Month Two of the Math Mights Show!
For the first four weeks, number talks were the focus of the warm-up for the shows. As we move into shows 303 and 304, we’re switching our focus, and introducing Professor Barble, a Mathville citizen that loves to solve story problems, to help us warm up our math brains. In fact, Professor Barble really helps students conquer word problems by using visual models. The goal of visual models, also known as model drawings, unit bars, tape diagrams, and lots of other names, is to create a picture that helps students understand word problems. It helps them go slow to go fast!
We’ve talked about word problems with visual models in many different blogs and have so many resources available (check them out here!), and so it is wonderful to see all that come to fruition in the warm-ups for these episodes that you can use right in your lesson!
February Focus: Word Problems
In these warm-ups, we’ll use a step-by-step visual model process, which will vary slightly depending on the grade level and what type of problem that we’re working on. Professor Barble helps students slow down, think about what the word problem is asking, and organize the information it conatins before they jump right into solving it. Yes, we even do this in Kindergarten! See sections below for more specific information about how word problems and model drawings are used in each grade.
“I Can” statement: I can compose and decompose numbers to five. / I can compose and decompose numbers and show 2 parts with an expression.
Extension Activity: 5-Frame Shake / Shake Those Discs!
In Kindergarten show 303, we’re going to be working with the “I Can” statement: “I can compose and decompose numbers to five.” In our warm-up for this show, we’re going to be using Professor Barble and our Kindergarten journal template, to do a part-whole addition problem. On their Math Work Mat, students will actually act out the word problem with Mrs. Gray, create that quick draw that we’ve been practicing in the previous episodes, but also show it in different modalities by doing a quick model on the 10-frame, showing a number bond, and finally doing the computation.
It’s important here to notice that at the bottom of the page, we did not put in the addition sign or subtraction sign. We want students to go through that process of determining the operation on their own. We still do the chunking and reading the problem, and create that sentence form as kids understand it, however, it’s very scaffolded so that Kindergarten students can be successful with this.
Then, we invite students into this lesson by looking at two pictures of diamonds in different arrangements. Asking students, What do you notice? What do you wonder? really helps kids activate their understanding of what we’re trying to accomplish in the lesson. Students can notice different layouts of the diamonds, seeing five together, but also seeing five spread out into three and two. They can wonder why the shapes are laid out that way. And we create an addition sentence like 5 + 0 = 5 or 3 + 2 = 5 to match the pictures.
We start off the main part of the lesson using linking cubes for a train activity where we have a bowl, and students put some of the cubes in the bowl, then we have to figure out how many cubes are in the bowl. We used a bowl because we don’t have actual students in our studio, so you could certainly play this game where a student puts a few of the cubes behind their back instead! If I have three unifix cubes in front of me, and there was a total of five, how many of them are in the bowl? This helps kids understand the part-part-total of decomposing the number of five, seeing it in an algorithm, as well as being able to do it in a number bond. By playing this game, we show that you can create five in multiple ways. If 5 is the total, maybe 2 and 3 are the parts, or maybe 5 is the total with 4 and 1 as the parts. You’ll hear how our fictitious students in the show are thinking as they decompose the number 5.
Towards the end of the show we play one of my favorite games, called 5-Frame Shake. This is a fun activity where students put five two-sided counters in a cup or bowl, shake them out, and then make the different combinations of five on a five-frame mat, again, using the number bond and number sentence to go with it. If you’re at home, you could do this with pennies and use heads and tails instead of the red and yellow of the counters. This game is also the extension activity so students can play at home with parents or in a classroom in a Math with Someone station.
As we move into show 304, the objective is for students to take what they learned in the previous show and apply it: “I can compose and decompose numbers and show 2 parts with an expression.” We use Professor Barble as a warm-up in a very similar problem, this time using lamps to get kids to understand the part-part-total, but more so using all the components of the kindergarten journal: a quick draw, the 10-frames, the number bond and then the computation.
