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Central Idea
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"Addition, subtraction, multiplication and division, as well as fractions, are related to each other and are used to help solve problems."
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Place Value
An inquiry into...
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Tuning In
Number Systems Prior Knowledge - What do we know about different number systems? Can we think of examples of different systems? When might we use one system instead of another? What are the advantages and disadvantages of each system? What other questions might you have. Create a mind map to show what you know. Exploding Dots - A crazy machine that can help us find out more about Place Value and other number systems! Try machines created by James Tanton (our friend in the video to the right) or create your own machines. What patterns do you see? How do these machines relate to number systems in the real world? |
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Place Value Prior Knowledge - We may know more about number patterns than we think. Use the sheet provided to share what you already know about patterns in numbers and number systems. You can also share any questions that you have going further.
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Rounding Whole Numbers - Using the calculators / money / base 10 blocks, or any other place value material, practice making numbers and rounding them to the nearest ten / hundred or even thousand. Complete the sheet attached to show what you know about rounding and how we use rounding in our day to day lives.
Number Hunt - There are numbers all around us. Using the sheet provided, explore the class and the surrounding area to see if you can find any numbers that fit the description. Don't forget to say where you saw the numbers and to try and find numbers from as many places as possible. How Much is A Million? - As we begin to recognize bigger and bigger numbers, we may start to lose track of just how large the numbers actually are. Read the book, How Much is a Million, by David Schwartz. This book shows a lot of real world examples relating to the extremely large number - 1,000,000! |
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Finding Out
Investigating Other Number Systems - Take a look at the watch to the left. Is there something different about the numbers on the face of the watch? What is interesting about the number system on the face of the watch? Are there other number systems? What do they have in common? What is different about them? Patterns in the Base 10 System - Using some of the concrete materials below, explain what patterns exist in the Base 10 number system. How are the tools used? What do they have in common? |
Skip Counting - Skip counting is easy, right? 2, 4, 6, 8, 10.... No problem! With a partner, practice your skip counting and see how you can make things more challenging. Forwards, backwards, by 10's, 5's, 12's, 10,000's? Start at random numbers, include decimals and fractions - negative numbers... yes, skip counting can be quite a challenge. What patterns can you find when you skip count while looking at a hundreds or thousands chart? Try to plot some of the numbers on a hundreds chart to see if there are visible patterns.
Website Search - to find out about place value and rounding, it is important that we use different tools to learn more. There are videos on YouTube, experts we can speak to, and there are websites like Math is Fun and BrainPop Jr. where we can find out more.
Calculator Go Fish - Check out our games page for more interesting games relating to Place Value and while you're there, be sure to check out the instructions on how to play Calculator Go Fish.
Say a Number / Write a Number - To continue to practice our numbers, we can use the calculators to play the game, Say a Number, Write a Number. With a partner, one person says a number out loud while also typing the number into their calculator. Your partner also punches the number that was said into their calculator (without looking at your screen). When both of you have put in the number, check the screens to make sure you have written in the same thing.
This game can get more challenging depending on how large the number you are saying. To create more of a challenge, try larger numbers, numbers with decimals, numbers with lots of Zeros spread throughout, and you could even try saying numbers with number nicknames, or in another language.
Numbers - It is important that we know how to spell the numbers that we use. In many languages, the names for numbers follow a pattern that make it very easy to remember.
Website Search - to find out about place value and rounding, it is important that we use different tools to learn more. There are videos on YouTube, experts we can speak to, and there are websites like Math is Fun and BrainPop Jr. where we can find out more.
Calculator Go Fish - Check out our games page for more interesting games relating to Place Value and while you're there, be sure to check out the instructions on how to play Calculator Go Fish.
Say a Number / Write a Number - To continue to practice our numbers, we can use the calculators to play the game, Say a Number, Write a Number. With a partner, one person says a number out loud while also typing the number into their calculator. Your partner also punches the number that was said into their calculator (without looking at your screen). When both of you have put in the number, check the screens to make sure you have written in the same thing.
This game can get more challenging depending on how large the number you are saying. To create more of a challenge, try larger numbers, numbers with decimals, numbers with lots of Zeros spread throughout, and you could even try saying numbers with number nicknames, or in another language.
Numbers - It is important that we know how to spell the numbers that we use. In many languages, the names for numbers follow a pattern that make it very easy to remember.
Concrete
All of the concrete materials below can be used to help solve problems when working with numbers. Some are better for working with larger numbers, while others are better for working with smaller numbers. Listed below are some example games and lessons where you might use these tools.
All of the concrete materials below can be used to help solve problems when working with numbers. Some are better for working with larger numbers, while others are better for working with smaller numbers. Listed below are some example games and lessons where you might use these tools.
