This step by step checklist that we use helps ensure that students will start to become more independent with this process as they go. We introduce it a little bit in Kindergarten as a more teacher-led process, but it is integrated into the first grade classroom as well where kids have a step-by-step visual model process to solve. As you can see in the examples, we have Professor Barble explaining the steps.
Read the problem. Then, have someone read it and repeat it, and every time a new piece of math information is presented, we’re going to put a chunk. So, as kids are reading the problem, they start to learn how to dissect what’s being asked. Not all first grade students will be able to read the story problem, but this process is modeled day after day after day in the first grade classroom, so eventually the child will become independent. I’m going to read a story problem: Mark has 9 strawberries, 6 of them are small. The rest are large. How many strawberries are large? Then, I’ll go back and read it in chunks: Mark has 9 strawberries. This is a new piece of mathematical information, so students will repeat that statement back and highlight or put a line there. The students also like to say chunk! Then we continue reading: Six of them were small. I’ll stop, repeat it, and the students say chunk! as they mark that chunk in their journals. Now we have two pieces of mathematical information. Let’s continue: The rest were large. Repeat and then chunk! So, we’ve got three sections of information that the problem has given us that we need to replicate in our visual model. Finally, How many strawberries are large? Repeat that and then chunk! By going through the problem slowly and methodically, students can really see these sections that they’re reading, and, as they’re going on to the subsequent steps of solving the problem, they can actually check off that they’ve included all the chunks of information in their visual model. Create a sentence form. What is a sentence form? Simply put, it is stating what the problem is asking in a complete sentence. I can’t tell you how many times I’ve seen students solve story problems, and actually forget what they’re being asked because they’ve gotten so into the arithmetic and figuring out what they’re doing! So this step keeps them focused on what the problem is actually asking. In our problem, it asked me how many strawberries are large? To put it in a sentence form, I would say: Mark has ____ large strawberries. I like to say Hmm for the ____ as we’re reading it out loud. In Kindergarten, we provide the sentence for students, leaving the blank space for their answer. But in 1st grade, we take some of the scaffolding away. It might say “There were _____ large ____” and the students have to fill in the blanks. The sentence form is a great way to make sure that kids are comprehending what they’re reading. Generally, students in first grade have a difficult time trying to create a sentence form, because they aren’t yet developmentally ready to give you a complete answer in reading. But students will be required to do a sentence form in 2nd through 5th grade so we can be sure they understand the problems being asked, so it’s really great practice to start in 1st grade with the scaffolding. Proportional model. We start the 1st grade year with a proportional model. We may scaffold here for the who or the what, and students will eventually start to learn what goes in that visual model. In this case, we’re talking about all of Mark’s strawberries, even though the question itself is only asking about how many of them are large. In a proportional model, you might see the 9 squares. This is a missing addend problem so that title is going to have PWMA at the top, and there will be exactly nine squares. Some people might think that’s giving it away, but remember the goal of visual models? It’s not to solve the problem but understand what’s going on in the problem, so we’re more concerned about whether or not the student can label the drawing correctly. In this example, the student would total the bar at 9 and check off the first chunk of the problem that we read earlier – Mark has nine strawberries. The next part says “6 of them are small.” In 6 of my boxes, I’ll make six Xs, or I might make small circles, and at the top I can either write small or abbreviate with an s. Then it says “the rest are big.” I could label that other section of the boxes B for big, or write the whole word if I wanted. Then, I need to put a question mark above that section between 9 (the total number of strawberries) and 6 (the number of small strawberries). That section represents the large strawberries, which is what my sentence form reminds me that I’m looking for. Technically, a student could just look at this easy proportional model and say there are 3 large strawberries because it’s right there in front of them. So some people might think this journal is just too easy, but at the end of the day, students are solidifying the process. They’re going back up to the problem and putting a check when they add Xs or circles for the six small strawberries. They’re putting in a check when they’ve talked about putting in the large strawberries. Then they put a question mark to show what we’re looking for. There’s a lot of detail that we’re looking for kids to have to interact with the text in math to show the comprehension. In some of our schools, we will do a unit bar at the bottom of the page. In the 1st grade journal we’ve created for Math4Littles, we’re going to leave the bar off and introduce the non-proportional bar a little bit later in the year. There is nothing wrong with having a model of the proportional bar and then underneath it having the non-proportional bar. In our journal, we plan to show the proportional bar, and then bring in both types of bars so that kids could see the relationship between the two. If where about this non proportional bar, where would I slice it to put the nine in? And then where’s my question mark? is it labeled? etc. The integral parts of visual models are: labelling the who or what, taking the bar and adjusting it based on the information that’s given, and writing in their question mark. Then it’s time to solve! Computation. Although this step might not seem necessary because our sample problem is so simple, and to first graders after they do so many, it seems simple and both teachers and students might wonder why they’re even doing it, but I can promise that these problems will become more complex, very quickly. In our 1st grade journal, we will feature this look at the proportional bar, and then transition to having proportional and non proportional models, and then eventually just leaving it blank and having the student put in a non proportional bar to see that they can develop this progression.
As we take this next step in the developmental journey, concrete, pictorial and abstract means still anchor the child’s mathematical understanding as they work with a blank math work mat to act out problems using items from the Math Salad Bar. The more practice a student has with understanding word problems as they’re being applied in the pictorial and abstract means, the better off they are.
To help students take the next step of their word problem solving, we’ve created the My Math Word Problem Journal, which is developmentally appropriate for Kindergarten students at the end of the first semester or the beginning of the second semester.
This journal contains 75 days worth of journal experiences that reinforce concrete, pictorial, and abstract means and helps students slow down through the various types of word problems. We begin with part-whole addition problems, move into subtraction, and finally missing addend problems. It helps students turn the corner in their early childhood experience to prepare them for what word problems will start to look like as they get older.
In the corner of each journal page is an image that will relate to the story problem. This helps students hold a concrete image in their head as they work with all the other abstract things (writing and number sentences) on the journal page that might not be as familiar to them. Additionally, we are still building oral language, so if students aren’t as familiar with one of the objects, they can use the story they’re reading to help them picture what might be happening.
Begin by reading the problem out loud – all of it, without stopping. After this, most students will want to jump right in and fill in the number bond, or start acting it out with concrete objects. They’ll grab the two numbers in the problem and, because we’re adding this week, they’ll add them. Or subtract them, or whatever we’re working on. However, we don’t want kids to look at story problems as things to be dissected – “circle the numbers” and “underline the important words,” etc. because that really in the end isn’t always going to work for students at this age.
We really want them to slow down and engage with the problem using something called chunking. Chunking has students put a line or highlight or underline a section in the story problem that is bringing in some new information. Even though kindergarten students might not be able to read the word problem themselves, we do want them to get into the habit of “reading” the problem, interacting with the words in the story, and repeating back different parts of the problem. It slows them down enough to really comprehend and visualize what the problem is asking, just like we do in reading. The same process we use for reading a book and trying to understand the author’s message holds true for reading story problems.
Let’s take this problem:
We could chunk our example by saying, “John made a paperclip chain.” And then the students repeat that back. Now you say chunk because there’s a new piece of information within that story problem that we just read. The next part says he put on five paper clips. chunk The kids repeat that section back. He added on three more paper clips chunk. How long is John’s paper clip chain now? chunk
The hard part about doing story problems in a slow, methodical, repetitive way, even though we know it helps build kids’ skills, is that students really want a quick and easy way to solve the problem. Spending a lot of time really looking at that story problem and getting that frame of reference using our template will be really helpful to our littles in kindergarten.
The second step in our step-by-step visual model process is to create a sentence form. In the kindergarten journal, we provide the sentence form since we felt that putting the question and a complete answer might be a little bit too difficult for some
For our example, our sentence would be, “John has ___ paper clips on his chain.” When we read the sentence form out loud and get to the blank, we usually say “hmm” and maybe shrug our shoulders because that’s the part we know we’re going to be solving for.
In Kindergarten, it’s really important to make a note of this sentence form because, as students get into 1st grade, they’re going to start to do more closed sentences with more blanks that require students to supply names and other information. By the time a child gets to 2nd grade, they should be able to read a story problem and be able to repeat back in a complete sentence exactly the way the problem is asking.
Some people ask why we would bother putting the sentence form in a Kindergarten journal. It all goes back to wanting to really slow the students down and keep them from jumping right into numbers and operations. Having students work with the sentence form encourages good decoding and reading skills as well, and guides students in their understanding of what the problem is asking. Also, I’ve seen many students solve story problems, but then forget to write in the sentence form, so a sentence form is a really great way to help train students’ brains so that when we get that final answer, it is going into the blank, which will complete the thought.
