In this unit, students will explore number concepts through a variety of money contexts including trading ($1, $10, and $100 notes) and extended trades and operations into using thousands, millions and billions of dollars.
- create groupings of tens, hundreds, thousands using play money and place value mats
- make sensible estimates for sums that are greater than / less than $10, $100, $1 000
- find the sum of groups of 10s, 100s, 1000s and mixtures of these denominations
- read and explain the meaning of large numbers with confidence
- begin to explore powers of ten
In this unit students will explore the meaning of digits in whole numbers as well as developing deeper understandings about our place value system within the context of money problems.
The focus of the Number learning will be on developing better understandings about the fundamentals of our place value system, namely that:
- Our whole number system involves groupings in tens and trading collections of ten. (Powers of ten are therefore an important concept for learners to develop).
- The same digits are used in different positions for different values. (This is often referred to as the distinction between the "face" of a number, the "place" of a number and the " total value" of a number.)
- Our system is called a base ten place value system. (Which is reflected in the name decimal where "deci" means based on ten.)
- There are only ten digits (0 – 9) that we use in our system but there an endless number of values that can be given to these digits – tens, hundreds, thousands – depending on their placement.
- Understanding the role of zero is critical as it serves as an important place holder in our place value system.
These concepts will be developed as learners participate in many trades involving play money. They will use the play money in conjunction with place value house mats upon which they will place the money. It is important to note that students will use play money that only involves powers of ten - $1, $10, $100, $1000, $10000 (no $2, $5, $20, $50 notes will be used in this series of lessons). Learners will develop their ability to read amounts of money in story problems and flyers. Most tasks will emphasise the use of flexible mental thinking strategies rather than recording answers on paper. Grouping, rounding and using compatible number combinations will be encouraged through the emphasis on the numbers that "go together" to make 10, make 100, make 1 000.
As students develop further confidence with making equitable trades involving ones, tens and hundreds, they will be encouraged to solve problems where they need to estimate, round off and carry out addition and subtraction to solve money tasks. The emphasis is on students using their knowledge of groupings of tens to help them round off and estimate amounts that are more than / less than $10, $100, $1 000.
Place Value House masters - Material Masters 4-11
Advertising circulars for sports gear / items for sale.
Plastic bags or envelopes to create pay packets
Play money. Material Masters 4-9
Base 10 / Multi-base / Place value blocks (cubes, longs, flats, large cubes)
It is best if the following activities are carried out by a group of 6 –10 students who are at a similar stage of mathematical thinking in terms of their addition / subtraction strategies and fluency reading 1-digit, 2-digit, and 3-digit whole numbers.
Hand out play money to a group of students and have them play "The Great Money Sort!" where they are asked to sort random piles of play money into logical piles. For the first day, students will be involved in exploring number patterns and relationships that they see in the play money, making amounts with the money by grouping it into ones, tens, hundreds and so on.
- The Great Money Sort!
As students take the mess of play money and begin to sort the money into piles, have them discuss patterns that they observe as they look at the numbers in the play money. Have students sort as much of the play money as possible. Have them discuss how they are choosing to organise the money into piles, and why.
Possible questions / prompts as students are sorting the money into piles:
- "What do you notice that is the same about all of the money?"
- "What do you notice that is different?"
- "What patterns do you notice in the numbers on the money?"
- "Can you find another way to describe the patterns that you see?"
- "How do you know which group of notes to put in the next pile?"
- "Why is that a logical choice?"
- "How could you record the pattern that you see using numbers?"
Eventually, all of the money should get sorted into piles that replicate the place value house designations:
- Encourage students to offer suggestions about patterns that they see in the numbers and to describe the same pattern in as many different ways as they possibly can such as:
I see the 1 repeating at the start of each number.(1, 10, 100, 1 000, 10 000…)
I see zeros repeating. (10, 100, 1 000, 10 000…)
I see the zeros growing at the ends of the numbers… 1 zero here – 10 (tens), 2 zeros here - 100 (hundreds), 3 zeros here - 1 000 (thousands) etc.
I see 1, 10, 100 and that keeps repeating in the thousands (1 thousand, 10 thousand, 100 thousand), the millions (1 million, 10 million, 100 million) etc.
Students can record their patterns on paper to share with each other during the discussion.
- Once students have communicated their patterns with the group, turn their attention back to the piles of money. Ask questions that will help students focus on the pattern that can be described as grouping by tens (ie. that 10 ones makes 1 ten, that 10 tens makes 1 hundred). This can also be discussed and recorded as powers of ten (100 = 10 x 10 = 102; 1000 = 10 x 10 x10 = 103).
