In case you’ve missed it, we’re a *little* excited about our Math Mights show that launched last week! Two new shows from every grade level (K-3) per week – it’s incredible!

Don’t let the cute, colorful cartoon characters fool you! There is a lot of careful thought and planning that goes into the development of each show! We purposefully incorporate the 8 Mathematical Practices, and each show is tied to specific standards that you’re teaching in your classroom (maybe even this week!). We also partnered with Illustrative Math, one of my favorite resources, and were able to use a beta version of their program, which is coming out in July. Many of the aspects of their program complement and support what we do at SIS4Teachers.

Check out all the details and supporting resources at mathmights.org.

Additionally, we want to help you understand the pedagogy behind the shows so you can better incorporate them into what you’re teaching. Just like with our students, *we want you to understand the WHY behind the HOW of each episode.* So, each week, we’ll bring you into our planning process a little bit, help you learn what each show is about, what strategies we use and why, and even what manipulatives we choose and why.

Episodes 213-214

**For the month of January, the number/numeracy talks are the key!** We hope you use them to really provide an opportunity for engagement for your students! USE THEM IN YOUR CLASSROOM! Show the video, let the teacher on the screen do your numeracy talk that day, and just pause it where you want your students to stop and talk, then resume – it’s that easy regardless of your teaching format!

**Focus: **213: Addition Word Problems / 214: Subtraction Word Problems

**“I Can” statement:** I can use objects or drawings to show addition/subtraction problems.

**Extension Activity:** math work mats and quantitative pictures.

The beginning part of the Kindergarten show really focuses on a number talk review or a numeracy review with conservation to 10 to see if we can tell how many without counting.

If you’re not familiar with our number talks and our numeracy talks in the lower elementary, we have some really great resources that will really help.

In these episodes, we’re working in the 10-frame, getting the idea of flashing numbers, the idea of subitizing, and helping kids get into the character of Dotson, the Subitizing Superhero from the Deck o’ Dots cards. The show gives different kids’ thought processes, and we’ve already heard from people that have used the number talks in their classrooms that their students were relating to the kids on the show and how they thought of solving the problems!

After the warm up, we have an engagement, where all kids have access to the lesson. We ask open-ended questions such as – Which one doesn’t belong? What do you wonder? What do you notice? The goal is to get kids to engage with the objective without really knowing it, which is similar to what I usually call a Lesson Launch. With word problems, we want kids to have the imagery in their heads, to be able to close their eyes and imagine the story – “On a warm, sunny afternoon, I saw two red birds in the tree and three blue birds join them.” We want to give kids a picture of what those birds look like, which will help when they’re acting out the problem.

Of course, the kids know that we can’t actually go grab real birds and put them on their Math Work Mat to act out the story, but we can use counters, or anything that you might have in your house or at school, to represent those birds in that problem. During this part of the lesson, students are using auditory understanding as they hear the word problem, and then act it out.

I think it is really important for kids to do where a story doesn’t just end as soon as you say it, but that a story can continue and that you can continue acting it out on a mat. In Episode 214, Mrs. Gray does a great job with using a forest mat and having different animals come to and leave the forest, which helps kids apply the beginning parts of addition and subtraction.

At this level, we’re not necessarily focused on kids writing an equation yet. We just want them to get an idea of what’s happening in the story. Approaching word problems like this is developmentally appropriate for Kindergarteners, as well as preschoolers.

We have more on using math work mats in our SIS4Students series, Week 3: Real Life Math, which you might find helpful!

Another helpful resource is the *Math Talk* book, which we reference for quantitative pictures, or pictures with plenty of action and objects that provide opportunities for students to create and engage in a story problem.

**Focus: **213: Identifying 10s and 1s: Part 1 / 214: Identifying 10s and 1s: Part 2

**“I Can” statement:** I can tell how many 10s and 1s are in a number.

**Extension Activities:** Place Value Concentration with Value Pak, Value Pak practice pages

In these shows, we’re really working on identifying 10s and 1s on two different levels. At the beginning of the first grade number talks, we’re starting off with our numeracy talks in the double 10-frame. Eventually, we’ll work into actual number talks. Here, we do a lot of subitizing with Dotson the Subitizing Superhero. We want the kids to tell how many they see and why, but we also want them to realize that there’s more than one reason why. It’s not 16 just because you saw a 10 at the top and a 6 of the bottom, but another student noticed that there were 4 empty in the double 10-frame, or another student saw it as 5, 5, 5 and 1.

We really focus on the 8 Mathematical Practices here, especially Practice 3 – getting kids to construct viable arguments and critique the reasoning of others. In the “I Can” statements of these shows, we really try to use that philosophy to help kids understand why 10s and 1s are so important. We start with a collection of base 10 blocks or a spill of unifix cubes and present the students with questions to get them to arrive at the objective on their own. We let them do the thinking! *How are we going to count all these things? What do you notice? What do you wonder? *Some kids might say we can count by 2s or 5s, but is that the most efficient way? Eventually, we want them to understand why grouping things in 10s is really important.

I think one of the biggest things that we stress in these shows is that there isn’t just one modality – you don’t just have to use unifix cubes! Bring in your 10-frame because kids are already familiar with that structure. Use the abacus, one of my favorite tools, to help students see our number system in 10s and 1s. And, of course, the Value Pak, which wears their values on their bellies.

While the shows are Part 1 and Part 2 of the same topic, I would say the difference between them comes down to manipulatives and models. Part 1 uses the abacus, unifix cubes, and 10-frames, but as we really get into the second show, we start to understand the idea of 10s and 1s with place value blocks, putting them into 10s and into 1s and really relating how Value Pak is so important. We can write it or see it in different ways, too – you have base 10 shorthand, you have writing (three 1s and seven 10s), you can show the 73 with Value Pak, with the value of 70 and 3, or you can show it in your actual base-10 blocks.

**Focus: **213 – Subtracting on a Numberline / 214 – Open Numberline

**“I Can” statement:** I can subtract with a number line.

**Extension Activity:** Springling practice pages

This is my show! I start with actual number talks reviewing the idea of being able to mentally add a two digit and a single digit number, using our friend D.C. I want kids to try this mental strategy! Pause it and see if your kids can come up with the ways of solving that we did! What’s important here is getting kids to talk about their mental strategies and show that, while we might have the same strategy, you might have a different way of going about it.

I will tell you I struggled in the idea of the way Illustrative Math uses a number line with actual numbers. As you know, I’m much more of an open number line person, but after some consideration, I decided to use the numbers on the number line at the beginning of the show as kids might be just becoming familiar with the idea of number lines. I wanted to have the numbers on the number line to get the idea of spacing between numbers, how to count up and count back and how to hop in friendly numbers and what our friendly numbers are.

In the second show, we stress the idea of the distance between two numbers. We take a string and show kids that subtraction really is not “take away,” but we’re looking at the distance between the two numbers to figure out what that difference is. Here, we use the open number line instead of the numbered number line because kids are able to start and stop that number line wherever they want to and I think that an important piece about subtraction. We also help kids relate to the character of Springling. Second graders love her! They really get into telling her to “Hop, Springling, Hop” as they’re able to visually see that distance by counting up or back.

**Focus: **213 – “groups of” / 214: “fair shares”

**“I Can” statement:** I can understand multiplication by thinking about groups of objects.

I can understand division by thinking about how one group can be divided into smaller groups.

**Extension Activities:** Multiplication Tetris, Division Bump (student-approved math games!)

I feel like we can never spend too much time reviewing multiplication and division concepts. In these episodes, we give a light introduction to multiplication and division to help you see where we’re going in future shows later on in the season.

We do start these episodes off with a number talk focusing on subtraction and reviewing the ideas of Springling. We’re doing a lot with addition and subtraction review and the number talks, as we’re learning multiplication and division. I think it’s okay to not have multiplication/division number talks when you’re just learning about it. Let kids keep their addition and subtraction skills sharp through review and see if they can construct viable arguments and critique the reasoning of others.