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Base 10 Blocks
- the ultimate tool for learning about place value - can be adapted for use in add and sub, mult and div, and even fractions |
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Expanded Notation Cups
- a fun tool to make to show how the digits in a number can have different values - also great for ordering and comparing numbers |
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Tens Frames
- great for subitizing numbers to 10 and 20 - helpful for seeing how numbers combine to make facts that are easier to mentally add and sub |
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Place Value Disks
- like the tools above, these are helpful for seeing pV and for working with numbers - useful for seeing how digits differ in numbers |
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Money
- great for learning about pV in a real world context - a good first step into learning about how there are numbers in between whole numbers on a number line |
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Place Value Chart
- great to use so that we can see patterns in larger numbers - helpful for naming larger numbers and when working with number nicknames |
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Hundreds Chart
- great for looking for patterns in skip counting - helpful for finding numbers when add and sub - can be used with counters (clear) - also helpful when using repeated add and sub (aka mult and div) |
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Calculators
- Calculator Go Fish - Broken Calculator - Rounding Whole Numbers |
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Pictorial
Alien Numbers - There are other number systems out there - I mean... WAY out there! If there was an Alien species that visited Earth, what Number System do you think they would use? Create an Alien number system and an explanation of how it is used so that our little Earthling friends can comprehend it! Place Value Monster - Create a place value monster using thousands, hundreds, tens and ones from our base 10 blocks. See the examples below for inspiration, but try to create something new. Name your monster and find out what special number they represent. |
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Abstract
Online Practice - Khan Academy and Prodigy have lots of different questions relating to Place Value and Rounding. See if you can achieve Mastery in these areas. If you can find another site that you enjoy, share it with the class. Place Value Video - Watch the video to the right to find out more about Place Value. Use our Research Sheet to track your understanding and to write any additional questions you have. Number Nicknames - The Base 10 place value system is used in many different countries and in many different languages. What are the difference and similarities between the way we name the numbers in English versus the other languages we know? Are some languages easier to see patterns within the number system we use? How can we use number nicknames to make finding patterns in our number system easier? |
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Representing Numbers - What are the different ways we can represent numbers? We have written numbers in English, as well as other languages, but are there other ways to represent numbers? What is expanded notation? How is it different from expanded form? What are the advantages and disadvantages of representing numbers in different ways?
Rounding Practice - with a partner, try rounding different numbers that you see to the nearest ten and the nearest hundred. Try to find a partner who can challenge you to round more difficult and larger numbers.
Using Rounding to Add and Subtract - So, how can rounding help us in other areas of mathematics? If there is no purpose for rounding, then why would we do it? I am sure that you use rounding to help you with computation, so how do you do it? Is there more than one way?
Rounding Practice - with a partner, try rounding different numbers that you see to the nearest ten and the nearest hundred. Try to find a partner who can challenge you to round more difficult and larger numbers.
Using Rounding to Add and Subtract - So, how can rounding help us in other areas of mathematics? If there is no purpose for rounding, then why would we do it? I am sure that you use rounding to help you with computation, so how do you do it? Is there more than one way?
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Applying
Finding Large Numbers in the Real World - Go on a number walk to find numbers in the real world. Numbers are all around us. Some big, some small. We can find numbers in everyday objects, even if the number isn't written out. Use the sheet attached to keep track of the numbers you find. You can also use your iPad to take photos. You can even put your collection together using Book Creator. Representing Numbers in the Real World - In addition to looking for large numbers, try and find different ways numbers are represented in the world. Be sure to check for other languages and number systems. |
Place Value Refresher Video - Above you will find a video going over some of the strategies we have worked on for representing numbers. Also, at the end of the video are some suggestions for how we might get better at mental addition with numbers to 20.
Rounding SeeSaw Video - There are many different strategies when it comes to Rounding numbers and there is no perfect way to explain how to do it. We have thought about the reason why we round, but can we explain how we round. Using the iPad, try to explain to someone your age the strategies that you use to round. Upload your videos to SeeSaw
Tuning In
Theories and Questions - as a class, create a list of questions and theories about the Central Idea and Concepts to add to the board. As time progresses, add to the board, or check to see if previous questions have been answered, or if theories have been proven or disproved.
Assessing our Prior Knowledge - To find out what we already know about mental and written strategies, it is important for us to show what we know. Using a mind map and some suggested questions from the teacher, show a variety of strategies for solving the problem given.
Venn Diagram - with a small group, create a poster with overlapping circles, where one circle shows what addition is all about and the other shows subtraction. Where the circles overlap, write down the similarities, if any, between the two. Once your small group has completed the task, join with the rest of the group to put the class ideas together.
Theories and Questions - as a class, create a list of questions and theories about the Central Idea and Concepts to add to the board. As time progresses, add to the board, or check to see if previous questions have been answered, or if theories have been proven or disproved.
Assessing our Prior Knowledge - To find out what we already know about mental and written strategies, it is important for us to show what we know. Using a mind map and some suggested questions from the teacher, show a variety of strategies for solving the problem given.
Venn Diagram - with a small group, create a poster with overlapping circles, where one circle shows what addition is all about and the other shows subtraction. Where the circles overlap, write down the similarities, if any, between the two. Once your small group has completed the task, join with the rest of the group to put the class ideas together.
