If you are a 3rd grade teacher, you’ve probably been reviewing addition and subtraction from 2nd grade and working on concepts of place value, and now it’s time to help kids understand their multiplication facts and patterns!

If you’re a 4th or 5th grade teacher, it might be that time of year for you too, as you review what your students learned in 3rd grade and determine where they might have weaknesses in their understanding of multiplication.

I can remember when teaching multiplication usually started off with a little bit about arrays, knowing how many groups are in a column versus a row, and then it just seemed like my book said, *Okay, now the kids start doing those timed tests because they should magically get those facts!*

As the title of the blog indicates, that’s not the pattern we want to continue as we teach multiplication to our students today. We want to go slow to be able to go fast. We want kids to really be able to develop a depth of understanding around multiplication, but in a way that’s developmentally appropriate.

The sequence we’ll talk about today, the same one we always talk about when we look at multiplication, is great to follow when you’re helping students in 3rd grade who are just learning, but could certainly benefit 4th, 5th, and even 6th grade students that might need to solidify their skills.

Of course, I can’t talk about this without talking about my favorite book – Time’s Up on Timed Tests, from my friends Kristin Hilty and Eliza Thomas Sorte, which we have available in our store as a digital download! These two fabulous ladies created a book that is truly amazing and filled with all kinds of activities for addition and subtraction, as well as multiplication and division. But what makes this book so incredible, in my opinion, is that it really looks at the foundation of fluency, which (spoiler alert) is not knowing facts quickly and being able to regurgitate answers under pressure.

Additionally, Kristin Hilty also designed one of my most favorite products – the Patterns of Multiplication journal. The sequence she outlines in this journal really helps 3rd graders to understand their facts in a different way.

When I taught multiplication in the classroom, we would do the x2, x3, x4, x5 and start to be really proud of how well our class was doing! But then we rounded the corner of x6, x7, x8 and x9, and things started to fall apart.

Instead of going in numerical order to learn the multiplication facts, the Patterns of Multiplication Journal looks at the patterns, or the families, that we find within multiplication to help students understand it more fundamentally.

We start with x10 and x5 – of course those go together! They have lots of things in common and you could easily and rhythmically count and hit those numbers based on the patterns.

The next set of family patterns would be our x2, x4 and x8. If kids learn their x2, the x4 will make more sense, and then the x8 will come along with it.

Next, we look at x3, x6, and x9, which also hit that same pattern.

The x7 is kind of the odd ball, which we usually teach separately. We decompose it into x5 and x2 because students can then anchor to a fact that they know. Sometimes kids have a fear of the x7s, but if we were to think of seven decomposed into 5 and 2, it’s a lot easier to figure out 7 x 7. I know that 7 x 5 is 35. And 7 x 2 is 14. When I add 35 and 14 together, it gives me that answer of 49.

In this tutorial video, we’ll walk through the progression to help you get the idea of how patterns of multiplication are really developed.

Some teachers like to do this as a station, taking a fact per week. Some teachers prefer to do a fact all in one day. In some of the classrooms in our project schools, we actually plan a break from our math book to take a good 15 days and use the Patterns of Multiplication journal to create a solid foundation that students can use going forward.

The journal has five different sections that we go through with each fact. Want to see the part of the video for a specific section? Check the resource page!

We have the students use an abacus or unifix cubes to talk about the patterns they see within each number. Teachers that use this sequence every year tell me, “Shannon, I really thought the kids would have gotten the idea of these patterns” and every year, they’re blown away at the things you and I might think is completely obvious but that students just don’t see.

In combination studying patterns, students write out the equations. They combine the concrete tool (maybe an abacus) with a grid that they can color in to reveal patterns, and then put it into words with equations and practice using appropriate language.

Let’s say we’re working on our fours. I have no groups of four, what does that look like? 0 groups of 4 = 0. So students could be writing 0 x 4 = 0, but their verbiage when they say “times” really should be “groups of.” If students understood the concept of “groups of” early on, man would it help them as they get older! Especially as we start to look at the multiplication of fractions.

As they start concretely, they might push over four beads on an abacus, and say, “Now I have one group of 4. 1 x 4 = 4.” So they color that in, Look at it on the abacus, and then create that equation.

Now I’m going to push over another 4 beads. Now I have 2 groups of 4. 2 groups of 4 is 8 or 2 x 4 = 8. Push over another group of 4 beads. I now have 3 groups of 4. I know that total is 12.

Really spend time here! This is not busy work to give your 3rd graders to complete at their seat or at home (if they’re virtual). WIthout the purposeful process, students will notice enough patterns to complete the chart (“fill in this one, skip the next two, fill it in, etc.), but they won’t really “get” multiplication.

It was brilliant of Kristin Hilty to put area models into her journal, because this really helps students to really understand the idea of the commutative property. As I draw 1 group of 4, and I’m kind of going across, I ask myself – how many groups do I have? I have 1. How many are in each group? 4. So 1 x 4. When you turn that paper around, or turn it 90 degrees, you would look at that differently. How many groups do I have now? I have 1 group, 2 groups, 3 groups, 4 groups. How many are in each group? 1. Well, that’s now going to be written differently – 4 x 1 or four groups of one. The same thing can be done for all the different parts is they’re creating the area models up to x10.

If I were to make six groups of four, I could go ahead and color in six groups of four. How many groups do I see? I see 6. How many are in each group? 4. Turn it 90 degrees – same idea. I now see 4 groups, but there’s 6 in each. So it’s 4 x 6.

This page is a great reference for students and their understanding of that area model. It will help them connect things like square numbers, and to find things like missing parts of an area.

Day four is another one of my favorites where students create their own real-world problems. We want them to use the patterns of multiplication they’re learning, and this helps us see if they are really understanding multiplication as they’re trying to apply it.

There’s also an area for a visual representation, where we’d like students to be able to make a visual model, an area for the answer, and one for the answer sentence.

You have kids work in pairs where one student might do write the story problem and the other creates the visual model to go with it. Maybe one student creates a visual model and the other student has to write the story problem. Either of those options are really good ways to engage students’ understanding of visual models.

This step is easily differentiated with three levels. The first level is a very basic problem, multiplying your number by anything up to 10. Level 2 usually takes it from 11 groups to 99 groups of the number. And then level 3, the most advanced level, has kids coming up with two or three digit numbers times the number you’re working on.

This is a fun day where students can apply the pattern that they’re working on. Here are some ideas for extensions:

- Fact Flap Cards. Check out the video on creating a multiplication and division fact flap card, which is actually 20 flashcards in one!
- Multiplication Bump. Watch the video and download the corresponding game boards.
- Multiplication Speed! This game helps students practice their speed if they’re working on the patterns. Check out the Blog Post!
- Number Bond Cards. Practice number bond concepts with multiplication and division. The video explains different ways that you can do multiplication and division number bonds. We have a downloadable, printable version in our store, and if you’re an M³ member, you can download it for free!
- Skip Count. Help your students get comfortable skip counting forwards and backwards by the number they’re working on. If they’re working on x4: 4-8-12-16-20-24-28-32-36 and then 40. Then ask your students if they can skip count backwards? They may want to utilize the abacus to do this. Showing 40 beads. Push 4. Now I see 36. Push 4. Now I see 32. Push 4. Now I see 28. Push 4. Now I see 24, and so on all the way back to zero.

The Time’s Up on Timed Tests book has a ton of resources where you can really look at how to collect data on fluency as you’ve gone through the different sequences. After you’ve completed your x5 and x10, are kids starting to understand this pattern? Are they able to understand the multiplication or the division within those patterns?