Language is a big focus in these episodes. What does the “x” mean in a multiplication problem? It’s not “times” but “groups of.” We can show four groups of three, but how would that look different if I showed three groups of four? We spend a lot of time with this concept and using the place value discs as a really valuable tool. You can use the virtual manipulative, or download the pdf and print/cut them out to use physically.

Then we spend good time on the concepts of division. What is a dividend? What’s a divisor? What’s a quotient? We want kids to be able to use the language and be able to act out the problem in a step-by-step process with place value discs so they understand the concept behind what we’re doing.

If you are a part of our M3 Molding Math Mindsets Membership, start yourself some folders and download the PowerPoints! You have access to ALL the teaching materials and animations that go with these lessons!

(valid M³ Membership login required)

**Welcome to Shannon’s 1st ever VLOG! **

Get the inside scoop on the Math Mights show from the creator herself – from how the show came about, the characters, to where you can watch it and how SIS4Teacher will continue to support the show. Most importantly, see how the show, and related resources on mathmights.org, gives you everything you need to get started using these strategies in your classroom right away!

Hi my name is Shannon! Thanks so much for joining me this week! Instead of doing my normal blog this week, I’m doing a blog to talk about an exciting opportunity that we’ve been working really hard on at SIS4Teachers.

0:15

As many of you know, I’m the creator of Math Mights, and they were brought to life in 2017. I’ve always wanted the Math Mights to be able to have a great outreach to help children in all different areas understand math strategies “three ways plus the traditional.” Many of you have seen me present, or maybe even have been a part of some of my online things to show how to use the Math Mights. We were offered a really exciting opportunity to partner with PBS and the new Michigan Learning Channel in Michigan, which is providing educational partnerships with schools and parents all around the state of Michigan. We are now able to connect with them and are able to do our Math Mights Show which is releasing two shows every week on Kindergarten, First, Second and Third Grade.

1:02

Each of these shows have been really thought out and really developed around the philosophies that we use at SIS4Teachers. We talk so much about the idea of Concrete-Pictorial-Abstract (CPA), getting kids to be able to explain their mathematical thinking, and to really be able to create a common language in math. Throughout the thoughtful process of each of the shows, we have used lots of different rich material, from number talks to get kids to communicate their thinking, to visual models for word problems, and being able to explain math where we can invite all students to the math lesson, getting them to have a preview to think about the content.

1:41

The exciting part about what we’ve been working on is each show that we create really aligns to what you’re teaching in your classroom in real time. So the January show comes out, and you might be working on place value in your first grade room, or you might be finishing learning about the basics of multiplication and division. The show will kind of grow with you throughout the school year, featuring different lessons that you can use right away.

2:06

I think some of the coolest news that I’ve gotten this week, with it being our launch week, is that we’ve already had teachers that have told me they’ve used it in their classroom, hit pause from the michiganlearning.org website, had their kids do the number talk that we were talking about in the show, and then turned it back on and kids who are relating to our characters in the Math Mights Show, not only with the strategy, but also the way that the other kids solved the problem. And so I think that connection is really important. For teachers to have accessibility to lessons that they can do virtually, whether they’re teaching virtually, in a hybrid situation, or even face-to-face.

2:45

This show also helps to branch out for parents, it’s accessible on the PBS channel in all different areas of Michigan. They can go on to their TV, even if they don’t have a computer, they can get the idea and the philosophy behind the lesson to do with their child. You also can get the lessons on demand on the mathmights.org website that we’ve created or the Michigan Learning Channel. For each episode we created an extension activity that was thoughtfully planned with the standards that we were delivering so that we could get those activities to be used and readily available for teachers or parents.

3:22

We also want the show for students to be able to enjoy and have fun. The Math Mights, if the kids can get in the world of the Math Mights thinking and Mathville, and thinking about all the ways that our characters solve the problems. It is truly amazed me at the students that I have seen that remember those characters, they remember the strategies that they’ve learned, and they’re able to explain their thinking.

3:46

The neatest part about the Math Mights is that if you learn the addition strategies, let’s say in first grade, and you get the idea of D.C. and Value Pak and maybe even Abracus, you can do all of those strategies when you’re doing multiplication. So you can take teen numbers and decompose them to make it easier to multiply with D.C. You might be familiar with doing partial products, which is part of decomposing by place value, which features our friends from the Value Pak. You also can use Abracus, which is compensation, as we do different things in our compensation for multiplication. And of course, T-Pops just always fits in because there’s always that traditional method. But students start to understand that there’s more than one way to solve the problem.

4:31

The show has been incredible on my end from developing. As you know I’m a presenter and a classroom teacher. I’m used to actually having an audience versus looking at a camera. Our crew at SIS has really pulled together, from everyone that we’ve worked with behind the scenes, to producing it in our studio, as well as doing all of the edits and all the outreach.

4:53

We have three fabulous teachers that have joined our team. Rhonda Askew is our third grade teacher, that’s doing an amazing job right now with multiplication and division. We have our first grade teacher, Tiffany Markavich, who is doing a great job with a lot of different things in place value. And then we also have Alicia Gray is our Kindergarten teacher. I play the second grade teacher in the show, so I’m able to use a lot of the different Math Mights and sort of be at the front of showing all the different portions to it.

5:22

I’ve created and developed all of the shows, which has taken a ton of time, and kind of rethought about the way I think about delivering things. But I think the most important part is is that we’re getting this information out to parents, not just in Michigan, but all throughout the United States because you can go to the show on the internet, which is great.

5:43

So in the coming weeks, I’m going to dedicate my time and my blog around the show. I’m going to be giving you tips and tricks for the different strategies that we use in each of the shows. Using those standards that you’re already on, really helping you figure out the philosophies of why we use certain tools and why we’re instructing in a certain way. So I hope that you’ll look forward to more of our blogs that have to do with that.

6:08

If you’re a part of our M^{3}: Molding Math Mindset, you have a special ticket! You are going to get all the behind-the-scenes footage, even the PowerPoints that I have developed, that you can use in your classroom! And the big secret is all the animated pieces that we use in the show come in the PowerPoint! So the kids are gonna love it! For those of you that are part of our M^{3}: Molding Math Mindsets membership website.

6:34

I hope that you’ll take time to join us on our social media channels that are Math Mights with our Facebook, we also have Instagram and Twitter. There’s going to be lots of great announcements as we do different shows, as we’re airing shows all through the second semester of this year and I think you’re going to really enjoy them.

6:51

If you want to check out more, you can go to MathMights.org. Or you also can see things on the michiganlearning.org which is the vehicle of which all of the on demand shows are. On our website, we’ll have lots of different tutorial videos and ways for different free downloads. I even matched all the Virtual Manipulatives that you can use from the items we’re using in the show. I truly hope that the show makes a difference for you, for your families and your students. Thanks so much!

It’s time to help students connect what they’ve learned from their kinesthetic experience to the concrete by creating an exploratory experience using concrete objects!

As most of you know, one of my favorite tools to use (for almost any subject!) is place value discs, and multiplication is no exception. They’re so versatile – you can use physical discs if you’re with students, or you can use the virtual discs (check out Virtual Manipulatives page under Multiplication) if you’re virtual.

Regardless of your teaching situation, we have a really amazing PowerPoint presentation (if you’re part of our M³: Molding Math Mindsets membership, download it free here or buy it in our store). I always say “go slow to go fast” and that rings especially true when we’re talking about the understanding of multiplication. This presentation walks you through the different problems and the steps that you want students to go through to make sure they have a thorough understanding of the concept. Want a sneak peek of the product? Check out this video!

Even though you might do this activity with third graders as they’re starting to understand the beginning concepts of multiplication, you might also be able to do this with higher level numbers as we start to talk about “groups of.”

