Illustrations: Beehive (Nadene Naude, Andrew Pagram)
Photos: Shutterstock
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Maths and Me – Innovative Digital Resources
Maths and Me provides access to an extensive range of over 2,500 fun, innovative, easy-to-use and engaging FREE Interactive Digital Resources including editable planning documents and a full range of printables. All resources are central to the programme and integrated into lessons, providing rich learning opportunities for children, by encouraging active participation and positive engagement. Designed specifically for Maths and Me, with the key pedagogical practices in mind, the resources promote maths talk, allow for formative assessment, include cognitively challenging tasks, mathematical modeling and lots more.
The extensive range of digital resources include:
n Rich multimedia resources –engaging videos and animations
n Interactive classroom activities
n Virtual manipulatives
n Editable planning documents
n Printables
Explainer videos help pupils to understand key maths concepts and strategies.
Classroom activities such as Concept Cartoons, Quick Images and Three-Act Tasks engage learners, support key pedagogies such as maths talk and playful learning, help pupils develop skills such as reasoning and allow for formative assessment.
access to a dedicated app with digital sources for pupils to use at home or in school on devices
Engaging animations feature the Maths and Me characters in relatable maths-based real-life scenarios.
An extensive range of colourful printable resources can also be accessed via the platform, including Maths Language Cards and Manipulative Printables.
Digital resources are accessed through an easy-to-use web platform.
Digital Resources for 5th Class, Unit 8: Fractions, Decimals and Percentages
11
22
Fractions as Decimals and Decimals as Fractions
Converting Fractions to Decimals and Decimals to Fractions
33 Percentages
44 Percentages as Fractions and Decimals
55 Converting between Fractions, Decimals and Percentages
66 Comparing and Ordering Fractions, Decimals and Percentages
77 Calculating Percentages of Amounts
88 Percentage Increase and Decrease
Same But Different –Fractions as Decimals
Benchmark Fractions
What Value Does this Represent? (1)
Converting between Fractions and Decimals
Same But Different –Percentages
Introducing Per Cent
What Value Does this Represent? (2)
Can 3 4 Be Expressed as a Percentage?
Benchmark Percentages
Converting between Fractions, Decimals and Percentages
Reason & Respond, with Think-PairShare; Write-Hide-Show
Reason & Respond, with Write-HideShow
Slideshow
Slideshow
Quick Images, with Write-Hide-ShowQuick Images
Reason & Respond, with Write-HideShow Video
Reason & Respond, with Think-PairShare Slideshow
Reason & Respond, with Write-HideShow Video
Quick Images, with Write-Hide-ShowQuick Images
Concept Cartoon, with Think-PairShare
Concept Cartoon
Quick Images, with Write-Hide-ShowQuick Images
Reason & Respond, with Write-HideShow Video
Order Them Would This Work?, with Build it; Sketch it; Write it
Would This Work?
Cycling Competition Three-Act Task Three-Act Task
Calculating using Related Percentages
Boys in the School
Number Strings, with Write-HideShow Slideshow
Would This Work?, with Build it; Sketch it; Write it
Test ScoreWould This Work?, with Build it; Sketch it; Write it
Sale Time
Percentage Increase and Decrease
Concept Cartoon, with Think-PairShare; Build it; Sketch it; Write it
Would This Work?
Would This Work?
Concept Cartoon
Reason & Respond, with Write-HideShow Slideshow
Scan the QR code below to access a selection of our new digital resources for Maths and Me 5th and 6th Class:
FOR A DEMO OF THE NEW DIGITAL RESOURCES
Pupil’s Book Contents
Note: The contents shown indicate what is included in the full Pupil’s Book.
Unit 1: Place Value
Unit 2: Operations 1
Unit 3: Operations 2
Unit 4: Expressions and Equations
Review 1
Unit 5: Shapes and Angles
Unit 6: Fractions
Unit 7: Operations 3
Review 2
Unit 8: Fractions, Decimals and Percentages
Unit 9: Measuring 1
Unit 10: Data
Review 3
Unit 11: Location and Transformation
Unit 12: Patterns and Rules
Unit 13: Time
Review 4
Unit 14: Money
Unit 15: Ratio
Unit 16: Measuring 2
Unit 17: Chance
Review 5
Let’s Try More
Glossary
The Maths and Me Pupil’s Pack includes a Progress Assessment Booklet and the following inserts to support learning: mini-whiteboard with open-number line on reverse, spinners (for playing games), cut-out Place Value Counters and a cut-out Fraction Wall.
Numbers Beyond 10,000
Let’s talk!
With a partner, take turns to read these numbers aloud:
What is the value of the digit 4 in each of the numbers?
In which number does the digit 5 have (a) the greatest value, and (b) the least value?
Write each number using digits (standard form).
1. Thirty-two thousand, six hundred and fifteen
2. Eleven thousand, two hundred and thirteen
3. Fifty-one thousand, two hundred and sixty
4. Sixty thousand and seventeen
5. Ten thousand, one hundred and one
6. Fifteen thousand and forty-one
7. Thirty-nine thousand and nineteen
8. Seventy thousand and thirty-eight
9. Fifty-six thousand, five hundred and two
10. Eleven thousand, one hundred and twelve
Build it! Write it! Sketch it!
Choose a number from above Build it using materials. Sketch it in your copy. Write it in different ways. Now choose two more numbers.
Example:
Don’t forget to use a comma. Also, use zero as a placeholder where necessary.
Standard form: 13,759 Word form: thirteen thousand, seven hundred and fifty-nine
Try this!
Write a 5-digit number to fit in each section of the Carroll Diagram to suit the given rule.
D What number is represented? Write the answer in different ways.
With a partner, take turns to read these numbers aloud: Where might you come across numbers like these?
Estimating and Rounding Whole Numbers
Look at the table and answer the questions.
1. Round the population of each county to the nearest thousand.
2. Which county has…
(a) roughly half the population of Roscommon?
(b) roughly double the population of Longford?
(c) a population of roughly one hundred thousand?
(d) a population of nearly 50 thousand?
Estimate the population of these counties.
1. Sligo:
2. Offaly:
3. Carlow:
4. Monaghan:
Let’s talk!
Which point on the number line below is the best estimate for 34,432? Explain why.
D Round the numbers and complete the table.
Try this! Two 5-digit numbers have a difference of 5. When they are both rounded to the nearest thousand, the difference is 1,000. What could the numbers be? and
Sligo
Carlow Monaghan
Thousandths
What (a) decimal fraction, and (b) fraction (thousandths) is coloured?
In above, what (a) decimal fraction, and (b) fraction of each shape is uncoloured?
1. (a) (b)
3. (a) (b)
5. (a) (b)
In above, what is the…
2. (a) (b)
4. (a) (b)
6. (a) (b)
1. largest decimal fraction that is (a) coloured? (b) uncoloured?
2. smallest decimal fraction that is (a) coloured? (b) uncoloured?
Let’s talk!
If you add the number of coloured parts to the number of uncoloured parts in each shape in above, what do you notice? Can you explain why this is so?
Write each answer as a fraction and a decimal number.
1. There are 1,000m in 1km. If Mia’s dad has run 234m of a 1km race, what fraction of the race has he (a) run? (b) left to run?
2. There were 1,000 paper clips in a box. If 750 of the paper clips have been taken out, what fraction of the paper clips… (a) has been taken out? (b) is still in the box?
1,000
Place Value in Decimal Numbers
Dara used base ten blocks to represent decimals. Write the numbers he built in (a) fraction form, and (b) decimal form in the table below.
=
Concrete form (a) Fraction form (b) Decimal form
Mia also used base ten blocks to represent numbers. Write the numbers she built in decimal form. 1. 2 tenths and 5 hundredths
3. 9 hundredths and 2 thousandths
15 hundredths
6 tenths and 8 thousandths
12 tenths
27 thousandths
Try this! Jay used six base ten blocks to build a decimal number. If he did not use any red blocks, but used at least one of each of the other block colours, what was the…
1. greatest decimal number that he could have built?
2. smallest decimal number that he could have built?
0028
Let’s talk!
Read aloud each of the expanded numbers in above What is the value of the digit 2 in each of the numbers?
In which number does the digit 8 have (a) the greatest value, and (b) the least value?
E Let’s play! Wipeout!
Number of players: 2–6
You will need: calculator, mini-whiteboard and marker per player
● To start, each player inputs a 5-digit number (any digits except 0) with 3 decimal places (to thousandths) on their calculator and writes the number on their mini-whiteboard.
● A caller (perhaps the teacher) calls out a digit, for example, 6.
● Any player with a 6 on their calculator display wipes out only this digit, so that there is a 0 in its place. They record how they did it on their mini-whiteboard.
● Only one digit can be wiped out each time.
● The player who wipes out all of their digits first wins the game.
Let’s talk!
With a partner, take turns to read these numbers aloud: Where might you come across numbers like these?
Try this!
Read aloud each of the expanded numbers. For example, 2 tenths, 8 hundredths and 3 thousandths.
Using only the digits above, make a decimal number with three places in which:
1. 6 has the least value
2.
3. 8 has
Estimating and Rounding Decimal Numbers
For each number line below, write (a) the number the arrow is pointing to, and (b) the whole number to which it rounds.
For each number line below, write (a) the number the arrow is pointing to, and (b) the tenth to which it rounds.
Let’s talk!
1. Which of the numbers in A and B were trickiest to work out? Explain why.
2. If a number is half-way between two numbers, to which one does it round?
Hint: To round to the nearest whole number, look at the digit in the place
3. How might you round decimal numbers without using a number line?
4. What does it mean to ‘round a number to one decimal place’?
5. Which of the answers in A and B have only one decimal place? Which of them have two decimal places?
(a) (b)
(a) (b)
3.
(a) (b)
(a) (b)
D Round each price to (a) the nearest whole euro, and (b) the nearest 10 cent.
Use these digits to make a number with three decimal places that…
8 0 4 2
1. when rounded to the nearest whole number is 5.
2. when rounded to the nearest tenth is 0·3. ·
3. when rounded to one decimal place is 0·9.
Try this! For each clue, write a number with three decimal places that is (a) the least number possible, and (b) the greatest number possible.
When I round my number to the nearest whole number, I get 4. When I round it to the nearest tenth, I get 3 7.
My number is between 5 and 9. When I round it to the nearest tenth, I get the same number as when I round it to the nearest whole number.
My number contains the digit 2. When rounded to one decimal place it is 8 3.
Comparing and Ordering Whole and Decimal Numbers
Write each decimal number marked by an arrow on the number line. 1.
2.
(a) (b) (c) (d) (f) 7·97
(a) (b) (e) (c) (e) (d) (f)
Write <, > or = to make these true.
1. 8433 844 2. 3307 34
4. 39,460
7. 65.306 5,356 8. six thousand and six 6,060
In your copy, write each group of numbers in order, starting with the smallest.
3. Write a number to make this true. 60,740 < < 65,970 4. Write <, > or = to make these true. (a) 515m 515cm (b) 4,506m 461km
Estimating and Checking Calculations
A Look at these calculations. Which answer looks most reasonable? 1.
+ 48 =
=
3.
+ 1,349 =
4. 6,264 285 =
B Let’s talk!
=
.
Use your calculator, if needed, to help you check.
Compare your answers for above with those of a partner Take turns to explain which answer was most reasonable and why. Then, look at the unreasonable answers. Take turns to suggest how you might arrive at those answers.
Complete this table.