So, if you’re in 3rd grade or (as I sometimes say) if you’re doing a “clean up on aisle 5” and having to start by determining which facts your students might still have difficulty with in 4th or 5th grade, using the Patterns of Multiplication or any of the extension resources we talked about, will really help students to “get” the patterns of multiplication.

]]>Some teachers are confused by additive and multiplicative comparison problems, thinking of them as just part-whole multi-step problems. The comparison bars you use in the visual models for this type or problem are sometimes called additive comparison bars, but that doesn’t mean you might not add or subtract in the problem. We also have multiplicative comparison problems, which look just as intimidating as they sound, but which also might feature multiplication, division, or even addition and subtraction! Makes sense, right?

The comparison genre of problem solving is not just a part-whole multi-step problem. While they might feature multiple steps, **the mark of a comparison problem is that you’re going to be comparing two or more entities. **

Additive comparison problems begin as early as 1st grade! Students start off by looking at an amount compared to someone else who has more or less than them, and then try to figure out how many more or less you have than they do.

The language of an additive comparison problem is very confusing to young learner. When you ask *How many more does Jane have than Bill? *They look at you like “what??” Am I supposed to add? Or subtract??

No! This is the signal that we’re supposed to compare! And, if we understand visual models, we can help the language of additive comparison problems become much more clear. We start with proportional bars, line up the blocks, and actually count to see how many more or less there are.

Take this sample problem:

Jane had five apples. Bill has two apples. How many more does Jane have than Bill?

There would be five squares to represent Jane’s apples. Bill’s apples go in a second bar underneath Jane’s, not on the same bar. If they were on the same bar, there’s no way we could compare the two numbers. Then, we put a question mark in the space below the three extra squares or apples to indicate that we need to find out what that amount is.

You could also ask that question in a different way.

Jane still has five apples. Bill has two apples. How many less apples does Bill have?

This could be really confusing as students are listening to that language, but drawing the bars can help clarify. **Remember, it’s about matching up the two amounts belonging to each character to see where they have the same amount, and counting how many less Bill has.** The question mark really goes in the same place, but what it represents – what it means – is different if the question is different.

Traditionally, at the basic level of problem solving, we might have made a t-chart of vocabulary that might go with addition problems – more, some, all together – or subtraction problems – difference, how many more. That kind of cheat sheet just isn’t useful when we are solving comparison problems because the language is much more complex.

As comparison problems progress, the language starts to change from just comparing one entity to another and looking for more or less, and we start to look at solving for different parts, given one of the parts or the total. For example: Jane and Bill have seven apples in all. If Jane has five, how many does Bill have? This gives the total and a part and students have to solve this differently. With this progression in language, we also want kids to progress to using non-proportional representation as well.

The Additive Comparison Bundle (M³ Members – Click Here!) that we have as a download for you to use in the classroom follows this same kind of progression – we start with the idea of more/less and then we move students a bit further.

Once we’re beyond just comparing more or less, I think the best way to help kids make the transition to non-proportional representation is by getting them used to drawing a bar of the same length for each character in the story problem, then reading the problem to work their way through it. Once you have those bars of equal size, you can manipulate them to set up the problem.

Let’s say John and Connor had 11 golf balls. John had 3 more golf balls than Connor. How many golf balls did Connor have?

The two bars belonging to John and Connor are “*x”* (thinking in algebra terms) – we don’t know how many they had. We know that John had three more than Connor, so we’re going to add a non-proportional unit that will make John’s bar grow by three. We have a bracket that shows the total is 11, and then we will put a question mark in Connor’s box.

Once this is set up, I can look at the information I actually have. I know the total is 11. John has 3 *more*, so I can subtract those 3 from the total, which leaves me with 8. I know I have two units that equal 8, so, I can determine that each of them is worth 4 and Connor has 4 golf balls. If you think about algebra, this problem really looks more like 2x + 3 = 11.

This is one of the biggest epiphanies that I’ve ever had while doing additive comparison problems – the connection to algebra. In middle school, I memorized procedures for concepts I certainly didn’t understand, which is partly why I do what I do today, and I remember writing down formulas, storing them in my graphing calculator, because I thought if I remember that formula, I’d be able to get the answers correct.

Nobody ever drew pictures, helped me use algebra tiles, or made any connection that what I was doing with my second grade problems was actually algebra!! For a problem like 2x + 3=11, she would say to “do the opposite” and subtract the three. Or I “needed to get the X alone” *But why? *I wondered! I never understood it, but I just did the math as I was told and got the answer.

It was not until I understood additive comparison models that I realized that we were solving for that was the X! It all made sense – why do I need to subtract the three? Why do I need to get X alone? So I can figure out how many golf balls Connor had!

Of course, that’s what my teacher was saying, but visually, I had zero understanding of what she was talking about. I’m sure I would have understood algebra a whole lot better if I had a picture in my head, which is certainly why algebra tiles and things like visual models are really helpful for kids to understand this type of problem.

The whole premise for additive comparison problems is to lay the foundation for early algebraic understanding. Students can actually write algebraic equations, or, as I’ve done with my audiences, they can create a model drawing based on the context of a story, or they could solve the problem based on seeing the visual model.

Enter: Professor Barble.

He was designed to help your students understand the different types of problems and make sense of them. Having a Professor Barble Additive Comparison Poster hanging in your classroom would be a great reference for students to use, especially since additive comparison problems are probably the number one word problem genre that stumps kids because they don’t understand what is being asked.

If you thought the language for additive comparison problems was tricky, just the name of this next type of problems is enough to make you scratch your head. Most people, especially students, have no idea what “multiplicative comparison” means.

However, if kids have a foundation of understanding additive comparison problems, they know more than they think!

Multiplicative comparison bars actually start as early as third grade, and a problem will read something like this: If A is 2, and B is 2 times as big as A, how big is B? Kids hear that kind of language and think *what?? *

One of the best tools that you can use to help students understand the language in multiplicative comparison is to use one inch square tiles, which you’ll see in the tutorial video of our Multiplicative Comparison Bundle (M³ Members – Click Here!).

The most effective visual I’ve used with 3rd, 4th or 5th graders to help them really get this concept of multiplication is a copy machine. We take a class field trip to the copy machine (which they think is pretty cool!), and we talk about how I have a unit that has 2 and I want to see two times as many. I put two one-inch square tiles on the glass, close the lid, and type in 2 copies, which I explain means 2 times. I ask them what they think will come out – some said 2, some just didn’t understand. But when I pushed the button, and they saw the copies come out, it was like a light bulb went off!

We weren’t talking about two more. We were talking about two TIMES as many, or two copies as many.

We did another example with 1 one-inch square tile on the glass. Let’s say A is 1, and B is 3 times as big as A. How many copies do I want? 3! So I hit the button three times, and the kids were able to predict right away that three units would be created.

Making sure your students understand the language of the problems before we start to enter into scenarios is super important. As students start to get this, they’ll start to understand it with the language and, of course, they can always use the one-inch square tiles if they need to.

Let’s look at another problem: Molly has three times as many books as Julie. All together, they have 20 books. How many does Molly have?

Kids have to be able to think about what that problem looks like – how can we make Molly have three times as many? Well, I have to be able to put in one unit for Julie, but then replicate that three times from Molly. So Molly has 3 units and Jane has 1. That represents easily what the words are saying.

Now, I know that the total is 20. I know that all 4 of those units equal 20. Therefore, how much does one unit equal? One unit must equal five. So, pretty easily, we can see that Molly has three units, each is worth 5, and so she has 15 books.

I always love to milk the problem for all it’s worth.