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Finding Out
Subitizing - How do you see the number of dots on the right? Do you see groups of three on the outside with one dot in the middle? Do you see four dots, like on a set of dice, with three dots running through the middle? Draw a picture to show how you see the dots. Share your pictures with others. Does anyone else see things the way you do? How many ways do you think people might see these dots. More Subitizing - Being able to see numbers quickly, especially by finding common groups is a very important skill that takes practice. Seeing 10s and making tens is very important to being able to add and subtract quickly. Click on the title to find a whole host of great games and lessons that you can work on using subitizing cards as well as 5 and 10 frames. |
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Number Talks - Number Talks are short, daily exercises aimed at building number sense. Number sense is the ability to play with numbers meaning you can visualise problem solving, perform calculations quickly, and are flexible with your mathematical strategies. What are some strategies that we know when adding and subtracting numbers within 20? What are some strategies we know when adding larger numbers? How are they the same? How are they different?
Creating Strategies - As a class, do number talks and discuss strategies you are using to add and subtract numbers. Come up with some names for different strategies. Rank your strategies in different situations. Are some better than others? What makes them better? If you come up with a new strategy, you could name it after yourself.
The Basics - A strong understanding of mental computation starts with a strong understanding of strategies for adding and subtracting to 20. The strategies below will continue to help you when numbers start to become larger. So, practice these strategies until you feel confident.
Creating Strategies - As a class, do number talks and discuss strategies you are using to add and subtract numbers. Come up with some names for different strategies. Rank your strategies in different situations. Are some better than others? What makes them better? If you come up with a new strategy, you could name it after yourself.
The Basics - A strong understanding of mental computation starts with a strong understanding of strategies for adding and subtracting to 20. The strategies below will continue to help you when numbers start to become larger. So, practice these strategies until you feel confident.
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Counting On - If you have a question like 8 + 3 and you need to count using your fingers, or in your head, which number would you count from? Even when we count, we can make things easier.
Doubles - Every number has a double. Knowing our doubles can help us when adding and multiplying. What is double 4? or Double 44? Or Double 444? Can we find patterns to make doubling easier? Near Doubles - It's not quite a double, but it's close. Maybe it's 8 +7, or 8+9. If we know our doubles, we should be able to add one more or one less to that answer to help us find near doubles. Making 10s - When I say, 6, you say 4. When I say 3, you say 7. Knowing what number adds to another to make 10s helps us add so much. Not just 10, but any nice round number, or "Tidy Ten," is much easier to add. Commutative Property - It doesn't matter if it's 6 + 4, or 4 + 6, the answer is going to be the same. This is not the same for subtraction or division. Knowing that the order doesn't matter means we can rearrange what we add to make things easier. Jump - Very helpful if you have a number line. Sometimes, we can jump by numbers we feel comfortable with to make addition easier. What numbers do you think are easier to jump by? |
Counting Back - If you have a question like 12 - 3, would you want to count up from 3 to find the answer, or count back from 12? Again, when we count, we can count efficiently.
Counting Up - On the other end of the spectrum, what if we had a question like 9 - 8? Would you count back from 9, or would you count up from 8 to find the difference? Halves - Just like with doubles, every number has a half, the only difference is not every number has a half that is a whole number. Can you find the pattern? Which numbers have a half that is a whole number and which ones do not? Making 10s - You can make 10s when subtracting as well as when adding. How does making a 10 help you when subtracting. If I have a question like 12 - 5, how would I make 10 and how would it help. Related Facts - There is no commutative property in subtraction, but it is helpful for us to find related facts. If I know that 7 - 5 = 3, then how will it help me with a question like 7 - 3? Jump Back - With or without a number line, we can visualise jumping back in the same way we jump forward. What numbers make it easier for you to jump back? What is different about jumping back on a number line compared to jumping forward. |
Mental Addition and Subtraction of Larger Numbers - Below are a series of videos that show strategies we might use for adding and subtracting larger numbers using mental math. The students in these videos are showing their thinking in writing, but by practicing with larger and larger numbers using some of these strategies, we can start to manipulate numbers in a way that makes them easier to work with. Subtraction strategies are on the left, addition on the right.
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Round the Subtrahend
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Round and Adjust
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We might use this strategy when we come across numbers that are close to 10's or 100's.
Try these: 13-9, 24-8, 61-8 63-28, 71-39, 84-59 134-99, 247-98, 315-97 |
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We might use this strategy to make things easier when we would otherwise have to "carry." Helpful for decimals and fractions as well.
Try these: 13+9, 24+8, 68+8 63+28, 74+39, 84+59 144+99, 248+98, 468+84 |
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Decompose the Subtrahend
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Take and Give
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This strategy is helpful for removing smaller numbers in manageable chunks.
Try these: 32-6, 21-8, 43-7 43-17, 62-16, 47-28 83-37, 91-26, 84-36 |
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A helpful strategy if there is enough to give to create nice, round numbers. Also helpful for decimals and factions.
Try these: 16+8, 18+6, 29+7 46+98, 89+65, 146+197 3.76+2.89, 8.9+0.56, 3.8+1.4 |
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Add Instead
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Start from the Left
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This strategy is helpful when rounding doesn't make sense. Plus, it's all about adding!
Try these: 23-19, 51-48, 34-27 223-219, 351-348, 435-427 |
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Going from Left to Right is a much more natural way to add. It's also easier when adding multiple numbers.
Try these: 43+56, 54+35, 24+67 376+523, 274+432, 117+356 23+31+48+42+29+35 |
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Same Difference
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Break one Addend Apart
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My personal favourite - a great strategy if we understand moving set distances around on the number line.
