicitly model the correct use of equations and algorithms and discuss the meaning of the symbols in context.
  • Alter the complexity of the numbers that are used. More places, e.g. three digits or two digits, tends to increase complexity though much depends on the renaming that is required. Addition tends to be easier than subtraction though that depends on the nature of the numbers.
  • Encourage students to collaborate in small groups and to share, and justify, their ideas.
  • Use technology, especially calculators, in predictive, pattern-based ways to estimate sums and differences, e.g. Is the answer to 83 – 47 closer to 20, 30, 40, or 50? How do you know? Allow use of calculators where you want students to focus more on the process of getting a reasonable answer than on practising calculation skills.

The contexts used for this unit include occupations, finance, and collections of objects. Adjust the contexts to everyday situations that your students are likely to be interested in, and relfect their cultural identities. Addition and subtraction problems can easily be framed in term of objects that are meaningful to your students, such as natural objects (e.g. pine cones), people in their kura or whānau, or points earned in games or other competitions. Encourage students to be creative by accepting a variety of strategies from others and asking students to create their own problems for others to solve, in contexts that are meaningful.

Required Resource Materials

 

Activity

Session One

Begin each session in this unit with an addition/subtraction grid. This session gives you detail about how to introduce the activity. Grids for the other sessions are provided in PowerPoint One. This activity and discussion of how students found the answers should take no more than ten minutes.

  1. Show Slide One of PowerPoint One.
    grid
    Discuss how to complete the grid. The answers to the additions go in the body of the table. For example, 7 + 6 = 13 so 13 can be placed in the second cell in the third row.
    Some of the addends (numbers to add) are missing.
    How will you workout what numbers they are?
    Students might recognise that 2 goes in the bottom of the left-hand column because 7 + 2 = 9.
  2. Let students complete the grid. Some students enjoy the challenge of bettering their time from day to day, but others may not. Students can record their time each day if that motivates them.
    Use the animation of Slide One (mouse clicks) to show the answers and how those answers can be determined from the information that is given.
  3. This lesson is taken from Book 5 of the Numeracy Books: Teaching Place Value, Addition and Subtraction. Go to Key Idea 3 on pages 56-58 and select problems that are appropriate for your students. Make sure that you have physical materials that represent the place value structure of whole numbers, such as Place Value Blocks, beaNZ in canisters, iceblock sticks in bundles, or Place Value People.  Money is not a proportional representation, and one cent coins are no longer used. Use problem stories other than those involving money, if possible.
    Represent each calculation using both the physical materials and empty number lines. For example, 56 + 37 = 93 can be represented as:
    image            image
    56 equals 5 tens, and 6 ones                     Adding 30 gives 86, 8 tens, and 6 ones
    image           image
    Adding 7 gives 8 tens, and 13 ones           10 ones make 1 ten so the answer equals 93
    numberline

Session Two

  1. Use Slide Two of PowerPoint One to give students an addition grid to solve. The animation provides the logical sequence to find the addends and sums (Use mouse clicks).
  2. This lesson is taken from Book 5 of the Numeracy Books: Teaching Place Value, Addition and Subtraction. Go to Key Idea 4 on pages 58-60 and select problems that are appropriate for your students. The key idea is about using tidy numbers with compensation to solve addition and subtraction problems. Use proportional place value materials rather than money. Change the contexts of the problems to be about items that are of interest to your students.
  3. Pay particular attention to the inverse compensation needed for subtraction.
    For example, consider 72 – 38 =  (in an appropriate story context).
    With place value blocks the problem might be represented as:
    image          image
    72 equals 7 tens, and 2 ones                    Subtract 40 (4 tens) leaves 3 tens, and 2 ones (32)
    image
    Compensating for removing 40 instead of 38 gives 3 tens and 4 ones (34)
    The same operation on an empty number line looks like this:
    number line
  4. Pose problems with three-digit whole numbers as well, such as:
    There are 725 trout in Lake Kahurangi at the start of the fishing season.
    297 of the trout are caught.
    How many trout are left?
  5. Let students practise from Copymaster One.