(a) Estimate using front-end (b) Estimate using rounding (c) Calculator answer
1. 23,875 + 5,946
2. 3,189 + 42,204
3.
4.
5.
D Let’s talk!
In above, which estimate (a) was faster? (b) was closer? (c) do you prefer?
Try this! Look at the table and answer the questions.
1. Do you think the figures are approximate or exact amounts? Why?
2. About what number is the population of Bangor and Bray in total?
3. About how many more people live in Drogheda than in Tralee?
4. What is the approximate difference in population between Newbridge and Ennis?
Populations (2022 Census) Town
Bangor, Co Down
Ennis, Co Clare27,923 Bray, Co Wicklow33,512 Newbridge, Co Kildare24,366
5. With a partner, write three more questions about the information in the table. Then, swap with another pair and solve.
Adding using the Column Method
A Let’s talk!
With a partner, take turns to estimate a reasonable answer for each of the calculations in and below. Explain your reasoning
B Solve these.
In your copy, rewrite these using columns, and solve.
1. 73,854 + 15,367
D Let’s talk!
Review your answers to and above Do they look reasonable? Check the answers using a calculator, but without using the + button.
Look at the table and answer the questions.
1. Anna is buying a City Hatchback and a single charger About how much money will she need? €
2. The Kelly family are buying a 7-seater and a double charger. About how much money will they need? €
3. One weekend, a salesperson sold one of each type of car and charger. Approximately, what was the total value of her sales that weekend? €
4. Is it better to round up or down when estimating how much money is needed? Why?
5. Now, work out the exact cost of 1, 2 and 3 above. (1) € (2) € (3) € Write in the unnecessary zeros to help.
Solve these.
1. Jay bought the following items at the shop: butter (0.227kg), cereal (0.75kg) and lasagne (1.5kg). If the cloth bag weighed 009kg, what was the total weight of the bag of items? kg
2. Mia bought the following items: orange juice (175l), shampoo (0.35l) and fabric conditioner that was double the capacity of the shampoo What was the total capacity of the three items? l
Try this! Re-write these in your copy. Insert the missing digits. I think there might be more than one correct answer for number 3.
G Let’s play!
Number of players: 2–6
Chance Calculation – Addition
You will need: mini-whiteboard and marker per player, 0–9 spinner, pencil and paper clip
● Each player should draw these boxes for a calculation on their mini-whiteboard.
● Each player, in turn, spins the 0–9 spinner and writes the digit spun into one of their empty boxes.
● When all the boxes are full, each player calculates their answer.
● The player with the greatest total scores a point.
● The player with the highest score when the time is up wins the game.
Variation
● At the beginning, choose one of these alternative layouts instead, or agree on your own chosen layout.
1. O· th th
Adding using other Strategies
A Complete these to arrive at an answer.
1. 171 + 225 = 2. 23,995 + 15,146 = 24,000 15,141
3. 1545 + 125 =
B Solve these without using the column method.
1. Mia has three parcels weighing 125kg, 164kg and 181kg. Jay has three parcels weighing 1.64kg, 1.81kg and 1.35kg Whose parcels weigh more?
2. A factory made 15,140 phones in January, 13,000 phones in February and 17,000 phones in March. How many phones did the factory produce in total over the three months?
3. What is the total weight of two pieces of meat if one weighs 0.697kg and the other weighs 1248kg? kg
4. A concert venue has 34,780 seats and standing room available for 2,000 more people. What is the total capacity of the venue?
5. Three planks of wood measure 17m, 23m and 1.4m. What is their total length? m
Try this! Total any two of the numbers shown. Show the total to a partner Without doing a calculation, can they work out which two numbers you totalled? Swap roles.
Variation: Total any three of the numbers shown.
Subtracting using the Column Method
A Let’s talk!
With a partner, take turns to estimate a reasonable answer for each of the calculations in and below. Explain your reasoning
B Solve these.
Write in the unnecessary zeros to help.
In your copy, rewrite these using columns, and solve.
D Let’s talk!
Review your answers to and above. Do they look reasonable? Check the answers using a calculator, but without using the – button.
Try this! Look at the table and answer the questions.
1. What was the difference between the fastest lap and the slowest lap? secs
2. What was the total time for laps 5 and 6? secs
3. What was the time difference between laps 1 and 2? secs
4. Approximately how many minutes did the six laps take? mins
5. With a partner, make up two more questions about the information in the table. One must be addition and one must be subtraction. Then, swap with another pair and solve.
E Let’s play!
Play ‘Chance Calculations – Subtraction’ on page xx.
Subtracting using other Strategies
A Complete these to arrive at an answer.
43,845 8,950 =
12·25 8·65=
B Let’s talk!
In what other ways could the questions in A above be solved?
C Solve these without using the column method.
1. Over Saturday and Sunday, 82,457 passengers travelled through Dublin Airport. If 31,956 passengers travelled through the airport on Saturday, how many travelled through on Sunday?
2. The distance from Ballybeg to Athmore is 18.65km. Sarah has cycled part of the way, and she has 5925km left to go. What distance has she cycled so far? km
3. The total weight of a box of cereal bars is 0.78kg If 003kg of this weight is made up of wrappers, what is the weight of the actual cereal bars? kg
4. 48,783 people attended an open-air concert this year. This was 3,560 more than attended the previous year What was the total attendance for the two years?
Try this!
1. Each letter to the right represents a different digit: 0, 1, 2, 3, 6, 7 or 9. What is the calculation?
2. Make up a story to match this model and solve it.
An aeroplane started a flight with 89,000 litres of fuel. If the aeroplane used up 13,580 litres of fuel for every hour of the flight, how much was left after two hours? litres
2. Explain to a partner how you thought of an answer. 7,000 200 80 26,000 ?
I will use rounding to the nearest thousand.
Properties of Operations
A Use one of the phrases below to make each statement true. Give an example for each. subtracted multiplied added subtracted from multiplied by divided added to divided by
Statement
A When two whole numbers are , the order does not affect the answer
B When two whole numbers are , the order does affect the answer.
C When three whole numbers are in a given order, the way I group them does not affect the answer
D When a number is zero, the answer is the same whole number.
E When a number is 1, the answer is the same whole number
F When a number is zero, the answer is zero.
G When a number is the same whole number, the answer is 1.
H When a number is the same whole number, the answer is zero.
I A number cannot be zero.
I think there might be more than one correct answer to some of these.
Example
B Which statement in A describes…
1. the commutative (turnaround) property of addition/multiplication?
2. the associative property of addition/multiplication?
3. the identity property of multiplication?
4. the zero property of addition?
5. the zero property of multiplication?
6. a characteristic of subtraction?
7. a characteristic of division?
C Solve these.
If is any number, place a symbol (+, −, ×, ÷) in each to make these true.
D Let’s talk!
Use a statement from A to describe each of the expressions in C above. Explain why the commutative or associative property does not apply to subtraction and division.
I think that some of these could have more than one correct answer
Try this! Find the value for each symbol below. Each time, choose only one of these numbers: 0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 12.
Factors
A What is the missing factor in each of these? 1. 420 2. 432 3.
B Build it! Sketch it! Write it!
A factor is a number that divides evenly into another number
Express each number below as multiplication sentences of its factors. Example: To express 12 as multiplication sentences of its factors, we would write: 1 × 12, 2 × 6, 3 × 4 1. 25: 2. 32: 3. 44: 4. 45: 5. 54: 6. 72:
C Dara was asked to list the factors of each of the numbers below. Ring the mistake in each group.
1. Factors of 14: 1, 2, 4, 7, 14 2. Factors of 10: 1, 2, 5, 10, 20
3. Factors of 18: 2, 3, 6, 9, 18 4. Factors of 32: 1, 2, 3, 4, 8, 16, 32
D Let’s talk!
Discuss as a class or in groups. Are these always, sometimes or never true?
● Every number is a factor of itself.
● 1 is a factor of every number
● The factors of a number are smaller than the number.
● 2 is a factor of numbers with 2, 4, 6, 8, or 0 ones.
● 3 is a factor of numbers with 3, 6 or 9 ones.
Try this! Sometimes factors can be factorised further to create a multiplication sentence with more than two factors. Look at the examples for 12 and 16. Then, in your copy, factorise each of the numbers below.
This is often called a factor tree. Why do you think this is so?
Multiples
A Mia was asked to list multiples of each of the numbers below. Ring the mistake(s) in each group.
1. Multiples of 10: 30, 55, 90, 240, 671, 890
2. Multiples of 2: 36, 74, 83, 130, 421, 758
3. Multiples of 5: 25, 54, 80, 195, 552, 915
4. Multiples of 3: 42, 57, 68, 144, 313, 468
B Let’s talk!
A multiple is the result of multiplying a whole number by another whole number.
How could you use your calculator to check your answers to A above?
C Answer these.
1. Identify and list the first five multiples of each number (a) 2: 3: (b) 4: 5: (c) 6: 9: (d) 8: 12:
2. Then, find the lowest common multiple (LCM) for… (a) 2 and 3: (b) 4 and 5: (c) 6 and 9: (d) 8 and 12:
D Use your understanding of the lowest common multiple to help you solve these.
1. Using cubes, find out the smallest number that can be arranged in equal groups of 2, equal groups of 3 and equal groups of 4.
2. What is the smallest number that is divisible by (a multiple of) 4, 9 and 12?
3. Burgers come in boxes of 4. Burger buns come in packs of 6. What is the least number of each that must be bought to ensure that every burger has a bun? burgers buns
4. Jay, Mia and Lexi went cycling on the first day of the month. If Jay then cycled every second day, Mia cycled every third day and Lexi cycled every fifth day, what was the next date on which they all went cycling? The day of the month
Try this! Dara was arranging his books on shelves. If he arranged the books in groups of 4, there were 3 left over If he arranged them in groups of 5, there were 4 left over
1. Exactly how many books could Dara have had, if there were… (a) fewer than 20? (b) between 20 and 50?
2. Using the answer to (b) above, in what way could he organise that amount of books into groups with exactly the same number in each? groups of
Prime and Composite Numbers
Prime number: a number that has exactly two factors –itself and one.
Which number is neither prime nor composite?
Composite number: a number with more than two factors.
A Read and follow the instructions.
This activity is called the Sieve of Eratosthenes, after the Greek mathematician who used it to identify the prime numbers up to 100.
1. Cross out 1.
2. Ring 2, but colour in all of the other multiples of 2.
3. Ring 3, but colour in all of the other multiples of 3.
4. Ring 5, but colour in all of the other multiples of 5.
5. Ring 7, but colour in all of the other multiples of 7.
6. Continue with this process for the remaining uncoloured numbers.
7. The ringed numbers are all numbers.
8. The coloured numbers are all numbers.
B Let’s talk!
Look at A above
● Why do you think it is called a sieve?
● What patterns do you notice?
● Why do we cross out 1?
● Why do we ring some numbers?
● Why do we colour other numbers?
C What is...
1. What is the only even prime number?
2. What is the smallest odd prime number?
3. What is the smallest odd composite number?
Let’s talk!
Without referring to the Sieve of Eratosthenes, can you use another strategy to work out…
● which of these are not prime numbers? 2357 11 13 17
● which of these are not composite numbers?
E Is each of these numbers prime or composite? (✓) 1. 311 2. 352
You can use your calculator to help you.
Solve this.
In a school, there is a class in which the teacher cannot divide the children into equal groups on days when everyone is present. If the amount of children in this class is greater than 20 and less than 30, what are possible amounts?
Mystery number! Use the clues to work out Jay and Dara’s numbers.