Try these: 93-28, 76-39, 57-18 236-188, 3456-687 |
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Adding any number to a multiple of ten is easier. Works with all problems and in flexible ways.
Try these: 15+23. 25+36, 37+49 237+314, 456+238, 183+276 0.23+0.57, 1.07+0.68 |
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Break Apart by Place
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Add Up
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This strategy is helpful as many of us naturally like to subtract numbers from left to right. You may need a good grasp of integers to do this.
Try these: 72-56, 81-27, 63-28 337-159 |
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Similar to break one addend apart, but with this strategy, we are usually breaking the addend into multiple parts. Helpful for larger numbers
Try these: 258+39, 547+34, 546+28 351+439, 1348+143, 1985+1245 1.09+0.83, 7.06+0.48 |
Addition and Subtraction Games - There are lots of different addition and subtraction games. Too many to list here. Check below under our Concrete Materials section and on our Math Games page, to find more videos and games.
Practice - Practice makes perfect. In some way, this is true. But there is a big difference between good and bad practice. What are some tips that you have for making sure that your practice is the best it can be?
Practice - Practice makes perfect. In some way, this is true. But there is a big difference between good and bad practice. What are some tips that you have for making sure that your practice is the best it can be?
Concrete
Tools - With a partner, show how certain tools can help you to visualize and model addition and subtraction in a concrete way. What tools will help you most? Are there certain tools that are better, or worse than others? What happens with the tool you are using as you get into larger and larger numbers?
Many of the tools listed above (in the Place Value section) can also be used when modelling addition and subtraction. Check out the tools below and think about how they can also be used.
Tools - With a partner, show how certain tools can help you to visualize and model addition and subtraction in a concrete way. What tools will help you most? Are there certain tools that are better, or worse than others? What happens with the tool you are using as you get into larger and larger numbers?
Many of the tools listed above (in the Place Value section) can also be used when modelling addition and subtraction. Check out the tools below and think about how they can also be used.
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Rekenrek
- very helpful for visualizing numbers to 20 - see what is missing, see what is there (subatizing) - helpful with a teacher / parent / partner that can help you improve your facts to 20 |
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Dominos
- another great tool for practicing add and sub numbers, especially numbers within 10 - many different games that can be played and changed to increase or decrease difficulty |
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Cards
- always great for add and subt games - easily adaptable for making games more challenging / easier - helpful for practicing facts without things getting boring |
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Dice
- just like cards, dice are great for fast practice of facts for add and sub - change up games often to get better at all areas of add and su - use different types of and numbers of dice |
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Tens Frames
- helpful for showing patterns in add and sub - great for visualizing Making 10, an important part of mental add and sub |
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100 Beads
- also great for showing patterns, especially with repeat add and su - like Tens Frames, also good for making tens, but further along the number line |
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Pictorial
Part Part Whole - When we are adding and subtracting, things can get a little confusing if we do not see numbers lined up in the familiar way. Most of us would have no problem finding the answer to 7 + 2 = ___, but things can get confusing when the question looks more like 7 + ___ = 9, or 9 = 2 + ___. Part / Part Whole is a way of modelling addition and subtraction questions without the need for pluses and minus. Simply identify whether the number you are trying to find is the part or the whole. The picture on the right shows several questions where the Part is missing. Number Bonds - Number Bonds look similar to Part Part Whole Models, but we would use them for different reasons. It is important that we can break apart numbers in ways that make them easier to add and subtract. Watch the video below to see how using number bonds can help with addition and subtraction. |
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Bar Modelling - Bar Modelling is an excellent way of visualizing problem solving questions. Watch the videos below to find out more.
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Abstract
Kakooma - This game is a great way to practice your addition. The way it works, as seen on the right, is that a puzzle is shown. In that puzzle, there are a series of numbers. One of the numbers in the group is the sum of two other numbers. That is the number you have to click on. Once you have done that for each of the puzzles, the final puzzle (the answer to each one of the sums you made before), is also just like the other puzzles. The goal is to find the sums as fast as possible. Some of the puzzles have 4 in a set, some as many as 9! The great thing about Kakooma is that you are doing LOTS of mental addition to find each answer and sometimes you don't even realize it. There are many more great games on Greg Tang Math. |
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Khan Academy - You will see Khan Academy on here many times. It is my favourite website for math learning. The reason being, it keeps getting better and students from Year 2, all the way through University can use it. When doing the addition and subtraction questions, Khan Academy asks questions in so many different ways - and unlike sitting down and doing a worksheet - you get instant feedback, you have videos and hints to help you along and there are rewards and points for mastering different skills. Although things look different on each, Khan Academy can be used on both the computer and the tablet. I recommend the tablet, because it is easier to use the scratch pad to work out your answers.
Math Trainer - With the Math Trainers (which can be used for all four operations), we can track our progress and focus our learning on the areas we need to improve. Here is a direct link to Addition and here is a direct link to Subtraction. I recommend setting the timer to 1 minute, with a cut off time of 8 seconds. The Math Trainer really needs a computer to work, as you need to be able to type in numbers quickly. Take a screen shot of your records and share with others! The table at the bottom will show you which areas you are answering well and which you may need to work on.