Session Three

  1. Use Slide Three of PowerPoint One to give students an addition grid to solve. The animation provides the logical sequence to find the addends and sums (Use mouse clicks).
  2. This lesson is taken from Book 5 of the Numeracy Books: Teaching Place Value, Addition and Subtraction. Go to Key Idea 5 on pages 61 and select problems that are appropriate for your students. The key idea is about using the inverse relationship between addition and subtraction, particularly solving subtraction problems by adding on. Use proportional place value materials rather than money. Change the contexts of the problems to be about items that are of interest to your students.
  3. It is important that students understand why it is possible to solve a problem like 95 – 68 = □ by solving 68 + = 95, as well as developing the procedural fluency in doing so.
    image               image
    95 equals 9 tens and 5 ones                        If 68 was subtracted there would be ? left
    image
    That means 68 + ? = 95
    In empty number line form the problem is represented like this:
    image

    number line
  4. Let students practise from Copymaster Two.

Session Four

  1. Use Slide Four of PowerPoint One to give students an addition grid to solve. The animation provides the logical sequence to find the addends and sums (Use mouse clicks).
  2. This lesson is taken from Book 5 of the Numeracy Books: Teaching Place Value, Addition and Subtraction. Go to Key Idea 7 on pages 63 and select problems that are appropriate for your students. The key idea is about students making sensible strategic choices about how to solve addition and subtraction problems. Encourage students to use mental and written methods. Only use proportional place value materials when students need additional support. Change the contexts of the problems to be about items that are of interest to your students.
  3. Discuss the criteria for deciding which method/s are the best for certain calculations. Encourage students to make up problems that each strategy (place value, tidy numbers, inverse thinking) would be useful for.They might conclude:
    • Place value works on any numbers but is more difficult where renaming is required.
    • Tidy number strategies work best when the one or both numbers are close to a tidy number (usually a decade or century).
    • Inverse thinking strategies work on any numbers but are especially useful if renaming is needed for subtraction.
  4. Equal adjustment is another strategy given in Key Idea 6 on Page 62 of Book 5 of the Numeracy Books: Teaching Place Value, Addition and Subtraction. The addition form of equal adjustment is conceptually easy, but the subtraction is not. Choose whether, or not, to offer those strategies to your students but do not invest too much time in doing so. Here are examples:
    • Addition 48 + 34 =  
      image       
                            image
      48 add 34                                     has the same sum as 40 + 32 = 72
      number line
      number line
    • Subtraction 81 -58 =
      image          image
      The difference between 81 and 58 is the same as the difference between 83 and 60.
      83-60 = 23.
      number line

      number line
  5. To practice applying addition and subtraction strategies ask students to work on these Figure It Out activities:

Session Five

  1. Use Slide Five of PowerPoint One to give students an addition grid to solve. The animation provides the logical sequence to find the addends and sums (Use mouse clicks).
  2. This lesson is taken from Book 5 of the Numeracy Books: Teaching Place Value, Addition and Subtraction. Go to Key Idea 8 on pages 64 and select problems that are appropriate for your students. The key idea involves use of standard written forms for addition and subtraction. Students should also use calculators as another way to find answers to addition and subtraction problems. Choose problems from the given examples but use the simplest written form for addition rather than the expanded form given in the lesson.
  3. Use proportional place value materials when developing the written forms so students understand what is occurring with the quantities. Alongside learning for understanding give students sufficient practice to develop fluency. Here are examples:
    • Addition
      image
    • Subtraction (using decomposition)
      image
  4. Let students practise from Copymaster Three.

Combos

Purpose

This unit introduces students to combinations as a type of situation that multiplicative thinking can be applied to.