My friend, Char Forsten, author of Step by Step Model Drawing, always says “milk the problem for all it’s worth.” I could extend that question about Molly and her friend and ask how many more books does Molly have than her friend? That’s putting the question mark in a different spot before you put in the question mark at the end of Molly.

Now we’re really comparing Molly and her friend to see how many more she has. In this case we, know the answer is 10.

Show parts of the video tutorial included in the word problem bundles to your class, then have them use the one-inch square tiles and the exercises that are all laid out for you. Make the word problems come to life in your classroom! You can download the PowerPoint presentation and even edit it to put in different problems or problems that meet the needs of your particular students. If you’re part of the M^{3 }Membership, you’ve got it made – just download the bundles from the Membership resource library.

Teaching virtually? One-inch square tiles are super easy to create if your students don’t have them at home. Or, have them use virtual manipulatives, like the square tiles from Didax. Or, have them use cereal, crackers or even post-it notes – anything so that they can replicate the understanding of multiplicative comparison.

The last two genres of word problems, when we combine additive and multiplicative comparisons, and then fractions, will be featured later in the school year due to their complexity. For now, download the Additive Comparison and Multiplicative Comparison Bundles – and go build your students’ foundation of problem solving!

]]>If you haven’t watched our Visual Model Basics video yet, stop – right now! – and go watch it! It will explain the thorough and effective step-by-step process for creating visual models that we use, and it will help you understand why we use it to help students comprehend word problems.

This group of word problems earned the family name – Part-Whole Problems – because we’re taking a piece and subtracting from it, or adding to it, or trying to find the missing part. After students have a grasp on part-whole addition, subtraction, and multi-step problems, **we also want them to understand part-whole multiplication and part-whole division, which sometimes look a little different than our regular part-whole operations.**

In addition and subtraction, we use a non-proportional bar to represent an amount. If it’s a larger number, like 54, it would be a larger bar. For a smaller number like 24, it would be a smaller bar. When we move into part-whole multiplication and part-whole division, we are still using a non-proportional bar, but we’re chopping it up into units.

Here’s an example problem: *I saw 5 spiders, each with 8 legs. How many legs were there all together?* Instead of making a large bar, we’re going to look at the intricacies of the individual pieces of information that are in the problem.

Since I have 5 spiders, I might show 5 units (spiders) and inside each unit, I would put 8 (legs). At this point, in part-whole multiplication and division, we start to label our bar diagram a little bit differently. The total of that bar will represent the total number of legs, which we’ll write to the left of the starting point. The question mark at the end of that bar shows that we’re looking for the total number of legs. We’re talking about individual spiders that are being counted in the problem, so, across the top of the bar, we label it to show that.

It’s the same idea if we were to think about maybe a child having money that they might go to the candy store and they have $10 and they want to buy candy that is, maybe 50 cents each. How many pieces of candy could they buy with the $10?

We’re now talking about that whole entire unit bar equaling the money the child had, and I want to break that up into 50 cent pieces to see how many total pieces they could get out of the $10.

As the problems get larger, you might find yourself thinking *I have to draw *how* many boxes??* Professor Barble gives us the answer for both multiplication and division types of problems. He can help us understand when things are quantifiable for drawing units on the bar, or when its just too many.

Imagine if I saw 48 spiders, instead of 5. That would be way too many to draw! Instead, I can use an ellipses ( “…”) to represent space between the first and last unit. We can do the same thing if we were dividing something large as well. If we had 426 apples and we wanted to put them into bags of 20, it would be too many to draw! But we can use the ellipses to still give us the visual, of dividing the bar into units, without using all our time drawing boxes!

Study the anchor posters (included in the Visual Models for Word Problems bundles!) from Professor Barble to see more about how and when to use the ellipses.

When students have a solid grasp on the part-whole family of problems – the first set when they can execute problems with a mixture of part-whole addition, subtraction and multi-step and then the second set when they’re adding in multiplication and division – then they really have a handle on a large portion of the types of problems they’ll encounter in elementary school.

One of the areas I’ve seen students really start to get tripped up in as they do part-whole word problems is when the two “sets” of operations cross paths in a multi-step problem.

Let’s look at this problem: If I have $250 and I wanted to buy new ski equipment. I bought skis for $100 and gear for $75. How much money do I have left?

That’s a part-whole multi-step problem that students might find pretty easy since they’re just adding and then subtracting. Some students might also find problems that combine multiplication and division to be easy. But when we cross operations, and have to complete two or three steps in a problem, students tend to revert to guess/check tactics and appealing to you for help (“Do I just add? Do I just multiply?”)

**However, if we’ve taken our time to lay in the foundation of our step-by-step visual model or reading comprehension process will help students slow down and really figure out what the problem is actually asking.**

We’re getting ready to go on a field trip and we have 152 students signed up to go. Ten teachers are going and we also have 8 parents/chaperones. We want to load the vehicles, and each can only fit 10. How many vehicles do we need?

This kind of problem is clearly going to involve higher order thinking. You can’t just quickly add or subtract to find an answer. Students really have to draw out that bar model to help them understand what’s going on!

We start by drawing a unit bar that will equal the total number of people going on the field trip. That bar shows a section for students, parents and teachers. Then, the information from that part of the problem is needed to solve the next portion of the problem, which is how many vehicles we’ll need. Again, I’ll draw a bar to represent the total number of people going, but this time, I need to divide it into groups of 10. Now this is starting to look more like a part-whole division problem!

Drawing a multi-step bar model helps students see that they have to do a little bit of transferring in this process. They take that bar which shows the answer they found in the first part of the problem and re-represent it in another bar below so they can go on to the next part of the problem.

Multi-step problems really just involve a little bit more grit!

When students can become efficient in this step-by-step process for solving word problems with visual models, It’s going to help them when they start to go in to the idea of comparison, which is what we’re going to talk more about next week!

]]>Sometimes in math, each book or teacher takes a name for a really great idea and twists it around to call it something else. Because this is often the case, and **because we work with so many different math series, I’ll be calling this concept visual models. **

Visual models are a comprehension strategy that is amazing for helping students solve word problems. I think back to how I taught word problems to my students. I remember thinking it made perfect sense – circle the numbers, underline the words, box important information, right? For the most part, that procedure really worked out well when students were doing part-whole addition, part-whole subtraction and part-whole missing addend problems. But I vividly remember thinking of problems that were a lot more complex, which might be what I know now as multiplicative comparison problems or additive multiplicative comparison and realizing that my fancy procedure didn’t hold up.

When we started to get more of our state testing, and the standards started to become more firm than they were when I first started teaching 20 some years ago, we had a few strategies – guess and check, make a table. There were lots of different icons at the top of the problems and I remember thinking, *Gosh, how am I helping kids to know which strategy to use for which problem? *

Fear of word problems is also a very real thing, for students and even adults. In fact, when I took the GRE to get into my master’s program at Oakland University in Michigan, there was a question on the test that said something like, *Fran drove eight more miles than Sam to work but three miles less than and twice the amount …*my eyes glazed over and I thought, “Oh no! Not one of these problems!” I tried to draw diagrams and find the best (longest!) way possible to solve the problem.

To be honest with you, most of the time, I felt like it was just kind of a guessing game and the result was that a lot of kids began to subscribe to the “guess and check” method. We also tried the chart method, where you have a T-chart with key vocabulary that indicates the type of operation the problem requires. If it says *sum* it will be addition, if it says *difference” *it will be subtraction. Kids just aren’t learning to think and analyze the problems. Usually, when the going gets tough, because they don’t understand, students resort to an appeal to their teacher for help.