To help show “groups of” in this activity, students could use bowls or plates to make clear groups with their place value discs, just as we did with the hula hoops at the kinesthetic level.

**A word of caution:** make sure you choose numbers for your problems in such a way that you won’t run out of discs as students are acting it out. Manipulating the discs creates another imprint on the brain, similar to the memory of the kinesthetic activity, which will help as we move into the pictorial/concrete level later on.

Start this off with something simple: ask students to show you 3 x 12 or 3 groups of 12. Give the students their discs, and allow them to begin exploring. Let them experience productive struggle as they begin to think through how they’re going to show you 3 groups of 12. Some students might start to count out 1s, but you want them to realize that they can make 12 with 1 red ten disc and 2 white one discs. They also need to remember to make 3 groups, not 12 groups of 3, because they won’t have enough discs for that!

In the presentation, we talk through each step of the problem as we want students to think about it – what are the two factors in the problem? How many groups do we need to make? How many are in each group? How are we going to find the product?

As students act out 3 x 12, making 12 with 1 red ten and 2 white one discs, and doing that three times, they can put those in the defined groups to make that statement come true. When the numbers become higher, students can pull those discs, as I demonstrate in the tutorial video that comes the PowerPoint, as they’re starting to add the groups (repeated addition of 12 plus 12 plus 12) to figure out the product. They can count the ten discs – 10, 20, 30 – and then the ones – 2, 4, 6 – and get their answer of 36.

There’s nothing wrong with having students write the corresponding statements next to this problem – 12 plus 12 plus 12 equals 36. We know that 3 groups of 12 equals 36. You might even have some students that understand how division is the opposite – so if I did 36 divided by 12, I would end up with 3. Letting students play with the place value discs in this way to discover those kinds of connections is so valuable! You might want to limit exploratory problems to 4 “groups of”, but you could ramp up the number in each group as students develop more understanding. Maybe you do 4 x 23.

Your third graders might be hesitant, thinking *I’m just learning how to do 8 x 5, why am I doing things with a double-digit number??* But, acting it out with place value discs will really help them to visualize what that problem is asking.

So, I need 4 groups of 24. I’ll get out 2 tens and 4 ones and do that three more times to make 4 groups. Because of how I set up my place value discs in my Math Salad Bar (check out that video tutorial here!), I usually have about 15 sets of place value discs and should have enough ones in each set for students to create this problem. Of course, if we’re doing this virtually with the place value discs from Didax.com, you won’t have to worry about this! Students can be creating their groups along with you as you demonstrate using the guided PowerPoint so they can still gain that understanding virtually.

You also can create different extension activities on Google Slides, with a virtual Math Salad Bar (get that free template here!) or replicating discs on slide, where students can model their thinking as they’re working through a problem.

But don’t stop there – ramp it up to the 100s! When we model this, the third graders get really excited! I might ask to see 3 x 122, so students start building with 1 orange hundred disc, 2 red tens, and 2 white ones. Doing that three times, they’ll be able to figure out the product by manipulating the discs and adding it all up.

You might also create a problem where you have multiple tens, which create a hundred, which is one of the steps I demonstrate in the presentation and the tutorial video. This can sometimes be a brain-stretcher, but if kids can see it, the light bulbs tend to start going off!

I’ll start with 142 and ask students to show me 3 x 142, or 3 groups of 142. They start to add those parts together – 3 hundreds is 300, but then I have 4 tens + 4 tens + 4 tens, which will eventually create another hundred. Helping kids see how their addition and place value skills relate to multiplication is really powerful!

Obviously, we’re going to go back to really teaching the ideas of multiplication as we start to look at the multiplication journal and all those things, but this kind of exploratory experience lets kids experiment with what multiplication means as they work with concrete objects that connect with what they’ve done kinesthetically.

There’s nothing wrong with students taking it to that next level and showing their problems pictorially as well. They might be making groups and lining them up pictorially to show they have 122 and another 122 and another 122. Kids group their discs – can put their hundreds with the hundreds and the tens and the tens and the ones with the ones, but then put circles in there to demonstrate those groups as you’re adding it together.

I think that pictorial model really helps students solidify and understand what we’re doing in the concrete. As students start to make multiplication problems a little bit harder, they’re going to really start to understand this concept. Creating a pictorial model also really connects down the road as students start to decompose numbers. If they see that, when you do 12 x 3, it’s really like 10 x 3 and 2 x 3, they will be able to make that connection for how they can add or multiply things by decomposing them into parts. This all leads them to the idea of partial products which, again, really helps kids to really understand multiplication on a higher level.

Of course, we are only at the beginning of multiplication by helping students explore the concrete connections, and partial products is further down the road, but it’s never too early to help students begin to make those connections by “playing” with place value discs. I think that you’ll find that students who approach multiplication in this way will not just have an understanding of multiplication that’s a mile long, but they’ll have it a mile deep. As you start to apply some of those concepts in multiplication gradually over time, your students are going to have a much more well rounded way of understanding what the meaning of multiplication is

We hope that you enjoy the presentation that we’ve created for you to use in your classroom to help your students build on their knowledge of multiplication at the concrete level!

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Multiplication isn’t a new topic for our blog – we’ve talked about our multiplication journals and lots of games like Multiplication Bump! or Multiplication Speed! But, today, we’re going to rewind to the very beginning – how to introduce multiplication!

**To really introduce multiplication correctly, I think it’s important to add the K (kinesthetic)** to the CPA or Concrete, Pictorial, and Abstract. Today, I want to show you how to do that at the beginning of your multiplication unit as students are just starting to learn the concept.

First and foremost, we want students to really understand what multiplication is, but just telling them robs them of the productive struggle of figuring it out! Before you bring any kind of lesson or even a mini lesson on multiplication, we want students to discover for themselves, in an inquiry-based activity, what multiplication is. I like to do that kinesthetically by having students act out what multiplication means with their physical bodies. This is far more meaningful to students than being told exactly what things are.

With more of us going back to virtual school rather than in-person school, we always want to have options in case doing this activity face-to-face isn’t possible. I created a video and a PowerPoint presentation that help you be able to do this activity in either setting. M³ Members – download it here!

Start with 12 students and have them come to the front of the classroom, the gym or even outside on the playground. You can have more than 12, it would just add to the inquiry because not everyone is going to fit into a group with the problem we’re about to solve. You might also want some hula hoops to help you mark your “groups of” later, but this can also be done with no equipment.

Write a statement on a board or piece of paper, we’ll use *3 x 4*, and tell students you want to see what they know about multiplication. Here are some things students might say:

- “I know you want me to do 3
*multiplied by*4.” - “I know the answer is 12.”
- “You want me to do repeated addition.”

Then have the students physically show you what that problem would look like, using themselves as the units. You want them to show you the idea of “groups of,” even if they haven’t said those words yet. Honestly, it might be better that way because saying “groups of” kind of gives it away!

As I’ve done this with students, it’s so interesting to watch how they process! The hardest part is not directing students on where to go. They have to figure it out on their own!! While they’re working, students might realize that they need 12 total, but they might not have the four groups of three. Instead they might have three groups of four. Stay strong and don’t point it out!

Ask the class if they agree that the way that the students are organized is representational of *3 x 4*? This conversation will be a great way to help that language emerge as students start to understand that, instead of saying *times* or *multiplied by*, the *x* actually means “groups of.” This will help them understand what multiplication really means, and it will be so much more memorable because they got to physically act it out.

If you look out and there are three groups of four, you might ask the students to try it again a different way. See how they react, and if they’re stumped, you can prompt them by saying, *Can you show me the way the statement is written – 3 x 4?*

We can continue the inquiry if we change the sample problem to *4 x 3*. Ask, *Would the answer change? *Some will say no and recognize that it’s still 12, some won’t. This is where we can start to see the commutative property come into play. We want students to understand rows and columns and arrays and all those great things that come with multiplication, and letting students play with this, switch up their groupings, will help solidify this principle. Side note: I wouldn’t go higher than four in a group, otherwise it gets a little chaotic!