Whether you use our horizontal or vertical math work mat, maybe a dry erase board or a purple piece of construction paper, this step is all about concrete tools. The student can go to the Math Salad Bar and choose manipulatives they can use to act out our story problem. They might get five of something, then show three and then count them all together to show how many paper clips John had.
Eventually, students will work more in the Quick Draw Box. We’ve made this section a little bit smaller in this journal because we want students to truly make it a “quick” draw using Xs or dots or something else small to represent the quantities in the problem. In our case, the students might do five large Xs, three small Xs, and then count them all together. ’
It’s very important for students’ mathematical understanding to keep things organized when we’re counting. We don’t want to always be one-to-one counting, so we use a 10-frame to guide us here. Students can use pencil, crayon, marker, or anything else to fill this in. I might choose five red circles for the large paper clips. Then I can represent the three remaining paper clips with blue circles. The idea is, when you look at that 10 frame, not only can you find the answer, but you also see it in an organized way based on what the problem is asking. Over time, we want the quick draw to be more organized as well
This next representation is what students are really learning a lot about in Kindergarten – part/part/whole. The number bond helps students visualize the relationship between those two numbers. In this case, the student fills in the two parts, the two bottom spokes, with 5 and 3, and then shows the total of 8 in the middle. The number bond would read 8 on the top, 5 on the bottom and 3 on the other side to complete the thought.
Number bonds lead into the algorithm where students can show their computations. This is the part they usually want to do first, but we’re only just now getting to it as we follow this developmental process.
We’ve given lots of thought to each part of the journal page so far, and this algorithm should be no exception. It would be easy to just grab the numbers out of the problem and plug them into the slots and call it good. But let’s start with that first box. In our word problem, we started with the 5 large paper clips and now we need to change that number in some way, whether we add or subtract. In our example, I’m going to be counting up the chain, so I know we need to put a plus sign in the circle. I write the 3 in the other square, because I’m changing the original number by that many. The equal sign is in another box to the right, and finally there is a place to write the total.
Every journal page takes students through this entire process, so by the end of 75 days, it should be ingrained in their brains! This will help our Kindergarten littles as they head into 1st grade, where they’ll continue to build on higher levels of understanding.
We have three different videos to help you to see the different types of problems that can be solved. We also use a coding system on all of our journals to help teachers understand the different types of problems that we’re using. For this journal, we have PWA (Part-Whole Addition), PWS (Part-Whole Subtraction), and PWMA (Part-Whole Missing Addend).
Why do we code problems for students this young? To a parent or teacher, or even a student, It might feel like there are endless types of story problems. But in fact, we actually have families of story problems, like we have genres of books. Coding the story problems compartmentalizes these different types of story problems. If we are very clear on the type of story problems, we can help students understand the characteristics of the types of problems they will encounter so that problem solving isn’t so scary.
Part-whole problems are the first genre of story problems, and K-1 graders spend a large majority of time working in this group. However, there are lots of different types of part-whole problems – part-whole addition, part-whole subtraction, part-whole missing addend, etc. The next family of problems are the additive-comparison problems, which students will get to later in 1st grade.
In the bottom right-hand corner of each page, there is a number, up through 75. You can download the PDF, print it (or have it printed), and staple it together to make a journal for an individual student. They write their name on the front and it becomes something they work on every day, four or five days a week. Of course, we still do our number talks in the classroom, but this way we make sure we’re bringing our numbers into words and words into numbers on a daily basis as well.
It’s a really important skill for a child to be able to construct a story problem based on an algorithm or even a missing addend. Let’s take: 4 + ? = 7 If I gave a student that problem, could they think of a story that goes with that? Maybe there were 7 sheep on the farm, but only 4 of them were in the barn. How many were outside? This type of thinking really works on Math Practice 2, which is to which is reason abstractly and quantitatively, to help bring numbers and words together.
Join us next week to find out what problem solving should be looking at with a child after the Kindergarten year. We’ll be working on taking our quick draw into a proportional visual model, and then into a non-proportional visual model.
]]>Whether you’re working with your student at home or in a traditional classroom, it’s important to understand the progression of how a young child’s mind starts to comprehend word problems.
We start with real objects in the physical world, providing students with language to explore and manipulate objects as they begin to understand the relationship between numbers and words and words and numbers. Then, we move into quantitative pictures, counting things and creating lots of conversation through purposeful questioning.
As students master the first two developmental stages, we can move them into the next stage which is acting out story problems using CPA. This has nothing to do with the person who does your taxes, but everything to do with getting kids to express their knowledge of mathematics concretely, pictorially, and abstractly.
Story mats are an excellent way to help children solve problems with different types of real-world scenarios. We are so excited to bring you these Word Problem Story Mats – 10 story mats, 10 sample problems for each = 100 problems you can use with your littles for only $3.99! There are three great video tutorials that will show you the mats in action and help you learn how to use them most effectively.
These mats are designed to guide students through exploring their thinking by providing a visual image and allowing students to act out the problem. The visual picture might be a barn, or a house, or a dinner table – something that is familiar to the child that they can use to put counters on to represent a quantity.
Sometimes it’s difficult for our littles to use an inanimate object, like a counter or an M&M, to represent something that isn’t real. As we’ve talked about before, many of our children are really skipping over the imagination stage in their childhood. Because kids are so plugged in these days, very few children have imaginary friends anymore, and they might even have difficulty imagining how to put on a puppet show or play something like kitchen where pretend something is real. I vividly remember sitting in a play kitchen in a preschool classroom and pretending to pass out pizza to everybody that was sitting at the little table. A little preschool boy said to me, “You didn’t put anything on my plate. There isn’t any pizza.” I said, “Well, yes I did! I just put it right there! It’s the pretend pizza!” He was skeptical, and he represents many of our kids today that, unless it’s a real pizza or a rubber pizza that looks like real pizza, they have a hard time using their imaginations to represent real objects.
Depending on their developmental stage, it might take littles some time to see concrete tools as something that can mathematically represent something entirely different than what it actually is. When you start to use objects like counters or beans, or maybe even cereal if you’re at home, to represent things, our littles sometimes have a hard time transitioning to that. For example, if we’re using counters to represent eyes for an owl, we may have to explain that owl eyes don’t really look like that, we’re just representing how many there are. Or if we’re using one-inch squares to represent people sitting on a playground bench, we may have to help students envision a person in each square. Realistically, if we’re looking at a beach or a garden, we know we can’t really turn our counters into fish or flowers.
So, concrete representation is the first stage in this next level. We need to help kids start to understand that an object can be related to something. We can provide different scenarios – a tree, a bucket, a sandbox, a lake, a road, etc. – and have them pretend that an inanimate object represents something quantitatively from the story problem. For example, if we’re working with our “In the Tree” story mat, we might talk about birds or owls in the tree. Students need to be able to hear the story problem read aloud, and then use the counters to represent the birds or owls, even though they look nothing like birds or owls, as they “act it out” on the mat.
As students become more and more familiar with concretely acting out the story problem that they’ve heard read aloud, they are going to get more and more fluid within their understanding of bringing math into the real world.
Much like with concrete representation, we have to retrain our students to think about pictorial representations differently. In math, and on our story mats, we have what we call a Quick Draw. In traditional writing and liberal arts, students are often told to use all the white space, and to accurately represent what they’re seeing – correct colors for hair, legs at the bottom, etc. Students end up with more color and description in this type of drawing. In math, less is more, especially in a quick draw.
If I’m counting ants on a playground (one of our story mats), I’m not going to draw individual ants out in my Quick Draw, because that’s not very quick. I’d have to draw their legs and then their antennas, etc. We don’t have time for that!
After we teach kids to transition from a real object to an inanimate object that represents something, we have to transition them to being able to draw a bird as an X or a circle, instead of a bird with a beak and eyes and feathers. I call it ringing and labeling – using an X or putting in a dot or whatever is going to represent pictorially what we’re talking about in the picture. Obviously, depending on the age of students that you’re working with, they may start by doing more detailed drawings. This is certainly okay because kids will naturally transition into a quick draw as they realize they can represent the object more quickly with simpler shapes. For example, if there were six ants crawling on the playset and two of them crawled away. How many ants are left? We could draw six circles, cross out two, and find out our answer.