I’m thinking of a prime number greater than 50 and less than 100. When this number is divided by 10, there is a remainder of 9. When this number is divided by 9, there is a remainder of 8.
I’m thinking of a prime number between 20 and 70. When this number is divided by 5, there is a remainder of 3. When this number is divided by 4, there is a remainder of 1.
1. The number is
2. The number is
Multiplying and Dividing by 10, 100, and 1,000
A Use the moving digits strategy to help you solve these.
1. 0.82 × 100 =
B Let’s talk!
Look at Lexi and Dara. What do you think?
Explain why.
C Solve these.
I notice a pattern based on the number of zeros in the multiplier or divisor.
D Use the moving digits strategy to help you solve these.
I think this strategy could be used to convert measures, such as cm to m, or g to kg.
Try this!
1. A toy shop sold 100 toy cars at €2.25 each. What was the total sales amount? €
2. A farmer has harvested 2,500 apples. He wants to divide them equally into 100 boxes. How many apples will there be in each box?
3. How many kilometres are there in a 10,000m race? km
4. A factory produces 1,000 batteries and each battery weighs 0·015kg What is the total weight of the chocolate produced in kg? kg
Multiplying and Dividing with Multiples of 10, 100 and 1,000
Multiplication is the inverse of division, so I can solve the division sentences by thinking about the opposite multiplication sentences.
1. 160 ÷ 2 = 2. 360 ÷ 4 = 3. 4,000 ÷ 8 =
4. 5,400 ÷ 9 = 5. 480 ÷ 6 = 6. 1,500 ÷ 5 =
7. 6,300 ÷ 7 = 8. 270 ÷ 3 =
D Solve these.
1. Newtown Primary School has about 300 pupils. Newtown Secondary School has about five times that amount. About how many pupils attend the two schools altogether?
2. A tablet costs €400. How much would it cost for 40 such tablets? €
3. If a grant of €35,000 for new IT equipment was divided among seven schools, how much would each school get? €
4. The Camera Club bought six new cameras, and had €200 left out of €2,000. How much did each camera cost? €
5. Dara has a collection of 2,000 stamps. He has four times as many international stamps as Irish stamps. How many Irish stamps does he have?
Try this!
1. If a 3-digit number is multiplied by a 2-digit number, how many digits will there be in the answer? digits
2. If a 4-digit number is multiplied by a 2-digit number, how many digits will there be in the answer? digits
I think there is more than one answer for these.
Estimating and Checking Products and Quotients
A Let’s talk!
Sometimes I use front-end estimation, and just look at the digit at the front.
I prefer to round the numbers to the most significant place value.
If there is an obvious friendly number, I will use that.
What estimation strategy do you think was used for each of the calculations below?
Do you think it was the most efficient strategy for the numbers involved? Explain why.
B Estimate a reasonable product for each of these.
4 × 478
16 × 85
9 × 7 2
4 × 7 143
31 × 3.48 8. 53 × 8.746
C Estimate a reasonable quotient for each of these. 1. 536 ÷ 6 2. 453 ÷ 7 3. 39 5 ÷ 5
8 76 ÷ 4
D Calculate the exact answers to B and C above using a calculator.
Let’s talk!
With a partner, compare the estimation strategies you used for and above
Do you think you used the most efficient strategy each time? Explain why.
Try this! Look at the table. Approximately what would be the total cost of…
1. 16 laptops? €
2. 16 laptops, a networked printer and a laptop trolley? €
3. 16 tablets? €
4. 16 tablets, a networked printer and a tablet charging cart? €
With a partner, take turns to estimate a reasonable answer for each of the calculations in C below.
Explain your reasoning.
C Solve these in your copy. 1. 7 × 9,758 = 2. 6,867 × 3 =
How could you check your answers?
5 × 7,006 = 4. 4.63 × 8 = 5. 9 × 8.71 =
7 × 26 18 =
D Model and solve these.
18.67 × 3 =
5 × 3 067 =
1. A teaspoon holds 0.005l. What is the capacity, in litres, of nine teaspoons? l
2. Moran’s Printing Press is buying three new printers for their business, each costing €1,325. What will be the total cost of the printers? €
3. Horan’s Hotel is buying new couches for the hotel lobby, each costing €2,135. If they have a budget of €10,000…
(a) how many can they afford to buy?
(a) how much money will be left? €
4. A jar of jam weighs 0.454kg How much do 8 jars weigh? kg
5. How much change did a school get out of €3,000 when it bought 7 new laptops at €385 each?
€
E Use a calculator to check your answers to the questions on this page, but without using the × button.
Dividing by 1-digit Numbers
A Try to solve these mentally. 1. 32 10 ÷ 8 =
B Let’s talk!
With a partner, take turns to estimate a reasonable answer for each of the calculations in below. Explain your reasoning
C Solve these in your copy.
D Model and solve these.
1. The children are painting a mural on a school wall that is 8.52m long If the wall is divided into six sections, one for each group, what length of wall is allocated to each group? m
2. Lexi and her mum walked from her house to the end of the town and back, every day for a week. This was a total distance of 18.2km.
(a) What distance did they walk each day? km
(b) What distance is it from Lexi’s house to the end of the town? km
(c) If they walk a further 0.5km past the end of the town every day the following week, what will be their total distance walked that week? km
E Use a calculator to check your answers to the questions on this page, but without using the ÷ button.
Try this! What digit does the letter in each of these calculations represent?
Expressions and Equations
A Ring the number sentences below that are equations.
An equation is a number sentence with an = symbol showing that both sides (expressions) have the same value.
B Which of the number sentences in A above are expressions? Write each expression and then use it to write a true equation.
1. Expression: True equation:
2. Expression: True equation:
C Complete these equations.
D Is each equation below true (T) or false (F)?
Try this!
1. If 48 is the answer, what is the question? Write four equations using a different operator (+, –, ×, ÷) each time.
2. The school needs some new equipment: 12 footballs, 9 basketballs and 1 pump If footballs cost €5 each, basketballs cost €8 each and pumps cost €6 each, write an equation that represents the money needed to purchase the new equipment. How can you prove it?
Unknown Values and Variables
A Let’s talk!
At Dee’s Diner, the orders are written in code to make the ordering system more efficient.
Look at the menu. Can you work out what each table below ordered?
● Table 1: B + F + A
● Table 2: L + F + T
● Table 3: 2C + F + R + 2A
● Table 4: 3B + 3F + 3W
● Table 5: 3C + 2R + P + 3M
● Table 6: 4C + 2R + 2F
Estimate how many people were dining at each table.
B Using the menu in A above, write each order below in code and calculate the cost.
Table 1: burger, salad, tea
Table 2: potatoes, salad, milk
Table 3: 2 curries, rice, fries, 2 waters
Table 4: 2 lasagnes, 2 salads, 2 apple juices
C Write an equation to represent each of these.
1. Mia multiplied the length (l) of this rectangle by the width (w) of this rectangle to find out that it had an area of 48 sq cm.
2. Dara multiplied the side length (l) of this square by another side length (l) to find out that it had an area of 36 sq cm.
3. Jay totalled the length of the sides of this scalene triangle to find out that the perimeter was 24cm.
4. Lexi totalled the length of the sides of this equilateral triangle to find out that the perimeter was 24cm.
D Look at C and answer these in your copy.
1. Which of the unknowns are constants? Write ‘C’ beside them, and solve them to work out the unknown constant.
2. Which of the unknowns are variables? Write ‘V’ beside them, and work out one set of values that would suit in each.
A constant is a fixed value.
A variable is an unknown value that can change or take on different values.
E Which equation below can be used to calculate the… (✓)
1. number of days (d) in a number of weeks (w)?
d= w ÷ 7
d= w × 7
w = d× 7
w = d+ 7
2. number of hours (h) in a number of days (d)?
h= 24 × d
d= 24 × h
d= h× 24
d= 24 ÷ h
F Write an equation to represent each of these.
1. The cost of a hurley (h) is 9 times the cost of a sliotar (s).
2. Ben (B) is 2 years younger than Cáit (C).
3. In an input-output table, the output number (o) is the input number (i) when doubled and 4 is added.
4. When sharing items, the amount (a) that each person gets is the total number of items (t) divided among the number of people (p).
5. Mo (M) is twice as old as Frank (F), who is three times as old as Emily (E).
3. number of minutes (m) in a number of hours (h)?
m= h 60
m= 60 ÷ h
h= 60 × m
m= 60 × h
Try this! Calculate the value of the remaining unknowns in above, if…
1. the cost of a hurley (h) is €45
2. Cáit (C) is 13
3. the input number (i) is 5
4. the total number of items (t) is 21 and number of people (p) is 3
5. Emily (E) is 10
I think the word ‘is’ could be swapped with an = symbol in these sentences.
Solving Unknown Values
A Work out the value of each expression if z = 10.
B Now, work out the value of each expression from A above, if z = 15.
C Write two different equations for each of these and solve for y.
D Visualise or draw a branching bond to help you solve for w in each equation.
E Write two different equations for each and solve for z.
Model and solve for y in each equation.
Functions, Inputs and Outputs
A What are the missing inputs and outputs?
B In your copy, create an input/output table to represent each of these scenarios. Record the function rule clearly at the top. Choose three different inputs each time.
1. You are baking cookies, and each tray holds 12 cookies. Work out how many cookies (output) you will bake based on a certain number of trays (input).
2. A snail is crawling at a speed of 3cm per minute. Work out how far the snail has crawled (output) after a certain number of minutes (input).
3. You are saving €5 each week. Work out how much money you will have saved after a certain number of weeks.
4. You are ordering pizzas for a party, and you estimate that one pizza is needed for every 4 people. Work out how many whole pizzas are needed for a certain number of people.
Try this! In your copy, create a suitable input/output table to represent each of these scenarios. Record the function rule clearly at the top
1. Tom’s Taxis charges €5 at the start of the journey, and then €2 per km thereafter Work out the cost of three different journeys measuring a certain number of km.
2. A plumber charges a standard €50 call-out fee, and then €30 per hour thereafter Work out the charge for three different jobs lasting a certain number of hours.
3. Mia is collecting cards, and she gets 4 new cards every day. Work out how many cards she will have after a certain number of days, if she already had 10 cards to begin with.
Fractions as Decimals and Decimals as Fractions
A Express each coloured amount as a fraction and a decimal.
B Use your thousandths grids (PCM 1) to represent the fractions below. Write the equivalent decimal form for each.
C Write the position of each letter as (a) a decimal and (b) a fraction.
D Complete the table.
(a) Decimal (b) Decimal expanded form (c) Fraction (d) Fraction expanded form
Try this! Jay made a 4-digit decimal number: . 0
The other digits he used were 2, 4 and 6.
Write in (a) decimal and (b) fraction form…
1. the greatest number that he could have made (a) (b)
2. the least number that he could have made. (a) (b)
Converting Fractions to Decimals and Decimals to Fractions
A Complete these in order to express the fractions as decimals.
B In your copy, do these in a similar way to the ones in A above.
C Complete these in order to express the decimals as fractions.
D In your copy, do these in a similar way to the ones in C above.
E Match each decimal to its most likely equivalent fraction. Use a calculator to check your answers.
F Let’s talk!
Not all fractions convert easily to decimals Some become recurring decimals.
1. Which fractions in E convert into recurring decimals? Why do you think this is?
2. Recurring decimals are often written with a dot over the repeating digit(s) to show they repeat. Why do you think this is?
Try this!