"How many tens do you need to make one hundred dollars?
"How many hundreds do you need to make one thousand dollars?"
"If you add 1 more thousand to 9 thousand dollars, how much will you have?"
If I have 10 hundred thousand dollar notes in this pile, how many have I got?"
At this stage, some revision of forward and backward counting may be required. In particular, students will often struggle when counting on the last ten/hundred/thousand in any group. You may need to use number lines and / or hundred charts to revise counting forward and backward in ones, tens, hundreds, etc. before you continue.
piles of ten: …50, 60, 70, 80, 90 and 10 more makes… many students will incorrectly say the next number is 20
piles of hundreds: … 700, 800, 900… many students will simply get "stuck" here because they don’t know the language of 1,000 or connect it to 900 + 100 more.
Where students struggle, have them actually count out the money in tens, hundreds or thousands saying the numbers aloud as they add on more money. Once students have had experience with counting on in tens ($30, $40, $50…), hundreds ($600, $700, $800…), thousands ($2 000, $3,000… $4,000…) stop the students at key times and have them predict how many notes of one denomination will be required to make the next amount required without counting on to find out.
I see you have counted out 7 thousand dollar notes which make $7,000 so far, how many more thousand dollar notes do you need in order to make $10 000 altogether (3 more)? How do you know? How did you work that out?
I see you have counted up to $50 000 using ten thousand dollar notes… how many $10 000 notes have you counted so far (5) and how many more do you need to make $100 000? How do you know without counting on?
Can you use an easier pattern/problem to help you solve it?
And other similar questions.
In the subsequent three days, students will continue working with the play money, practise making pay packets in "Pay Packet Play", carry out "fair trades" in the "Go Fair Trading Game" and explore number problems by grouping money on Place Value Houses, thinking about rounding, grouping, and estimating to solve money story problems.
- Pay Packet Play
For the first day of exploration, provide pairs or small groups of students with plastic bags or clear-faced envelopes into which they can put play money piles. Discuss a scenario with the students where they are in charge of banking and payroll at their school. It is their job to put together teachers’ pay packets or keep track of the total amount of money received to pay staff for the week. Have students challenge each other to solve how much money is in their pay packet by providing them with information such as:
I have put in 10 $10 notes. How much money do you have?
Can you work it out without counting?
I have put in 10 $10 000 notes. How much money do you have? How do you know?
I have put in 20 $1 000 notes. How much money do you have? How do you know?
Have students take the play money out if they need to use it to work out the problems they create for each other. However, ultimately the goal is to have students use their place value understandings that 10 tens means 100 without them having to count forward in tens to work out answers.
Throughout the pay packet play, have students discuss what patterns they are observing and how they are working out how much is in their pay packet. Students take turns to make pay packets for each other. Have them record their problems using words and numerals on pieces of paper.
You have 10 $100 notes. How much money do you have altogether in your pay packet?
You have 30 $1 000 notes. How much money do you have altogether in your pay packet?
You have 55 10 000 notes. How much money do you have altogether in your pay packet?
You have 100 $100 notes. How much money do you have altogether in your pay packet?
Students can discuss possible ways to record the problems using only numerals and symbols.
10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 + 10 = $100
10 x $10 = $100 etc.
Discuss which ways of recording are the most efficient and why.
Finally, have students "win" extra lotto money to add to their pay packet or "lose" tax money that they have to pay at the end of the week. If a student has worked out that their pay packet of 20 $10 000 equals $20 000 pose problems such as:
How much will you now have altogether if you win another 10 x $1 000 with lotto this week?
How much will you have left over if you decide to give away $5 000? Which notes, and how many of each, will you have left?
Have students use the play money if they need to in order to solve the problems.
- Go Fair Trading Game
For this second day of exploration, the activity can be played in a small group or as a whole class. Have the students sit in a circle. Tell them you are going to give them some money in a pay packet that is theirs to keep and trade with throughout the "Go Fair Trading Game". Their aim is to not lose any money as they play each round of the game. Hand out "packets" of pay. They will need to find the total of their pay packet of money and, without talking, find someone else in the group who has an equivalent amount of money made from different denominations with whom they can make a fair trade for their money.
For example, give one student twenty $10 and another student in the circle two $100. They can make a fair trade as they do not lose any money in the transaction and they both have different denominations of money to make their $200.
Play a number of rounds with students receiving different pay packets each round. Money returns to the teacher at the end of each round.