Once students can represent an object concretely with an unrelated object, and draw a picture of their thinking, we can add in abstract thinking. This can be a number bond or a number sentence – something using numbers to show what they’ve done concretely and pictorially.
This is not the right level for all students right away, and we don’t want to build to this level too early. In my opinion, it’s more important for students to understand the concept behind the digit, or what the subtraction sign represents, than to just be able to write it. If kids can’t make the connection between words and numbers or number and words, or are pushed through the standards too quickly, it can be detrimental to their math foundation. That’s why quantitative pictures and story mats are such great tools as kids can interact with these concepts in a playful, real-world way.
Working through story mats hit all three aspects of this phase of development as students build problem solving skills. We have a set of Word Problem Story Mats, a Math4Littles publication, and some great tutorial videos that will help as you apply CPA to real life situations. Whether you’re a teacher or a parent, you can use these with your littles! Check out the previews in the videos below and get all 10 mats for $3.99!
For each story mat, there are 10 sample problems. The first three are focused on part-whole addition. Numbers four and five focus on subtraction, six and seven are missing addend problems, and the last three are deeper thinking challenge problems. These problems ask questions a little differently to encourage kids to think outside the box and use higher-order thinking skills in order to find a solution. For example, I saw six eyes. How many frogs were in the pond?
You can easily differentiate the level of any of the problems simply by changing the quantity based on the skill level of the students you’re working with. If you’re working on quantities to 10, you can adjust the numbers to make sure the total is 10 or less. If you’re working with younger students, you might want to keep the total at 5 or less.
You might also notice that the problems are pretty generic. We use lots of personal pronouns and general nouns. Insert the student’s name into the story problem, or the names of family members or teachers that are familiar to make the story problems more personal.
Our goal, as teachers and parents, is to help kids become deeper thinkers. To do this, we have to remember that we aren’t the givers of all the information! Yes, we know the answers and how to solve the problem, but we need to let them figure it out for themselves because we know that they will remember it better if they do. Our natural instinct, if we notice kids beginning to act out a problem incorrectly, is to stop and fix it so they don’t end up “wrong.” But we don’t need to autocorrect students as they are solving problems!
Think about how we would handle this in reading, maybe in a guided reading group. If a student comes to a word they don’t know or says a word incorrectly, they’ll typically appeal and wait for you to give the word. As good teachers, we don’t do that though! We want them to engage in a little productive struggle and use the strategies they’ve learned to help them decode the word on their own. We might walk them through a few strategies – let’s look at the picture, let’s reread the sentence and look for context clues – but ultimately, you want them to decode the word. We want them to do the work because, if they don’t, the next time they come to that word, they’ll be right back asking for help again.
Math works the same way. Obviously, we don’t want to tell them the answer. We want students to have a little bit of productive struggle as they begin to solve the problem. But we also want to be careful about our questioning. We can technically not give away the answer, but still walk students through exactly how to solve the problem just by asking leading questions, which practically give away all the information they need. This is what we don’t want: “There were six frog eyes. Ok, everyone get out six counters. Now, let’s put them in groups of two because you know that frogs have two eyes. How many frogs were there? Ok, let’s count our groups of two to get the answer.” No learning took place here!
Instead, we want you to ask questions that are open-ended: Can you tell me more about what you’re thinking? I noticed that you have out counters…how many counters do you have? Why did you pick that many counters? The more we can let kids explain their own thinking, the better!
Additionally, when you begin questioning a student, most of them think they’re wrong because “you wouldn’t question me if I was right!”. So we encourage you to question students as they’re using these story mats, even if what they’re doing is correct. At the very least, it will have them going back and rethinking themselves Did I do it wrong? We want kids to be able to defend their answers and the way they found it.
Because we are encouraging deeper thinking, there isn’t one right answer in how to solve the word problems we’ve included in our story mats. If a student’s work is different than you expected, that doesn’t mean it is incorrect! When students are working, I encourage you to ask those open-ended questions to get them to do more of the talking and the deeper thinking. This is how we will get kids to get better at problem solving as they get older.
After students can easily move through all three levels of CPA with a story mat, it’s time to give them a blank story mat where they can work similar problems without the scaffolding of a picture. Developmentally, students at this stage can envision that scenario without the picture because they’ve worked through all the levels and built their foundation for math. They’ve done it with real objects, they’ve done it with quantitative pictures, and now they’re doing it on a story mat. We have one you can download for free – either in portrait or landscape.
You certainly don’t have to use the story mat from our website, but we do recommend having a designated mat or space to work the math problems. These mats are designated space where students can act out the story in a concrete way, and it helps eliminate some confusion as they switch back and forth between pencil and paper. It could be a dry erase board, it could be purple construction paper, but it just needs to be something to hold manipulatives!
This works well in a group setting, maybe with kindergartners, to have students come and sit in a circle with you with their blank work mat and a bowl of counters. Read the story problem out loud to them, and once they hear the problem, they can use their manipulatives to act it out the way they’re hearing it. This is a great way to know that the child is at that top point of being able to start to move into the early stages of using a visual model to solve story problems.
Next week, we’ll talk about a way to scaffold students in kindergarten and first grade to move through the progression into more traditional story problem format that will help them create success in applying mathematics.
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Last week we focused on providing some of those real life experiences for littles through intentional, math-focused conversations embedded into your daily routines. Even as you progress through the stages of developing early problem solving skills, don’t neglect those conversations! The topics of conversation might adapt as your child masters certain skills, but every math-focused conversation you have strengthens that child’s understanding of math and adds to their body of experience with real objects in the physical world.
The next stage of problem solving involves using what we call quantitative pictures. It sounds fancy, but quantitative pictures is simply a picture that contains lots of things to count! You could use a picture from the Hidden Picture section of the Highlights magazine (if you have it, or you can get a few for free online: https://www.highlightskids.com/games), or you could use a page from a coloring book or favorite picture book. National Geographic even has pictures of animals and things in nature that could be counted.
One of my favorite resources to use is a book called Math Talks by my friends Char Forsten and Tori Richards. So many of our teachers use and love this book. Sadly, it has been out of print for a while, but we are super excited to announce that we have the exclusive rights to the digital version of this book! You can download the eBook from our SIS4Teachers store and use it right away!
The title of this book catches people off guard sometimes because “math talks” sound a lot like the “number talks” that we talk about. Of course, both types of talks deal with numbers, but in different ways. Math talks use real pictures in a quantitative way and focus on using different types of questions (from beginning to challenging) to help students think about numbers.
For our littlest learners, questions about a quantitative picture could be very basic: What do you notice? How many children do you see? Our advanced students could look at the same picture and we could ask more complex questions: If two more children join these children on the beach, how many children would there be?
In Math Talk, not only do you get a great selection of full-color images and a range of questions to ask about it, you also get a black and white version of the image that is almost like a coloring sheet. This is great if you want students to literally interact with the picture by coloring the different objects you are asking questions about.
In these videos, I’m going to show you how you can use quantitative pictures to help make word problems come to life for students of all ages! I’ll be using two of the examples from Math Talks, At the Apple Orchard and At the Beach, and you’ll be able to see how you can use one picture for a really wide range of students just by varying the questions from basic, to intermediate, to advanced, and finally to more challenging.
Download At the Apple Orchard: bit.ly/AttheOrchard
Download At the Beach: bit.ly/SISAttheBeach
I like to display my quantitative pictures on a document camera or even a smartboard so we can interact with them in different ways, depending on the level of students. Younger students might just need to come up and be touching and counting. Older kids might start to draw or circle objects in the picture if you’re doing this in a whole class setting.
This is a great opportunity to double-dip and work in some vocabulary words as well! Depending on what grade you’re teaching, directional words work really well with quantitative pictures. Talk about objects that are above or below, next to, behind, etc. Use plenty of adjectives to describe physical characteristics of objects you’re looking at – large, small, round, flat, etc. – as well as comparison adjectives – big, bigger, biggest.
Even though Math Talks lists sample questions, those aren’t the only questions you can ask about the picture! By applying different mathematical concepts – part/part/whole addition, multiplication, division, fractions, etc., you can extend the use of a quantitative picture beyond the early childhood years. With a little practice (and a few sample questions as a guide!), you can apply whatever concept you’re teaching in math to a quantitative picture!
Once students are confident and comfortable examining quantitative pictures of math in real life situations, we want to look at how we can take this progression of problem solving to a more hands-on level of actually where children act out the problem with concrete objects, create a pictorial representation, and then record it in an abstract equation. Join us next week to talk about how we do this with story mats!