In the warm-up, we do a little bit of a numeracy talk by showing kids sets of two-sided counters and asking them How many do you see? How do you see them? We’re using a total of six here, but we want kids to see this in an unstructured way, where it’s not always in the 10-frame. They might see a scatter or a dice/Domino pattern. Using two-sided (red/yellow) counters, we show three and three, equaling six, we show four counters that are red and two counters that are yellow where the set of four looks a lot like a dice pattern, and then we do another one with five and one. We’re having students use their numeracy to apply this concept of part-part-total, but also asking them to see what is the same and what is different. All of those pictures equal six, but they’re composed in different ways.
In this episode, we play a game called Shake Those Discs, which is really similar to the 5-Frame Shake, but with more counters. We want students to be able to have the expression and represent it in a drawing. So, if we put seven counters in a cup, shook it, and then got two yellow and five red discs, could we do the drawing? The students would draw the first two circles, and then put the partition line before drawing the additional five to show that two and then five would equal seven. Obviously, students could use colors (the yellow and the red) to show the two parts of the expression, but we want kids to understand how that drawing might be created.
We go through several different examples of Shake Those Discs, and we play a round with eight counters. Maybe I have three red and five yellow, and I write down the expression 3 + 5 to match it.
We also talk a lot about how two expressions can be represented with the same amount of counters. If I have three yellow and five red, I could also show five red and three yellow. In first grade, we call this “flip flop backs” because the numbers are just changing spaces, but the total is the same and the numbers are the same. We want kids to understand the commonalities between part-part-total as they look at it.
We don’t always have room for everything we plan in our PowerPoint to be in our final show, and this is one of those times! Members, download the PowerPoint for this show because we talk about different ways to make seven with the discs.
As an extension activity, kids can play Shake Those Discs. We suggest playing with nine counters, but this game is easily differentiated so that any student can play. Ultimately, we want students to create the expression to match the discs they shake out!
“I Can” statement: I can add tens and ones to two-digit numbers. / I can add two-digit numbers and find matching equations.
Extension Activity: What Did I Add? / Four-in-a-Row
Professor Barble kicks off episode 303 for first grade! We start with a scaffolded journal for first graders. Obviously, we’re presenting this in the second half of the school year, so if students were doing visual models, hopefully they’ve been doing them with both proportional and non-proportional models. In this show, however, we decided to introduce visual models as we might do for students seeing it for the first time. We do a part-whole missing addend problem with proportional manipulatives or proportional bar so you can see the individual units.
I think the most important thing here is to get kids to NOT solve on the visual model! We see that happen so often in first grade! It seems like first graders also like to guess I think we should add! Or I think we should subtract! But we want them to actually look at the model, fill in the information that’s needed and put the question mark above the section we’re looking at. We want them to play close attention to the visual model checklist (download one here – in English AND Spanish!) which outlines the seven steps of the visual model process that we use in grades 2-5. This checklist breaks down that process to be more developmentally appropriate for a first grader to follow the steps with the teacher.
We go through checking and chunking, making sure that we have these pieces of information in the word problem that are chunked out, checking those off and putting them into the visual model BEFORE we compute and just jump to solving. As always, we provide the sentence form as well.
In the meat of the lesson, the engagement looks a little bit like a numeracy talk, where we have 10 frames showing 30 – three 10 frames that are filled completely with red counters. We ask students How many do you see? How do you see them? Obviously, you see 30 and you see full 10-frames, but as we change the image, we leave that 30, and then we add more on. You could think of this as an extension of the numeracy talks, but working with higher numbers here. If kids understand that conservation to 10 within a 10-frame, can we now extend that knowledge to see three full 10-frames and another 10-frame with three red counters and two yellow counters and know instantaneously that it’s 35? Or 30 + 3 + 2?
We then extend that further where kids see three full 10-frames, and then a 10-frame with five red counters and four yellow counters. Can they tell us how many we see? Enter Value Pak, the character that clicks together and shows their values on their belly. If I do see three full 10-frames, it’s three groups of 10, but the value of that is really 30. And then if I were to see another 10-frame with nine in it, I know that total is 39.