Encourage them to avoid counting on or back to work out the total of their pay packet. Push them to adopt more efficient strategies such as:
I have thirty $100 notes. I know that ten $100 notes makes $1 000 so three groups of ten $100 would be $1 000 + $1 000 + $1 000. That makes 3 x $1 000 or $3 000 altogether that I have.
Continue discussing the trades increasing the complexity and number size with each round. Have students share and record their strategies for each round.
The game can be made more challenging by using more than one denomination in each pay packet. For example two $100 and ten $10 notes.
- Place Value Houses and Problems
In this third day of exploration students will make further links between the play money and our whole number place value system by placing money on the Place Value Houses. Students will gain confidence reading multi-digit whole numbers using the Place Value Houses to help them make sense of large numbers. Finally, they will begin solving addition and subtraction problems by using their knowledge of grouping play money by tens/hundreds and by making sensible estimations that they can check with the money.
Pull out A3 laminated Place Value Houses along with the piles of Play Money that have been used throughout the past two days. Ask students to decide how to organise the Place Value Houses based on the play money sorting task they carried out on the previous day.
Eventually, working together, students should be able to connect the Place Value Houses in order starting with the "Trend setter house" on the far right connecting to the "Thousands House" on its left, the "Millions House" to the next left, the "Billions House" next and finally the "Trillions House" on the far left hand side.
Have students place the play money into piles in the appropriate spaces in the place value houses. Discuss how and why they know where each pile belongs in each house. Ask them why they think the first house on the right is named the "Trend Setter House" (because it sets the trend of ones, tens and hundreds that keeps repeating in our whole number place value system).
Once students have discussed the names of the houses and reviewed the patterns and connections between the groupings within the houses (each pile to the left is equal to ten of the denomination to its right…. 1 hundred is equal to 10 tens.) have them place the piles of money just below the Place Value Houses.
On the whiteboard or on a piece of paper, write amounts of money such as:
$1 200 300
With each amount you write, have students first try to visualise what piles of money they would place on the appropriate places within the Place Value Houses. Have them describe to a partner what ones/ tens/ hundreds of play money etc they would place on the houses to make the amount $5, 450.
As students discuss what they are visualising, if they get stuck, have them actually count out the required money, placing it in the appropriate spaces of the Place Value Houses. Again, encourage them to use their groupings of tens knowledge to count out money rather than counting on to add up money.
[Option: As students create money involving ones, tens and hundreds, you can also place Base 10 / Multi-Base / Place Value Blocks alongside the play money to show the students you are dealing with the same concept of 1/10/100 but in the context of money.]
Have students challenge a person beside them to tell them what piles of money they would place on the Place Value Houses for an amount that they write on paper.
Help students decode the reading of large numbers by always having them begin reading numbers at the far left and saying first the "ones, tens and hundreds" and then the Place Value House name and so on.
In this way, the number $45 874 230 would be explained and read as:
(Always start reading with the far left hand side of the number):
(ones, tens, hundreds) 45 (millions house) million, (ones, tens, hundreds) 874 (thousands house) thousand, (ones, tens, hundreds) 230 (trend setter house so we just say the number of ones, tens, hundreds).
Remind students: The spaces remind us where a new house begins! When we record this the convention is that we use a space between houses for amounts of money or a comma between houses for other whole numbers.
Practise reading each other’s large numbers written on paper and making them on the Place Value Houses.
Extension: Begin simple addition (that doesn’t require renaming at first) and subtraction (that doesn’t require decomposition) problems.
For the final day, students will be given an open-ended task involving Advertising circulars (such as a Warehouse, Rebel Sport or Farmers advertising flyer) and play money.
The teacher can discuss the learning task with all of the students and then move from student to student observing the students’ strategies. Students can use the play money, Place Value Houses, number lines and hundred charts to assist them as they work and reflect on the following task.
Open-Ended Task: This is your lucky day! Your Grandmother gives you an envelope for your birthday and when you open it you find ten $100 notes inside! Your task:
- Work out how much money you have to spend.
- Look through the advertising flyers provided and decide on the items you would really like to purchase, what you'd like to share with others or buy as presents, what you'd like to save etc.
- See if you can combine 2 or more items that you would like to buy and estimate if you will have enough money to buy them.
- Try to work out the maximum number of items that you can buy with your money.
- Record your preferred purchases on paper.
- Be prepared to discuss how you have worked out your thinking.
- You may use play money, place value houses and any of the number lines or charts to help you.