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This is one of the math practices that really stumps many of the teachers I work with, and rightly so! The practice itself is a little more abstract than we’re used to when it comes to math! But what it really means is that we’re looking at numbers connected to words and words connected to numbers, which we do when we apply mathematics into real world scenarios, or when we do word problems.
Just like fractions (which we explored in the previous blog series), word problems tend to strike fear into the hearts of teachers and students alike. In this blog series, we’re going to demystify the dreaded “story problem” and explore real life math so we can help our students learn to “reason abstractly and quantitatively.” We’ll look at a process to help students with reading comprehension skills using all kinds of different problems, from part/whole addition to part/whole subtraction, part/whole missing addend, part/whole multi step, part/whole multiplication and division, additive comparison, multiplicative comparison, fractions and more.
Story problems can help students learn the basic arithmetic of math – the mechanical adding/subtracting/dividing/multiplying of numbers. But when students are asked to apply that arithmetic to real world situations or to think more deeply about those mathematical concepts, they get confused because they don’t have a deeper understanding of the mathematical processes or what the problem was actually asking.
We have the simplistic story problems like, “3 frogs jumped in the pond, and then two more joined them. How many frogs are in the pond?” But we also need to have more complex problems such as, “I baked four dozen cookies last night. 1/3 of the cookies that I made were chocolate chip, 3/4 of the remainder were gingersnaps, and the rest are peanut butter. How many peanut butter cookies did I make?” Answering a question about frogs is much different than answering a fraction multistep comparison problem.
The complexity of that kind of problem can stop a student dead in their tracks. Their first response is usually a high-pitched questioning voice wanting to know Do we just add (or subtract or multiply or divide)? They watch the response to their questions very carefully – if the teacher doesn’t seem happy with one suggestion for an operation to use, they’ll quickly suggest another until they figure out what they think their teacher wants. When it comes to story problems, students just don’t seem to have the perseverance they need and they aren’t interested in “productive struggle.” They just want a teacher or adult to tell them what to do and how to get the answer and save them from the story problem.
However, that approach isn’t really going to help kids in the long run. Everything that we’re doing with 21st century mathematics is really about applying it to situations in the real world. I’ve talked about this many times – I can just pick up my phone and ask Siri What’s 25 times 35? And Siri will regurgitate that answer without a problem. But if I can’t apply it in a real world situation, that’s where the struggle begins with the application of math.
We’ll start this journey through the development of word problem skills at the beginning: with our littles. I always talk about peeling the layers back to look at where a child is coming from in the early years of problem solving. How do we do that?
Ask almost any teacher where we start with reading, and they’ll say something about oral language, phonemic awareness, all those great things that happen to allow a child to put sounds together in order to read words on a page. Decoding in reading is a lot like calculation in math. As a student progresses with the mechanics of reading, we start to think about reading comprehension, which is another skill altogether. The same thing follows suit with math. As students become more competent with math calculation, we move into what’s called math reasoning.
If you’ve learned with us before, either in an on-campus workshop, or through our M³ coaching, or even just reading our blog, you’ll know that we really emphasize peeling back the layers in math calculation with Math Mights and concrete, pictorial, abstract (CPA). Clearly, math calculation gets a lot of attention because it’s so foundational for the rest of mathematics. But, if you’re like many of the teachers we work with, you might feel a little less confident helping students with that next layer of math reasoning. The good news is that we can start building math reasoning skills very early on!
As we know, the first part of this involves real objects in the physical world. We want students to have thousands and thousands of experiences with real objects in the physical world to give them the opportunity to explore real life mathematically. This doesn’t have to be a formal math lesson with counters and flashcards for your 2-year-old! It can be picking up two objects and talking about which one is heavy and which one is light. It can be making a tower out of red blocks and building a tower of equal size out of blue blocks. It can be setting the table for a family dinner and knowing how many plates and forks will be needed. This is where real life math begins!
We just finished an amazing #sis4students Virtual Math Series (check out the archive here!), where we partnered with Making Math Make Sense to help parents, teachers and families work with their students to deepen their understanding of math. One of the things that we created for that Virtual Math Series is something called Math4Littles, since littles is where this all starts!
Creating a math-friendly home is the topic of the first Math4Littles series of videos. It is a room-by-room guide to helping math come alive and become concrete. Each video has five specific ideas or activities you can use to have math-focused conversations with your little as you go about your daily routine, and its all written out on a one-page printable so you don’t have to remember everything!
The kitchen is a great place to begin to bring math into reality! Next time you’re eating breakfast with your little, count together to see how many tablespoons of Cheerios it takes to fill up your cereal bowl.
When you head outside to explore, let your little pick up different types of leaves and rocks and sticks to carry home. Then, work together to sort them by an attribute like shape, color, or texture to help them see mathematical concepts with real things in their life.
After lunch, play “I Spy” in the family room, but describe the target object with mathematical vocabulary – shape, quantities, etc.
Then, after dinner, before your little jumps into the bathtub, take a second to observe the waterline, maybe even mark it with a bathtub crayon. Then, when your little is in the tub, help them notice what happened to the water line. You don’t need to go into a complicated explanation of volume and displacement, but simply calling attention to the change plants the seed for understanding volume.
It might seem insignificant, but strategically incorporating math-focused conversation will lay the foundation of early problem solving, which feeds mathematical reasoning skills later on! These concepts are crucial to our littles’ development of oral language with real objects in the physical world. Even if you don’t have a math degree, even if math “isn’t your thing,” all it takes is a little guidance and some intentionality in your conversation to help your little develop early problem solving skills in the comfort of your own home! Check out the tutorial videos and let us know how it goes!
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When it comes to subtracting fractions, this operation is very similar to adding fractions in how students understand it. We always believe that the CPA approach is the best way to help students gain an in-depth understanding of what we’re conceptually asking them to do so they don’t just learn a procedure for a concept they don’t understand.
Pattern blocks are one of the concrete tools we like to use to help students understand this concept. I found some new pattern blocks that I’m really excited about because they have a brown piece in the set that represents ¼! There’s also a purple piece that represents 1/12! You can get those on Amazon here!
When using the pattern blocks, remember that one whole is the hexagon, the trapezoid is the ½, the rhombus is the ⅓, the triangle is the ⅙, and now we have a brown shape that is ¼ as well.
Let’s look at ¾ – ¼, and we’ll start by building ¾ on top of the whole. If three brown pieces are put onto top of the hexagon, and it becomes very easy to see how, if ¼ piece is taken off, then 2/4 will be left.
Of course, that is not the simplest version of that fraction. Building the problems and visualizing them with pattern blocks also makes it easy for students to be able to simplify their fractions. They can match up the pattern blocks and see what piece would cover the total. In this case, it is easy to see that the trapezoid, or the ½, fits over the 2/4 pieces, so ½ is equivalent to 2/4.
The same process works for subtracting fractions larger than one. These “top heavy” fractions where the numerator is larger than the denominator, are often called improper, even though they are still legitimate fractions. They’re just larger than one!
For this example, let’s take 10/6 and build it with pattern blocks. We’ll use two hexagons to show that the fraction is larger than one, and we’ll layer on the other blocks so we can represent sixths. Once I count out all 10 of my ⅙ pieces on top of the hexagons, I can see that one hexagon is full, and I have 6/6 and 4/6. I can see that 10/6 = 1 4/6.
Now that we have that number built with pattern blocks, if I wanted to take 7/6 away from 10/6, I could easily see how that would work. 6/6 fills up one whole, and then we’re left with taking ⅙ from the rest, which gives us 3/6. I can see the three triangle pieces, compare it to two brown pieces to see that 3/6 is equal to 2/4, or put the trapezoid on top and see that 3/6 is equal to ½.
You could use pattern blocks for working with this type of fraction, and I always love using patty paper with fraction examples, but the ideal tool here is going to area model paper, and that’s the tool I’ll use in this video.
With area model paper, my whole is going to be red, my ½ will be orange, ¼ will be yellow, ⅛ is pink, and 1/16 is green.
My problem will be ⅞ – 4/16. Let’s pause for a second and make sure that students really understand what ⅞ is and what it’s asking. So let’s build the ⅞ by stacking 7 pink pieces on top of the whole. At a glance, I can notice that one more ⅛ would cover the whole. But I want to take away 4/16.
At this point, we are often quick to teach children a procedure such as “just skip count to find the common denominator.” However, I love taking an inquiry-based approach to problem solving, especially with fractions like this. I want students to think about how they could solve this problem in multiple ways.