1. In your copy, express each test result on this report card as... (a) a fraction in its simplest form (b) a decimal number
2. In which subject did this pupil perform... (a) best? (b) worst?
Report card: test results
English 60 out of 80
26 out of 40
40 out of 50
18 out of 20 Geography21 out of 24
17 out of 20
3. When Lexi converted a specific fraction to a decimal her answer was 0·6. Write down three possible fractions.
Recurring decimal: has infinite (never ending) digits
What model could you use to prove your answers?
Percentages
A Write the amount that is coloured as per cent and hundredths (fraction form and decimal form).
2. 12% of the people surveyed wear glasses. How many people wear glasses, if the number of people surveyed was (a) 100? (b) 200? (c) 50? 17 17
B In your copy, write the amount that is uncoloured in A above as per cent and hundredths (fraction form and decimal form).
C Solve these.
1. Jay has read 79 pages of a 100-page book. What percentage has he (a) read? (b) left to read?
2. There are 100 children in a club. 37 of them are boys. What percentage are (a) boys? (b) girls?
3. There were 100 trees planted, but 13 were damaged during a storm. What percentage of the trees were undamaged?
Try this!
1. Of the 10 bananas in a bunch, three of them are not yet ready to eat (unripened).
(a) What percentage are ready to eat? (b) What percentage are not ready to eat?
Let’s talk!
Read the statements. What do you think they mean? When might people use them? What other per cent phrases do you know?
I believe in you 100%. I’m 100% sure that’s right.
Percentages as Fractions and Decimals
A Write the amount that is coloured as a percentage and a fraction.
D Fill in one of the values below to make each expression true.
F Solve this.
The children were practising taking shots. Out of 25 shots, Jay scored 15 points. Out of 20 shots, Mia scored 13 points. Out of 10 shots, Lexi scored 7 points.
Whose point-scoring was the most accurate?
Try this! Who has read the most of their book?
I’ve read 243 pages of a 300-page book.
I’ve read 210 pages of a 250-page book.
I’ve read 182 pages of a 200-page book.
A Solve these.
Calculating Percentages of Amounts
1. In a club, 60% of the members were boys and the rest were girls. If there were 120 members, how many were (a) boys?
(b) girls?
2. Of the cars in a car park, 25% were black, 35% were white and the rest were silver If 56 of the cars were silver, how many cars…
(a) were there altogether?
(b) were black?
(c) were white?
3. There are 80 members in a chess club If 30% of the members were absent last week, how many were
(a) absent?
(b) present?
4. Mia has saved 60% of the cost of a new computer game. If she has saved €30 so far, how much…
(a) does the game cost? €
(b) more does she need to save? €
5. Dara has read 40% of his book. If he has 96 pages left to read, how many pages are there in the entire book?
Use the bar models to help you.
B Use a calculator to check your answers to A above.
C Model and solve these. Use a calculator to check your answers.
1. In a class of 30 pupils, 60% said that their favourite sport is Gaelic football. How many pupils in the class chose Gaelic football?
2. James has saved €35, which is 50% of the money he needs to buy a new sports strip What is the total cost of the sports strip? €
3. 75% of the players on a soccer team played in the final. If 12 of the players played in the final, what is the total number of players on the team?
4. 20% of the 185 flowers planted in a garden had bloomed. How many flowers were yet to bloom?
5. So far, a school has raised €200, which is 10% of their total fundraising goal. What is the school’s total fundraising goal? €
6. The O’Neill family are travelling from Cork to Dublin, a distance of 260km. When they have travelled 90% of the journey, how far have they still to go? km
7. In a test in which every question was worth the same marks, Toni scored 75%. If she answered 36 questions correctly, how many questions…
(a) did he get wrong?
(b) were there in the test altogether?
8. Zofia wants to send letters to 16 friends. 50% of the letters will need one sheet of paper each and the rest will need two sheets each. How many sheets will be needed altogether?
9. Sam’s smart watch is 80% charged. If the fully charged battery lasts 5 days, how many days are left?
Try this! Look at the image and answer the questions.
1. How long more until this device is fully charged? mins
2. At what time will that be?
3. If this device was allowed to completely discharge (i.e. for the battery to run out), for how many hours and minutes would it need to be plugged in to fully recharge? hrs mins
4. If this device was plugged in 1 hour and 10 minutes ago, what percentage would it be at then?
Percentage Increase and Decrease
A Let’s talk!
Look at B below. Is each special offer an increase or a decrease? Could you solve it mentally? If yes, how? If no, how else might you solve it?
B Model and solve each of these (without a calculator) to find out the current value.
Was €260 Now 50% off Was €80 Now 50% off Was €300 Now 40% off Was €480 Now 50% off Was 1.5l Now with 10% extra free Was 0.5kg Now with 30% extra free Was 20l Now with 5% extra free
Was 800ml Now with 20% extra free
C Use a calculator to check your answers to B above in a different way. In your copy, record the inputs you made on the calculator.
Try this! Use a calculator to work out the…
1. total cost of a meal that was €95 before adding a 10% service charge. €
2. total cost of a holiday that was €1,785 before adding a 4% booking fee. €
3. total cost of cinema tickets that cost €48 before adding a 3% bank card fee. €
4. amount of fibre in an apple that contains 8% of the recommended daily fibre intake of 75g. g
Round your answers to the nearest whole number as needed.
5. amount of sugar in a cookie that weighs 23g and is made up of 26% sugar g
Teacher’s Planning Book Extract Contents
The Maths and Me Teacher’s Pack contains a Teacher’s Planning Book and a Teacher’s Resources Book. Along with the plans for each unit, the Teacher’s Planning Book also contains a comprehensive introductory section featuring the following sections:
● How Maths and Me Aligns to the Primary Maths Curriculum
● Your Guide to Maths and Me
● Supporting Maths Learning in Your Classroom
● Yearly Overview
The Maths and Me Teacher’s Resources Book includes all additional materials outside of planning. A range of photocopiable materials (PCMs) will be available, including general PCMs. For each unit the following are provided:
● Let’s Strengthen PCMs
● Let’s Deepen PCMs
● Let’s Strengthen Suggestions for Teachers
● Maths Journal Prompts PCMs
● Home/School Links PCMs
● Games Bank
Additionally an Answer Booklet and various printables (such as manipulative printables, Maths Language Lists and Equipment List) are also available.
Str and(s) > Str and Unit(s) Number > Fr actions.
Outcome ( s )T hr ough appr opriately pla yful and engaging learning experiences childr en should be able to: explor e (model, compar e and con ve rt ) the re lationships betw een fr actions, decimals and per centages; in ve stigate pr opor tionality and ra tios of quantities (sets).
Learning
Learning Experiences Assessment
Lesson Fo cus of Learning (with Elements) CM
Intuitiv e Assessment: re sponding to emer ging misconceptions Planned Inter actions: re sponding to insights gleaned fr om childr en’ s re sponses to learning experiences
D P Re ason & Re spond L1–3, 5, 7–8
D Think -P air -Shar e L1, 3–4, 8
D Wr ite-Hide-Sho w L1–5, 7–8
C P Exploring Other Fr actions L1
D Quick Images L1, 4–5
C Modeling Pe r Cent L3
D Concept Car toon L4, 8
C Let’ s in ve stigate! Benchmark Fr actions as Pe rc entages and Decimals L4
D Wo uld This Wo rk? L6–7
D C P Build it; Sk etch it; Wr ite it
L6–8
D Thr eeAct Ta sk L6
D Number Strings L7
Assessment Ev ents: inf ormation gather ed fr om completion of the unit assessment in the Pr ogr ess Assessment Booklet page XX
1 Fr actions as Decimals and Decimals as Fr actions: Models and expr esses fr actions as decimals and vice ve rsa (C ); Identifies benchmark fr actions as a decimal (R); Identifies and re pr esents equivalent fr actions of a giv en fr action, including tenths and hundr edths (C )
2 Con ve rt ing Fr actions to Decimals and Decimals to Fr actions: Con ve rt s fr actions to decimals and vice ve rsa, without manipulativ es and other re pr esentations (U&C )
3 Pe rc entages: Re cognises the per cent symbol (%) and re lates this to ‘number of par ts per hundr ed’ (U&C ); Identifies per centages as a fr action [with denominator 10 0] and as a decimal (R)
4 Pe rc entages as Fr actions and Decimals: Demonstr ates an understanding of benchmark per centages (F or example: 50%, 25%, 10 %, 1% ) in the context of fr actions and decimals (C ); Identifies benchmark per centages as a fr action in its lo we st fo rm and as a decimal (R); Models and expr esses per centages as fr actions and decimals and vice ve rsa (C )
5 Con ve rt ing betw een Fr actions, Decimals and Pe rc entages: Explor es the re lationship betw een fr actions, decimals and per centages (R); Con ve rt s betw een fr actions, decimals and per centages, without manipulativ es and other re pr esentations (U&C )
Print re sour ces Pupil’ s Book pages 76 –84 PCMs XX
6 Comparing and Ordering Fr actions, Decimals and Pe rc entages: Orders fr actions, decimals and per centages, of the same and of various whole units, by their compar ativ e value (R)
7 Calculating Pe rc entages of Amounts: Solv es pr oblems in vo lving oper ations with whole numbers, fr actions, decimals and benchmark per centages (A&PS)
8 Pe rc entage Incr ease and Decr ease: Solv es pr oblems re lating to per centage incr eases and decr eases (A&PS); Completes pr oblemsolving tasks in vo lving per centages, fr actions and measur es, explaining methods and re asoning (A&PS)
9 Re view and Re flect: Re views and re flects on learning (U&C )
CM: Cuntas Míosúil: please tick when yo u ha ve completed the fo cus of learning. Learning Experiences: C concr ete activity; D digital activity; P activity based on printed materials, fo llo we d by lesson numbers.
Ke y: Elements: (U&C ) Understanding and Connecting; (C ) Communicating; (R) Re asoning; (A&PS) Applying and Pr oblem-Solving.
Progression Continua
Maths Language
Equipment
Inclusive Practices
Additional information for planning
See ‘5th Class Maths and Me Progression Continua Overview’ for a detailed breakdown of how all progression continua are covered.
See ‘5th Class Maths and Me Maths Language Overview’ (Appendix 2) and individual lesson plans and Unit 8 Maths Language Cards.
See ‘5th Class Maths and Me Maths Equipment Overview’ (Appendix 3) and individual lesson plans.
● See Let’s Strengthen and Let’s Deepen suggestions throughout lesson plans.
● See Unit 8 Let’s Strengthen Suggestions for Teachers. (These address the Common Misconceptions and Difficulties listed below.)
● See Unit 8 Let’s Strengthen PCM.
● See Unit 8 Let’s Deepen PCM.
Integration See individual lesson plans.
Home/School Links See the Unit 8 Home/School Links PCM and Pupil/Parent App.
Background and rationale
● This unit is a two-week block of content that builds upon the children’s understanding from Unit 1 Place Value (decimals, in particular), Units 2 and 3 Operations 1 and 2 (multiplication and division, including multiples and factors) and Unit 6 Fractions. This unit also lays the foundations for future exploration of fractional understanding in Unit 15 Ratio.
● In this unit, the children will be formally introduced to the concept of percentages, but it is likely that they will already have encountered this concept informally in a variety of real-world contexts.