Math Trainer - With the Math Trainers (which can be used for all four operations), we can track our progress and focus our learning on the areas we need to improve. Here is a direct link to Addition and here is a direct link to Subtraction. I recommend setting the timer to 1 minute, with a cut off time of 8 seconds. The Math Trainer really needs a computer to work, as you need to be able to type in numbers quickly. Take a screen shot of your records and share with others! The table at the bottom will show you which areas you are answering well and which you may need to work on.
Applying
Creating Word Problems - There are many things we can do to create word problems that are more challenging than the standard. As a class, investigate the different types of addition and subtraction word problems, then create some of your own. Rank the word problems that you have created from least to most challenging.
What are the variables that you can change in a word problem to make it more challenging? What problems have you experienced in your day to day life that represent the different types of word problems. Click on the title of this activity to find a blank Problem Solving template to help create your questions and to answer questions from the World Problem Generator.
Creating Word Problems - There are many things we can do to create word problems that are more challenging than the standard. As a class, investigate the different types of addition and subtraction word problems, then create some of your own. Rank the word problems that you have created from least to most challenging.
What are the variables that you can change in a word problem to make it more challenging? What problems have you experienced in your day to day life that represent the different types of word problems. Click on the title of this activity to find a blank Problem Solving template to help create your questions and to answer questions from the World Problem Generator.
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Word Problem Generator - From the same person who brought you Kakooma, here is a site that can help you to come up with some word problems that are not all just "What's the Total?" type questions. Check the images out below to see the different types of questions you might have. Being able to take the question and write it in an abstract way may be difficult to do at first. Try using models to help you figure out what type of question it is.
Problem Solving Recap Video - To the right you can find a quick video reviewing how and why we do problem solving the way we do. This video goes well with the addition situation diagrams located below. |
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101qs.com - Addition and Subtraction are all around us. Often we can find math in everyday situations. On the website 101qs, there are videos and pictures that people have made where they are looking at problems through the lens of mathematics. Try to come up with your own questions in these three act math tasks. What information will you need to know to answer your question?
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Multiplication and Division
An inquiry into...
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Tuning In
Theories and Questions - as a class, create a list of questions and theories about the Concepts to add to the board. As time progresses, add to the board, or check to see if previous questions have been answered, or if theories have been proven or disproved.
Assessing our Prior Knowledge - To find out what we already know about mental and written strategies, it is important for us to show what we know. Using a mind map and some suggested questions from the teacher, show a variety of strategies for solving different types of multiplication and division questions.
Venn Diagram - with a small group, create a poster with overlapping circles, where one circle shows what multiplication is all about and the other shows division. Where the circles overlap, write down the similarities, if any, between the two. Once your small group has completed the task, join with the rest of the group to put the class ideas together. How can these circles overlap with the Venn Diagram you made for addition and subtraction?
Theories and Questions - as a class, create a list of questions and theories about the Concepts to add to the board. As time progresses, add to the board, or check to see if previous questions have been answered, or if theories have been proven or disproved.
Assessing our Prior Knowledge - To find out what we already know about mental and written strategies, it is important for us to show what we know. Using a mind map and some suggested questions from the teacher, show a variety of strategies for solving different types of multiplication and division questions.
Venn Diagram - with a small group, create a poster with overlapping circles, where one circle shows what multiplication is all about and the other shows division. Where the circles overlap, write down the similarities, if any, between the two. Once your small group has completed the task, join with the rest of the group to put the class ideas together. How can these circles overlap with the Venn Diagram you made for addition and subtraction?
Finding Out
Number Talks - Number Talks can be done with multiplication and division as well. Below you will find different strategies for breaking apart numbers to help us complete multiplication and division questions in our head. In our abstract section, we can see how we can find patterns within the times tables. With practice, patience and finding patterns, we will be able to improve our mental computation.
Number Talks - Number Talks can be done with multiplication and division as well. Below you will find different strategies for breaking apart numbers to help us complete multiplication and division questions in our head. In our abstract section, we can see how we can find patterns within the times tables. With practice, patience and finding patterns, we will be able to improve our mental computation.
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Break a Factor into Two or More Addends
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Multiply Instead
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To start, we might want to practice with smaller two digit factors, or single digit multiplied by two digit,
12 x 6 13 x 8 14 x 12 12 x 13 |
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When we multiply instead, we should have a good understanding of our multiplication facts.
15 ÷ 3 21 ÷ 5 17 ÷ 3 19 ÷ 6 |
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Factor a Factor
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Chunk Out
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When we Factor a Factor, how do we choose which number to factor? How does this make the question easier? And why does it work? Can we do this with all numbers, or just specific examples?
14 x 25 25 x 16 51 x 14 18 x 26 |
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Chunking out is a very useful strategy, as you can use estimation and work with friendly numbers when dividing. Also, you don't need to be "perfect" in your guesses the first time. This is especially helpful with multi-digit division questions.