The most important part of this activity is that you, and any other adults that are present, say NOTHING! Don’t swoop and save the students, let them have that productive struggle. Let them have conversations with each other and you’ll be able to see which students speak up and which students agree or disagree. This kind of engagement also helps kids communicate their thinking and promotes the idea of being a community of math learners.

Now, some of us are saying, *Sounds like a great idea, Shannon, but ‘m teaching virtually. How am I going to go about doing all that? *Well, the great part is, if you’re a part of our M³ (Molding Math Mindsets) Membership, we have presentations and videos that you can use in your (virtual) classroom right away! If you’re not a member, these resources are really inexpensive and you could get them really easily.

If you can’t have students acting out multiplication, the PowerPoint presentation does it for you so you could still go through this same activity on Zoom, or maybe even on Google Meet, and kids would still be able to understand the kinesthetic part of multiplication.

Have students look at the pictures of the 12 heads or the 12 students that we show in the PowerPoint, which is designed to provoke deeper thinking in the way that the students are arranged. In one picture for *4 x 3*, I might show three groups of four. To make this open-ended and inquiry-based in a virtual setting, as *What do you notice?* Some students won’t even notice and will say, *yeah, there are 12 kids there so that’s right! *Some will say, *There are three groups of four. That’s the same as four groups of three, so the way I see it on the slide is correct. *

On the slide that says *What do you wonder? *you really want to dig deeper and have them talk more about what they’re seeing. This is also the time to find out the misconceptions they might have. When doing this virtually, you have to be careful not to just give it away, but to really provoke kids to turn their camera on or type in the chat and engage. Maybe ask *What are you thinking about the way this is?* Making sure to point out that there’s not a right or wrong answer.

Of course, in the progression of the PowerPoint, we eventually show *4 x 3* (four groups of three) done correctly, and so we can, of course, elicit conversations about what it really means when you see that *x*.

We go through about four or five different problems on the PowerPoint so that we can really help students to be able to understand this concept. As you work through the problems together, clarify misconceptions as needed, but really remember that self-discovery is what is going to help students remember this concept the most.

As you start to see what students are understanding about what multiplication means, you might create a T-chart or anchor chart of their definition(s). Some kids might connect with the idea of repeated addition – *If I’m doing four groups of three, it’s like three plus three plus three.* Some students might realize that multiplication is the opposite of division – *If I had 12 and divided it into four equal groups I’d have three in each one. *Others will see that it means “to multiply” or showing multiples of a number.

We do want students to be able to use the correct language, which we show in the PowerPoint. What are the numbers 4 and 3 called in *4 x 3*? They’re the factors. What does that *x *mean? “Groups of.” What do we call the answer, 12? It’s not the sum, it’s the product.

The more that you demand that students use appropriate mathematical terms, and the more you use that language and model it, the more students are going to really start to understand those different terminologies.

As students deepen their understanding of multiplication and start using rows and columns to make area models, like we do in our multiplication journal, it’s really important that kids understand the commutative property. They need to understand that, if I’m looking at it one way, I have four groups of five, and the other way, it’s five groups of four.

Building Arrays is a really fun activity (check out our video tutorial here!) where students practice building arrays by rolling a die. The first number they roll will be the number of rows, and the second will be the number of columns in each row. Then, students will draw the array, using the pictorial, and then abstractly write the possibilities for a certain problem. So, if they rolled a 5 and a 4, they could look at the repeated addition. Students would know that it’s five groups of four, but it also could be written “four groups of five. This really helps students to just slow down a little bit before we start to get really fast into the ideas of multiplication.

You could help students with the idea of rows and columns by relating it to a football game or a concert. Do I want to sit in row 105 or row 1? And then how many are in my row? How many friends do I have with tickets in that row? Relating it to something like building an array will help them to understand it a little better.

In our next blog, we’re going to talk about how to bring multiplication to the concrete level as we bring in place value discs. Being able to do a kinesthetic activity like this, and then follow it up with concrete place value discs is so power for third graders! Don’t miss it!

]]>As I always say, the book isn’t your curriculum; the standards are your curriculum. The book, whether it’s Math Expressions, GoMath, My Math, Everyday Math, Ready Common Core, Eureka Math, etc., that was purchased for you is a really great resource that helps you build a common language for math instruction. Each of the series has great components that can help as you develop a curriculum for your school.

I also say that curriculum shouldn’t be stagnant. It should change from year to year. But when you open your book from last year, it looks the same as it did last year, doesn’t it? Does it have some of the guiding principles that you’re looking for with student engagement? Does it integrate the 8 Mathematical Practices? Does it help students to be able to construct viable arguments and critique the reasoning of others? Is it less about just getting into the direct teaching of the lesson?

At SIS4Teachers, we’ve done lots of curriculum development to help teachers have some kind of roadmap for their math curriculum as they apply the 8 Mathematical Practices and help students access the curriculum in various ways. Without a plan of some kind, we end up with a “binder series” kind of school, where one teacher has something from TPT, one teacher found something cute on Pinterest, another teacher stole an idea from someone else…although they’re all following the same standards, the glue of a common language to hold it all together is missing. Don’t get me wrong, I’m a huge fan of things being free. I’ve done tons of work with Eureka Math and Engage New York, and I love having an online platform that you can mold to meet your district’s needs.

For all these reasons, I’m so excited about the opportunity to do some work with Illustrative Math, from KendallHunt. You might have heard that Illustrative Math is for the grades 6-8 and grades 9-12, and they are known for that, but they’re releasing a K-5 series in 2021!

Each grade level contains 8 or 9 units that contain anywhere from 8 to 25 lesson plans each. Each lesson plan is designed to fit into a 60-minute math period. I know what you’re thinking – *yeah right, those “60 minute” lesson plans are totally unrealistic with actual 2nd graders!* But these lesson plans from Illustrative Math are the real deal! They give you flexibility because they offer optional activities that you can include for additional student practice or centers, or just on certain days when you have more time.

Some of the grade levels have a pre-unit practice within each section, which I think is really important. Sometimes, we get eager to dive right into the content of the lesson, but our students aren’t ready because it might have been a while since they used that skill. This pre-unit helps make sure we’re all ready at the same time.

Each of the lessons and assessments are aligned to center activities that support unit content and ongoing procedural fluency. Many of the activities they have require Blackline Masters and include recording sheets. There are some game boards and cards that might require some teacher prep (printing, cutting, plastic baggies), but a lot of the materials you need are already done there for you.

One of the most important parts of this Illustrative Math series is the design principles used to create a problem-based curriculum that really fosters the development of math learning with the community in a classroom. The 5 principles really align with what SIS4Teachers is all about, and the result is a resource that gives access to mathematics through a really cohesive progression, that provides an opportunity for students to get that deep understanding in mathematics, and helps promote student thinking.

The first guiding principle is that **students are capable as learners of mathematics.** This principle is really important in a world where the teacher is viewed as the giver of all the information in math. We know that students are capable of learning math, that they can make use of learning communities to make math meaningful and meet their own unique needs. Students can practice through equitable structures that provide experiences that are accessible to their particular grade level. The resource includes classroom structures that support students taking risks. It is really important for kids to take risks because, so many times, they don’t even want to raise their hand and engage for fear that they’re going to not be correct. The ideas of math discourse and productive struggle come into play in this “students as capable learners” idea, which are two of the main underlying concepts that we talk about at SIS, come into play in this particular design principle.

Another guiding principle is **learning mathematics by doing mathematics.** Within their learning communities, this looks like students learning mathematical concepts and procedures while engaging in the 8 Mathematical Practices. Can kids make sense of problems and persevere in solving them? Reason abstractly and quantitatively? Create viable arguments and critique the reasoning of others? Can they model with mathematics? Can they use the appropriate tools? Can they attend to precision and using the language that they’re supposed to? And can they look and make sense of structure and express regularity in repeated reasoning?