● It is essential that calculator use is developed as a tool, alongside other strategies and models, such as pictorial representations (e.g. thousandths and hundredths grids, number lines, bar models) and written calculations. Once the children grasp the concept, and if demonstrated and used meaningfully, the use of calculators can help them with checking answers and encourage them to estimate. It can also allow them to focus on the process in an efficient way, rather than focusing on doing the calculations only
● In Maths and Me, we have consciously avoided focusing too much on memorising rules for converting between percentages, fractions and decimals, such as ‘to convert percentages to decimals, move the decimal point two places to the left’, or ‘to convert fractions to percentages, multiply the numerator by 100 and divide by the denominator’. Emphasising rules such as these can be counterproductive for several reasons, including: hindering conceptual understanding (children may not understand the underlying rationale); discouraging exploration and identification of patterns and relationships, leading to confusion and mistakes (they might mix up the rule, or apply it correctly to the wrong context); limiting problemsolving flexibility (rigid application of rules does not foster adaptive reasoning).
Common misconceptions and difficulties
Difficulties in understanding the connection between percentages and their equivalents may stem from misconceptions about fractions or decimals. Therefore, refer also as required to the Common Misconceptions and Difficulties for Unit 1 Place Value and Unit 6 Fractions for further insight.
● The children may correctly identify that a percentage such as 25% means 25 100 , but struggle to simplify this fraction to its lowest form of 1 4 due to misconceptions about fractions and factors.
● When converting a percentage to a decimal, they may misplace the digits. For example, they might write 50% as 50.0 instead of 0.50 or 0.5. It is important to constantly reinforce that since per cent is the same as hundredths, we typically are using the first two decimal places (i.e. tenths and hundredths) when converting percentages to decimals.
● They may be able to recall the equivalent benchmark percentages and fractions (knowing that 50% is 1 2 , 1 4 is 25%, etc.), while not understanding why (e.g. that 50% equals 50 100 , which simplifies to 1 2 ).
● They may struggle to convert fractions to percentages when they do not have a denominator of 100 (e.g. converting 1 5 to a percentage). They may not realise or recall that they need to find an equivalent fraction with a denominator of 100 first (e.g 20 100 ) before converting to the equivalent percentage (e.g. 20%).
● When comparing and ordering fractions, decimals and percentages (e.g. 1 2 , 0.4, 6%), they may try to compare them in their different formats. This in turn may lead to other difficulties, such as incorrectly assuming that the value with the larger digit/number is always greater (e.g. thinking 6% is the greatest because 6 is greater than 1, 2 or 4). Instead, encourage them to convert the values to hundredths and/or to use their reasoning and ‘fraction sense’ to justify the ‘size’ of each amount. For example, 0.4 is 4 tenths, whereas 1 2 as tenths is 5 tenths (greatest), and 6% is 6 hundredths, which has no tenths and is therefore the least.
● When calculating a percentage of an amount, they may apply a procedure without knowing why or how it works. For example, when finding 75% of 60, they might apply the ‘divide by 100 and multiply by the per cent’ procedure (i.e. divide 60 by 100, then multiply by 75), which works, but they may not understand the underlying concept that they are finding the value of 75 100 of the total amount. In doing so they may also fail to appreciate that since 75% is also 3 4 , it would have been easier to divide 60 by 4, then multiply by 3.
● When calculating percentage increase and decrease, they might confuse the meanings of increase and decrease. Furthermore, they may incorrectly add or subtract the percentage itself. For example, if an item costs €50 and has a 10% discount, they might say that the new price is €40, having subtracted 10 instead of 10% of €50 (i.e. €5).
The Unit 8 Let’s Strengthen Suggestions for Teachers address the common misconceptions and difficulties listed above.
Mathematical models and representations
● Thousandths and hundredths grids
● Diagrams of 2-D shapes divided into various parts/fractions and/or with various parts coloured
● Bar models
● Physical materials that can be used to build bar models, such as number rods (e.g. Cuisenaire) or interlocking cubes
● Number lines and other similar linear representations, including metre stick
● Physical materials to represent hundredths (e.g. base ten money, place value counters)
3.
Thousandths grid
Write the position of each letter as
Circle divided into hundredths
Teaching tip
The following manipulative printables are available to support the unit: Thousandths Grid, Hundredths Grid, Open Number Lines, Base Ten Money, Place Value Counters. Click on the resources icon on the Maths and Me book cover on edcolearning.ie
Day 1, Lesson 1
Fractions as Decimals and Decimals as Fractions
● Models and expresses fractions as decimals and vice versa (C)
● Identifies benchmark fractions as a decimal (R)
● Identifies and represents equivalent fractions of a given fraction, including tenths and hundredths (C) Focus of learning (with Elements)
Learning experiences
Digital activity: Same But Different – Fractions as Decimals
MAM Routines: Reason & Respond, with Think-Pair-Share; WriteHide-Show
Digital activity: Benchmark Fractions
MAM Routines: Reason & Respond, with Write-Hide-Show
Concrete activity: Exploring Other Fractions
Pupil’s Book page 76: Fractions as Decimals and Decimals as Fractions
Maths language
Warm-up
D Digital activity: Same But Different – Fractions as Decimals MAM Routines: Reason & Respond, with Think-Pair-Share; Write-Hide-Show
Play the slideshow and, using Think-Pair-Share, ask the children to propose reasons for why the images are the same and why they are different.
Using Write-Hide-Show to gather feedback, ask:
● How might you write these amounts?
● Can they be written/shown in multiple ways? Show us.
Teaching tip
If not suggested, prompt the children to consider equivalent fraction forms, when appropriate (e.g. 7 10 = 70 100 = 700 1,000). While decimal equivalents (e.g.
D Digital activity: Benchmark Fractions
MAM Routines: Reason & Respond with Write-Hide-Show
0.7, 0.70, 0.700) are not required at this stage, if suggested, acknowledge them and ask the children to suggest reasons for why they are also valid answers.
Teaching tip
Encourage the children to use both decimal and fractional language when verbalising decimal notation, i.e. expressing 0.75 as ‘zero point seven five’ and also as ‘seventy-five hundredths’. Using fractional language to read decimal numbers reinforces the value of the digit(s) in the decimal place(s). However, when using decimal language, emphasise that it is incorrect to say ‘zero point seventy-five’, because the suffix -ty means tens.
Main event
Teaching tip
Using PCM xx Thousandths Grids placed in reusable dry-erase pockets, the children could copy each of the fractions being displayed each time.
Display the slideshow. Using Write-Hide-Show to gather feedback, ask:
● What fraction of the grid is coloured?
● How might you represent this fraction in writing?
● How might you represent it in fraction form?
● How might you represent it in decimal form?
For each slide (excluding the first slide), repeat the questions above, but this time, ask:
● What fraction of the grid is uncoloured?
Let’s strengthen
The children may find it beneficial to review the video titled ‘Decimal Fractions’ from Unit 1, Lesson 3.
Let’s deepen
Challenge the children to suggest how they might prove that the fraction form and decimal form are equivalent, without referring to the image or using any visual or concrete supports. For example:
● 1 4 multiplied by 25 25 (a fraction equivalent of 1) = 25 100 = 0.25
● 0.05 = 5 100 divided by 5 5 (a fraction equivalent of 1) = 1 20 (simplest form)
Explain how the first step must always be to change the fraction or decimal to its equivalent fraction form, where the denominator is 10, 100 or 1,000. This will be explored in more depth in the next lesson.
C P Concrete activity: Exploring Other Fractions
Distribute a copy of PCM xx Thousandths Grids placed inside a reusable dry-erase pocket to each pair. Tell the children to turn to activity B on page 76 of their Pupil’s Books. Ask them to colour or highlight the given fractional amounts on the PCM, and record the amounts as decimal form.
Unit 8: Fractions, Decimals and Percentages
Teaching tip
Depending on the availability of digital devices, online virtual manipulatives could also be used to model fractions. See the Optional Consolidation and Extension Possibilities below for suggestions.
The following questions could be posed during or after the activity to assess understanding:
● Show me your representation. How do you know you are correct?
● How might you do this in an efficient way (e.g. for 129 thousandths, instead of colouring in and counting 129 individual parts, colour instead one full row/column and 2 hundredths, and 9 thousandths)?
● What value is represented by the uncoloured parts of the grid? How do you know? Do you need to count all of the uncoloured parts? Explain why.
Let’s deepen
Challenge the children to suggest strategies for identifying the value of the uncoloured amount, and to come up with a way to justify their approach. If not suggested, prompt them to first consider how many parts in total, and then use their understanding of bonds of 10, 100 and/or 1,000 to help identify the uncoloured amount.
Teaching tip
Maths Journals: The children could record what they have learned in this lesson, and/or paste in their completed thousandths grids from PCM xx.
P Pupil’s Book page 76: Fractions as Decimals and Decimals as Fractions
Optional consolidation and extension possibilities
Maths Journal Use the Lesson 1 Maths Journal Prompt from the Unit 8 Maths Journal Prompts PCM.
Display Set up a display for Fractions, Decimals and Percentages or add to an existing display for Fractions in the classroom. This could include examples of the children’s work from this and subequent units, alongside appropriate labels (see the Unit 8 Maths Language Cards).
Digital Online Tool These virtual manipulatives could be very useful to model and compare tenths, hundredths and/or thousandths: ‘Decimal Squares’ at edco.ie/7any
Games Bank Play ‘4 Spins to 1’ (Decimals). The children could use PCM xx Thousandths Grids to colour in what they choose each time. Review and Reflect Use the prompt questions in the Review and Reflect poster.
Day 2, Lesson 2
Converting Fractions to Decimals and Decimals to Fractions
Focus of learning (with Elements)
● Converts fractions to decimals and vice versa, without manipulatives and other representations (U&C)
Learning experiences
Digital activity: What Value Does This Represent? (1)
MAM Routines: Quick Images, with Write-Hide-Show
Video: Converting between Fractions and Decimals
MAM Routines: Reason & Respond, with Write-Hide-Show
Pupil’s Book page 77: Converting Fractions to Decimals and Decimals to Fractions
Maths language
Teaching tip
● There is no equipment needed for this lesson. Equipment
In the previous lesson, the children learned to identify and represent equivalent fractions and decimals using thousandths grids, whereas in this lesson, they will use alternative strategies that do not depend on manipulatives or other visual representations.
Warm-up
D Digital activity: What Value Does This Represent? (1) MAM Routines: Quick Images, with Write-Hide-Show
Teaching tip
While this is a Quick Images resource, due to the level of detail in the graphics, these slides may need to be displayed on screen for longer than usual.
Display the slideshow. Click to briefly reveal and then hide the image on the first slide, which shows a representation using a thousandths grid. Ask the children to record the value represented on their MWBs, and to ‘show’ their proposed answer when called upon. Record all of the proposed answers on the board, being careful not to give any undue weight to the correct answer. Ask:
● Which answer are you going for?
● What proof do you have?
● Does anybody have different proof?
● Are there any proposed answers that are equivalent forms, representing the same amount or value (‘same value, different appearance’)?
If any unreasonable answers are suggested, ask:
● Are there any answers that are unreasonable/ unlikely because they don’t make sense? Which ones?
● Why do you think this?
When there are no new strategies to discuss, reveal the image again and confirm the answer using a variety of possible strategies. Repeat for the remaining images.
Teaching tip
For every image, prompt the children to suggest multiple different ways to write the amount (e.g. 1 4 as 25 100 , 250 1,000, 0.25, 0.250). As the children suggest various forms (or as they become more confident with identification), identify each as decimal or fraction form.