When most kids read a story problem, they want to get right to the nitty gritty. They look at the problem and decide – add or subtract? Typically, these kids are really more interested in solving it and figuring out what operation they’re going to use, than really understanding what the problem is asking.

For the majority of my teaching career, up until about eight years ago, I felt like we needed a strategy to help all students with all the different types of problems. I felt like we needed a common language around problem solving, which didn’t really exist at the time.

**Is there a solution? Is there a way that we could tailor problem solving for students as early as kindergarten all the way through eighth grade? There is: using visual models to solve word problems. **

Much of the information for visual models comes from research that’s been done in Singapore. The process works for everything, all the different types of problems that students are going to encounter: part-whole problems, additive comparison, multiplicative comparison, additive multiplicative, fractions, and as you go up even higher, ratios and proportions.

Sometimes, people in Michigan or the US think, *wait a minute, why are we doing something that is from another country?* It’s just good math!! Using visual models to give students a comprehension strategy that helps them solve word problems is golden!

I really wanted a process that can be consistent from teacher to teacher, grade level to grade level, that could follow a student all the way up. In most of our M^{3} Building Math Mindsets project schools, we proudly display this step-by-step poster that walks students through being able to break down and answer problems, both the simple and the more complex.

This tutorial video, Word Problems with Visual Models: Basics, will explain the need for a process for students to be able to follow with visual models, from as early as first grade, students can be doing visual models with proportional units to help them understand part-whole addition, part-whole subtraction, part-whole missing addend, and even additive comparison. As students start to get a little bit older, they start to no longer be able to use a proportional model because we’re not just talking about five jelly beans any more. We might be talking about 29 jelly beans, and we’re certainly not going to proportionally draw out 29 boxes.

Many students that are in first grade feel very frustrated, maybe thinking that there’s no purpose in being able to do a visual model because they know, from reading the problem, that we add or subtract. It’s “easy peasy”! But, what they don’t realize is that, the second a more complex problem comes up, they will stop in their tracks.

For example, if we were to say that Karen brought 48 ice cream cups to the soccer game. One third of those cups were chocolate chip ice cream, three fourths of the remainder were strawberry, and the rest were vanilla. How many vanilla cups did Karen bring to the soccer game?

Students’ reactions to a problem like this are probably pretty similar to yours: Huh? Three fourths of the remainder – what’s the remainder? Should I find a common denominator? Okay, I circled all the numbers and I underlined the words, but I don’t really know what this is actually asking!

Quite quickly, problems can go from very simplistic adding problems to really more complex problems like this one. **If we teach this step-by-step process of using visual models while the problems are simple, the students will be able to use it most effectively when the problems do get more complex. **

This spring, we started talking about the journey to help kids connect proportional and non-proportional thinking with our Math4Littles series (catch up here!). Beginning with real objects in the physical world, moving to quantitative pictures, math work mats, and finally to journals beginning in Kindergarten and then 1st grade, where we really start to make the transition official.

So what comes after we have this understanding? Part-whole problems with a non-proportional representation!

For your instructional convenience, we have created some really amazing videos that you can use during face-to-face or virtual instruction that will help students gain understanding about part-whole problems! The videos come with a PowerPoint tutorial that you can use in the classroom, a reference poster, a student journal that mirrors the presentation, and a blank journal template that you can customize based on the types of problems that you’re working on with your students. Check out our sneak peeks to see what you can expect!

It’s all at your fingertips! Our M^{3 }Members have access to download each of the six bundles at no cost, or you can buy the bundles individually in our store for less than a cup of coffee.

**…you hate coming up with problems.**Me too! And usually, the sample problems in our math books don’t flow the way we want our kids to learn. First, we want kids to learn part-whole addition, then part-whole subtraction, and then maybe part-whole missing addend. Next, I want to give them a mixed review before I add the next type of problem, which might be part-whole multi-step. You can completely customize the flow of problems by mixing and matching the bundles.**…you love a good anchor chart/poster.**Each bundle has a unique poster, featuring the studious Professor Barble, that can be blown up for a classroom wall, or printed small on a bookmark that the kids could have in their journals. The posters will help students learn to recognize, and develop familiarity with, the types of drawings and problems they’ll encounter in elementary school.**…you don’t love creating PowerPoints.**That part is done for you! Of course, we include the original PowerPoint file in the bundle, so you could always edit the presentation if you wanted to add your own problems, but if you don’t, no problem! It’s ready to go into your classroom – virtual or face-to-face – tomorrow!

**Check out our new Word Problems with Visual Models series and let us know how it goes! **

**Coming soon** – look for journals you can use in 1st and 2nd grade, all the way up through 5th grade eventually! We can’t wait!

**Next week** – we’ll look at visual models that go with comparison word problems! See you then!

Ever felt like banging your head against a wall as you’re teaching this one? You’re in good company!

Usually, we go to that old procedure we were taught where you underline the place you’re rounding to (if you’re rounding to the nearest 10, you underline the 10s place), then look at the place next to it – if it’s 5 or higher you round up, if it’s 4 or lower, you round down. Oh, and don’t forget to add zeros and leave all the other numbers the same! Crystal clear, right???

This definitely falls into the category of kids memorizing a procedure for a concept they don’t understand. I was in a third grade classroom where the teacher had students in a Math with the Teacher station, and they were learning about rounding up or down. The students seemed to be doing well while they were in the station because she was there guiding them, but when they went back to their seats, they were really struggling.

Some kids could use a number line and plot the two points, find the midpoint, and then determine whether the number is closer to 150 or 160. But not all students have the number sense to be able to make that decision.

An abacus is another great option for this situation, and so I grabbed an abacus (which I unwrapped because it was still shrink-wrapped at the back of the classroom!), and sat down with one student to take a closer look at his number – 156. We were supposed to round to the nearest 10, so would it go up to 160 or down to 150?

Let’s think of the number 156. You could build it with two abaci – one to show the full 100 and the other to show 56 – but we just used the one abacus to show 56 as a number between 50 and 60. Once it was built, I asked, “Is it quicker for you to push all 6 beads back and go to 50? Or should we just push 4 beads forward and go up to 60?” Obviously, he told me that it would be much quicker to push 4 beads and go up to 60, so he knew that he would round 156 up to 160.

I did a couple problems with him in this way, and finally let him take the abacus to do some problems on his own. Using the visual representation on the abacus, he totally got the idea of rounding and was able to explain to me *why* he would round up or down.

The best part? Afterwards, the student said, “Is that what they were trying to explain with the *five and higher four and lower round up keep the same* stuff?? This way is way easier, Miss Shannon!” Some kids can’t do it abstractly, or pictorially (which is what rounding on a number line would be), so they need the concrete visual of the abacus to be able to see exactly what they’re thinking in their head.

What about students who can already do rounding? They need to be able to do this too! It’s important for them to be able to articulate the *why* behind what they’re doing and why they made a decision to round the way they did.

This set of PowerPoint presentations and video tutorials on rounding ranges from rounding to the nearest 10 within 100, to the nearest 100 (2nd grade), to the nearest 10 within 1000 (4th grade), and rounding with decimals (5th).

In these videos, we’ll offer both the abacus and the number lines as options for students to use as they work to understand the concept and visually see how they’re rounding.

Using a number line, as you’ll see in the videos, is all about finding the midpoint. Students will plot their two points, say 150 and 160 on the number line, find the midpoint (you can use the abacus to help them find this!), and then plot the number in question. Then, it’s a simple decision of which endpoint is closer.