32 ÷ 3 43 ÷ 4 87 ÷ 8 76 ÷ 7 |
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Round a Factor and Adjust
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Make a Tower
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When we round a Factor and Adjust, we always want to try and take "messy" numbers and make them more "friendly." Rounding to a multiple of ten when a number is pretty close is always a helpful strategy. This can be done with all numbers, regardless of the size.
12 x 9 6 x 19 21 x 7 8 x 13 |
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Making a Tower can really help when you're chunking out. Sometimes, we can make a good guess as to what number to try and chunk out, but other times, it is not so easy. When we make a tower, we can see more clearly the friendly numbers that we can work with.
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Halving and Doubling
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Halving and Halving
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Halving and Doubling can be a useful strategy as well, but does not always work for every number. How do we know when halving and doubling might be a good strategy to use? Why does it make some problems easier to think about?
8 x 13 4 x 17 6 x 13 8 x 25 |
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Just like Halving and Doubling, Halving and Halving does not work for all numbers. That said, when we do find some numbers that might be useful, we need to be able to identify them.
26 ÷ 4 52 ÷ 4 128 ÷ 8 192 ÷ 24 |
Concrete
Tools - As with addition and subtraction and place value, there are many tools that overlap with multiplication and division. Check out the tools below to see how they might help you with gaining a better understanding of "repeated addition" and "repeated subtraction."
Tools - As with addition and subtraction and place value, there are many tools that overlap with multiplication and division. Check out the tools below to see how they might help you with gaining a better understanding of "repeated addition" and "repeated subtraction."
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Hundreds Chart
- helpful for finding patterns in repeat addition and subtraction - can be used with counters / bears / etc for patterning |
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Times Table Chart
- in some people's minds, the only thing that matters when it comes to multiplication (not true) - great for showing what you already know and for identifying what you need to work toward |
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Cards
- the classic. Great for games like war and snap - easily adaptable for games we have played before |
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Dice
- just like cards, dice can be used for many games to practice our multiplication and division |
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Calculators
- checking answers and playing games - skip counting forward and backward (repeated addition and subtraction) |
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Base 10 Blocks
- helpful for showing multiplication of larger numbers in a visual way - showing how we share and trade when doing division (even LONG division) |
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Bears / Shells / Buttons
- Anything that can be sorted, really - useful for making equal shares when doing division |
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Plastic Counters
- helpful if they are clear in colour - can be used on the hundreds chart and the multiplication chart - also good for sorting and sharing |
Pictorial
Bar Modelling - We can also use bar models to help us solve multiplication and division questions. They aren't that different than our Part Part Whole drawings. The only difference is that we might have more parts. Try different simple multiplication and division questions using a bar model to show your understanding. When would you not want to use a Bar Model? What could you use instead?
Area Models - Just like bar models, an Area Model can be used to represent, or show a picture of a multiplication or a division question. The more we understand what the math looks like, the easier it will be for us to understand how all the numbers go together. When would you use an Area Model? When should it not be used?
Bar Modelling - We can also use bar models to help us solve multiplication and division questions. They aren't that different than our Part Part Whole drawings. The only difference is that we might have more parts. Try different simple multiplication and division questions using a bar model to show your understanding. When would you not want to use a Bar Model? What could you use instead?
Area Models - Just like bar models, an Area Model can be used to represent, or show a picture of a multiplication or a division question. The more we understand what the math looks like, the easier it will be for us to understand how all the numbers go together. When would you use an Area Model? When should it not be used?
Abstract
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Going Beyond the Grid - If we look at the times table grid, with all of its numbers, it is very easy to get overwhelmed. All told there are more than ONE HUNDRED numbers to memorize! But there is so much more to multiplication than just what is on this grid. Ask most of your parents what 6 x 5 or 9 x 8 is, and they will give you the answers with no problem, but ask them a question that goes outside of the grid and most will lose their confidence and speed.
So, when we look into "memorizing" our times tables, we are actually going to try and find strategies that will help us answer questions quickly up to 10 x 10 and beyond. |
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Imagine if we started with nothing. Where would you begin? Would you start with your 6 times tables? Would you start with your 7's? Would you go in order from 0 to 10? Remember that from 0 x 0, up to 10 x 10, you will have 121 numbers to remember. So where do you start?
I think most students would agree that the easiest thing to multiply is ANYTHING x Zero! Why is it so easy to remember, because it doesn't matter what the factor you are multiplying is, If you multiply it by 0 than the answer will be 0! 5 x 0.... 0! 56 x 0.... 0! 1,234,045 x 0.... you guessed it... 0! If we know this, than we can actually start filling in our times table grid and achieve our goal of Going Beyond as well! Let's see what that looks like: |
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Notice that we have now actually filled in 21 digits on our times table. This is because we also know that 0 x 5 and 5 x 0, as well as all other examples where we switch the order of the factors, will not change the product.
So, we started with 121 and now we have 100 to go. But what do we do next? Most students would probably suggest that our ONE times tables is the next easiest, because ANYTHING x One is just repeating the number. What's ONE group of Twenty? Twenty! 1 x 7.... 7! 9 x 1.... 9? 432 x 1.... 432! 1 x 8,123,324.... 8,123,324! Let's see what that looks like: |
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Again, notice that we have now filled in 19 more digits. Why not 21 like the last time? Well, because we already new what 0 x 1 was, and we already knew what 1 x 0 was.