Remember, it’s not about you doing it as the teacher. We go through the math practices every day! But it’s about allowing the students to really engage with the practices with peers or others, which really helps them have the opportunity to see themselves, that their thinking is really worthwhile and their ideas have perspective.

The fact that this principle is incorporated into a program, instead of you having to integrate these practices yourself, is amazing!

The third one principle is **problem-based lesson structure**. Students learn mathematics as a result of solving problems. To support our students in productive struggle, giving them problem-based instructional frameworks to help them understand their mathematics is really important. At SIS, we talk a lot about performance tasks and three acts tasks – getting kids to have a higher level of thinking. The teacher’s role in this kind of framework is that of a listener, a facilitator, a questioner, a synthesizer. As we’ve talked about in so many past blogs, it’s about taking a back seat and becoming a co-learner with your students. Illustrative Math includes these tasks as part of their guides, which I think is awesome!

I also think that teachers can guide students in understanding problems by asking questions. We can help students think in a more productive way as they approach problems. Using question stems like *What did you notice? What do you wonder?* Is a pivotal piece of what we talk about at SIS.

**Balanced rigor** is the fourth designing principle. Those two words are really great together, right? The three aspects of rigor are really essential in math. You have conceptual understanding, procedural fluency and the ability to be able to apply the concept and skill in math problems in the real world. The NCTM talks about this in their Principles to Action where they state that interconnections support students’ understanding. Hello! This is our idea of CPA (*concrete, pictorial, abstract* or *concrete, representational and abstract*). If students don’t have the conceptual understanding, along with the procedural understanding, it’s going to be a misfit.

The materials in Illustrative Math really offer all three aspects of the rigor by helping students access the new mathematics, really engage in rigorous routines, and connect to new representation in math language from prior learning.

I think it really requires students to apply their knowledge and really helps them to understand it. There are specific grade level expectations for procedural fluency, which come with warm ups and centers and practice problems, but there’s also continued opportunities for students to really apply their understanding in situations to give them practice of the new materials that are being presented.

The last design principle talks about **coherent progression**. This is so important and I’m a huge advocate for this. At SIS, I talk about a “vertical zip,” which is the basis of the materials that support all learners through a cohesive progression of mathematics, both by the standard and research-based learning trajectories. If you don’t have this progression from grade to grade great, you’re on islands! The islands might look nice, but they don’t allow students to view mathematics as a connected set of ideas to make sense as they get older.

This kind of support helps to bring in students’ prior knowledge for the upcoming grade level work, and it’s also important for teachers to understand the progression of the materials. At SIS, we talk a lot about looking at the trajectories from grade to grade to help teachers see a flow and gain an understanding of mathematics and how it’s connected to the prior or the upcoming grade level.

Each unit begins with an invitation to mathematics. The first few lessons provide an accessible entry point for all students. We do this a lot with number talks, with games like Steve Wyborney’s SPLAT!, with an engaging Math Talk picture. The invitation to come to mathematics is something that we’ve always talked about at SIS4teachers. We want to give kids an entry point where they feel like they can get going instead of just jumping into the lesson.

Beginning with a warm up, which I often call a lesson launch, also helps students activate their prior knowledge. This part isn’t always about success, but just getting the ideas going or percolating, because they will then be followed up with the instructional activities which are an introduction to the new concepts, procedures or representations and making the connections between them.

The lessons end with a synthesis or coming together to talk about our learning goals, so each lesson includes a cool down that helps students apply what they’ve learned, which I love. Often, with a performance task or three acts task, we’re trying to cram in so much, but Illustrative Math has done an amazing job of integrating a lot of the important components of math instruction.

So, each activity starts with a launch to help students get to their task, followed by some independent work time for students to have what I call “productive struggle” with problems individually, before they work at small groups. And the activity ends with students coming together and really talking about their work.

Having a community in your classroom is something we talk a lot about in our SIS trainings. We want to create an environment where students feel safe, where they have a productive disposition about math, and are able to engage in mathematical practices. It’s really important us, as teachers, to start the school year by creating this safe math community that allows students to express their ideas. For most of us, that’s really hard to do because we didn’t necessarily grow up with that in our own classrooms. Additionally, kids can sometimes have an idea of how math looks and it doesn’t necessarily include conversation around mathematics.

Each of the units in each of the grades provides a lesson structure that helps us establish this math community, establish the norms and invite students to make mathematics accessible to them. Each lesson offers opportunities for students to learn, and really go more in depth with the math language as they become familiar with the curriculum routines. Being able to create a community is what is going to help get kids feeling really tied to the different parts of what’s going on.

As we know, instructional routines are what really creates a structure to help elicit the math conversations. They help students understand that there’s a predictable flow from day to day,.and that we have high expectations for learning. In the Illustrative Math materials, they chose a small set of instructional routines to really ensure that they’re used frequently enough to become truly routine. That small set of routines was also chosen carefully in this program so as to not overload teachers. You’re not a magician! You can’t make all of these things happen at once! So having a small set of strategies can really help teachers focus their energy on the structuring of the activities while students have thinking time and mathematical ideas play out. So each of the routines brings in a lot of the different types of parts to help build a collective understanding of the structure. These routines are aligned with the unit, the lesson, and the activity /learning goal.

If you can’t tell by now, Illustrative Math is such a great resource!! To add a cherry on top of this resource, it also includes videos of the routines in action for teachers to watch. So many times we wonder how is it even possible to get all this done, but now you can watch it happen in a real-life scenario! There are also professional learning materials for teachers to use to practice and reflect on their craft, or maybe use as part of a PLC.

This is a really great book (find it on Amazon) which talks about the ideas of creating task complexity, purposed representations, establishing teaching structures and practices, as well as teaching learning through curriculum materials, and then really being able to model with mathematics, K-5. And so, if you’re familiar with this book, I think that the ideas or the principles are great and they tie into this portion on what they have in Illustrative Math.

Parts of the Illustrative Math are currently out right now! If you go to https://im.kendallhunt.com/, click on the K-5, and you can see the teacher and the parent resources. As of the writing of this blog, I believe unit one is all loaded, but they will be intermittently loading all the different parts, as the full launch will be in 2021.

This program truly does mirror so many of the things we are always talking about with our schools, which is why I wanted to share the designing principles with you. I think they are so important, and it makes me so excited to be able to share this resource with you! To be able to have all this strategically designed, standards-aligned material available in a ready-to-use format that is free…it’s amazing!

Our next blog will really talk more about how the materials are used and some of the things that will help me understand the layout of how Illustrative Math has created lessons that I think that you’ll find helpful in your classroom.

]]>This is a theme for me this week as I’ve been helping my son, Connor (now in 6th grade), with his math. I watched him on his journey through elementary math, and I’ve been able to help with all kinds of things, but this week solidified for me the importance of the vertical approach to math strategies that we teach at SIS4Teachers in our Molding Math Mindsets training. I got to see, first-hand, that when students don’t have the benefit of a framework, or a vertical zip, in a school district, they might end up having holes in their math comprehension as they get older.

Model drawings, visual models, bar models, unit bars – by now you know this is one of my most favorite ways to teach kids how to do word problems. Understanding this whole approach helps relate back to the concepts that students should be learning to help them develop algebraic understanding.

The high school where we live has 7000 kids, and our middle schools are large as well, so we are still 100% virtual learning due to COVID (though we hope to be able to go back in-person in the second semester!). Both of my children are doing online learning at home, and I’ve been able to watch my kids’ teachers work their tails off, as all teachers are right now, to help make accommodations for what we’re facing with COVID.

It’s been a rare opportunity to be there with my son Connor as he’s learning math concepts in sixth grade, and it’s really helped me to diagnostically see where misconceptions of students’ mathematical understanding can really come to the forefront as they get older.