If rounding with whole numbers is confusing, rounding with decimals can quickly become really confusing for students. If you ask them to round 5.26 to the nearest 10th, kids have to think the fact that we’re trying to round between two tenths and three tenths. The big question is, do your students know where the midpoint is in a problem like this? If they put 5.2 and 5.3 on a number line, could they identify the midpoint as 5.25? Many times, students don’t have the number sense for this and so, we can use the place value discs as a really great way for kids to visualize how those numbers go.

Once they find that midpoint, they then have to decide: Is 5.26 closer to 5.2 or 5.3? Again, kids being able to find that midpoint and be able to plot that number will really help them to be able to make that decision. There’s certainly nothing wrong with using the abacus as an alternative to represent what they’re trying to figure out. Make the single row of the abacus beads represent 10ths, so each of those beads would be worth one 10th.You could also use the entire abacus and have each bead be worth one 100th.

Place value is a very abstract concept for kids, and these tutorial videos will help make both more/less and rounding concepts more concrete. It’s so important for students to have a firm grasp on both concepts. If a student can get the idea of rounding, it will help them when they come to estimating and some of the other higher level math concepts. So often, math books only give you a few days for rounding, but in reality, kids need so much more repetition.

Try a rounding exercise as a warm up, even if you’re teaching virtually! Use a Google Doc or Slide to have students plot points on a number line. Or use a Flip Grid video to have students explain why they chose to round one way or another.

Don’t forget that having students explain their thinking is just as important as getting the right answer, if not more so!

Check out our Place Value: Rounding series (M³ Members – you can download it for free!) and let us know what you think! Use the videos to help kids (and even parents!) understand how to explain the rounding process!

K-5 Math Teaching Resources has some great games in the 2nd and 3rd levels to help kids get some extra practice with rounding, and **M³ Members**, check out the tutorial videos and accountability sheets that go with them!

Got place value basics? Great! Now, let’s apply those concepts as we go deeper into place value with the idea of more and less!

When thinking about “more or less,” we want students to be able to manipulate the place value strips in a way that helps it make sense. For many students, “more or less” is very confusing. If you ask a student to put their finger on 46 on a hundreds chart and then add 10 or show 10 less, most often, they’ll just start one-to-one corresponding to find their new number (47, 48, 49, 50 and so on), instead of looking for the pattern and going down a column to get to the new number.

Typically, the person who actually goes down the row to get to the new number is actually the teacher, pointing, guiding, whispering until the student gets it right. In actuality, a student may learn the procedure, but not completely understand the idea of adding on that 10 more without counting on because they’d haven’t had the experience. **We want to help students become comfortable manipulating numbers in different ways, without really having to give much thought to it. **

“More or less,” while we might start it in early first grade, is a very predominantly second grade concept. You might start by having students add on 10 more, and then 20, or 10 or 20 less, then 100 more or 100 less. The video tutorials for this week (FREE for M³ Members, along with the already done-for-you presentation that you can use for virtual instruction or face-to-face) will engage students in five different exercises (based on grade level).

For example, you might have your students build 528 and then add on 10 more. This activity, similarly to the place value basics activities, can be done with students in pairs – one person being in charge of the 100s and the 10s, the other person in charge of the 1s and serving as the Clip Captain. Students have to be able to show the change that’s taking place if you add 10 to the number, so the person in charge of the 10s would bring over an extra 10 and act out the creation of a new number with the place value strips so they can see the overall value.

In our Place Value: More or Less PowerPoint presentations, you’ll find step-by-step instructions for how the students will build the number using concrete manipulatives and adjust it based on what you ask (either adding more or taking away less). We even show some complicated problems where we might stretch over a decade or a century. For example, if we had 592 and we asked kids to add on 10 more, they’d be in the 600s. Trust me, these videos and presentations will show you and your students exactly what that looks like to be able to manipulate that! Remember, they are free for M³ Members! Not a member? Buy the lessons here or click here to become a member today!)

This lesson launch is set up to help kids really understand place value through the use of concrete tools. Students that need the tools will have them readily available, but students that need less structure will be able to use the tools on their own to show how they solved the problem.

To stretch this concept up into Math4BigKids (upper elementary), we can look at applying the “more or less” idea to decimals. Do your students know what to do when they have .4 and need to add on .3? Do they know that 10 tenths is equal to one whole? You can ask students to show you 10 more or 10 less within hundredths or even into the thousandths place. Helping our big kids manipulate their number to build the mystery number we’re asking for is fun, and you can stretch it even more to include more than just 10 more or 10 less. Depending on the student’s level, you could take it up to 100 more or 100 less, etc.

Some students might just be able to do the arithmetic in their head to add or take away 10, but if they aren’t able to explain why they got their answer, they don’t really understand the concept. This becomes especially important when students advance in the area of decimals. Students could simultaneously build their numbers with the place value discs so they relate the quantity to what they’re doing with the place value strips.

]]>As we move from numeracy into number sense, one of the most important things we need students to understand is our place value system.

Typically, we start off our year learning about place value, reviewing from last year and then building on that. This year might look different for you, especially if you are extending your review of last year’s standards that students may have missed during COVID.

Even if you find yourself reviewing, I think number sense is at the forefront of what students need to understand, and you can absolutely make a connection from numeracy going into place value. Yes, even in 5th grade!

Let’s start at the very beginning – place value basics. This means the terminology we’ll use as we explain place value to our students. It is vitally important to have a universal language in your building as you approach this concept. There are three main terms, and our Place Value Basics video will help as we ensure that we’re naming things properly.

What is a “digit”? Many students will say it’s a number, but that’s not completely accurate because a number really goes to infinity. A *digit* represents the 10 numerals that we have in our number system: 0,1, 2, 3, 4, 5, 6, 7, 8, and 9. When discussing place value, we want to be careful to use either *digit* or *numeral *when we’re referring to the parts of a number. For example, if we had the number 542, you would not ask a student “What *number* is in the 10s place?” Instead, you would ask, “What *digit or numeral* is in the 10s place?” and in this case the answer would be 4.

What does *place *mean? The *place* is where a digit or numeral lives. Does it live in the 100s place? The 1000s place? The 1s place? When we’re talking about decimals, is it living in the 10ths place? The 100ths? Or the 1000ths place?

What does *value* mean? It means how much the digit is worth. To help students understand this concept, I thoroughly enjoy using the place value strips because the strips allow students to show or prove the value of their digit.

Let’s go back to 542. What is the value of the 4? Using place value strips, students could show that the 4 is actually worth 40.

Think back for a second to the tools that you’ve used in your teaching career. I vividly remember teaching place value with a spiral-bound flipchart type thing that sat propped up in a triangle on the table. It was labelled at the top – 1,000,000s, 100,000s, 10,000s, 1000s, and so forth. I had all the digits in each place, and I could flip them around to build a number to show the students. if I wanted to say add 10 more, I would just flip a number in that place.

BUT…as I look at that chart now, can you see what may not be the most useful tool for helping kids understand the WHY before the HOW? The all-in-one place value flip chart certainly was a novel concept. And, yes, I can use it to point out the digit in a specific place, and even to help students see the different places, but when it comes to value, the old-school flip chart falls flat. If If I’m asking students to find the value of the 5 in the 100s place, they can’t separate that 5 to see the value is actually 500 like you can with place value strips.

That’s why I feel that student size place value strips are probably one of the most valuable tools that you can use during your place value unit. Giving students the ability to manipulate digits, build numbers, and apply vocabulary is vitally important. Talking about the digits and values helps students to be able to explain their thinking and start to conceptualize the idea of place value. Without this experience, place value it’s all very abstract.