I’ll put 16ths on top of the ⅞, and when I do I can easily see that ⅞ is equivalent to 14/16. Then, it’s an easy subtraction problem because I’ve made them all 16ths. I just have to take 4/16 away from 14/16 and I end up with 10/16. Although 10/16 is not in the simplest form, it is definitely correct. I’m not worried about simplifying as much as I am about helping students discover an answer and solve the problem using concrete tools.
Let’s look at the same problem again: ⅞ – 4/16. Some students might notice that I could have used 8ths instead of 16ths. So, I can take the 4/16 and put the ⅛ pieces on top to show that it is equal to 2/8. When the problem becomes ⅞ – 2/8, we can solve it much more quickly and find that we have ⅝. In this case, that answer is in the simplest form, and we eliminated the step of having to reduce or simplify.
When students start to learn about subtracting mixed numbers, it’s important to make sure they understand it in a conceptual way. We’ll switch back to pattern blocks for this example. While I like using the area model papers, pattern blocks represent whole numbers much more effectively and efficiently, as you don’t need multiple sets to do so.
Let’s take 2 ⅔ – ⅓. Let’s build the two wholes with two hexagons, and then we’ll have 2 rhombuses to show the ⅔. I can simply take off ⅓ and see the answer will be 2 ⅓.
This is pretty simple, but where students sometimes get tripped up is when they have to rename numbers. Sometimes we call it “borrowing” but borrowing really means we’re going to give it back, and that’s not what we’re doing when we’re subtracting!
To demonstrate concretely how we rename fractions, we’ll use the example 2 ⅓ – ⅔. I can’t take ⅔ away from ⅓ so I’ll have to decompose (like D.C. does) and rename that 2 ⅓. We’ll do this problem concretely and pictorially in the video.
I’ll change one of the wholes into 3/3, add it to the remaining 1 ⅓, so it becomes 1 4/3. By building it with pattern blocks, students can clearly see what’s happening to the numbers when they are renamed. We can also ask students to prove that 1 4/3 is equal to 2 ⅓ by building the numbers as we just did.
Once the number is renamed, students can see how it then becomes possible to take ⅔ away from the new amount of 1 4/3. I prefer to show this horizontally as opposed to stacking it in a traditional algorithm at first, because I want to promote their number sense and what they’re doing, versus just having them memorize a procedure.
In our last example, we’ll look at how we can take away 1 ¼ away from 3. I know that taking away that one will kind of be easy, but how do I take away the ¼ when I have three wholes? Again, using D.C., I’ll rename the 3 to be 2 4/4. This will make it a lot easier visually to take away 1 ¼ because I’m just going to take away one hexagon and one ¼ piece, leaving a nice easy answer of 1 ¾.
We can’t leave subtraction without a visit from one of our favorite Math Mights – Springling! The fanciest Math Might, she was born with a coily tail and fancy eyelashes. Fractions on a number line is a really important concept for students to learn, especially because using an open number line to find the distance between two numbers is something that they will use into their middle school years.
For Springling’s example, we’ll use 2 ⅓ – 1 ⅔. Springling wants to hop to find the distance between the two numbers, so we’re going to start by putting the subtrahend (the second number in a subtraction problem) on an open number line. Then, we’ll draw a line and put the minuend at the other end.
As we look at the distance between 1 ⅔ and 2 ⅓, it should remind students of looking at the distance between whole numbers. If students aren’t familiar with Springling, however, there’s nothing wrong with taking upper grade students back to whole numbers to understand the concept of the open number line. Watch the video of Springling subtracting whole numbers on YouTube.
Starting at 1 ⅔, we can see we only have to go ⅓ to get to the whole number 2. From two, we want to get to 2 ⅓, so we can see the distance between the two is ⅔.
Let’s take an example with higher numbers – 12 ¼ – 6 ½.
I recommend using friendly fractions with Springling as kids start to develop their understanding before moving on to more complicated ones! Friendly fractions might be ¼, ½, ¾ – things kids might be able to relate to time or even quarters and dollars.
In our example, Springling will start at the subtrahend, the 6 ½, and look ahead all the way to 12 ¼. She can see that 6 ½ is only ½ hop away from the friendly number of 7. We can draw a curved line, like a small hill, to represent a part of a whole.
Once Springling gets to 7 on the number line, she wants to travel to the next highest whole number, so she hops all the way to the number 12. We’ll draw a large peak here to represent a whole number. The same idea applies if you’re using Springling for elapsed time, money, decimals, or even fractions – use a curved line for part of a whole, and a large peak for whole numbers jumps so kids can easily tell the difference.
The distance between 7 and 12 is 5, and then Springling only needs to hop to 12 ¼, which is ¼ hop. Once I add all Springling’s hops together (½ + 5 + ¼) we can see the answer is 5 ¾.
Using Springling to solve subtraction problems will help to build the number sense that we want children to have as they can visualize exactly what is happening as we work with the different numbers.
The big takeaway message for teachers is that we really need to slow down the process of subtraction to help kids build conceptual understanding for what we’re doing with fractions. We hope these videos will help you do just that.
You can use these videos in two ways. One is as a lesson launch, making sure students have concrete tools in hand as they watch so they can explore the concepts simultaneously. After watching the video, see if students can create other problems like the ones we showed in the video to demonstrate their understanding. They can solve their problems with a partner or share out in a small or whole group.
You also can use the videos in a flipped classroom model where students listen to the video either at home or maybe in another station in your classroom as an introduction to what you’ll be talking more about in your Math with a Teacher station. It’s really helpful for students to have more of a background knowledge of what they’re learning about before you actually try to teach them, especially if we’re teaching with an inquiry-based approach. When we’re not serving as the givers of all information, kids can develop a deeper understanding because they can grapple with the concepts they’re learning and how to apply them.
In our next series, we’re going to feature visual models for word problems that you can apply to any of the math programs that you’re working with. We’re excited about bringing you really great reading comprehension strategies to help your children become more proficient in understanding what story problems are asking!
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I remember my teacher telling me to add the numerator and just keep the denominator the same. I also vividly remember asking why? Why am I adding the top number, but not the bottom number?? The teacher answered “because that’s just how you do it.”
Think about this problem: 5/7 + 6/7. The rule says add the numerators and keep the denominator the same, so I go across and add 5 + 6, keep the bottom number the same and the answer is 11/7.
Many students faced with the same problem forget about the rule for the denominator, so you might also get 11/14 as an answer. While that looks like it should be right (after all, you added the top numbers and then added the bottom numbers, right?), of course, we know that isn’t the answer at all. Teachers will tell students to go back and “do it right,” meaning “go back and follow the procedure.”
But does this procedure we’re supposed to follow actually help students understand the why before teaching them the how? Do students in your classroom really understand what’s happening when you’re adding fractions? Do they understand what that whole fraction is? Do they understand benchmark fractions as they’re adding?
In this blog we’re going to see how we can use CPA (concrete pictorial abstract) means to help our students to understand what’s going on with the addition of fractions.
Let’s go back to that problem of 5/8 + 6/8. When the procedure is followed correctly, we get 11/8. The answer is what we might call an improper fraction.
I try not to use the term improper fraction, however, because it isn’t truly improper, is it? It’s still a fraction that we could visualize – if I was really hungry and I wanted to eat 11/8 of the pizzas that were out, I could do that. Instead of calling them improper fractions, we need to make sure that students understand that an improper fraction just means a fraction larger than one. Consequently, students need to know what one whole is in any given problem to help them be successful as they continue solving.
Back to our “fraction that’s larger than one” – 11/8. The next thing the teacher says is usually, change this improper fraction into a mixed number. The student has been taught a procedure for that, so he starts to think Oh yeah, I have to figure out how many times 8 goes into 11. I think it might go in there twice, no wait, that’s too high. Let me think about it… it goes in there once, and that would be 8 and then if I count up 9, 10, 11. I’m going to put there’s a remainder of 3.
But it’s a fraction, so the teacher tells the student to write it that way. The student puts that remainder over the fractional part, and gets the answer of 1 3/8.
What we see here is that, if the student follows all the procedures, they get the answer. But I’m not convinced that student, or any students today, will really understand the why behind what they’re doing with fractions by just following procedures.
Many adults have fraction phobia as well because they weren’t comfortable using fractions when they were younger, and therefore, when they go to teach it, it comes out procedural, just like they learned it.