● denominator, powers of ten, fraction equivalent to 1, simplify, greatest factor
Main event
D Converting between Fractions and Decimals
MAM Routines: Reason & Respond, with Write-Hide-Show
Play the video and, using Write-Hide-Show, ask the children to respond to the questions on their MWBs, giving reasons for their responses as appropriate.
Teaching tip
While the children are learning to discuss and identify the greatest or highest factor possible, they will not be expected to use the terminology of highest common factor (HCF) until 6th Class.
Let’s strengthen
The children may struggle to identify the common factors of numbers. They may benefit from compiling a personal reference chart (e.g. in their Maths Journals) in which they list the factor pairs for the composite numbers up to 25, and other composite numbers that commonly occur (e.g. 40, 50, 100). See the Unit 8 Let’s Strengthen Suggestions for Teachers for more information.
P Pupil’s Book page 77: Converting Fractions to Decimals and Decimals to Fractions
Optional consolidation and extension possibilities
Maths Journal Use the Lesson 2 Maths Journal
Prompt from the Unit 8 Maths Journal Prompts PCM.
Display The children could contribute samples of their own work from this lesson and label them.
Games Bank Play ‘Spin and Keep’ (Decimals).
Online Game Play this game, choosing ‘convert fractions to decimals’: ‘Maths Invaders’ at edco.ie/cqr6
Review and Reflect Use the prompt questions in the Review and Reflect poster.
Percentages
MAM Routines:
Pupil’s
Warm-up
D Digital activity: Same But Different –Percentages MAM Routines: Reason & Respond, with Think-Pair-Share
Play the slideshow and, using Think-Pair-Share, ask the children to propose reasons for why the images are the same and why they are different.
Teaching tip
If not suggested, prompt the children to consider equivalent fraction forms, when appropriate (e.g. 9 out of 10 = 9 10 = 90 out of 100 = 90 100 = 0.9 = 0.90).
Main event
D Video: Introducing Per Cent MAM Routines: Reason & Respond, with Write-Hide-Show
Play the video and, using Write-Hide-Show, ask the children to respond to the questions on their MWBs, giving reasons for their responses as appropriate.
C Concrete activity: Modeling Per Cent
Distribute a copy of PCM xx Hundredths Grids placed inside a reusable dry-erase pocket to each child. Ask the children to use each grid to represent a random value as a percentage (e.g. 36%, 54%, 19%, 1%, 7%).
Teaching tip
The digital version of the PCM could also be displayed on the IWB in order to demonstrate each representation using the inbuilt pen/ annotation features.
The children shade/colour in the amount and record the values as per cent and hundredths in both fraction and decimal form below the pictorial representation (e.g. 36 per cent as 36%, 36 100 and 0.36).
Teaching tip
Depending on the availability of digital devices, online virtual manipulatives could also be used to model per cent. See the Optional Consolidation and Extension Possibilities below for suggestions.
The following questions could be posed during or after the activity to assess understanding:
● Show me your representation. How do you know you are correct?
● How might you do this in an efficient way? (e.g. For 36%, instead of colouring in and counting 36 individual squares, three full rows/columns could be coloured, plus six more squares).
● Express the uncoloured parts of the grid in different ways.
● How do you know? Do you need to count all of the uncoloured parts? Explain why.
Teaching tip
Maths Journals: The children could record what they have learned in this lesson, and/or paste in their completed hundredths grids from PCM xx.
Let’s deepen
Challenge the children to suggest strategies for identifying the value of the uncoloured amount, and to come up with a way to justify their approach. If not suggested, prompt them to realise that if there is 100% (100 hundredths) in total, they can use their understanding of bonds of 10 and 100 to help them identify and express the uncoloured amount.
P Pupil’s Book page 78: Percentages
Optional consolidation and extension possibilities
Maths Journal Use the Lesson 3 Maths Journal
Prompt from the Unit 8 Maths Journal Prompts PCM.
Display The children could contribute samples of their own work from this lesson and label them.
Integration Geography: Interpreting data about the world around us that uses percentages (e.g. ‘What percentage of the Earth’s surface is covered by water?’ or ‘What was the percentage of the different types of waste collected in our class clean-up?’).
STEM: Technology could be used to create spreadsheets and graphs of data collected for the aforementioned Geography activity, as well as to analyse and communicate findings using percentages.
Maths Eyes Use food packaging, supermarket flyers, brochures, etc., to collect images that express percentages. Add these to the class display (see above). Ask the children to find examples of
percentages in the classroom, school and/or at home. They could draw the examples or take photos to share digitally with the teacher and class.
Digital Online Tools These virtual manipulatives, while not specific to percentages, can be used to represent them:
● ‘Number Frames’ (counters and customisable frames; drag a 100 frame out onto the workspace) at: edco.ie/s89u
● ‘Number Rack’ (virtual rekenrek; add rows of beads until there are 10 rows of ten) at: edco.ie/3jff
● ‘Beadstring’ (horizontal 100-bead string; use the beads to model decimals to hundredths) at: edco.ie/7285
Review and Reflect Use the prompt questions in the Review and Reflect poster.
Day 4, Lesson 4
Percentages as Fractions and Decimals
Focus of learning (with Elements)
● Demonstrates an understanding of benchmark percentages (For example: 50%, 25%, 10%, 1%) in the context of fractions and decimals (C)
● Identifies benchmark percentages as a fraction in its lowest form and as a decimal (R)
● Models and expresses percentages as fractions and decimals and vice versa (C)
Learning experiences
Digital activity: What Value Does This Represent? (2)
MAM Routines: Quick Images, with Write-Hide-Show
Digital activity: Can 3 4 Be Expressed as a Percentage?
MAM Routines: Concept Cartoon, with Think-Pair-Share
Concrete activity: Let’s investigate! Benchmark Fractions as Percentages and Decimals
Pupil’s Book page 79: Percentages as Fractions and Decimals
Maths language
Teaching tip
● Reusable dry-erase pockets (if available)
● PCM xx Equipment
● benchmark fractions/percentages, equivalent, ‘same value, different appearance’, simplest (form)
While the children are not yet expected to calculate conversions of per cent to fractions and decimals and vice versa (this is for the next lesson), it is anticipated that they will understand and recall the per cent form of the benchmark fractions, especially 1 2 , 1 4 , 3 4 , 1 10 and 1 100
Warm-up
D Digital activity: What Value Does This Represent? (2) MAM Routines: Quick Images, with Write-Hide-Show
Teaching tip
While this is a Quick Images resource, due to the level of detail in the graphics, these slides may need to be displayed on screen for longer than usual.
Display the slideshow. Click to briefly reveal and then hide the image on the first slide, which shows a representation of per cent as hundredths. Ask the children to record the value represented on their MWBs in multiple ways, and to ‘show’ their proposed answer when called upon. Record all of the proposed answers on the board, being careful not to give any undue weight to the correct answer. Ask:
● Which answer are you going for?
● What proof do you have?
● Does anybody have different proof?
● Are there any proposed answers that are equivalent, representing the same amount or value (‘same value, different appearance’)?
If any unreasonable answers are suggested, ask:
● Are there any answers that are unreasonable/ unlikely because they don’t make sense? Which ones?
● Why do you think this?
When there are no new strategies to discuss, reveal the image again and confirm the answer using a variety of possible strategies. Repeat for the remaining images.
Teaching tip
For every image, prompt the children to suggest multiple ways of writing the value (e.g. 63 hundredths as 63%, 63 100 , 0.63).
Main event
D Digital activity: Can 3 4 Be Expressed as a Percentage? MAM Routines: Concept Cartoon, with Think-Pair-Share
Display the Concept Cartoon, in which the characters are discussing whether or not 3 4 can be expressed as a percentage. Click to hear each character’s thoughts. Then, using Think-Pair-Share, ask:
● What do you think? Explain why.
● (Point to a specific character.) Do you agree with their idea? Explain why.
● Do you think something different? What do you think? Why do you think this?
If appropriate, record the children’s responses to these questions on the board. Allow the children the opportunity to respond to (agree/disagree with or query) others’ responses, but do not confirm or reject any of the ideas. Ask:
● How can we find out who is correct?
Encourage the children to present their suggested approaches and/or solutions.
C Concrete activity: Let’s investigate! Benchmark Fractions as Percentages and Decimals
Teaching tip
Benchmark fractions are common, easy-tovisualise fractions that serve as reference points for estimating and comparing other fractions. Commonly accepted benchmark fractions include 1 2 , 1 4 , 1 5 , 1 10 , 1 20 , 1 100 and their multiples. The fraction equivalents of 0 and 1 are also often included, since they provide context for the fractions that fall between them.
Distribute a copy of PCM xx Hundredths Grids placed inside a reusable dry-erase pocket to each child. Display the digital version of the PCM on the IWB. Ask/say:
● The common fractions, that are often easiest to work with, are sometimes referred to as ‘benchmark’ fractions. Write some fractions that you think are benchmark fractions. (Record the children’s suggestions on the board.)
● Let’s work with one of these benchmark fractions (e.g. 1 4 ). Colour in one of the grids on your PCM to represent 1 quarter. Show me. (Encourage the children to do this in different ways, and then demonstrate these on the IWB.)
● What do you notice about the parts of the grid that have been coloured in?
● How many hundredths have been coloured? Record this in both fraction and decimal form. (25 hundredths = 25 100 = 0.25)
● What is the amount of hundredths when expressed as a percentage? (25%)
● What do you notice about the parts of the grid that are not coloured? How might you express this portion?
Repeat this process with other fractions, including
and
Unit 8: Fractions, Decimals and Percentages
Let’s deepen
Challenge the children to identify the per cent equivalent for various amounts of tenths, fifths or twentieths. For example, if they know the per cent equivalent for 1 10 , 1 5 and 1 20 , can they use this knowledge to identify the per cent equivalent for 3 10 , 4 5 and 7 20 ?
P Pupil’s Book page 79: Percentages as Fractions and Decimals
Optional consolidation and extension possibilities
Maths Journal Use the Lesson 4 Maths Journal
Prompt from the Unit 8 Maths Journal Prompts PCM.
The children could also create their own personal fractions, decimals and percentages reference grid for the new equivalent forms they identified in this lesson in their Maths Journals, and paste in their completed hundredths grids from PCM xx.
Display The children could contribute samples of their own work from this lesson and label them. Games Bank Play ‘Pairs’, using PCM xx Pairs (1 of 2).
Online Game Play the game ‘Decention’ (which involves making groups containing a fraction, a decimal and a percentage of equal value) at edco.ie/59cw
Story Read Piece = Part = Portion: Fractions = Decimals = Percents by Scott Gifford, which showcases real-world examples of fractions and their equivalents. A reading of the story is also available at edco.ie/ndhw
Review and Reflect Use the prompt questions in the Review and Reflect poster.
Day 5, Lesson 5
Converting between Fractions, Decimals and Percentages
● Explores the relationship between fractions, decimals and percentages (R)
● Converts between fractions, decimals and percentages, without manipulatives and other representations (U&C) Focus of learning (with Elements)
Digital activity: Benchmark Percentages
MAM Routines: Quick Images, with Write-Hide-Show
Video: Converting between Fractions, Decimals and Percentages
MAM Routines: Reason & Respond, with Write-Hide-Show
Pupil’s Book page 80: Converting between Fractions, Decimals and Percentages
● There is no equipment needed for this lesson. Equipment
● There is no new maths language for this lesson.
Warm-up
D Digital activity: Benchmark Percentages MAM Routines: Quick Images, with Write-Hide-Show
Teaching
tip
While this is a Quick Images resource, due to the level of detail in the graphics, these slides may need to be displayed on screen for longer than usual.