Not sure where to start? This week’s collection of PowerPoints is going to show you a series of activities for second grade (whole numbers in the 10s and 100s), third grade (whole numbers up to 1000s), and then for fourth and fifth grade that will use whole numbers (up to 1000s in 5th grade) and decimals (10ths and 100ths). Both the PowerPoints and the video tutorials will walk you through the process of getting students to be able to build a number, and then to manipulate it. I love watching students problem-solve through building a number and asking questions about the number and their process – it truly shows their depth of understanding!

When using place value strips in your classroom, we like to have students work in pairs. Designate one student to be the clip captain. This student will be in charge of manipulating the mini binder clips that serve as a stand for the place value strips once the number is built. This same student will also be in charge of the 1s and the 10s. The other student will be in charge of the 100s and 1000s. As you call out numbers to build, they will work together to create it using their places.

If your students are virtual, students can do this on their own and show you their place value strips on their screen. Many of our schools have created mini math toolkits that students can take home, and one of the things the kits always include are the place value strips!

Many of the activities we talked about in our blog last week can get things rolling and create that bridge from numeracy to place value. You might show your number on an abacus, or two abaci to show numbers into the hundreds, and have students build with the place value strips.

Once the number is built, you can ask students questions such as “What place is the 3 in?” Then, using the strips, students can show that digit and talk about the place. You could ask, “What is the value of the 4 in your number?” and students can prove the value by pulling apart their place value strips to show you what it looks like.

Place value discs, the whole numbers and the decimal tiles, would also work well to get kids to see the quantity of how to build numbers.

I find that showing expanded form is one of the most fun parts about using the place value strips. If you ask students to read a number that we’re talking about and then show the expanded form, many times they have trouble visualizing that. But, if you are able to pull the number completely apart and see that it is really 4000 + 500 + 30 + 2, which equals 4532, the lightbulbs usually go off! The video explains the whole process of how we can help students understand that correspondence.

Our tutorial videos – featured on our M3 Membership site or available as a bundle in our store – will walk you through the entire process for each grade level, but you can do any of the activities with any of the place values! The goal is for students to be able to build their number and articulate their reasoning as they manipulate it.

Getting kids comfortable with playing with place value will help as we start looking at the idea of “more or less” – taking a number and building it a certain amount more or less. It will even be useful when we get to that really tricky topic of rounding a few weeks down the road! Join us next week to see what we have in store for you as you continue your place value instruction!

]]>As our numeracy talks start to lean towards actual number talks, let’s look at conservation to 20!

Conservation to 20 numeracy talks are appropriate towards the end of the kindergarten year (the last two or three months). In 1st grade, we would start conservation to 20 around October, after starting with numeracy talks with conservation to 10 for the first few months of school. In 2nd grade, I like starting off the school year, just for the first month or two, with numeracy talks before moving into actual number talks in order to make sure that students have a really good foundation of being able to see the quantity of 20 in multiple modalities.

Just as we’ve talked about with conservation to five and conservation to ten, conservation to 20 numeracy talks help kids learn different ways that they can talk about how they know what they know in a seated number talk at the carpet area and then transition to the table, where students will use manipulatives and replicate the quantities they’ve identified.

Before we begin, as always, you first want to figure out where students are in their understanding of conservation to 20. ESGI has a really great screener that looks at conservation to 20 to see if students are able to identify quantities without counting one-to-one, looking at a double 10 frame, as well as a rekenrek.

There are three different ways that you can look at the quantity of 20: a double 10-frame with four stacks of five, a rekenrek (linear) with one row of five and five, and then below it, another row of five and five, and a Counting Buddy Sr., another linear representation which is not featured in the screener, but is something that we’ll use in the numeracy talks.

The ESGI screener is very simple and you can see in this tutorial video that very quickly, you can figure out which students in your classroom have a solid grasp of conservation to 20. Can they see 13 on a double 10-frame, and then see it in a different modality on a rekenrek and still identify it and transfer their skills?

Perhaps even more importantly than identifying the quantity is the reasoning behind the students’ answers. How do they know what they know? How did they know it was 13? If kids are still saying things like, “I know it’s 13 because there’s ten at the top and three at the bottom,” that’s okay, but we want to encourage more outside-the-box thinking at this stage. For example, “I know that it’s 13 because there are seven spaces empty on the double 10 frame, which means that the filled ones would be 13.”

You also want to look for kids to be able to tie in the part-part-total idea as you’re listening to their reasoning for understanding quantity. They have 13, but can they see that one part is 10 and the other part is 3? Can they see it in different configurations based on how they are decomposing it?

Doing a scatter arrangement at this level is a little bit over the top. It’s too much for the brain to be able to produce back how many they see when we’re dealing with conservation to 20, so we don’t present the scattered arrangements at this level.

After you conduct the ESGI Conservation to 20 screener, you’ll have a good idea of where your students are with this level. If a student, either at the beginning of 2nd grade or who has passed conservation to 10 in a scatter, fails this screener, you want to go back and ensure that their conservation to 10 is rock solid. Go back and do the ESGI Conservation to 10 screener to double-check. If the student does amazing with that screener, then their instructional level is right at conservation to 20. However, if you go back to check conservation to 10, you might find hidden areas of weakness within conservation to 10, so that is where you would concentrate your efforts.

When kids are ready for conservation to 20, what can we do? How can we help students develop skills in this area? Numeracy talks are one of the greatest things you can do, as they help students see quantities in different ways. Our Numeracy Talks Progression cards are designed to do just that! Progression Cards #1-12 will help you get started with numeracy talks in a systematic way.

Each set of Progression Cards comes with tutorial videos that walk you through how to do numeracy talks in your classroom, whether you’re face-to-face or virtual, and includes a classroom-ready PowerPoint presentation and a companion that goes along with it. We want to see if a student can really understand a quantity in a double 10-frame, in the linear Counting Buddies Sr. (which you can get in our store), as well as a rekenrek (which you can find in our store as well or make your own with these directions!). Get a freebie of Card #9, with all the activities and tutorial videos you’ll need to get started. Members have free access to Cards #9-12 with their M3 membership login, or you could purchase the Green Numeracy Talks Bundle in our store!

Card #12, the last in the progression, really mixes the modalities for students, as you can see in the video. Students will see the quantity in one way, say in a double 10-frame, but will then replicate it with another tool – the Counting Buddy Sr. for example.

Once students reach the end of the progression, we can extend it and help students start to gain number sense. Place value strips are a fan favorite at SIS4Teachers. They’re a great way to help kids associate what they’re seeing on the double-10 frame, the Counting Buddy Sr., and the rekenrek as “10 and some more.” Making a strong connection between how many 10s and how many 1s is really a precursor for what we’re going to be talking about next in our blog series, which is all about the importance of place value.

I recommend using the place value strips by starting with the numeracy piece. Say you flash the number 17 as a quantity in a double 10-frame and you want the students to build it with their place value strips. They’d use a 10 and a 7. Knowing how to look at a teen number and isolate the 10 will help students be more successful when the start expanded form, as well as partial sums and decomposing by place value in second grade.

Students could also build the quantity on non-proportional or proportional manipulatives. Proportional manipulatives would show the quantity in base 10 blocks so for 17, students would show me one 10 stick and 7 individual ones. T-Pops’ Place Value Mat and place value discs would be a non-proportional manipulative option where students could show their 10 and then 7 more.

Don’t stop when you get to 20! After this stage of conservation to 20, it’s really important to continue that growth of getting kids to understand numeracy as it starts to relate to number sense. I love going to dreambox.com/teachertools, and using their 1st grade tools, Numbers to Forty on the TenFrame to continue your numeracy talks (especially in first and second grade) so kids can start to understand the numeracy and connect the number sense.