In elementary school, Connor did really well. He was able to keep up with most of the math concepts, but there was never really a connection of the concepts being taught from one grade level to the next, there was never a common language, which is one of the things we at SIS4Teachers think is so vital.

I made a connection today while helping Connor with his homework. As he was reading the story problems, I immediately knew they were working on ratios, but he was really struggling with some misconceptions. I realized that, if Connor had had a series of multiplicative comparison understanding when he was in 4th and 5th grade, and if he really understood the step-by-step visual model process, this concept of ratios would have been a piece of cake.

If we teach kids this early on in first grade, imagine Connor having that process reinforced year after year after year. Imagine him being able to independently use this process on tests to help him apply his learning. That’s something that Connor didn’t get in his school because they didn’t really have a cohesive way of doing word problems, there wasn’t really an adopted way to help students do this. And although he did fine in elementary math, we can see that the work that we’re doing in elementary will make a difference for students as they get older.

I’ve gone into 6th grade classrooms where students have grown up in their school system. They do visual models religiously with the journal that’s coded. Everyone uses a common language, whether you’re in 2nd grade, 3rd grade, 4th grade, or 5th grade, you learn the step-by-step model. As problems start to become harder, students are able to rely on this process as a way that can help them problem solve through different types of problems. I’ve watched these sixth graders say things like, “These ratio problems are so simple!” because they have the foundation of word problems and understand the concept of multiplicative comparison.

Today, we are featuring our Model Drawing video tutorial, and PowerPoint companion for multiplicative comparison so you can see how ratio problems are similar. **If kids could learn this process of reading the problem, it would be simple for them too!**

As you’ll see in the video, we use one inch square tiles to help students understand this concept. Quite frankly, that’s what Connor needed to help with his homework this week so he could understand what the words were saying.

One of Connor’s biggest problems was that it was difficult for him to not just plow through the problem and want to know exactly what the answer was, because that’s what most kids do. I had to force him to slow down and “chunk and check” a problem. As soon as he heard the ratio, I had him just stop for a minute – not think about the total, but to think about how many dark chocolates were there compared to white chocolates? How many green apples were there to red apples?

He had 12 or so word problems for homework, and he felt like he’d never get through because he didn’t understand the concept! But as soon as he agreed to my pace (SLOW!) it seemed to start to click. He put in his labels (white chocolate/dark chocolate, green apples/red apples), and then had to figure out how many one inch square tiles we would need to create the ratio. He needed to look at the details – were they talking about the ratio for comparison of green apples first? Or red apples first? That’s what the chunking and checking process is really about. (It’s also the first step in our Step-by-Step Visual Models Checklist! Download it here for free!)

As Connor continued to read the problem, I wanted him to think about where that total went. The total they give you – was it for the red apples? Was it for the green apples? Was that the white chocolate? Or was it the dark chocolate? Was it the white and dark chocolate combined? I needed him to understand, once he knew where those one inch square should go, where the total would go.

Connor can do simple division, and we know that ratios aren’t anything really hard in math, but when it’s applied in a word problem, kids panic because they don’t really know what to do with ratios.

He kept working through the problem. He told me, “There’s 7 total containers – 5 red and 2 green. I know there are 45 red apples total, so I’m going to divide – take the 45 divided by 5 boxes so then I know that each box is equal to 9.” So, then as he was figuring out how many green apples there were, two boxes worth, he said, “each of those boxes are equal, so there must be 18 green apples.” It was almost like a lightbulb going off! He realized what the problem was asking and really understood how simple it was!

Sometimes, the problem might ask a different question – something like* How many total apples are there?* Well now that Connor realized how much each unit bar was worth, he could figure that out pretty easily as well.

**It just hit me – with all the work that we put in to help kids learn this process – IT IS FOR A REASON!! **So many teachers say, “I don’t want to do the area model with partial products. I just wanna teach partial products because middle school just needs to do the traditional way so we’re just gonna do that.” I hear that a lot. But when kids get to polynomials and *X* becomes a value, it does matter if they understand the area model has partial products!

It’s the same thing with word problems. If kids went through the word problem process that we’ve talked about in past blogs, that we have in our store with our book, poster and Model Drawing Bundles, with the Bundles you have access to in the M3 Membership Library, if they understood the basics of part-whole addition, part-whole subtraction, part-whole missing addend, part-whole multiplication and division, and were able to do multi step, if they understood the idea of additive comparison, and the power of what we call multiplicative comparison, as they turn that corner in fourth to fifth grade into middle school, it IS going to stick so they can understand ratios and rates.

*Step-by-Step Model Drawing, *by Char Forsten, is a really great resource to use if you’re a sixth grade teacher. We also have an advanced model drawing book that takes these algebraic concepts to a higher level for older students. I also love our sixth grade model drawing book, because it’s filled with all of these things with fractions and decimals and ratios and rates – all the things that Connor is going to be learning this year.

Of course, I will continue to serve as Connor’s “virtual home para-pro” and help reinforce these concepts so he can understand math in a conceptual way, even if it might not be taught that way. It might not be being taught that way because we may assume that kids have these conceptual understandings, but we have to realize that math is all connected. When we look at schools today, it’s not about making sure each individual teacher has the same book purchased from the same publisher. It’s more about applying the strategies, which is where the Math Mights come in – can kids use D.C. with 8 + 5? And 0.8 + 0.5 =? It’s more about having concrete tools available with the Math Salad Bar – whether it’s virtual or physical. Get the Virtual Math Salad Bar template!

As I worked with Connor, struggling to understand ratios, it was powerful for me as an educator and as a parent, to be able to unlock that understanding for him and to watch him to start to look at the ideas of word problems differently.

One of the things he noticed was that he was really solving for *X*. We wrote the algebraic equation on the side a few times because I know what’s coming up next! We’re going to start to have x and y for numbers and algebra tiles are going to be needed – good thing I have some downstairs in our warehouse to help him!

As elementary teachers, sometimes we think about all the work we’re doing to help students get better, and unless you have a K-8 building, you don’t get to see the pay off of what happens in 6th, 7th, 8th grade.

So, for all the elementary teachers doing our application journals, using our videos, really trying to diligently implement visual models – **realize that your work is really laying a foundation for students to really have the understanding of math comprehension.** It’s important, and it’s worth it.

For middle school teachers, check out thinkingblocks.com, part of Math Playground. It has a whole section for ratios and proportions that can help students to understand this concept visually. I certainly am going to have Connor use it to practice more of these this week!

]]>Education has certainly been inundated with technology during 2020, and most educators have had to rewire their thinking about technology as they look for ways to deliver instruction through virtual learning platforms.

It’s funny to look back at the development of my Strategy Games product as education has shifted over the years. I vividly remember making this collection of games that brings in the eight mathematical practices. It was a really fun publication to work on, and was actually one of my first publications that I offered as a digital download. I had done a lot of things with hard goods, like the Counting Buddy and the Counting Buddy Book, and my Finger Funatics book, but when I decided I was going to produce my first digital games, I knew I wanted it to be strategy games. These games help students develop higher order thinking skills as they anticipate what their partner might do so they can stay a step ahead, and I felt they were a great (fun!) addition to the math classroom!

In those days, “digital” meant a disc. I remember actually burning the discs to make this product at the beginning! I would make the labels, print them on my printer, and one by one, I would stick a disc into the computer to burn the file on to it, then put the disc in a sleeve. Rinse and repeat…hundreds of times.

It got hard to keep up on the discs as the games gained popularity, so we ended up using a disc burner at one of our schools that would burn multiple CDs at a time. We did this for our Numeracy Screener and our Missing Parts screener that we do for first grade. Eventually, even that got to be too much, and I really felt like I needed some help to be able to get these done quickly! I finally went to a service called DiscMakers, and by then we got lots of discs sent to us.