So, we were at 100 and now we're down to 81! Just look how the number of boxes keeps getting smaller? But what do we do next? It would be easy to say that we just move on to our TWO times tables, and then to our Threes, but is our TWO times table really the next easiest to learn? What about the TEN times tables? Do you see any patterns? |
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Multiplying by 10 is pretty easy. It would appear that all we do is add a ZERO to the end of the other factor. However, this is not actually the case. We are actually moving the place value of the number to make it 10 times greater than before.
With questions like 7 x 10 or 10 x 84, this makes it look like we just add a zero, but when we multiply decimals, we can see that is not always the case. 34 x 10... 340! 10 x 9... 90! But what about 16.7 x 10? Our answer is 167. That doesn't end in a zero! We are actually moving our decimal place one spot to the right. This works for multiplying by 100 as well. Instead of moving the decimal one place... we move it two! 34 x 100 = 3400? 100 x 9 = 900 16.7 x 100 = 1670 How much would we move the decimal if we multiply by 1000? What about a million? And now, like magic, we've dropped 17 more from 81 to 64! Now let's look at the TWO times tables: |
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We have all been doubling since grade 1, so by now we should be pretty good at it. Up to 10, our doubles are not that difficult. Multiplying by 2 and doubling are the exact same thing.
So now, without breaking a sweat, we're down by 15 - and only have 49 left to go. That's less than half of what we had to remember at the start, and really, it hasn't even been that tough. Not only that, but all of the numbers we have worked on go BEYOND THE GRID and that means, not only have we sorted out more than half of the times tables up to 10, but we have figured out so much more. But what do we do next? Whether you're skip counting, or finding half of multiples of 10, the next best place to go might just be the FIVE times tables. Let's see what that looks like: |
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Skip counting by fives is quite easy, so many of us will rely on skip counting to answer questions with the five times tables within ten until we have them committed to memory, but there is another interesting thing about five that makes it great to multiply with.
Five is half of ten! Why does that matter? Well, it's easy to multiply by ten right? What's 44 x 10? 440, right? Well half of 440 is 220. So, 5 x 44 is 220! 10 x 8.... 80! 5 x 8.... 40! 10 x 86.... 860! 5 x 86.... 430! This can be a little bit tricky with odd numbers, but the pattern still works! Skip counting is fine, but no one wants to skip count 86 times to find an answer. And with an understanding of our 5 times tables, we're down to 36. Just 36 numbers to remember! And now we jump to 9. And I'll tell yah why. |
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There are all kinds of cool strategies for remembering your NINE times tables. Did you know that you can even figure out your nine times tables with your fingers? It's true. There are more ways than just that though. My favourite is to think about multiplying by 10 again. Just multiply the number by 10 instead of 9, then subtract the other factor from your total.
For example... 10 x 6 = 60. 60 - 6 = 54. So 9 x 6 = 54 There are more ways to multiply by 9 than just the ones listed above. See what works best for you. With that we are down to just 25! And now for something a little different: |
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Square Numbers are the product of multiplying the same factors together. So, 2 x 2, or 8 x 8, or 182 x 182. Why do you think they are called square numbers?
Either way, you can see the pattern of where square numbers show up on our times table chart and just like that, we are down to just 20 numbers left. You can see that there is a lot of overlap between the different times tables group, so the more we learn, the more we discover that we have multiple strategies for recalling each product. Almost there... What do you think we could do next? 3? 4? How about double doubles! Or the four times tables. |
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The four times tables are really just doubling a number (multiplying by 2) and then doubling the number again (times 2 one more time). So 4 x 7 is like saying 2 x 7 x 2. 7 x 2 = 14 and 14 x 2 = 28.
64 x 4 is like saying 64 x 2 x 2. 64 x 2 = 128 and 128 x 2 = 256. Also, a Double Double is two sugars and two creams in a Tim Horton's coffee in Canada. And then there were 12. And on to the Threes... |
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While there might not be as many interesting tricks for multiplying by 3, you can see that there is a lot less for us to figure out at this stage in the game.
Once there were 121 products to remember on our times table and now, there are only 6! I call these... THE GOLDEN TRIANGLE. |
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These questions always seem to be the ones that give even times table experts the most problems.
But when there is only three products left to remember, what excuse do you have? And there we have it. A times table chart that is completely filled in and a whole lot less overwhelming than before. |
Applying
Creating Word Problems - There are many things we can do to create word problems that are more challenging than the standard. As a class, investigate the different types of multiplication and division word problems, then create some of your own. Rank the word problems that you have created from least to most challenging. Use the blank Problem Solving template to help you create your own questions and to use solve the questions from the World Problem Generator below.
Word Problem Generator - Check the images out below to see the different types of questions you might have. Being able to take the question and write it in an abstract way may be difficult to do at first. Try using models to help you figure out what type of question it is.
Creating Word Problems - There are many things we can do to create word problems that are more challenging than the standard. As a class, investigate the different types of multiplication and division word problems, then create some of your own. Rank the word problems that you have created from least to most challenging. Use the blank Problem Solving template to help you create your own questions and to use solve the questions from the World Problem Generator below.