So let’s say I flash up three 10-frames that are full, with two on the fourth 10-frame (32). Students could either build that with the place value strips, place value discs, or with place value, by using the base 10 blocks.

You could go all the way to conservation to 100 using an abacus. The abacus is a great tool for students to be able to understand our number system, because it’s all in 10s. Don’t have an abacus? Get one from our store, or use a virtual abacus from dreambox.com/teachertools or the virtual app from the Math Learning Center. Show an amount, maybe 63, and have kids build it in another way. You could even ask students to show you 10 more or 10 less than that number.

Last, but not least, don’t forget the Deck o’ Dots! Our Yellow Level cards are made up of 10-frames, but don’t limit yourself to one! Maybe you flip over four full 10-frames and then you flip another next to it. Maybe you flip three 10-frames and then add another number to it. Play around with the different Deck o’ Dots games and activities and see how this kind of activity helps solidify the connection from numeracy to number sense.

What’s next? Numeracy talks lead us to number sense, which is a pivotal piece to the understanding of place value. Join us next week to find out how it all works!

]]>First grade is a big year – the transition from numeracy to number talks!

Beginning in PreK, we assess the kinesthetic level of our PreK students and begin to work on conservation to 5. In Kindergarten, we want to work with any students that need kinesthetic help and then continue to move through the whole progression of conservation to five. By first grade, we get to begin with conservation to 10!

Conservation to 10 is a lot like conservation to 5. We look to see if a child can see the configuration of 10 in different ways without recounting. It seems like kids have always counted one-to-one – 1, 2, 3, 4, 5, 6. If they have the one to one correspondence we looked at last week, they get it. But they can’t always rely on being able to count one-to-one. Instead, we really want to help them develop strategies that will build numeracy, which is where conservation comes into play.

With conservation to 10, we want students to be able to see a 10-frame and say, “I see that there’s 5 on the top, and 3 on the bottom. So I know it’s 8.” Kids might also say, “I know that a full 10-frame has 10, and there are 2 empty, so that means it’s 8.” Reasoning is really, really important as students build conservation skills.

However, you and I both know that our littles will eventually memorize the modality of the 10-frame – the boxes are always the same, they always see the same orientation of the rows and the counters. So, we need to mix it up! We want kids to be able to see structure with the 10 – it’s a foundational part of our number system, after all!

Maybe, instead of a 10-frame, we present 10 in a linear fashion that matches our Counting Buddy Jr. The beads are in a straight line, with 5 of one color and 5 of another. When we do this, it might throw kids off if they were used to the 10-frame patterns. So, what do they do? They go back to one-to-one counting.

We want kids to look at the 10-frame and immediately say that it’s eight, and then see it in a linear way on the counting buddy with five red and three blue, and immediately know that it’s eight, because it’s two less than 10. Being able to provide reasoning in the different situations is a great indicator that students are working towards mastering conservation to 10.

The third level of conservation to 10 is to present it in a scattered arrangement? Can they still identify the quantity without counting? We don’t use a traditional definition of scatter, where the pattern is truly random like if you had dropped a handful of beans on a table and counted them where they fell. Of course, in that situation, kids are going to end up one to one corresponding if the beans aren’t in any kind of clusters.

In our scatters, students might see the structure of a 5 that looks like the five on a dice, and then two more. Do they immediately know it’s seven? Can they break the scatter into groups and really understand what it is showing? Can students see that 8 is decomposed into 5 and 3? Or 4 and 4? If they can, along with understanding the 10-frame and Counting Buddy in a linear fashion, students are laying the foundation for part-part-total kinds of problems in the future.

This is the beginning of letting kids play with quantities without the digit, which is so incredibly important and what this blog series is all about! Take time to let kids develop this familiarity with quantities in different presentations. Don’t rush them into addition facts or memorizing concepts they don’t really quite understand.

In this progression, you want to make sure that you have screened students effectively to really understand where they are. One level in our ESGI screener looks at conservation to five in a linear fashion and then the 10 frame. Those quantities are always depicted the same, meaning that when you look at it, they’re always organized so that, if you can understand the structure of the part-part-total, or a five group and some more, you’re going to be successful.

In this tutorial video, I’ll show you how to use ESGI to screen your students for conservation to 10 with a 10-frame, but also with the linear modality mixed in. Obviously, it’s really important to only show the quantity to the students for a few seconds so they don’t have time to count one-to-one and you can accurately assess their conservation.

When kids get really good at identifying quantities within a structure, the scatter arrangement will help you take conservation to 10 to a higher level. The ESGI screener for this level is also a very quick screener that flashes different quantities for students to identify. At this level, you might see a pattern with some students as to their understanding. For example, some students may, understand the scatters up to 6, but once it gets to 7, 8, 9, it becomes a little bit too complicated for them. The screener will identify those patterns for you, which is why I love using a screener like the one ESGI provides!

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Additionally, what we want to ask: do they have a reason for their answer? When we ask why they responded a certain way, does the student say, “I just thought in my brain and I’m super smart,” or are they able to verbalize why they know that. The more kids can explain to you how they know the answer, the more we can be assured that they have a deeper understanding of the concept.

In addition to providing amazing screeners, ESGI has some other tools that you are going to love – that will make your life easier! After you complete a screener with the student, you can print out a parent report to help them know where their child is – if they have conservation to 10 in a 10-frame, and in a linear, or where they are with the scatter.

I think my favorite thing might be the personalized flashcards that ESGI has the capability of printing based on the results of a screener. Just like that! Think about intervention. Think about home support! We can figure out where kids are and really target what they need to improve!

As we understand where our kids are with conservation, we want to be able to support that in the classroom in a consistent manner. We can do that by continuing our numeracy talks that we started with conservation to five (check out the first four progression cards in the Red Level Bundle!), starting at the beginning of the year in first grade, later in the year in Kindergarten, and even some PreK classrooms towards the very end of their year.

Our Yellow Level of Numeracy Talks builds on the Red Level, with Progression Cards 5, 6, 7, and 8. Each card has activities A, B, C, and D, and they come with classroom-ready presentations and videos that will help you implement those activities in your classroom!

Of course, we have a sneak peek for you, our blog readers! Card #5 is yours to download so you can get an idea of how this all works. Watch the tutorial and use the presentation in your classroom with your students. If you love it (or if your students love it and demand more!), you can take on the whole series of the progression with the Yellow Level Bundle in our store. Or, if you’re an M^{3} Member, you get free access to the entire library of numeracy talks products! Members, check it out here!

Of course, we can’t forget our Deck ‘o Dots cards as a great support for this level of conservation! The yellow level of the cards corresponds to the yellow level of our conservation progression. The cards in the level are all 10-frames, but the green level of Deck o’ Dots cards has scatters (half are 0-5, the other half are 6-10) that you can use. Check out the Deck o’ Dots tutorial videos or the Deck o’ Dots games to see how you might implement some fun games into your classroom that will sneakily reinforce the conservation to 10 skills you want students to learn!

Don’t forget about parents as you’re working to build students’ conservation skills! I find getting a Deck o’ Dots into a parent’s hands can be powerful! You can have a math night using the Deck o’ Dots and talk about this concept of conservation. We want to help parents understand that we are focused less on the numerals but more on the numeracy, and what numeracy really is all about.

Conservation to 5, conservation to 10…can you guess what’s next? **Conservation to 20!** The last level of our numeracy talks, which is where we begin second grade, is up next week! Even if you don’t teach second grade, don’t worry – your students will get to this level too, so tune in next week for the final level!