Achievement Objectives
NA3-1: Use a range of additive and simple multiplicative strategies with whole numbers, fractions, decimals, and percentages.
NA3-6: Record and interpret additive and simple multiplicative strategies, using words, diagrams, and symbols, with an understanding of equality.
S3-3: Investigate simple situations that involve elements of chance by comparing experimental results with expectations from models of all the outcomes, acknowledging that samples vary.
Specific Learning Outcomes
  • Use tree diagrams and two-way tables to find all the combinations for a simple pairing situation.
  • Use multiplication to count all the possible combinations (and permutations).
  • Use models of all the possible outcomes, and experimental results, to compare the chances of different outcomes.
Description of Mathematics

The Cartesian product between two sets is the set of all possible ordered pairs, with the first element from the first set, and second element from the second set. For example, in this unit students consider the number of different outfits that Lucy can make from a set of 3 pairs of shorts matched with a set of 2 t-shirts:

Each arrow represents a possible pairing of shorts and t-shirts, with 6 different combinations possible. The number of combinations possible can also be found by multiplying the number of pairs of shorts by the number of t-shirts: 3 shorts x 2 t-shirts = 6 different combinations.

Cartesian products can also be represented as arrays. Arrays are arrangements of objects in rows and columns, and are fundamental to the measurement of area and volume. Two-way tables are also very important in statistics, logic and algebra. Each pairing above can be represented as a cell in this two-way table:

As with the arrow diagram the number of shorts and t-shirts make up the dimensions of the array, 3 x 2 = 6.

Finding all the combinations is part of a theoretical approach to probability. Experimenting is also very important. The results of an experiment are seldom an exact match to what might be expected from the theoretical model. Experimental results from selecting one t-shirt and one pair of shorts randomly are likely to vary considerably from the predicted frequencies, especially in the short term.

Observations of students during this unit can be used to inform judgments in relation to the Learning Progression Frameworks. Click for tables of guidelines.

Opportunities for Adaptation and Differentiation

The learning opportunities in this unit can be differentiated by varying the scaffolding provided or simplifying the learning tasks to make the learning opportunities accessible to a range of learners. Ways to support students include:

  • providing materials which support students to find possible combinations. This helps students develop an understanding of the situation before they think about how to identify all the possible outcomes systematically.
  • explicit modelling of drawing tree diagrams and tables, and of writing systematic lists of outcomes. Possible combinations shown on the diagrams can be linked to specific combinations found with materials.
  • simplifying the situations that students are asked to work with by reducing the number of items in the sets that are being combined. For example, removing a pair of shorts or a t-shirt from the items in session one, or reducing the number of choices available at the sandwich bar in session three.
  • allowing flexibility in the way students find all the outcomes.
  • using collaborative grouping so students can learn from each other.

The context for this unit can be adapted to suit the interests and experiences of your students. While the focus is on identifying combinations, the tasks can be made more engaging by providing sets of items that connect to the everyday experiences of your students. For example:

  • using clothing items in that are familiar or engaging, possibly t-shirts from favourite sports teams (sessions one and two).
  • using food options for that are familiar and appealing (session three).
  • using locations that are familiar to the students. This could include reducing the scale of the activity to look at travel around your local area, or within the school grounds (session five).
Required Resource Materials
Activity

Previous Experience

It is expected that students have an understanding of multiplication, and can solve problems involving equal sets. They should also have some experience in ordering everyday events by likelihood and using words such as “likely”, and “unlikely”, to describe chance

Session One 

In this session students learn about systematically organising pairings of objects, to find all the possible combinations. They consider efficient ways to count the number of combinations.

  1. Play PowerPoint One, slides 1, 2, and 3 about Lucy going on holiday. The scenario is that Lucy wants to wear a different outfit everyday of her 24 day summer holiday. She has packed 5 different t-shirts and 5 different pairs of shorts. Surely that should be enough!
     
  2. Ask the students to explore the different outfits Lucy could make in small groups. They could make Lucy’s coloured t-shirts and shorts using Copymaster 1 and mix and mix and match these to find different combinations, or they could colour the copymaster to record the combinations that they find. Alternatively, they could use the coloured t-shirts and shorts on Copymaster 2.
     