Display the slideshow. Click to briefly reveal and then hide the image on the first slide, which shows a representation using the hundredths grid. Ask the children to record the value represented on their MWBs in multiple ways, and to ‘show’ their proposed answer when called upon. Record all of the proposed answers on the board, being careful not to give any undue weight to the correct answer. Ask:
● Which answer are you going for?
● What proof do you have?
● Does anybody have different proof?
● Are there any proposed answers that are equivalent forms, representing the same amount or value (‘same value, different appearance’)?
If any unreasonable answers are suggested, ask:
● Are there any answers that are unreasonable/ unlikely because they don’t make sense? Which ones?
● Why do you think this?
When there are no new strategies to discuss, reveal the image again and confirm the answer, using a variety of possible strategies. Repeat for the remaining images.
Teaching tip
For every image, prompt the children to suggest multiple different ways to write the amount (e.g. 1 4 as 25 100 , 0.25, 25%). As the children suggest various forms, ask them also to identify it as being fraction, decimal or per cent form.
Let’s strengthen
The children may be able to recall the equivalent benchmark percentages and fractions as facts (knowing that 50% is 1 2 , 1 4 is 25%, etc.), while not understanding why. Encourage them to justify their reasoning (with models, as appropriate) in order to assess their understanding. For example, can they explain that 50% equals 50 100 , which simplifies to 1 2 ?
Main event
D Video: Converting between Fractions, Decimals and Percentages MAM Routines: Reason & Respond, with Write-Hide-Show
Play the video and, using Write-Hide-Show, ask the children to respond to the questions on their MWBs, giving reasons for their responses as appropriate.
Let’s strengthen
As mentioned previously, the children may struggle to identify the common factors of numbers, and so may benefit from compiling a personal reference chart (e.g. in their Maths Journals).
Additionally, when converting a percentage to a decimal, the children may misplace the digits. Constantly reinforce that because per cent is the same as hundredths, we typically are using the first two decimal places (i.e. tenths and hundredths).
P Pupil’s Book page 80: Converting between Fractions, Decimals and Percentages
Optional consolidation and extension possibilities
Maths Journal Use the Lesson 5 Maths Journal
Prompt from the Unit 8 Maths Journal Prompts PCM. The children could also create their own personal fractions, decimals and percentages reference grid for the new equivalent forms they identified in this lesson in their Maths Journals.
Display The children could contribute samples of their own work from this lesson and label them.
Online Game Play this game, choosing ‘convert fractions to percentages’: ‘Maths Invaders’ at edco.ie/cqr6
Games Bank Play ‘Pairs’, using PCM xx Pairs (1 of 2).
Fraction Tool To assess the children’s ability to recall the equivalent fractions, decimals, and percentages, go to the Manipulatives e-Toolkit and open the Fraction tool. Create a representation of a known fraction, (e.g. 1 2 ), and then ask the children to write on their MWBs, what the equivalent percentage and/or decimal for that amount would be. Click on the 1 2 symbol adjacent to the representation, to toggle through the decimal and the percentage numerical representations also.
Review and Reflect Use the prompt questions in the Review and Reflect poster.
Day 6, Lesson 6
Comparing and Ordering Fractions, Decimals and Percentages
Focus of learning (with Elements)
● Orders fractions, decimals and percentages, of the same and of various whole units, by their comparative value (R)
Digital activity: Order Them
MAM Routines: Would This Work?, with Build it; Sketch it; Write it
Digital activity: Cycling Competition
MAM Routine: Three-Act Task
Pupil’s Book page 81: Comparing and Ordering Fractions, Decimals and Percentages
Maths language
● compare, order, greater than (>), less than (<), equal to (=), common denominator
Warm-up
D Digital activity: Order Them MAM Routines: Would This Work?, with Build it; Sketch it; Write it
Display the activity and play the opening sequence.
Begin by asking the children to use Build it; Sketch it; Write it to model and solve this question:
● Put these amounts in order, from the least to the greatest.
Allow the children sufficient time to explore various ways to build, sketch and write solutions, and to provide feedback. Then, click to reveal the various
models and approaches chosen by the characters. Do they work? Encourage the children to discuss and consider the accuracy and efficiency of each character’s approach. Ask:
● Do the answers and/or approaches look reasonable? Explain why.
● How could we check the answers?
● Which is the most efficient way to arrive at an answer, in your opinion?
Unit 8: Fractions, Decimals and Percentages
Allow the children time to comment on each and justify if the methods/opinions work.
Teaching
tip
Encourage the children to also consider equivalent values for the benchmark values (e.g. 1 2 is the same as 50%, 5 10 and 50 hundredths). Given that the other amounts are both less than 50% and 5 10 , then 1 2 has to be the greatest.
Main event
D Digital activity: Cycling Competition MAM Routine: Three-Act Task
Teaching tip
Ensure that the children have calculators available to them for this task, should they choose to use them.
This is a Three-Act Task to work out which school won a cycling competition.
Act 1: Notice & Wonder
Play the animation, which shows a cycle-toschool competition for local schools. Using Think-Pair-Share, click to play or ask:
● What do you notice?
● What do you wonder?
Record the children’s responses to both questions on the board. Allow the children the opportunity to respond to (agree/disagree with or query) others’ responses, but do not confirm or reject any of the ideas.
● (Reveal the focus question.) Which school won the competition?
Act 2: Productive Struggle
Look at the image. Using Think-Pair-Share and Write-Hide-Show, click to play or ask:
● Write an estimate that is too high on your MWB.
● Write an estimate that is too low.
● Write a reasonable estimate.
The children work in pairs or small groups to answer the focus question. If necessary, prompt them by asking:
● Do you have enough information? What else do you need to know?
Once the children explain that knowing the number of cyclists is not sufficient information, and they need to know how many children there are in each school, click to flip the images and reveal that extra information.
Click to play or ask:
● What information do you have now?
● To get an answer, what needs to be done?
● What strategies can you use?
Using Build it; Sketch it; Write it, the children choose their preferred way to mathematically model their strategies/solution(s).
Teaching tip
If required prompt the children to express the cyclists in each school as a fraction of the overall pupils in that school, and to then convert the fractions to hundredths, simplifying where appropriate. For example:
A calculator could be used to explore possible factors for each initial fraction (e.g. exploring the common factors for 105 and 140). Another possible approach using a calculator would be to divide the numerator by the denominator to create a decimal (as shown in the video titled ‘Converting between Fractions and Decimals’ from Lesson 2).
Act 3: The Big Reveal
The children share and discuss their strategies, solutions and models. Click to play/ask:
● What answer did you get?
● What strategies did you use to get the answer?
● What do you think was the most efficient strategy?
Click to flip the card and play the final part of the animation to reveal the answer (Castlemore NS were the winners, as 90% of their pupils cycled all week).
Click to play or ask:
● Is this the answer that you expected? Why or why not?
● What ‘I wonder’ questions did you answer?
● Do you have any new ‘I wonder’ questions?
Teaching tip
Maths Journals: The children could record using images and/or words what they built, sketched or wrote above.
Unit 8: Fractions, Decimals and
P Pupil’s Book page 81: Comparing and ordering Fractions, Decimals and Percentages
Optional consolidation and extension possibilities
Maths Journal Use the Lesson 6 Maths Journal
Prompt from the Unit 8 Maths Journal Prompts PCM.
Display The children could contribute samples of their own work from this lesson and label them.
Games Bank Play ‘Percentage Pairs’, using PCM xx Pairs (2 of 2).
Review and Reflect Use the prompt questions in the Review and Reflect poster.
Days 7 and 8, Lesson 7
Calculating Percentages of Amounts
Focus of learning (with Elements)
● Solves problems involving operations with whole numbers, fractions, decimals and benchmark percentages (A&PS)
Learning experiences
Digital activity: Calculating using Related Percentages MAM Routines: Number Strings, with Write-Hide-Show
Digital activity: Boys in the School MAM Routines: Would This Work?, with Build it; Sketch it; Write it
Digital activity: Test Score MAM Routines: Would This Work?, with Build it; Sketch it; Write it
Concrete activity: Calculating Amounts Involving Percentages MAM Routines: Reason & Respond, with Build it; Sketch it; Write it
Pupil’s Book pages 82–83: Calculating Percentages of Amounts
● There is no new maths language for this lesson.
Warm-up
Teaching tip
It is not necessary to do all of the number strings. Choose a selection that best suits the ability of your class, on either or both days.
● Calculators
● Materials that can be used to build bar models (e.g. number rods or interlocking cubes) Equipment
D Digital activity: Calculating using Related Percentages MAM Routines: Number Strings, with Write-Hide-Show
This slideshow contains five number strings. Each slide builds up a number string, with one part revealed per click.
Maths language
Unit 8: Fractions, Decimals and Percentages
Play the slideshow, revealing the first part of the first number string. Using Write-Hide-Show, ask the children to record their proposed answer only on their MWBs. Emphasise that during the ‘hide’ stage they should consider their strategy and other strategies that might work.
Record all of the children’s answers on the board, being careful not to give away the correct answer. Ask:
● Are there any answers that are unreasonable/ unlikely because they don’t make sense? Which ones? Why do you think this? (e.g. Is the answer too big or too small because the incorrect operation was used?)
● Which answer do you agree with? Explain the strategy you used to get your answer.
● Did anybody use a different strategy?
Repeat with the remaining parts of the number string. Repeat using the other number strings.
Teaching tip
Encourage the children to identify the connections between each part of the number string (e.g. 25% is 1 4 and also 1 2 of 50%; 5% could be considered as 1 2 of 10% or 1 10 of 50%).
Let’s deepen
Challenge the children to suggest how they might calculate 17.5% of an amount using related percentages (e.g. 10% + 5% + 2.5%).
Main event
D Digital activity: Boys in the School MAM Routines: Would This Work?, with Build it; Sketch it; Write it
Let’s strengthen
To encourage the children in the drawing of bar models, it might be beneficial to review the digital resource titled ‘Slow Reveal Problems’ from Unit 6, Lesson 4.
Display the activity and play the opening sequence. Begin by asking the children to use Build it; Sketch it; Write it to model and solve this question:
● In a school of 350 children, 40% are boys. How many boys are there?
Allow the children sufficient time to explore various ways to build, sketch and write solutions and to provide feedback. Then, click to reveal the various models and approaches chosen by the characters. Do they work? Encourage the children to discuss and consider the accuracy and efficiency of each character’s approach. Ask:
● Do the answers and/or approaches look reasonable? Explain why.
● How could we check the answers?
● Which is the most efficient way to arrive at an answer, in your opinion?
Allow the children time to comment on each and justify if the methods/opinions work.
Teaching tip
Maths Journals: The children could record using images and/or words what they built, sketched or wrote above.
Teaching tip
Encourage the children to estimate. If not suggested, prompt them to realise that since 40% is less than half, the answer will be less than half of 350 – perhaps one hundred and something.
Afterwards, explore the various approaches further. Ask/say:
● When solving percentages, typically a fraction method or decimal method is used. Which of the children used a fraction method? Who used a decimal method?
● In what ways are the fraction methods similar? In what ways are they different?
● In your opinion, which fraction method was most efficient in this case?
● Will that fraction method always be the most efficient method? What might it depend on? (It will always depend on the numbers involved.)
● In what ways are the decimal methods similar? In what ways are they different?
● In your opinion, which decimal method was most efficient in this case?
● Will that decimal method always be the most efficient method? What might it depend on?