Take a second to brag on your student (or yourself!) in the comments or leave us a post on social media – Facebook, Instagram, or Twitter!

]]>Numeracy development is key at the beginning of the year! Within the first three weeks of starting school, whether you are a PreK or Kindergarten teachers, we want to find out where students are within their early math levels so you can plan for the rest of the year, just like we do in literacy!

In this back-to-school blog series, we’re going to go through the different types of conservation milestones that we want kids to hit, as well as how we will know they have the early foundations of numeracy in place before we start building number sense.

When looking at levels of numeracy, the first thing we want to do is screen students to find out where they are, just like you would screen students for phonemic awareness. We want to look deeply to see if students can play with number concepts – either with their whole body or within a certain level of number conservation, either to 5 or 10.

The first level we’ll look at is the kinesthetic level, and to help teachers do that quickly, the ESGI assessment system provides an assessment that you can customize on your own, or you can use mine, which is already loaded in and easy to implement with your littles!

PreK teachers want to start by looking at their students to make sure they have kinesthetic one-to-one correspondence. In this video, we screened some early Kindergarteners for this skill, and you’ll see that one of the students has great kinesthetic one-to-one correspondence, and the other does not. Have you screened your PreK students for this skill? If you’re a Kindergarten teacher, are there students in your classroom that still do not exhibit this skill?

ESGI makes it simple to find out! This tutorial video shows you how to use my kinesthetic one-to-one correspondence screener to quickly assess a PreK class or any student in Kindergarten that might not be at the level we’re looking for.

To perform the screener, you’ll need 10 poly spots, which we have available in our store. Before you begin the screener, label each poly spot with dots from 1 to 10 in a familiar pattern. For 1 through 6, you can use a dice pattern, and then for 7, 8, 9, and 10, I use a 10 frame format to keep the dots orderly and numerical. You don’t want numbers on these poly spots because we’re trying to see if kids have a skill called counting and cardinality.

Then, using the ESGI checklist, you simply ask students to stand at the beginning of the number line and to walk to a particular number while counting kinesthetically. When students can do this, we know they have kinesthetic one-to-one correspondence.

I’ve often described kinesthetic one-to-one correspondence like rhyming. If you ask a child to supply a rhyme for cat and hat, and they say *water bottle*, you know they don’t really have the phonemic awareness skills that are needed to start rhyming. It’s the same way with kinesthetic one-to-one correspondence. We can’t necessarily teach kids to have this skill, and it’s very obvious when students don’t have it, but as they practice counting and corresponding their body to the numbers, we will eventually notice that students can learn this skill quite quickly.

If students can walk on the poly spots, as you saw, with confident one-to-one correspondence, we mark *Yes* to the first question.

Then, we take the students off of the poly spots to ask the second question on our ESGI checklist. We want to see if they can maintain the same kinesthetic one-to-one correspondence without a guide. In the video, you can see that one student is able to do it, but the other student is not. Typically, if a student doesn’t have one-to-one correspondence ON the poly spots, she will not have it without the poly spots, but you do want to check both parts of this screener.

Kinesthetic one-to-one correspondence is an essential skill that we want to help our littles build. You can check out our series on math4littles, which highlights mathematical thinking all around us, to find lots of different activities that kids can do to help build kinesthetic one-to-one correspondence in their daily routines.

What about our students in Kindergarten? Is it necessary to assess every single Kindergartener on our kinesthetic one to one correspondence? At one time, we did that. However, it was kind of a lengthy process to find out that most of our students had the skill in place. Instead, we realized that, when kids don’t have the skill of conservation to five, they would more than likely not have kinesthetic one-to-one correspondence either.

It’s a matter of development. As you can see in this pyramid of math development, we want kids to have lots of opportunities to interact with real objects in the physical world, and then to be able to build kinesthetic one-to-one correspondence before going on to conservation to five.

In Kindergarten, we want to use our ESGI checklist to assess conservation to five. If a child fails this assessment, only then do we need to back our way out and screen that student to determine if kinesthetic one-to-one correspondence is in place, since that is a foundational part of math development.

In this screener you’ll see that we are really looking in depth at the conservation of five. We want to see if students can recognize a quantity, from 1 to 5, regardless of how it is presented. We’re going to show students the five frames vertically, as well as horizontally to make sure that students aren’t just memorizing the five frame, but actually processing and thinking through why they know what they know. The ESGI screener also includes dice patterns, again stopping at five, just to make sure students are solid on this concept of conservation to five.

Think about how long it would take you to screen your students. You have to get your five-frame cards and your conservation to five cards (either dice or domino pattern), then you have to go through your five frames horizontal, then vertical, and then through all your dice or dominoes. Multiply that by *how many* students??

I love how ESGI says “click click done” because it’s really that easy to use their screeners! It’s all digital, so your students can be anywhere while you’re working to assess their numeracy skills – they can be in the Playhouse or maybe out on the playground (kids will think it’s fun to come over to sit with you and do their assessment on your mobile device!)

Remember, if a student fails this and they don’t have 80% accuracy, or better, we want to go back and double-check that they have the kinesthetic one-to-one using the first ESGI screener.

Sometimes you’ll go back and find out that kids are doing great! Don’t be alarmed if you go back and find that they have pretty good kinesthetic one-to-one correspondence. That probably means that they’re really ready for being able to look at conservation to five and work on that level!

Once we’ve screened all of our students, let’s get them going with numeracy talks! This is a great way to help students as they begin to work on their conservation to 5 and early numeracy skills.

We have a great new product that is designed to help you implement these numeracy talks in your classroom! Our Numeracy Talks series follows a very systematic structure of that will help your students progress through numeracy development. Each of the four Progression Cards corresponds with a classroom-ready PowerPoint presentation that you can use to guide them through the activities.

For Kindergarten, these are great to start at the beginning of the year! In PreK, this series will come in handy in January, once you’ve built the kinesthetic one-to-one correspondence and are ready to move on to conservation to 5.

You can begin with Numeracy Talks Basics, a free video that will help you get numeracy talks going in your classroom. The first progression card is a freebie you can download here, which has everything you need to get started and see how it works in your classroom! As you start to see improvement in your students’ numeracy, you can get the whole bundle for conservation to 5 – Progression Cards #1-4, the corresponding presentations and tutorial videos – in our store.

If you’re one of our M^{3} Members, you have free access to this whole series and all kinds of other videos that we’ve produced in the area of numeracy! Check it out here!

Deck o’ Dots games are the perfect addition to numeracy talks to support numeracy development! Our red level is all about conservation to five, with vertical and horizontal five frames. Half of the green deck shows scatters from zero to five, which is perfect for conservation to five to help students build an understanding of scatters.

The Deck o’ Dots cards go hand-in-hand with the Progression Cards from our Numeracy Talks series, matching the horizontal view of the five frame, then the vertical view, the scatter, and then mixed modalities.

How are numeracy talks going? Do your students love Deck o’ Dots? How did you do on the screener?

Have you checked out ESGI to see the kinesthetic one-to-one correspondence screener and the conservation to five screener? Make sure you use the promo code SIS4Teachers to get a free trial, and if you like it, you can use it for your whole class! Just imagine being able to see, at a gland, where all your students are with this concept of conservation to five! It will save you hours and hours, I promise!

Up next week – let’s move through the conservation milestones to look at conservation to ten and check on our first graders as they are coming into the school year. I’ll show you another great ESGI screener and give you another sneak peek into our numeracy talk series – don’t miss it!

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