  1. After a suitable period of investigation bring the class together to discuss their strategies.
    Look for students to show a system for finding combinations, like:
    “I started with the pink shorts and matched the different tees one-by-one to it. There are five combinations with that pair of shorts…”
    Show students how a strategy like this one can be shown as a tree-diagram (see slide 4 of PowerPoint One).


    What ways can the students find to efficiently count the number of combinations?
    They may count in fives, for each grouping in the tree diagram or use repeated addition. Some students may know 5 x 5 = 25. Record each method using symbols, e.g. 5, 10, 15. 20, 25; 5 + 5 + 5 + 5 + 5 = 25; 5 x 5 = 25.
    Show students a two-way table as another systematic way to find all combinations. In the case of pairings this is a clear representation.

    So Lucy is able to create 5 x 5 = 25 different outfits for her 24 day holiday.
  2. Pose other scenarios to see if students can work with the tree diagram and two-way table representations.
    For example:
    If Lucy ripped one of her pairs of shorts and could not wear them, how many outfits could she make? (4 x 5 = 20) Relate this to losing one arm of the tree diagram or a row of the table.
    So Lucy goes shopping and buys two new pairs of shorts. How many outfits can she make now? (6 x 5 = 30) Relate this to adding two arms to the tree diagram or two rows to the table.
    Lucy buys a red t-shirt on sale. How many outfits can she make?
    Combinations can be made more complicated by adding options for shorts and t-shirts or simplified by reducing the options available.

Session Two

In this session students explore the relationship between probability and the set of all possible outcomes. Chance is involved if the selection of shorts and t-shirts is random, that is, if each item has an equal chance of being selected.

Part One

  1. Show PowerPoint Two about Lucy’s untidy chest of drawers. Since most of her clothes are in the wash she only has two pairs of shorts and two t-shirts to choose from. Point out that the clothes are messed up inside the drawer so Lucy could get any pair of shorts and any t-shirt.
  2. Ask: What are the chances of her getting both items of the same colour?
  3. Students discuss the chances of a same colour outfit occurring, in small groups justifying their ideas as they work.
  4. Bring the class together to share ideas.
    Do students recognise that only four (2 x 2) combinations are possible?
    Can they list all the possibilities?
    Can they organise the possibilities in a diagram, e.g. tree diagram or two-way table? 

Part Two

  1. Lucy washes a green pair of shorts and a green t-shirt and adds those items to the drawers. Now she has three different pairs of shorts and three different t-shirts.
  2. Ask: Have her chances of getting a same colour outfit improved now?
  3. Ask the students to act out Lucy taking two items to wear. They will need two opaque containers (e.g. plastic icecream containers) and paper copies of the shorts and t-shirts from yesterday. Without looking they reach into the containers one at a time and remove an item. Tell them to record what happened and place the items back in the containers for the next turn.
  4. Ask each group to carry out nine trials.
  5. Bring the class together to discuss the findings.
    Are your results what you expected? Explain
  6. Collate the results of all groups.
  7. What do you notice? Students should see that the sample results vary.
  8. Put the ‘Same Colour’ frequencies in a dot plot on the whiteboard, like this:

    Do students notice that three is the centre of the frequencies? Do they realise that 3 out of 9 equals 1 out of 3, or one-third?
  9. Discuss the number of possible outcomes that can occur with three t-shirts and three pairs of shorts. Add to the two way table or tree diagram from Part One. There are nine possible outcomes.
  10. Ask: Why would three out of nine be the most common? There is a one third chance of getting a colour match.

    Only three of the nine outcomes are colour matches (marked with x).
  11. If time permits pose other scenarios:
    • How can Lucy improve her chances of a colour match?
    • Could she ever get a 100% (certain) chance of a colour match? How?
    • Could she ever have no chance, zero, of a colour match? How?
    • If she had all five t-shirts and all five shorts available, what would be her chances of getting a colour match?