(While the calculator is probably most efficient, that will be dependent on a calculator being available.)
● Can a calculator be used alongside the fraction method or is it only appropriate to use with the decimal method? (It can be used with both, e.g. the calculator could have been used to divide 350 by 10 and multiply the answer by 4.)
● Another fraction method would have been to think of 40% as 40 100 . Would this have resulted in a more efficient or less efficient calculation in your opinion?
● When might it be more efficient to use a calculator? (if the fraction equivalent is not an easy one to divide/multiply by).
D Digital activity: Test Score MAM Routines: Would This Work?, with Build it; Sketch it; Write it
Display the activity and play the opening sequence. Begin by asking the children to use Build it; Sketch it; Write it to model and solve this question:
● How many questions were on the test in total?
Allow the children sufficient time to explore various ways to build, sketch and write solutions and to provide feedback. Then, click to reveal the various models and approaches chosen by the characters. Do they work? Encourage the children to discuss and consider the accuracy and efficiency of each character’s approach. Ask:
● Do the answers and/or approaches look reasonable? Explain why.
● How could we check the answers?
● Which is the most efficient way to arrive at an answer, in your opinion?
Allow the children time to comment on each and justify if the methods/opinions work.
Teaching tip
Maths Journals: The children could record using images and words what they sketched and wrote above
Teaching tip
While the approach with the calculator works, many children might find this difficult to conceptualise, i.e. how dividing results in a greater answer. Use a much simpler example, e.g. 6 divided by 0.5: this is the same as asking how many halves are in 6, which can be represented visually before arriving at an answer of 12.
Unit 8: Fractions, Decimals and Percentages
Remind the children that they should now know that dividing by a number smaller than 1 will result in a larger answer and multiplying by a number smaller than 1 will result in a smaller answer. It might suffice to explain this as a strategy that could be used to check solutions, after using the more visual approach of drawing a bar model.
Afterwards, explore the various approaches further. Ask/say:
● Can you describe the error made by Mia? What tips might you offer to avoid making this mistake in the future?
● What other fraction methods could have been used? (e.g. using 4 5 instead of 8 10 )
C P Concrete activity: Calculating Amounts
Involving Percentages MAM Routines:
Reason & Respond, with Build it; Sketch it; Write it
On pages 82–83 of the Pupil’s Book, choose an activity appropriate to the general ability of the class.
Ask:
● What are you being asked to do?
● How could you model this? What could you build, sketch or write?
● How could you check?
● Is there more than one way to do this?
● Estimate: Will the answer be greater or less than the number(s) given? Why do you think this?
Build it; Sketch it; Write it: With the children working in pairs or groups of three, allow them time to consider, explore, model, solve and present their approaches/solutions.
● Build it! If appropriate, provide suitable concrete materials for the children to build bar models (e.g. number rods or interlocking cubes to represent the bars).
● Sketch it! Draw a bar model to represent the quantities, relationships and missing value(s).
● Write it! Write an equation, representation or explanation of what was done.
● Use a calculator to check your calculation. After an appropriate amount of time, ask the children to show and share with the class their strategies and models.
Teaching tip
Maths Journals: The children could record the strategies and models they used.
P Pupil’s Book pages 82–83: Calculating Percentages of Amounts
Let’s
strengthen
The children should be encouraged to use a calculator to check their answers. Those who struggle with the required operations should also be encouraged to use a calculator for assistance with the calculations.
Optional consolidation and extension possibilities
Maths Journal Use the Lesson 7 Maths Journal
Prompt from the Unit 8 Maths Journal Prompts PCM.
Display The children could contribute samples of their own work from this lesson and label them.
Review and Reflect Use the prompt questions in the Review and Reflect poster.
Day 9, Lesson 8
Percentage Increase and Decrease
Focus
of learning (with Elements)
● Solves problems relating to percentage increases and decreases (A&PS)
● Completes problem-solving tasks involving percentages, fractions and measures, explaining methods and reasoning (A&PS)
Learning experiences
Digital activity: Sale Time MAM Routines: Concept Cartoon, with Think-Pair-Share; Build it; Sketch it; Write it
Digital activity: Percentage Increase and Decrease MAM Routines: Reason & Respond, with Write-Hide-Show
Concrete activity: Calculating Percentage Increase and Decrease MAM Routines: Reason & Respond, with Build it; Sketch it; Write it
Pupil’s Book page 84: Percentage Increase and Decrease
● increase, decrease, original Maths language
Warm-up
D Digital activity: Sale Time MAM Routines: Concept Cartoon, with Think-Pair-Share; Build it; Sketch it; Write it
● Calculators
● Materials that can be used to build bar models (e.g. number rods or interlocking cubes) Equipment
Display the Concept Cartoon, in which the characters are discussing the meaning of ‘25% off’. Click to hear each character’s thoughts. Using Think-Pair-Share, ask:
● What do you think?
● (Point to a specific character.) Do you agree with their idea? Explain why.
● Do you think something different? What do you think? Why do you think this?
If appropriate, record the children’s responses to these questions on the board. Allow the children the opportunity to respond to (agree/disagree with or query) others’ responses, but do not confirm or reject any of the ideas. Ask:
● How can we find out who is correct?
Encourage the children to present their suggested approaches and/or solutions.
Build it; Sketch it; Write it: Using materials and/or sketches, encourage the children to provide/create models as examples and/or counter-examples to prove/disprove the characters’ ideas.
Maths Journals: The children could record their findings and proofs.
Unit 8: Fractions, Decimals
Afterwards, present this problem:
● In the same sale, an item is now €24. What was its price before the sale? (Some of the children may calculate 1 4 of €24 = €6 and assume incorrectly that the original price was €30, instead of recognising that the sale price is 3 4 of the original price, which therefore must have been €32.)
Let’s deepen
Challenge the children to explain how finding a percentage of an amount or finding the whole amount is related to multiplication and division. In what order should the operations be used? Why?
Main event
D Digital activity: Percentage Increase and Decrease MAM Routines: Reason & Respond, with Write-Hide-Show
Play the slideshow and for each revealed part of the image, using Write-Hide-Show to gather feedback, ask the children:
● (After revealing the deal on the left-hand side) What do you notice? What do you wonder?
● Is the original amount going to increase or decrease and by what percentage?
● What percentage represents the original or whole amount at the beginning? (100%)
● What percentage will then represent the amount after the percentage increase/decrease? (In the case of the first one, it will be 90%.)
● What fraction is this percentage equivalent to? (Prompt the children to suggest the simplest form, if not suggested.)
● What fraction bar model could be used to represent the original or whole amount at the beginning? (Prompt them to think of fractions equivalent to 1, i.e. 2 2 , 4 4 , 5 5 , 10 10 , etc.). Sketch this on your MWB.
Let’s strengthen
The children may prefer to use number rods or interlocking cubes to physically build the bar models.
● By what fraction of the whole amount does this amount increase/decrease?
● What fraction bar model could be used to represent the amount after the increase/ decrease? Sketch this on your MWB.
● What other details can you add to your bar model? What do we know? What are we looking for? (e.g. original value, question mark on new value)
● (Reveal the right-hand side of image.) Is this what you expected? Would this work?
● What else do you need to do to arrive at an answer?
● How will you check your answer?
Repeat the questions above with the remaining pairs of images.
Let’s deepen
Challenge the children to consider ways in which their calculator could be used to check the answers (other than just checking the accuracy of the calculations required to perform the fraction method). For example, the following could be input directly:
230 × 0.9 or 230 – 10%
300 × 1.5 or 300 + 50%
C P Concrete activity: Calculating Percentage
Increase and Decrease
MAM Routines: Reason & Respond, with Build it; Sketch it; Write it
On page 84 of the Pupil’s Book, choose an activity appropriate to the general ability of the class. Ask:
Unit 8: Fractions, Decimals and Percentages
● What are you being asked to do?
● How could you model this? What could you build, sketch or write?
● How could you check?
● Is there more than one way to do this?
● Estimate: Will the answer be greater or less than the original amount given? Why do you think this?
Build it; Sketch it; Write it: With the children working in pairs or groups of three, allow them time to consider, explore, model, solve and present their approaches/solutions.
● Build it! If appropriate, provide suitable concrete materials for the children to build bar models (e.g. number rods or interlocking cubes to represent the bars).
● Sketch it! Draw a bar model to represent the quantities, relationships and missing value(s)
● Write it! Write an equation, representation or explanation of what was done.
● Use a calculator to check your calculation.
Teaching tip
Maths Journals: The children could record their findings and proofs.
After an appropriate amount of time, ask the children to show and share with the class their strategies and models.
P Pupil’s Book page 84: Percentage Increase and Decrease
Optional consolidation and extension possibilities
Maths Journal Use the Lesson 8 Maths Journal
Prompt from the Unit 8 Maths Journal Prompts PCM.
Display The children could contribute samples of their own work from this lesson and label them.
Online Game Use this quiz to test overall understanding of percentages and their application: ‘Mathopolis’ at edco.ie/fwzu
Maths Eyes The children could use websites, brochures, flyers, etc. (ensure websites are childfriendly and content/items advertised are appropriate), as inspiration for scenarios to create percentage-based problems (e.g. calculate the current value if there is 50% or 20% off the displayed price).
Review and Reflect Use the prompt questions in the Review and Reflect poster.
Day 10, Lesson 9
Review and Reflect
Warm-up
Carry
Choose from this menu of activity ideas, or choose your own way to best structure this last lesson to suit your needs and the needs of your class.
Let’s talk!
Classroom poster: Review and Reflect
Use Think-Pair-Share alongside the prompt questions to review the unit. Individual children could present examples of their own drawings/work/ constructions to the class, and talk about what they have learned.
Maths language
Ask the children to explain the following terms (perhaps using examples or drawings on their MWBs): fraction(s), decimal(s), fraction/decimal form, equivalent form, equal, tenth(s), hundredth(s), thousandth(s), denominator, powers of ten, fraction equivalent to 1, simplify, greatest factor, per cent, percentage, benchmark fractions/percentages, equivalent, ‘same value, different appearance’, simplest form, compare, order, greater than (>), less than (<), equal to (=), common denominator, increase, decrease, original.
Use the Unit 8 Maths Language Cards for this unit to revise the key terms. For example: if the image and text are cut apart, can the children match them?
Progress Assessment Booklet
Complete questions xx–xx on pages xx–xx. Alternatively, these can be left to do as part of a bigger review during the next review week.
Let’s strengthen
Identify children who might benefit from extra practice with some of the key concepts or skills in this unit. Consult the Unit 8 Let’s Strengthen Suggestions for Teachers and/or use the Unit 8
Let’s Strengthen PCM.
Unit 8: Fractions, Decimals and
Let’s play!
Play any of the fraction games from the Games Bank. Play or use some of the online digital resources referenced in the unit (see the Optional Consolidation and Extension Possibilities throughout).
Maths strategies and models
Ask the children to give examples of the strategies they used in this unit (e.g. how to convert between fractions, decimals and percentages, how to calculate amounts and percentages, including increases and decreases, how to use a calculator to check calculations). Ask the children to give examples of the models they used in this unit (e.g. what they built, sketched and wrote, how they used bar models or thousandths and hundredths grids). Which strategies and models did they prefer and why? What needs to be considered when choosing the ‘best’ strategy and model for each situation?
Maths eyes
Use examples of fractions, decimals and percentages in the school and/or local environment as well as in media (print and online) to create problems (e.g. If there are 320 children in a school and 45% are boys, how many boys is that?).