I think the visualization of the quantity is a really important connection for students to make, and then to also relate that back to what it looks like in the actual number form. So, seeing 30 + 9 in the digits will really create that number 39.
We play a game called What Did I Add? This game can be a little bit complex when you first look at it, but it’s a great game to play with kids! Partner A flips over a number made with place value strips – say, 54. Partner B draws a secret number to add to the original number. But the catch is that the secret number can be 10s or 1s! Partner B draws a 3 and can decide to add three 10s or three ones to 54, and then tells Partner A what was added. 54 + ? = 84 And then Partner A has to figure out what was added. It’s a great inquiry-based game to play for students to apply this concept of adding 10s and 1s.
For the extension activity, students get to play a spinner version of What Did I Add? to practice applying the concepts we learned in this episode.
In episode 304, we continue to work with Professor Barble and proportional bars at the beginning of the show, doing part-whole subtraction. In the later shows, we will eventually use a non-proportional bar, so this will be a springboard for students to understand this concept.
Posing an inquiry-based question that gets kids to think differently about how other students solve problems is a great way to point their attention toward what we’re focusing on in the lesson. In this particular warm-up we start off with Ming and Keshawn sharing their answers for 5 + 34. One student thinks it equals 39. Another student thinks it equals 84. This student made a common error when solving this type of problem – they stacked up the problem and lined up the five in the 10s column. Instead of adding 5, they actually added 50. We want kids to discover that the placement, or the place value, of numbers actually makes a difference when you’re adding.
In this show, we look at different ways that you could add two-digit numbers. Pay close attention to our sample problems, for example 23 + 45. Students often start with the number that begins the number sentence instead of actually thinking about which number is more or less. When we’re solving these problems, we show it with both place value blocks and then place value strips. One student starts with 23, adding in the 10s and then the 1s, where another student looks at it from the perspective of starting with the larger number. We can see that it’s less work to actually add by starting with 45 and adding two 10s and then three ones. We want students to notice the efficiency of these methods and equations, and explore what is the same and different about them. We also have students look at base-10 ways of seeing numbers so they can discover which equations match.
Finally, we do a somewhat complicated activity with drawing where a student adds two numbers together and has to analyze which equations actually match. In this particular problem, they’re adding together 63 + 25 – what are all the different equations that they could make with that one problem? You could do 63 + 20 = 83, and then add in the rest by adding 83 + 5 to get 88. This helps kids to analyze how they’re going about solving the problem, and really be able to break down the equation.
The extension activity for first graders to work on adding two-digit numbers is a Four-in-a-Row game. The kids will apply their understanding of how they can add two two-digit numbers and then apply their learning to see if they can get four in a row before their partner.
“I Can” statement: I can locate, represent and compare three-digit numbers on a number line. / I can use place value to compare three-digit numbers.
Extension Activity: Plot and Compare / Compare with Value Pak
In Episode 303 for second grade, we’re working specifically on helping students understand model drawings or visual models with Professor Barble. At the beginning, we’re not necessarily solving a problem, but we’ll have a picture of Professor Barble with a word problem already drawn out in a visual model and students will guess what word problem goes with the visual model.
I can’t stress enough the importance of kids having lots of fluid practice with this! Yes, everyone can break down a story problem, but can they look at a drawing and create the story problem it was based on?
For this warm-up, we do a several missing addend problems where students have to look at how Professor Barble created his visual model, and then come up with what that visual model says. For students who haven’t used visual models, this is a great introduction to help them understand why visual models are important.
Our “I Can” statement is “to locate, represent and compare three-digit numbers on a number line.” When we look at place value, we often just look at 100s, 10s and 1s, but kids use number lines for so many other things (fractions and higher level concepts), that we need to be sure to include number lines for place value.