How could we could help students to easily understand taking a fraction larger than one, and changing it into a mixed number? Let’s have them think about what the fractional parts are. If we ended up having this total of 11/8, we need to know what the whole is. Since we have 8 as the denominator, we know that the whole would be equal to 8/8. Based on that, I can use what I know about decomposing from our Math Mights friend, D.C. to smash or decompose that 11/8 into a whole, which is equal to 8/8. We have students circle the 8/8 to help them remember that 8/8 is equal to one whole. If our total was 11/8, and I’m showing 8/8, how many more 8ths are left? 3/8. So the answer is 1 3/8
When students think of simplifying fractions this way, they can decompose fractions that are larger than one and put them into a mixed number in a way that makes more sense. In this video we’re going to show you different examples of how you can use a concrete tool when students are adding two fractions with common denominators, coming up with a fraction larger than one, and changing it into a mixed number.
When adding fractions, it’s really important that we make sure that students are understanding what we’re talking about conceptually, pictorially, and abstractly. But we don’t want to teach those things in isolation. We want to teach through CPA, meaning that kids can show different ways of understanding what they’re doing.
I like using our area model papers for students because they aren’t stamped like the fraction tiles, and they help students to conceptualize what’s actually happening as they simplify, or reduce, a fraction down to show another name of that fraction. In this video, we’re going to show how to add fractions with common denominators. I’ll also show how you could have conversations with students around their thinking based on their prior knowledge, or what they’re doing as they’re using the different parts to it.
Note: When adding fractions, we might need two of the area model kits so students can see what’s happening, especially if the total is going to be larger than one.
Let’s use the example of 6/8 + 4/8. To show this in a concrete way, we want students to use their whole sheet of paper and stack their 8ths on top to show 6/8 and then add in 4 more 8ths. You could also have students work in pairs to do this simultaneously where Partner A builds 6/8 and Partner B builds 4/8.
We want students to look and see how many more pieces we need to make a whole, which in this case is 8/8. On one paper, Partner A has 6/8 covering the whole, and on the other Partner B has 4/8 covering the whole. With the problem laid out like this, it is easy to see how you can decompose the 4/8 into 2/8 and 2/8, take 2/8 and combine it with 6/8 to make a whole, and then have 2/8 left over.
When kids see the problem concretely, with the area model papers, they can use a kind of fraction number bond which is a really great pictorial representation of their mathematical thinking. Then abstractly, they can show the new problem that they created through decomposing and composing, which is 1 plus 2/8.
Using decomposing and composing is a great way to show kids how to understand how to add fractions with common denominators. Check out our DC video on how DC is adding together fractions by decomposing to make the problems easier.
You can also use the area model papers to help students to conceptualize adding fractions with uncommon denominators. The area model papers we use in the videos (get the free template!) are broken down into 1, 1/2 , 1/4, 1/8, and 1/16. These are friendly fractions to help students understand the patterns of these numbers and how they can add and subtract using different strategies.
This video will show the common types of problems that either have an even answer, or you could offer the opportunity for students to add them and then figure out what the simplest form would be or how to reduce the fraction.
For this example, let’s add 2/4 plus 4/16. First, we’ll build the 2/4 with area model paper, in my case, those are the yellow pieces. Then, add on the 4/16, which in my set are green. With both yellow and green pieces on the whole, then ask the students – how can we add this together??
When we learned how to do fractions (the traditional way), many times our teacher said to skip count and find the least common denominator. But, I want to pose the question: do we really have to find the least common denominator? Could the denominator, in this case, be 4ths? Could the denominator be 8ths? 16ths?
The opportunity for students to look at problems in an inquiry based way, by asking questions and grappling with concepts are what will help them truly learn the concept. Too often, we teach students procedures for concepts they don’t understand. Then, when they get to fractions that have uncommon denominators, they’re automatically programmed to start skip counting by that number and searching for that common denominator.
In our example, if I were to change the denominator for both fractions to 16, I could take both of my 4ths and put 4/16 on top of each of them. So, 1/4 = 4/16. The other 1/4 would equal 4/16, and then I would have my third set of 4/16. So I would have 12/16 all together.
That’s great, the teacher says, but can you reduce that? But what does it mean to reduce or simplify? When we’re asking students to do that, we’re really asking what piece of paper could I lay on top of all 12 of those 16ths to cover it? From there, students can use the papers and see that perhaps we could put 8ths on top of each set of 2/16. One side has 4/8 and the other side of my area model paper has 2/8, which is 6/8. Is that fully reduced? No.
Let’s try 4ths. The 1/4 piece fits over 2/8 , and we can do that 3 times, so the answer is equal to 3/4
Going through this process with concrete manipulatives will help students understand what you mean when you ask them to simplify or reduce the fraction to its simplest form.
If you ask me, 3/4 , 6/8, and 12/16 are all correct answers. You can find arguments that say a fraction has to be in its simplest form to be correct, and other arguments that say it doesn’t matter. I always tell students that, if a question asks you to put a fraction in simplest form, or if you’re looking at a multiple choice and you don’t see the answer you got, to use any equivalent fraction.
Do we necessarily have to teach kids to make all of those 16ths? couldn’t I let the kids think in an inquiry-based way about maybe a more efficient way of answering that? Could I rename my 4/16 as 2/8? Could I rename my 2/4 into 4/8? That’s a possibility because those are all equivalent to the original fractions of 4/16 and 2/4. Of course, then I would add together all of those 8ths and see that I have 6/8, even though it’s not necessarily reduced all the way.
Maybe a student looks at 2/4 plus 4/16 and thinks about adding it as 4ths. Could you rename it the 4/16? Is 4ths the least common denominator? No. So why do we have the rule that students have to skip count to find the least common denominator when in fact, with this example, we’re showing you that it’s actually more work!
Remember, these videos are great to use as a lesson launch for the instruction you deliver in your classroom. You can use these videos with the flipped classroom concept where students watch the video prior to meeting with you in the Math with the Teacher station. This allows them to bring some background knowledge to the table as you start instruction on the concept.
Concrete reinforcement in the classroom, after watching the video, is important, and you can use the area model papers, as we do in the video, or you could use pattern blocks, or even patty paper. The whole idea is to help kids understand what’s happening when they’re adding fractions. When students understand the why, the underlying concept, we can teach them shortcuts. Shortcuts in math are great, but shouldn’t be taught unless students really get the concept.
If you ask me, the “shortcut” of skip counting to find the common denominator isn’t much of a shortcut, especially when I understand equivalent fractions, which makes that process a whole lot easier.
We hope you’ll check out our other YouTube videos in our Working with Fractions series, including multiplying and dividing fractions. Join us next week as we start to look at subtracting fractions!
]]>Most of us memorized a procedure with a concept we don’t understand: “don’t ask why, just invert and multiply.” Maybe it’s time to ask why!!
When we look at dividing fractions with our students today, we can start with a lot of questions that challenge that memorized procedure: Is it possible to actually divide by fractions or can you only multiply by fractions? And why do we flip or invert fractions and then multiply??
I was in a classroom not too long ago, modeling the concepts we’re talking about today by using the pattern blocks to help students explore their knowledge, and I asked the students why do we invert and multiply (or “keep, change, flip” in some classrooms)?
A student raised his hand and said, So you can get more of an accurate answer. I responded by asking Is it going to be inaccurate if you divide??
In today’s blog, we’re going to explore how we should have learned about dividing fractions before we learned a trick to get us the right answer.
It’s really important that students actually understand really what’s happening in division with whole numbers before they jump into division of fractions, or numbers less than one. As we look at this dividing fractions, we can start by asking students about a statement for division: If I had 35 ÷ 6, what does that mean? A lot of kids, just like we talked about in multiplication, might say you’re dividing, or they might even tell you the answer. But what that question is really asking is how many groups of 6 are in 35, or how many 6 are there in 35? This helps students revisit their prior knowledge of division, but also helps them extend that knowledge as we start dividing whole numbers by a fraction.
Another thing to think about with division is the relationship with multiplication. Like we have fact family with whole numbers, we want to help kids see how, when you flip multiplication and division, there’s an inverse relationship there. If I have 12 ÷ 4, it goes in 3 times, just like if I have 3 × 4, I know that it equals 12. Understanding this relationship between multiplication and division with whole numbers will help students make further connections when we are looking at fractions.
A students begin to process division of fractions, we can start with asking them to think about the idea of division as applied to fractions: When we divide by a whole number, we end up with a smaller number. But when we divide by a fraction, it actually increases the size of the answer – why is that? Why is it that, when we divide fractions we can end up with a whole number if it goes in an equal amount of groups?