Technology marches on, however, and laptop computers started hitting the market without CD players! I remember getting a Mac and having to get a portable CD player to even run any of my own discs! I realized this trend AFTER I ordered 1000 copies of all three downloadable goods, so, we had to figure out how to change the format.

At this point, we started to transition to a jump drive loaded with whichever product you ordered as something we could sell. It was small, people could use it for other things as well, and it was pretty universal. We were able to give jump drives away at conferences, and it didn’t matter what type of computer you had, you typically had a USB port to plug it into.

Fast forward to 2020. I just got a new computer and I no longer even have a USB port! I have USB C that seems to require 18 different conversion tools to get anything to run! Instead, we have cloud space, and we rarely see jump drives anymore.

The 2020 lens of technology evaluates everything on how well it can translate into a virtual environment. In education, we have the added challenge of trying to figure out how to get QUALITY things into our virtual classrooms while trying to minimize interruptions to instruction that come with quarantines and hybrid situations.

I realized the impact of this new 2020 technology lens this week when I was responding to a question from a customer who purchased the Strategy Games on our Teachers Pay Teachers store. Back in the day, we were careful to put (digital) or (CD) next to anything that wasn’t a hard good, and as time has gone on it just stayed there. So the product was titled *Math Strategy Games (digital)*. The question from our customer was, “I can’t figure out how my kids will play this digitally. It just looks like a PDF.”

That was when it hit me. These days, “digital” is usually used interchangeably with “virtual,” which we expect to mean interactive and suitable for online learning.

One of the biggest challenges I see in classrooms that I’m working with this year is that we can’t have kids playing games together because that means touching manipulatives and not social distancing. Because of this, a lot of the application and what I would call “the fun of math” has been left out of many math lessons.

So, I started to think about how I could take my Strategy Games and update them so that kids can still use them virtually, and implement the eight mathematical practices as they become critical thinkers.

The answer: Google Slides. If I was on a Chromebook, and you were on a Chromebook, we could open up the same Google Slides presentation and play together.

I bring to you four of my favorite games that get kids to think: Rotten Apple, 9-Holes, Across the Pond, and Checkmate. They’re not flashy with all the bells and whistles of some online math game platforms, but sometimes, I think plain is just fine and allows students to focus on using their strategies. We put the game boards inside of Google Slides and put “counters” right on the game board so that students could play like they would if they were sitting on opposite sides of the table. Just make a copy for each pair of students, drop the link(s) into Google Classroom, and they can play!

**Visit the resource page for each game to make a copy of the Google Template and watch the video tutorial to learn more about how to play!**

Note: the counters won’t move in Present Mode, but students can have the presentation open in Work mode and be able to interact with each other..

So now, when you’re doing a math workshop, you could still have a Math by Myself station, or a Math with Technology station, where students might be on Zearn or Dreambox or a different website that you use, but then you could actually have kids go to Math with Someone to play their games. Yes, it won’t quite be the same – they’re going to have a computer in front of them instead of hands-on manipulatives, which I prefer, but they will still be able to play! They’ll be able to apply the strategies, communicate and enjoy themselves through Google Slides!

Check out these four games and let us know how you’re implementing games in your 2020 classroom within the varying constraints of virtual, hybrid, or face-to-face instruction.

]]>In the past, the thought of “virtual manipulatives” always overwhelmed me a bit. Maybe this sounds familiar: I had a collection of great sites bookmarked, but never had time to go through them to figure out which tool to use where? Which tools worked with my interactive whiteboard? Would I be able to interact with students? Would students actually be able to use it? And then I would try to use the tools on my Mac and it said I needed to download plugins…and then I just gave up. *I’ll come back to it some other time,* I told myself.

Well, fast forward to 2020 and we’re hanging on by a thread in the education world. We’re grasping at every possible resource available to help kids learn – it’s time to take another look at virtual manipulatives.

Virtual manipulatives can do many things for students, but if you know me and my early childhood philosophy kids discover concepts by using their hands to interact with real objects in a physical world. Hands down, hands on learning is so important (no pun intended!) This is why we have our Math Salad Bar concept, where we have a manipulative library that is full of the manipulatives that correspond to math concepts being taught. However, we are in the 21st century and Marzano has done a lot of research looking at the parts of a child’s brain that light up for different activities that are done through technology. He has concluded the conceptual part of the brain can be tapped into by interacting with a visual image in a technological way. However, we still want to have a balance of using virtual manipulatives versus actual, physical manipulatives that kids hold in their hands. But with how we’re teaching today, we need a variety of tools at our students’ fingertips.

Just giving you a list of websites isn’t going to magically help you incorporate virtual manipulatives into your classroom. I have a few places that I think of as my *go-tos*, but instead of sending you off on your own exploratory journey, I’ve gone through my favorites, like ToyTheater, Didax, and the Math Learning Center apps, and we’ve organized our list of virtual manipulatives by concept, right alongside all the other great resources you’ve come to expect from SIS4Teachers.

When you’re in the midst of teaching, you don’t have time to search your list to find what you need. So we put all the tools at your fingertips, just like you would do with your Math Salad Bar items, and it’s as easy as that. Of course, we didn’t add every single virtual manipulative to our library – you might have some of your own favorites (send them our way!) – but you’ll be able to see the many uses and related resources for each of the most common virtual manipulatives we use.

Place value discs, for example. We can use them for things like adding, subtracting, multiplying and dividing. I know that I’d want to have those place strips for teaching place value (obviously), but we would also use the place value strips for addition with partial sums, and with how we do subtraction with T-Pops.

As you’re looking for virtual manipulatives, don’t miss one of my favorite parts of the new website – you can search! By grade level? Sure! By concept? Sure! Maybe you’re looking at teaching numeracy. Let’s click the numeracy icon so I can pull up the five- and ten-frames. Click the icon to get to the resource page, and there you can download related pdfs, see related videos or products. You almost have your own private guide to using these virtual manipulatives!

For place value, you certainly could use base-10 blocks. You could use place value discs and place value strips, and you’ll find tools that work for this concept in many different categories on our website: multiplication, division, addition, subtraction and place value.

For numeracy, the rekenrek, the abacus, the 5-frame and 10-frame, even multiple 10-frames are all great tools to use.

The Partial Product Finder is an amazing tool, one that we were just talking about in a classroom the other day, to help kids understand the idea of partial products, why we decompose, and how the product is actually made up of the area.

Students are working on multiplicative comparison problems? They might need 1” square tiles. You can put that image and put that link to that image, so students at home doing that work can grab those manipulatives.

Number bonds also have their own manipulatives – just plug in your numbers and go!

What about kids that don’t have a dice or don’t have counters? I love the dice tool that I included in the list because you can play with two (or however many you need) 6- or 10-sided dice, and then they can use different types of dice that help them to play games, or to interact with the math concepts. Roll three dice, and then see if students can virtually round the number to the nearest 100, or maybe find the highest sum or the lowest product, or even the highest product.

One teacher put together a virtual Math Salad Bar for her students. She created a Bitmoji virtual classroom for her students and put links to the manipulatives on the shelves.

Of course, you could also post the direct link to the manipulative into Google Classroom. You can select different types of maniuplatives that you would want your kids to use based on what you’re teaching and put it inside of your Google Classroom with the hyperlink. I think this is a great way to help students to be able to access manipulatives when maybe we can’t have them in person, as we would prefer.

Of course, I never let go of the fact that I would much rather have the students using these tools with their hands. But the ability to show something conceptually in math is quite important, and virtual manipulatives make it possible for students today to have that imprint in their brain to understand mathematical concepts in a concrete way.

]]>I designed the Counting Buddy nine years ago as an organized way to have students use a math manipulative without getting out a ton of counters. I felt like I lost the mathematical concept I was trying to illustrate whenever students started playing with a bunch of counters – knocking them on the floor or making stacks instead of doing what I wanted them to do.