Session Three

In this session students consider combinations in a different context, Build Your Own Sandwich. They move beyond pairings to consider what happens with three or four event combinations.

  1. PowerPoint Three introduces the scenario. The Daily Bread Sandwich bar has an option where you can choose two fillings to go in your sandwich, one from the main selection of meats and one from the extras selection. The options are:
    Mains: Ham, Chicken, Beef
    Extras: Cheese, Lettuce, Tomato, Avocado, Peppers 
  2. Without any discussion, ask the students to work in groups to answer Lucy’s question:
    How many different sandwiches can be made with these choices?
  3. Copymaster 3 contains images of the fillings that students might use to find all the sandwiches that are possible. Note that extra fillings are included in the Copymaster for extension. Look for:
    • Do the students use a systematic strategy to find all the possibilities?, e.g. organised list, tree diagram, two-way table.
    • Do the students use multiplication to count the number of possible sandwiches or do they rely on one by one counting, skip counting or addition?
    • Do they consider the impact of changes to the fillings? e.g. Adding another extras topping would increase the number of combinations to 3 x 6 = 18. After a suitable period of investigation gather the class to discuss the points above.
  4. Useful extension questions include:
    • If Daily Bread wanted more sandwich combinations are they better to add another choice of main or another choice of extras to their menu? (One new main adds 5 new combinations because it can be combined with 5 extras, one new extra adds 3 new combinations because it can be combined with 3 mains).
    • Suppose Daily Bread wanted to advertise that you can make 28 different sandwiches. What could they add to their fillings so that is possible? Note that the extra fillings on Copymaster Three make it possible to model the solution to this problem.
    • If you were allowed two extras with the main filling instead of one, would that increase the number of possible sandwiches? By how much?
      This is a complex problem since duplications need to be avoided, e.g. ham with lettuce and tomato is the same sandwich as ham with tomato and lettuce. The number of possible combinations doubles to 30. 

Session Four

In this session students explore a calculator number game called Odds and Evens. The students consider what combinations are possible in the game and, therefore, what is each player’s chance of winning. They then consider how the game is different to a similar game where numbers are not replaced.

  1. Tell the students that Lucy has a couple of favourite holiday games for times when the weather turns bad, and one of these is the calculator game. Show students how to play the calculator game. You may like to use Video 1 and get two students to demonstrate the game in front of the class.  

    The calculator game:

    • Each player has a calculator. One player is designated as “evens” and one player is designated as “odds”.
    • Each player enters a number from 1 to 9 into their calculator, without showing the other.
    • If the sum of the two numbers is an even number then the “evens” player gets a point, if the sum is an odd number then the “odds” player gets a point.
    • Players keep a tally of the scores and play an agreed number of times.
       
  2. Students play the game in pairs, at least ten times and record who wins each time. Pairs work together to answer the question: Is the game fair to both players?
    Students justify their ideas as they work. Look for students to:
    • Record the results of their games systematically.
    • Use a strategy such as a table or a tree diagram to find all the combinations.
    • Knowing all the possible combinations, make decisions about which player, odd or even, has the best chance of winning the game.
  3. After a suitable time invite ideas about the fairness of the game. Some views may be based on experimental results. Players do tend to win about the same number of times. However if you pool the results across the whole class you might find slightly more wins for the even than the odd players. Expect some students to use systematic ways to find all the possible outcomes, for example “I thought there must be 9 x 9 = 81 possible combinations. It is like the shorts and t-shirts problem.”
    Note that if students are listing all possible outcomes rather than drawing an array they need to be aware that the combination of, for example, a 6 and a 5 is not the same as a 5 and a 6.
  4. Look for systems to find all the possible outcomes. A two way table is a tidy strategy, in which the even and odd outcomes can be shaded differently. Use PowerPoint Four to stimulate discussion.
    Seeing patterns in the table helps speed up the shading process, e.g. Every second cell is shaded. Ask the students if there is a quick way to count the number of shaded (even) cells. They might notice that number of shaded cells in each row going down is 5 + 4 + 5 + 4 + …
    Can they use properties of multiplication to find the total? (5 x 5 + 4 x 4 = 41)
  5. So how many non-shaded (odd) cells are there?
    Students might suggest 81 – 41 = 40, the total number of cells less the even cells. Others might notice the pattern in the number of blank cells going down by row, 4 + 5 + 4 + 5 +… which gives 5 x 4 + 4 x 5 = 40.
    So 41 out of 81 possible outcomes are even, and 40 out of 81 are odd. That makes the game close to fair. Do the students recognise that fairness? To realise that 41 out of 81 is very close to 50% requires proportional thinking.