We have three different number lines in the opening activity to get kids interested. There’s the number line written from zero to 10, another number line that ends at 100, and another number line that ends at 1000. All the number lines are the same length and the tick marks all match up, but really the values of those really depend on the total number that is at the end of it. We ask What do you notice? What do you wonder? about the number lines and see if students can locate the number three, the number 30, and the number 300. To do that, you have to look and analyze each of those number lines to see what each tick mark is actually worth. The first number line, the tick marks are worth one, in the second number line, the tick marks are each worth 10. And the third number line has tick marks worth 100 each.
We look at different number lines where we have located points on the number line. One number line might start at 620 and end at 630. Students have to understand that it is going up by one, and try to figure out where the number would be located on the line.
We end the show by comparing numbers. We build them with base-10 blocks and plot them on the number line so kids can put those two pieces together. When using the number line to compare numbers like 371 and 317, we can gauge which number is greater by looking down the number line – numbers to the right are higher numbers. This is a step further than just using the base-10 blocks like we have in the past.
So we have several different examples where kids are going to do that, and in the extension activity, students will plot and compare three-digit numbers to see if they can apply what they’ve learned in the show.
In Episode 304, the word problems with Professor Barble are similar to the previous episode, with the drawings presented to see if kids can come up with what the word problem actually said. This time, we’re focusing on additive comparisons, trying to really make sense of what those types of word problems are asking. This is quite tricky for students because they often don’t understand that the algebraic X is the same value. Even though a line in a drawing might be longer than the other, students may not realize that the X represents a constant value in additive comparison problems.
We’re continuing our learning about place value in this episode with the “I Can” statement of “I can use place value to compare three-digit numbers.” We have two people who have a quantity of base-10 blocks, and we’re trying to decide Who has more, and how do they know? We want students to look at the quantity of place value blocks in each place to see how many there are, which really helps break down what the number is. We look at the hundreds, at the 10s and then at the ones. Oftentimes, kids look at quantity over value. So, if they were to see two hundred blocks, two 10 blocks and then see some ones, versus seeing one hundred block, four ones and a variety of 10s, a kid is instantaneously going to think that the number with two hundred blocks in it is larger. This show takes into consideration that we can rename numbers different ways and that it’s not okay just to look at a quantity, especially just the hundreds blocks, because, in this case, there might be more than ten 10 bocks and so kids need to see that maybe those numbers are actually the same when they compare their values.
We also use the place value discs to address this concept, looking at it in a non-proportional way, which is important in second grade. As we go across, when the hundreds are the same in the discs, the 10s are the same, but the ones are different, they can break down those numbers in that way. Of course, we use Value Pak as a way to help students understand that we’re looking at the values of numbers.
As an extension, students get to play a comparison game with Value Pak, where they build different numbers and find out which one is greater and which one is less.
Focus: 303: Dividing with Larger Quotients / 304: Representing Division: Part 1
“I Can” statement: I can divide with larger numbers within story problems. / I can make sense of representations of division.
Extension Activity: Division Problem Journal / Representing Division Problems with Base-10 Blocks
In the third grade episode 303, Professor Barble presents a visual model for students to try and guess. The warm-ups will be on the topics that we’ll use in the next segments of the show, so it’s really just getting students’ feet wet by looking at a visual model. These problems in the warm-ups for third grade are focused on multiplication visual models. Professor Barble will have his drawing, out from his starting line, looking at a diamond collection, and there will be bags with seven diamonds in each. We want to see: Can a student construct a word problem based on that picture? We give them a couple of opportunities here to start to get that language of what word problems are looking at, which will eventually help them to actually do visual models. The “I Can” statement here is “I can divide with larger numbers within story problems.”
Anytime you can set the scene for a word problem for students, whether it’s with a small video or even a picture, it really helps for students to understand it. This one starts off with Maria’s class splitting up into groups to go on a field trip to the aquarium. What do you notice? What do you wonder? I’ve been to the aquarium before! So, what are things that you’re thinking about? I see that they’re splitting up into groups. I wonder how many kids are going in each group? So, we pose the question: There are 48 students going on a field trip to the aquarium, they’re put into four groups. How many students are in each group?