The Counting Buddy is a macrame beaded tool that is one of my favorites that we have in classrooms. I even have one on my golf bag for the times I’m really off on par! Counting Buddy Jr. is five beads of one color and five of another. The Counting Buddy Sr. has 10 of one color and 10 of another. Most teachers will put 30 Counting Buddy Jrs in their Math Salad Bar for Preschool and Kindergarten. First grade and second grade teachers use the Counting Buddy Sr.

The idea for the Counting Buddy came while I was on an airplane ride to Chicago for a presentation. I didn’t have any paper, and so I used the barf bag from the back of the seat in front of me and started drawing the idea for what it could look like. I knew I wanted it to be cute, friendly for kids, and that I wanted it to have the macrame beading to make it easy to use, but then I had to figure out all the different parts and see how I could make my doodling actually come to fruition. It’s funny to look back at this original design and see how we started, and then to look at what the Counting Buddy is today.

At the time, my cousin was doing a lot with bracelet-making so I asked her to make a few different prototypes. We ordered lots of different doll heads, and the Counting Buddy has definitely evolved over the years. I wish now that I had saved all the different types of heads, because as we look online, whether it’s at JoAnn Fabrics or on Etsy, or on Amazon, the different doll heads over time, go out of stock and I need them to have the right size hole drilled in them to move them to be big enough. Some of our heads have been larger doll bead heads that were different colors one time. we had a blue head and a pink and the yellow head, those are really cute. We ended up then having a little bit smaller of a head, but it was different ethnicities and so that was kind of fun to have for students just to see the doll heads resembling more of themselves. We have kept pretty consistent with our large rounded bead head with the smiley face on it, to kind of be the picture of the Counting Buddy.

The Counting Buddy has a plastic clip at the top, above the head, because I thought kids could clip it to either a notebook or their backpack and it would be a great way for them to be able to hang on to it. We looked all over and found the company, America’s Plastics in California, so we could get the clips as cheap as possible.

It took us a long time to figure out what was going to be the best lacing. We ended up using a black athletic lace because it had a nice smooth feel to it as the beads are moving on it in a macrame beaded fashion, they really slid nicely for students. Also, it didn’t fray. Now, we buy this lacing in bulk. We have a lacing company that will create seven grosses of lace at a time for us. Putting the tabs on the end of the shoelace is the most expensive part, so this company will just let the machine run all evening without cutting the lace or tabbing it, and we end up with large spools of lacing.

And then we ordered little bare feet to go at the bottom of the Counting Buddy, which are really hard to find. I ended up having to go and get them from China and have them sent to us. They’re actually charms that could go on a necklace like you might see in P.E. where students earn footprints for however many miles they’ve walked.

The hardest part of the Counting Buddy, however, is finding the larger pony beads. Pony beads are sold in the smaller version everywhere, but we needed 11 or 12 millimeters to be able to make the bead, big enough for a small hand to move in a macrame beaded fashion. Ironically, I have found most of the beads at bird shops or African American hair suppliers. We’ve worked with several different companies to try to get the beads in, and we just got some really pretty pastel colors and primary colors as well!

Now that we had these materials, meticulously sourced so we could keep the cost down as low as possible, we had to figure out how to actually make the Counting Buddies. I was presenting every day and had a family with small children, so there was no way I could add Counting Buddy Elf to my job title!

We ended up recruiting some high school girls that played on the basketball team, asking them if they wanted to make some extra money to sit around and make Counting Buddies while they watched TV. When the girls were done with a batch of Counting Buddies, they’d bring them over and we’d pay them. We joked that our neighbors probably thought we were drug lords – I mean, high school kids bring us brown paper bags crammed with stuff, and we paid them cash.

Honestly, though, it was one of the best factories we could come up with as we went. Now, Pat Carpenter, an aunt of ours that is just amazing, works with her husband to create them for me. Read more about her in this blog post here.

With all the different virtual conferences we’ve been doing, I must say this has been one item that we struggle to have in stock. I hope one day to be able to mass produce the Counting Buddy to get the cost down even lower and be able to get it out to more schools.

**As a fidget.**I could present and I could sit in the class and move the macrame beads back and forth, and not really be destructive with my attention.**Writing sentences**. Early on, when kids are beginning to write, they really struggle with recording their thoughts onto paper in a way that says what they want it to say. When kids are writing, they can use the Counting Buddy to count how many words are in their sentence. “I went to the park. They could pull over one bead for “I”, another bead for “went”, another beat for “to”, another bead for “the” and another bead for park. Then, when the teacher asks how many words are in your sentence, you can respond confidently!**Ear Spelling.**A child can write down the word “went” and begin to sound out the word. They can pull a bead across for the /w/, another for the /n/ sound, and the last bead for the /t/.**Phonemic Awareness.**I can say “say say sit, say sit, say the sounds you hear,” and set a kid to pull over the beads to say the /s/ sound, the /i/ sound and the /t/ sound. Similar to elkonin boxes, this is a great mnemonic device to help kids associate the sound they’re hearing with the word.**Playing with Sounds.**Take the word “lip,” have kids sound out the word on the Counting Buddy, and then ask if they can change the last sound they heard to a /t/. That would be the word “lit.” The Counting Buddy is a great physical reminder of which sound they’re going to switch and what the new sound will be as they’re really working on understanding phonemic awareness.

Of course, we use our Counting Buddy a whole lot in our numeracy talks as rekenreks. We clear all the beads to the right, over to Counting Buddy’s head, and then push beads to the left to have kids read the quantity.

The Counting Buddy Jr. is used in our yellow level of numeracy talks, our conservation to 10 series.We also use the Counting Buddy Sr. in the green level of our numeracy talks where we are talking about conservation to 20. If you’re part of our membership website, download all three levels for free, or you can get the freebie before you buy the product in our store.

There were so many uses for a Counting Buddy that ended up writing a book containing 50 activities!

Sometimes we just don’t have time for a book, though, so I also created a video tutorial that walks you through the basics of using a Counting Buddy and then demonstrates four activities using each level of Counting Buddy. There’s a free download for each level that you can use to help you remember some of these new ideas.

One of my favorite activities with a Counting Buddy is to push all of one color beads towards the head and the other color towards the Counting Buddy’s feet, basically separating the colors on either end. We’re always talking about part-part-total with students, and this helps them visualize all the parts of a number. Let’s take seven, for example. Kids can pull over five of one color and two of another, meeting in the middle to display their amount. Then, you can ask to see the same number a different way, so a child might pull over four of one color and three of another. This activity would correspond well with number bonds, so kids can see those different parts of that number.

“I Wish I Had” is a fun game to play, where you can display an amount on the Counting Buddy (let’s say four), and the say “I wish I had seven.” Then the kids have to figure out how many more you would need to make the number you “wish you had,” which would be the missing addend.

Another of my favorite activities on the Counting Buddy Sr. is being able to help students really understand the idea of making 10. We clear the beads towards the Counting Buddy’s head, push eight beads down to the left, towards the feet, and then leave a little bit of space to add five more beads. You can see this idea in action in our D.C. video. In this activity, students can see that there are 10 of one color and 10 of another, so instead of putting eight in their head and counting on, one by one, to go up, we want kids to see what “making a 10” really means. Saying that phrase might not mean much to a student, but when they can see it happening on the Counting Buddy, it makes so much more sense. They can see that, as they’re adding on, the beads are the same color. It’s a lot easier for me to push these two beads up with the other eight because they’re all the same color. Be sure to bring in the correct language here – you want to *decompose* the five into two and three, bring that two in with the eight to make a 10.

Check out the video that we did on D.C. to show that, as well.

This is just the beginning – we’ll continue to do some more videos on the Counting Buddy to help you find different ways to use it in your classroom! In the meantime, check out our video and download your free activity cards to help you remember what you learned today!

Don’t have Counting Buddies yet? You can get a class set, which comes with a download of our book for free!

It’s about that time of the year…

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?

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.

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