Word Problem Generator - Check the images out below to see the different types of questions you might have. Being able to take the question and write it in an abstract way may be difficult to do at first. Try using models to help you figure out what type of question it is.
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Fractions and Decimals
An inquiry into...
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Tuning In
Theories and Questions - as a class, create a list of questions and theories about the Concepts to add to the board. As time progresses, add to the board, or check to see if previous questions have been answered, or if theories have been proven or disproved.
Assessing our Prior Knowledge - To find out what we already know about mental and written strategies, it is important for us to show what we know. Using a mind map and some suggested questions from the teacher, show what you know about fractions.
Theories and Questions - as a class, create a list of questions and theories about the Concepts to add to the board. As time progresses, add to the board, or check to see if previous questions have been answered, or if theories have been proven or disproved.
Assessing our Prior Knowledge - To find out what we already know about mental and written strategies, it is important for us to show what we know. Using a mind map and some suggested questions from the teacher, show what you know about fractions.
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Finding Out
The Picnic - To the right is a picture of some items that two students intend to bring to a picnic. If just the two of those students attend and they want to share the food equally, can you tell us what each of them will have? What if there were 4 people? What if there were 8? Could you share everything equally? How would you do it? 
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How can we Make Half? - The images above are split in half, or are they? How might you check that each was correct? Can you think of more ways to split a square into two halves? Can you think of ways to split a square into quarters?
How about the images below? Can they be split into equal halves? How about into quarters? Can you make it so each of the halves look the same? If not, how can you tell they are the same size?
How about the images below? Can they be split into equal halves? How about into quarters? Can you make it so each of the halves look the same? If not, how can you tell they are the same size?
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Family Fractions - Not all fractions are going to be circles and squares. In fact, we can see fractions all over the place in the real world. Take a look at the funny family to the right. Can you think of different fractions to represent the family. For example, half of the family have noses. Can you think of any fractions that involve halves? How about quarters?
Paper Folding - Start with a square sheet of paper and make folds to make a new shape.
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Concrete
Tools - there are several tools above that can also be used with fractions. With fractions, there are a lot of specialized tools that we may have never used before. These are helpful for us when looking at parts of a whole. Remember though, we can not know what the fraction is unless we know what the part is and what the whole is.
Tools - there are several tools above that can also be used with fractions. With fractions, there are a lot of specialized tools that we may have never used before. These are helpful for us when looking at parts of a whole. Remember though, we can not know what the fraction is unless we know what the part is and what the whole is.
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Fraction Tiles
- very helpful when visualizing how fractions can be part of a whole - also great for seeing equivalent fractions |
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Fraction Blocks
- similar to fraction tiles, but can be used to stack and connect fractions together - in that way, they are good for showing equivalent fractions, as well as adding fractions together |
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Fraction Circles
- good for showing how equal parts do not always have to be rectangles - great if they do not have any numbers written on them, because 1 circle should not always equal 1 whole |
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Cuisenaire Rods
- a great little tool for showing how parts of a whole can change, depending on what the whole is - also good for showing equivalent fractions |
Pictorial
A Half is a Half is a Half - How many different ways can you draw a half? What does a half look like? Draw pictures to show the different ways you can see a half. Try to be creative. Fractions aren't all just circles and squares.
Fractions on a Number Line - All of the fractions we have been talking about fit between just two whole numbers on the number line - Zero and One! But where do they go? Using the number line below, try to plot the different fractions you know (using words, pictures, abstract numbers, etc.) and where they would go. Where does a half go? How about a quarter? Where is 3/4? What about a third? How do you know for sure?
A Half is a Half is a Half - How many different ways can you draw a half? What does a half look like? Draw pictures to show the different ways you can see a half. Try to be creative. Fractions aren't all just circles and squares.
Fractions on a Number Line - All of the fractions we have been talking about fit between just two whole numbers on the number line - Zero and One! But where do they go? Using the number line below, try to plot the different fractions you know (using words, pictures, abstract numbers, etc.) and where they would go. Where does a half go? How about a quarter? Where is 3/4? What about a third? How do you know for sure?
Abstract
Representing Fractions - How do we show fractions using numbers? What are the parts of a fraction called? What do they mean? Why is a half written as 1 / 2? Can we write the fraction half in any other way?
Take pictures of different fractions using the manipulatives we have in the classroom, or by drawing pictures of the fractions. After you've taken the picture, write down what fraction you think it is using the proper fraction notation. Explain why you think what you think.
Representing Fractions - How do we show fractions using numbers? What are the parts of a fraction called? What do they mean? Why is a half written as 1 / 2? Can we write the fraction half in any other way?
Take pictures of different fractions using the manipulatives we have in the classroom, or by drawing pictures of the fractions. After you've taken the picture, write down what fraction you think it is using the proper fraction notation. Explain why you think what you think.
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Applying
Cutting the Cake - I think if most of us were given a square cake and we had to cut it into four equal pieces, we would probably be able to manage. But is there only one way? Is the one way that we were thinking the same as the one way that other people were thinking? How many different ways can you cut a cake into 4 equal pieces. Using the template in the title and four different coloured pencils, see how many different, creative ways you can cut the cake into four. |
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