In this video, I model dividing a whole number by a fraction, and I use the pattern blocks to give students a visual manipulative to help solidify the concept for them. We’ll be using the hexagon as a whole, the trapezoid as ½, our rhombus as ⅓, and then our triangle as ⅙.
So, let’s take 3 ÷ ⅓ – what is that asking? It’s asking how many ⅓ are there in 3? I’ll start by building 3 with 3 hexagons so kids can see it represented, and then ask how many ⅓ do you see in there? Students can lay the rhombus on top of those 3 hexagons to see how many actually fit in 3. If I covered each whole with 3 rhombuses, I would know for every 1 whole, there are 3 rhombuses, so 3 ÷ ⅓ = 9.
Let’s relate this problem to multiplication. If I were to change the problem around and multiply, would I end up with the same answer? 9 × ⅓ is asking what is 9 groups of ⅓? The answer is 3, as we can see if we build it with the pattern blocks. This is the same kind of fact family relationship that we might understand with whole numbers with multiplication or division.
When we start thinking about dividing a fraction by a fraction, it’s important to keep the same language that you’ve been using with whole numbers in division, or even with a whole number divided by a fraction.
Some fractions, when you divide them, go in nice and evenly. I definitely suggest starting with this kind of friendly unit fractions at first to help students understand dividing a fraction by a fraction. Fractions that divide evenly without a remainder or something that is less than one will help students develop a more conceptual understanding at first.
Let’s start with ½ ÷ ⅙ . What that’s asking is how many ⅙ are there in ½ ? When using the pattern blocks, I like to put the hexagon down so kids always have that reference for the whole that we’re working with. I’ll start with ½ piece and ask how many ⅙ will fit into ½?
We can see that, if we took the ⅙ pieces and we tried to fit them inside of the ½, we know that three of those would fit. So we could say that three ⅙ pieces will fit evenly on the ½, which will give us a whole number (3) as our answer.
Other fraction examples might fit more than one. So let’s look at ⅚, and ask how many ⅓ can fit in ⅚? Or ⅚ ÷ ⅓ . First, we want to build the ⅚, so I put ⅚, 5 triangle pieces, on top of the hexagon. We know that if we added one more ⅙, it would be nice and even. But in this case, we’re working with ⅚. Then, I take my rhombus, which is equal to ⅓, to try and figure out how many times it will fit.
I see that ⅓ is equal to 2/6, and if I have another 2/6, I will have another ⅓. By using the rhombus, I can see that ⅓ goes into ⅚ two times, but with ⅙ leftover. This is where students often get confused in Division. They see that there’s another ⅙ left and think that’s the remainder. But we have to remember that we’re really dividing by ⅓, so we have to look at any fractional parts in relation to that amount.
If there’s ⅙ left, we still want to know how many ⅓ will fit into it. A whole ⅓ doesn’t fit, but a half of ⅓ does. So, ⅚ ÷ ⅓ = 2 ½.
The other thing we show in the video is a situation when something might be less than one.
If we were to ask how many ½ are there in ⅓ , which we’ve written out as ⅓ ÷ ½, we know that we might not be able to figure that out completely. In the video, you can see it’s especially challenging to look at this in 6th grade when they see that ½ doesn’t even go into ⅓ one whole time. The answer is less than one time.
In the video we compare the trapezoid with the rhombus and show kids that it doesn’t even go in one time. But if I took that ½ and divided it into three sections, how much of it does fit? This is a little bit more higher level thinking, and it could throw even some of the teachers that we work with for a loop. It’s very hard to visualize this because we didn’t learn this way as children.
Visually, we can stack that ⅓ on top of ½ so kids could see the ½ doesn’t fit completely. However, if we divide the ⅓ up into three equal sections, ⅔ of that ½ does actually fit. As confusing as it sounds, the video shows how kids could visualize or see that based on the pieces that they’re using!
The next thing you really want students to understand is why we flip or invert the fractions and multiply? Can I create a multiplication sentence that will be true for the fact family? Does that work only with a whole number and a fraction or can you do it with the fraction and a fraction?
That is one of those open ended things we want your kids to explore instead of you answering the question for them. I would rather your kids see why it works when you invert and multiply and then see it would still work the same way with division and using the language appropriately. In offering inquiry-based questions, you don’t necessarily have to be the giver of all the information. Getting kids to make that connection and deepen their understanding is what we’re looking for in the 21st century math instruction.
Using our Working with Fractions series videos – both multiplication and division so far! – your students will really start to understand the why before the how when it comes to fractions. If we can get kids to understand what they’re really doing and have a deep conceptual understanding for it, they’re better able to explain.
Today’s questions on 21st century tests no longer just as students to compute the answer. It often makes students draw a picture to show a problem like 4 ÷ ½, and show their understanding of what they’re doing, not just that they can regurgitate the information.
These YouTube videos are really great in a flipped classroom style setting. Whether you’re doing it in the launch of your lesson, or you’re actually having the students listen to this at home, or maybe even in a guided group with other peers, they could listen to this video before coming to see you. The 21st century mind needs that technological dip, even as much as I try to fight it, Marzano’s research really supports this idea that, if we could get kids to see something in video and interact with it hands on, and then apply it, it makes all the difference.
Next week we’ll be talking about the addition of fractions using a conceptual way as well to show how we can reduce fractions using one of our favorite friends, DC from Mathville. Check it out!
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Most fraction units are positioned in the school year so students can get as much exposure to fractions as possible. Developing a solid understanding of fractions is just as important as developing that foundational whole number sense. Students need a firm grasp on the concept of fractions in general so they can be successful as they start exploring the various facets of fractions: equivalency, comparing, add/subtract, multiply/divide, etc.
The big question for us as teachers is how do we go about teaching kids about fractions? We want them to get it quickly, but how do we do that with a conceptual understanding instead of just teaching them to memorize rules or tricks?
I often wonder why teaching fractions is so daunting for so many teachers, and I find that many of the teachers that I work with were not comfortable with fractions as a student themselves. Their development of whole number sense wasn’t strong and they have gaps in their own understanding of operations, rounding, estimating and other things with whole numbers. So then, when you start talking about doing those things with parts of a whole and it gets even more confusing.
As education is tending more towards the concept of a flipped classroom where students watch clips of how to do something as a precursor to going into a Math with a Teacher station, I wanted to dedicate the month of March to visually answer the question of how we go about teaching fractions.
As we continue to build our video library and YouTube channel, you’ll have on-demand access to the videos to use in a variety of different ways. You could have students watch the video on their own prior to meeting you in a guided math group, you could use it to help kick off a lesson in a whole group and then take students into a workshop model, or you could make it a warm-up to get kids’ brains going.
However you plan to use these videos, take advantage of the nature of technology and digital media! Research from Horizontals on brain development shows that, when kids learn with technology, they learn a lot faster. We definitely don’t want to turn into the “wah wah” Charlie Brown teacher and have information we’re imparting going in one ear out the other. With a video, students have the opportunity to really digest the content – pause the video, go back to rewatch anything that doesn’t make sense so they really get it. This could take the level of engagement up, and especially as you’re using CPA for concrete pictorial abstract, could engage more learners!
I always find myself introducing the concept of multiplication by asking a class about what they know about multiplication within a whole number. I’ll write a problem on the board – 5 x 6 – and ask – what does this mean? Typically, students will say things like, “multiply” or “times” or “the answer is 30.” So I keep asking the question – what does it mean? As the students engage and dig deeper, we hope they eventually arrive at the fact that 5 x 6 really means “5 groups of 6.” We also know that if we changed it to be 6 x 5, it would be “6 groups of 5.” If your students are doing the multiplication journals and playing the Speed! game, this concept of “groups of” is drilled into their heads, even if it might take a little prompting for them to say it.
Next, we might go to a larger problem. I’ll write something like 89 x 12 on the board. What does this mean? It means we have have 89 groups of 12, so how many do we have all together? We really want kids to understand this pattern and phrasing with whole numbers because, as we transition into multiplying fractions, it will make that transition much smoother.
So, next we write a problem with a fraction on the board – 4 x ½. What does this mean now? All of a sudden one of our numbers is a fraction! In this case, we could really say it means “4 groups of ½.” If I had ½, ½, ½, and ½, that would be 2. So, 4 groups of ½ equals 2.