  6. If time permits contrast the Odds and Evens calculator game with a variation that Lucy has made up using a pack of cards. Show students how to play the card game. You may like to use Video 2 and get two students to demonstrate the game in front of the class. 

    The card game:

    • The game uses 5 playing cards: Ace (which counts as 1), 2, 3, 4 and 5. One player is designated as “evens” and one player is designated as “odds”.
    • The cards are shuffled and placed in a pile, face down. Each player takes a card from the pile, and then the two cards are turned over.
    • If the sum of the two numbers is an even number then the “evens” player gets a point, if the sum is an odd number then the “odds” player gets a point.
    • Players keep a tally of the scores and play an agreed number of times.
  7. Ask the students to investigate:
    • How is the game the same and different to the calculator game?
      In the calculator game players choose a number, but in the card game players draw a number randomly. In the calculator game both players can choose the same number in a turn, but in the card game players always have different numbers.
  8. Students can cut up five pieces of paper and label them with digits 1-5. They can act out the game many times. Student may notice that the results tend to favour the odd player.
    • Is this game fair or unfair? Why?
      The game is unfair because there are 20 possible outcomes, and 12 of them result in an even sum.
    • How could the game be made fair?
      The game could be made fair by changing it so there is an equal chance of either an odd or an even sum. This can be done by have an even number of cards, and by each player having their own set of cards so that it is possible for both players to draw the same number. Your students may find other ways to make the game fair.

Session Five

This session gives you an opportunity to see if the students can apply multiplication to problems involving combinations, and to apply their knowledge of possible outcomes to think about chance.

  1. Begin by showing the students PowerPoint Five. Lucy’s family travel from Tauranga to Palmerston North, via Taupo. There are four possible routes leading from Tauranga to Taupo and five possible routes from Taupo to Palmerston North. So in total there are 4 x 5 = 20 different ways for the family to drive home. Pose the problem from slide 5 and let the students explore it independently using Copymaster 4. Look for them to:
    • Identify that multiplication can be used to count the number of different ways.
    • Systematically organise the routes using a strategy like a two-way table or tree diagram.
  2. The second part of the problem explores probability. As seen on slide 6, the family place the Highway numbers, 2, 28, 29 and 33, in a sunhat and randomly select one of those numbers as the route from Tauranga to Taupo. Similarly, they put the Highway numbers, 1, 5, 30, 38, and 43, in the sunhat to select the last leg from Taupo to Palmerston North. Pose the three questions from slide 7:
    • What are the chances that the route home will take the family through Taupo? (100% Certain)
    • What are the chances that the family will travel through Napier? (2 out of 20 or 1/10)
    • What are the chances that the trip from Taupo to Palmerston North will take them by the sea? It is easier to think about which trips are not by the sea at any point. Only the trip down Highway 1 does not touch coastline. So the chances of a coastal route are four out of five or 4/5.
  3. If any students are ready for extension ask: How many of the 20 routes from Tauranga to Palmerston North encounter coastline?

Printed from https://nzmaths.co.nz/user/75803/planning-space/early-level-3-plan-term-1 at 9:51pm on the 16th August 2022