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.
In this year
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.
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.
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
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:
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
Here is an interesting fact. You will probably blink about ten times per minute.
- 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 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.
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.
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?
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:
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
What fraction of our class is getting enough sleep on weeknights?
365 x 10 = 3 650 hours per year 3 650 ÷ 24 = 152 days per year
Food
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.
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?
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
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.
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.
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.
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.
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.
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:
Picton, Nelson, Kaikoura, Hanmer Springs, Franz Joseph Glacier, Lake Tekapo, Queenstown, Moeraki, Bluff.
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.
Do they think proportionally about the distance or time they are estimating and the reference distance and time?
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.
Dear family and whānau,
For the first week of school our mathematics unit is about setting goals for this year. We will investigate how many times we will do things this year (like blink), how long things will take, healthy eating and sleeping, and the distance and travel time to our favourite places. We will also test ourselves on some maths problems to set learning goals for this year.
Building a multiplicative view of whole number place value
This unit requires students to work with the number of ones, tens, hundreds and thousands in four-digit whole numbers.
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.
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:
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.
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
What is an efficient way to count the number of people?
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:
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:
For each turn:
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
Is there a pattern?
What is the pattern?
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.
- Develop an algorithm for multiplying a three-digit whole number by 11.
- Develop an algorithm for multiplying a two-digit number by 101.
- Develop an algorithm for multiplying a two-digit whole number by 111.
The solutions are:For example, 345 x 11 = 3 795. Note that renaming is needed when a + b and/or b + c equal ten or more.
For example, 68 x 101 = 6 868.
Session 3
In this session students investigate how many tens are in a whole number to four places.
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?
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?)
You need to make teams of ten for sports day. How many teams of ten can you make with 245 people?
How might we record what we have found in an equation?
Some students might suggest that five is half of ten or that there are 24.5 tens in 245.
How many people are in this stack? (1000 since there are ten hundreds)
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)?
What patterns do you see in dividing by ten and one hundred?
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.
What is the sum of 6 300 and 800? How do you know? (63 + 8 = 71 in units of 100, so 7 100)
880 + 60 =
390 + 90 =
940 + 80 =
430 – 60 =
910 – 80 =
540 – 90 =
820 – 70 =
Session 5
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 = ?
Family and whānau,
This week we have been exploring place value with whole numbers. We looked at the number of tens and hundreds in numbers like 4762 and used that knowledge to solve addition and subtraction problems. To help us we used a model called Place Value People so we could see what happened to the numbers we worked with.
Location, Location
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.
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.
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:
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:
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.
What can you tell me about this shape?
What could it represent?
How wide is our classroom?
How long?
Check this by measuring the actual classroom.
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)
If the door is 80cm wide (which is approximately the width of most doors) how large will it be on the plan?
Exploring
Over the next 2 to 3 days the students prepare detailed scale plans of the classroom to include desks and other major features.
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.
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.)
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.
What do you think is good about the way our classroom is arranged?
What would you like to change?
What do we need to remember when we do this?
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?
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.
Can you locate our school?
Your home?
The local shop?
The park?
The swimming pool?
What do the symbols on the map mean?
Dear family and whānau,
This week we have been working on a scale plan of the classroom and using it to suggest ways that it might be rearranged. We were also introduced to the use of a compass so that we can tell which direction we are facing.
We’d like your help for the children to make a scale floor plan of one of the bedrooms or some other room where you live. It does not have to be too elaborate but it should show the major features such as beds, chairs, tables, windows and doors.
We have been working with a scale of 1 metre = 10 centimetres.
Figure it Out Links
Some links from the Figure It Out series which you may find useful are:
Addition and subtraction with whole numbers
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.
There are three types of situation to which addition and subtraction are applied.
Example: You have 35 toy cars and are given 47 more. How many cars do you have?
Equation: 35 + 47 = 82
Example: You have 82 toy cars and give 47 away. How many cars do you have?
Equation: 82 - 47 = 35
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.
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
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
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.
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:
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.
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.
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.
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.
Represent each calculation using both the physical materials and empty number lines. For example, 56 + 37 = 93 can be represented as:
56 equals 5 tens, and 6 ones Adding 30 gives 86, 8 tens, and 6 ones
Adding 7 gives 8 tens, and 13 ones 10 ones make 1 ten so the answer equals 93
Session Two
For example, consider 72 – 38 = □ (in an appropriate story context).
With place value blocks the problem might be represented as:
72 equals 7 tens, and 2 ones Subtract 40 (4 tens) leaves 3 tens, and 2 ones (32)
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:
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?
Session Three
95 equals 9 tens and 5 ones If 68 was subtracted there would be ? left
That means 68 + ? = 95
In empty number line form the problem is represented like this:
Session Four
48 add 34 has the same sum as 40 + 32 = 72
The difference between 81 and 58 is the same as the difference between 83 and 60.
83-60 = 23.
Session Five
Dear parents and whānau,
In mathematics this week, our focus is on solving problems with addition and subtraction. We will learn several different strategies including working out answers in our head, on paper, and using a calculator.
You might give your student a problem to solve, such as 38 + 62 or 73 – 26 and ask them to explain their preferred way to solve it. When you went to school you may have learned a different method. Talk to your student about how your method works.
Figure It Out Links
Addition and Subtraction Basic Facts:
Applying Addition and Subtraction:
Combos
This unit introduces students to combinations as a type of situation that multiplicative thinking can be applied to.
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.
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:
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:
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.
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.
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
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
Are your results what you expected? Explain
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?
Only three of the nine outcomes are colour matches (marked with x).
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.
Mains: Ham, Chicken, Beef
Extras: Cheese, Lettuce, Tomato, Avocado, Peppers
How many different sandwiches can be made with these choices?
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.
The calculator game:
Students justify their ideas as they work. Look for students to:
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.
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)
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.
The card 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.
The game is unfair because there are 20 possible outcomes, and 12 of them result in an even sum.
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.
Dear parents and caregivers
This week we are studying combinations. We sometimes need to know in how many different ways objects can be paired, or even arranged in threes or other matchings. For example, we might need to do this if we are making up a match schedule for a sporting tournament or to think about the different ways to travel from one place to another.
Sometimes we can use multiplication to calculate the number of pairings for a situation. This can help use to work out the chances of something occurring. This kind of thinking is used to work out the risk of getting a disease, insurance premiums, and the odds of winning Lotto.