Using this progression of inquiry-based activities really helps pique the students’ interest, helping them understand the concepts behind multiplication. Usually, with fractions, we start with a procedure – do this step, this step, this step, and this step. Then we have students do the procedure with us, and then they do a worksheet on their own to make sure they’ve memorized the procedures. That isn’t really the level of engagement that 21st century students need. They need to engage in conversation and grapple with questions like what does that symbol really mean?
From here, there are lots of different activities you could use to help students deepen their concrete understanding. They need lots of practice! If I’m in 4th grade, I might stop here though, start to bring in manipulatives that can support development of the students’ understanding.
In this video, I’m going to show you how to use two of my fan favorite manipulatives: pattern blocks and area model papers.
I like to start with the pattern blocks because they are probably already familiar with pattern blocks from previous grades.
If I was using fraction tiles to model this problem, I’d have to get two sets of fraction tiles because I’d need 4 of the ½ pieces, so since pattern blocks come in a big bucket, we don’t have this challenge! Additionally, fraction tiles are also imprinted with the part they represent, so there is no flexibility for using those pieces to represent another value as there are with pattern blocks.
Another benefit of using pattern blocks is that, when we get ready to talk about a whole number divided by a fraction, kids will already be familiar with the process and will actually be able to see how multiplication and division of fractions goes together. It’s like the fact families of addition and subtraction and how, when we teach the missing addend we see that addition and subtraction go hand in hand.
Using the pattern blocks, I’ll model how to tackle a whole number times a fraction problem. If we’re using the correct language for this, “four groups of half,” it will make a lot of sense. Let’s use the hexagon as the whole, then we have the red trapezoid as the ½, a blue rhombus would be a ⅓, and then we could just use the triangle simply as ⅙.
To begin, we wanted to see 4 groups of ½, so kids can pull over their ½ pieces, the trapezoids, until they have 4 trapezoids. Kids can then easily see, with the one whole hexagon as a reference, that 4 groups of ½, 4 trapezoids, is the same as 2 whole hexagons. For some more at-risk kids, they might want to actually match the trapezoids on top of the hexagons to solidify that concept of quantity. Kids can then play several other examples as they explore multiplication of fractions.
We always want to start with concrete because it gives students what’s called an imprint for them to refer back to later. That imprint could be you modeling under a document camera or my demo on a YouTube video. It could be remembering how they built it with a partner in class. Regardless of how they get it, they now have a memory of what it was that we were talking about.
So, for our 4th graders, it starts with getting the language right to understand what we’re really doing with fractions – “groups of” – and then starting with whole numbers and transitioning into whole number times fractions. This will get them ready for 5th grade standards, where they will be multiplying two fractions.
Multiplying Two Fractions (5th Grade)
Again, it’s important to build as a strong foundation of conceptual knowledge as possible so kids have a plan for what they’re doing. We want to avoid teaching procedures that kids have to memorize without really understanding what’s going on.
In this video, we will walk through how to conceptually walk through an example of multiplying a fraction by a fraction. Let’s take ⅓ x ½. What is that really asking us? By now, our students shouldn’t be intimidated! They should feel confident knowing that it’s asking for ⅓ of ½.
You could definitely use area model papers for these example, or pattern blocks, or other materials, just make sure the manipulatives you choose match the fractions in the examples you’re using.
So, what is ⅓ of ½? I’m going to start with the half – the red trapezoid. Looking at that piece, I need to think about how I’m going to break it up into thirds or group of ⅓. They can’t think of the green triangle as ⅙ though, because we’re not talking about a whole piece, we’re looking at part of a whole. But looking at the trapezoid, students should see that we could fit three triangles onto the trapezoid and break that ½ up into three ⅓ pieces. Then, looking at the ⅓ pieces in relationship to the whole, we can see that ⅓ of ½ represents ⅙.
Throughout this process, I like to have the whole pattern block, the hexagon, visible to the students so they can know that they can connect that to the whole. As I break up the trapezoid, the triangles are ⅓ of the trapezoid, but when I give my final answer, it’s in relationship to the whole, which makes the triangles ⅙ again.
After we model this process of finding ⅓ of ½, we can ask students what they would do if we asked them to find ½ of ⅓? This gets students to think about the commutative property. We know that you’re probably still going to get the same answer of ⅙, but does the model look the same? No, it doesn’t. Just like if you had whole number multiplication problem like 5 x 6 and built an array. If I change it to 6 x 5, the commutative property says the answer will still be the same, but the picture will be different.
If we want to know what ½ of ⅓ is, we have to start with the blue rhombus, the ⅓ piece. When I look at that, I can take my triangle and lay it on top of the rhombus to see that it actually does represent ½ of the piece. But, when looking at the whole as a hexagon, the green triangle represents ⅙, which is the same as the previous problem.
I don’t think kids get enough time to explore different fractions with pattern blocks as we did in these videos. My encouragement to you as the teacher is to think of as many fraction multiplication problems as you can – not just in simple unit fractions of ⅓ and ⅙ and so forth, but fraction problems that will prompt your students to think in different ways as they use the pattern blocks in more versatile ways. Could you designate 2 hexagons as the whole? If so, that changes the pieces so that the trapezoid is worth ¼, the rhombus is ⅙, the triangles are 1/12.
Cover/Uncover is a simple fraction game that can be easily differentiated in your classroom and uses some great fraction tools – fraction tiles and area model papers. For more on other types of fraction manipulatives, check out our videos on getting fraction manipulatives student-ready and this blog series on fractions.
When students are using fraction tiles to find equivalent fractions, they often try to use the “guess and check” method. If presented with two ⅓ pieces and asked to find an equivalent fraction, they might start with the 1/10 and see how many of those would fit. Finding it to be uneven at the end, they’ll grab another set of pieces and keep trying until they find the equivalent. However, we want students to be more confident and strategic in identifying equivalent fractions, so a game like Cover/Uncover is great practice.
A note about fraction tiles: Fraction tiles might not be my favorite fraction tool because each tile is stamped with the fractional part, which makes them always predictable. They don’t allow students to use any kind of conceptual thinking about the size or the proportions of the pieces. However, I do think that, for students in 2nd, 3rd, 4th, and 5th grade that are new to fractions, that it’s important to present the fraction tiles as a useful tool.
Download the basic instructions as a PDF
Materials:
This introductory level is great for younger students, such as those in 3rd grade or just starting fractions. It could also be an introductory or review game in 4th or 5th grade.
Rules:
There’s not a lot of strategy going on in this level apart from students getting the idea of the parts of the whole. You’ll see kids will be “guessing and checking” to see if that piece they rolled will or will not fit. But they’re using friendly fractions because they’re all in the even pattern with using ½, ¼, ⅛, 1/12.
You could make the game more challenging by using different manipulatives, like the area model papers, that have different fractional parts.
Uncover uses the same materials and the same basic rules, except we go the opposite way here – instead of covering the whole with fractions, we’re trying to uncover it by removing fractional parts as they’re rolled.
Rules:
Important Notes:
● You may not remove a piece and exchange on the same turn; you can do only one or the other.
● You have to go out exactly. That means if you have only one piece left and roll a fraction that’s larger, you may not remove the piece.
In this version, we use the same materials and the same basic rules, but we bring in equivalent fractions!
Rules:
The first player who removes all pieces from the whole strip wins.
Option B is where students have to really think about what they’re doing. You want them to be more strategic in how they’re replacing their blocks. I could take my ½ and exchange it for 2/4, 4/8, or 6/12 (if I were using the fraction tiles – adjust fractional parts if using area model papers). I might want to do a combination of ¼ and 2/8. Kids will start to realize that there is more of a probability of rolling the ⅛ or 1/12 than the ½ or ¼. Because of this, do they go for more 8ths if they know they can always remove 2/8 if they roll a ¼? Of course, you want kids to pick up these strategies on their own (don’t tell them!) and start to have these realization of how to play better.
At the end of their turn, you want the students to check and see if their partner agrees with the way they’ve done it. When they’re done with the way they’re doing it, they say done and pass the die to their partner. The first player that removes all thier pieces from the whole strip wins.
To add complexity to Levels 2 and 3, you can start by playing with the fraction tiles, and then move to the area model papers as students start to think about other equivalents that they could pull off or they could exchange.
An accountability sheet is very easy for this game. You would want students to record what they rolled and show the equivalent fractions. It could be just a half-sheet of paper that they turn in so you can see that they’ve been working through that process as they’re doing it.
Next week, we’re going to start working on fractions and looking at some video tutorials that can help you in your classroom as you teach all the operations with fractions and knowing which manipulatives you can use!
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