We ended up being able to integrate the step-by-step visual model process for what Professor barble is doing with this idea and topic, even though we’re just in the introductory part of word problems with our show. It really fit nicely to have students go through the steps with this story problem, and then get at the concept of how to divide. We end up having 48 students, and there would be 12 in each group.
We use place value discs here to help students use the “multiplying up” strategy to figure this out. How many groups of four are there in 48? We can start off with the idea of at least five groups of four are in 48, which gives us 20. We want to build up to get to that 48, so we put in another group of five groups of four, to get another 20. Now we’re up to 40, with 10 sets of 4. We’re almost up to that 48!. We do two groups of four to give us eight, which gives us that target number 48. Then, it’s easy for kids to see: add up the five and the five and the two to get 12 groups of four.
We originally wanted to focus on three different ways to divide, but obviously we didn’t have enough time! (M³ Members – don’t forget to download the PowerPoint!) Shows sometimes get cut short, and so the big take home for this show is that they can use place value discs to find fair shares with division. Also, we wanted kids to start to understand that strategy of multiplying up.
In order to do a division word problem for the extension activity, we gave the students a journal, similar to what we use in the show, which walks them through the step-by-step process we want them to use.
In episode 304, we again do a visual model with Professor Barble, who already has the problem ready to go and ready for students to guess the word problem that goes with it. This time, we also do a little bit of multiplication and division to see if kids can pick out those parts. Our “I Can” statement is “I can make sense of representations of division.”
We have blocks to show how someone’s already solved the problem. So I can see three equal groups built with base-10 blocks, and in those three groups, I see 13. The key here is Can a student tell me what the division problem was based on the way I’ve separated it? This gets kids to look at the total, to see that there’s 39 altogether, and then how we put them into three equal groups. It’s the opposite of how we would look at it in a different way.
Next we practice doing 55 ÷ 5 with base-10 blocks where students can actually see 55 base-10 blocks and ask, how we could go about dividing those? Obviously we can do that with fair shares, like kids are used to doing, and we can even extend that knowledge as we start to look at 65 ÷ 5, by doing those fair shares.
In this show we want students to connect how you can break blocks apart, but that eventually doing fair shares almost becomes inefficient. We even take a hundred block and divide it by five, where kids figure out how to cash in that 100 block for 10 10s and are still able to divide. But then as we start to realize, when we get to something like 90 ÷ 15, this becomes less efficient. It’s not like I’m going to take one 10 and then break a 10 into five ones and continue to do this.
This is why we want students to focus on the idea that we introduced of “multiplying up.” We have this target number of 90. We’re trying to figure out how many groups of 15 are in 90. Students that don’t know their multiplication facts well can always use this strategy because you can anchor to lower numbers. Do they know what 15 + 15 is? Well if I know that, I know I can get two groups of 15 to make 30. Another two groups of 15 is another 30, and I’m at 60. I’m trying to get to 90. Another two groups of 15 is another 30. So if I add that up – two groups, two groups, and two groups – I know that six groups of 15 equal 90.
We want students to explore this idea and represent division, so some of the problems we know will work well with fair shares, But, as a third grader, when is it time to start to use multiplying up as a strategy? Not every book uses this particular strategy, but I find it really beneficial to help students understand the concept of multiplying up, because oftentimes, kids go into that traditional algorithm and are taught that really quickly, and don’t really fully understand that concept.
If you’re an M³: Molding Math Mindset member, remember that you can download these presentations! How great would it be to have visual models all mapped our, ready to show in your virtual or face-to-face classroom? Maybe you show the actual show, or maybe you just want to download the PowerPoint and alter the numbers in the problems to fit what you’re doing – either way, they’re all yours to file and use!
(valid M³ Membership login required)