It’s hard for me to believe that I just finished writing the 112th Math Mights show! My husband, Scott, and I just finished editing the next 24 shows so that we can go away on spring break! We’re super excited to bring to you another 48 episodes of the Math Mights show on PBS Detroit Public Television and the Michigan Learning Channel.

# Teacher's Guide

Episodes 311-312

**April Focus: Mystery Math Mistake**In our first eight shows per grade level, we did a numeracy or number talk. Our second run of eight shows featured Professor Barble and a visual model for a word problem. This set of eight shows will showcase a Mystery Math Mistake for our warm up. I love the graphic and animation that our animator created to show how all the Math Mights get their strategies all mixed up! Students have to be detectives to see if they can find the mathematical error as we work through the problem. You’ll have great fun watching the shows as students start to look with a critical eye to see if they can spot the Mystery Math Mistake!

### Kindergarten

**Focus: **311: Match Equations to 10-Frames / 312: 10-Frames Different Ways

**“I Can” statement:** I can match equations to 10-frames. / I can fill 10-frames in different ways.

**Extension Activity: **Represent the Equation with 10-Frames / 10-Frame Shake

In Kindergarten, we start with our Mystery Math Mistake, which features a delicious word problem about a plate of cookies. If there were 6 cookies on the tray, and someone ate 2, how many were left? You’ll have to be a detective to see if you can figure out where Mrs. Gray made an error while solving this. I bet Jordan and Carly, the two kids on this show, are going to help steer her right!

As we start getting into the main part of the lesson, we do a lot with those great phrases: *What do you notice? What do you wonder?* We show kids three different modalities of numbers put together to match equations. There are 10-frames, but we also have snap-cube towers, as well as the Counting Buddy Jr. that has five beads of one color, and five of another. We start off with what students notice within these three different types of models that they’re seeing. At the end of the day, all of the models equal 10, but they are composed of different numbers – 7 and 3 on the 10-frame, 8 and 2 on the snap-cubes, and 5 and 5 on the Counting Buddy.

We then start working with students being able to see an equation and see if they can match it to the 10-frame. We can do a process of elimination for students, but they see the equation with 10 = 7 + 3 and they have to decide if that 10-frame matches. This really helps students with their part-part-whole understanding, getting them to understand that 10 is the total, and it’s made up of two parts that are denoted by red or yellow counters.

We then flip over to a different tool by using our Counting Buddy Sr., which is one of my favorite tools that you can use to show different parts to numbers. Some people clear it by pushing the beads up to the Counting Buddy’s head. But another option is to put half the beads towards his head (10 beads) and half of the beads towards his feet (10 beads). This way you can pull the beads into the middle to come up with different combinations, which we do in this episode. Then, you can draw in the two parts of the 10.

We’ll show different equations on the Counting Buddy Sr. – 6 + 4, 3 + 7, and so forth. In the end, we want kids to talk about what they see is similar or different about the tools that we’re using in the show compared to a 10-frame, or even a Counting Buddy.

For the extension activity, it’s the students’ turn to represent the equation with the Counting Buddy Sr., so we actually have a spot on the extension page where students can fill in their different combinations as they’re looking at it, and then matching the Counting Buddy to the equation that they’re working on.

In show 312, we’re also leading with a Mystery Math Mistake. This time, however, we take a story problem that’s very similar to the one in the previous show about cookies. In this problem, there were 4 cookies on the tray and 2 more came out of the oven and were added to the tray. How many cookies are there now? I wonder if you can see where Mrs Gray’s mistake is…Did she add or subtract to find out the answer? Our friends on the show, Jordan and Carly, are going to be a great help for that.

Our “I can” statement is I can fill 10-frames in different ways. We have four different pictures (A, B, C and D) that we’ll show students to ask them which one doesn’t belong. These all are different structures of 10-frames with a different variety of red and yellow counters, some filling the whole 10 frame, some not. Kids will talk about which one doesn’t belong.

The best part of this activity is that each 10-frame, as they look at it deeply, might not belong for one reason or another. We pay particular attention to a 10-frame that looks like a 10-frame, but is actually two 5-frames. Students should be able to see how two five frames really does represent seeing 10, even though it’s two separate parts.

We then play a really fun game called the 10-Frame Shake. Students are going to shake 10 two-sided counters and spill them onto the table. They have their 10-frame mat and a really great recording sheet where students can write in their 10-frame, write in their number sentence and then write in their number bond.

In the last portion of show 312 for Kindergarten, we work on* how many more* counters are needed to fill the 10-frame. So we show 7 on the 10-frame and students can see that there are three empty. They get to match it to the equation 10 = 7 + 3, 10 = 8 + 2, or 10 = 5 + 5. As students work through this with Mrs Gray, they’ll start to see that they can look at the empty spaces to help them to be successful with this concept.

### First Grade

**Focus: **311: Compare Lengths / 312: Compare Indirect Lengths

**“I Can” statement:** I can compare and order objects by length. / I can use a third object to find longer and shorter objects.

**Extension Activity:** Compare the lengths of objects / Create comparison statements

We start episode 311 by working on the Mystery Math Mistake, but in first grade, we’ll look at it a little differently than in Kindergarten. For this problem, we’ll be revisiting a topic from a previous show (310) where we added 10s and 10s and 1s and 1s. Value Pak is helping with this problem, but is all upside down, confused and turned around, so he solves incorrectly. Students have to use their magnifying glasses to study the problem and see if they can figure out where Value Pak has gone wrong.

As we begin this length unit, students are going to be presented with three different pencils. The pencils are actually the same length, but they are positioned to start at different places. By asking students what they wonder and what they notice about the pencils, we give them a taste of what the lesson will be about. The students in our show, Dennis and Han, talk about why they think the pencils might be different lengths based on how they look. We also ask which pencil they think is the longest (trick question, remember!). This leads into a conversation about measuring things from endpoint to endpoint, and why that will help you be more accurate when you measure.

Next, we look at a pencil and a crayon and we create comparison statements about the two objects. Students will learn to use phrases like “longer than” or “shorter than” to describe different objects of length. Then, we add in a highlighter and snap cubes to spark even more conversation. For example, “Comparing the highlighter to nine connecting cubes, which one is shorter?”

We had so much fun comparing things that there’s a deleted scene from this part of the show! You can watch it here to see even more examples, show it in your class to let someone else do the talking for a few minutes, or even send let students watch it at home for extra practice! In the scene, Mrs. Markavich talks about how we can look or think about the size of your foot and your pointer finger and make comparison statements about objects that are shorter or longer.

In the episode, we continue to look at different objects, like scissors and a hole punch and a stapler, and decide if we can order them from shortest to longest. Then we take fettuccine noodles, a straw, and one base 10 block, and see if we can order it from longest to shortest. We also talk about those words of “taller” and “shorter” as we turn things around differently, and it helps us to think about those descriptors that show length.

Mrs Markavich does a great job talking about the idea of a starting line, like you’re going to start a race, but you’re going to start at the endpoints and everyone is going to begin in the same place.

For the extension activity, it’s the student’s turn to compare the length of different objects, just like we did in the show.

As we move on to episode 312, we do another Mystery Math Mistake, again with Value Pak. This is similar to the previous episode, but we want to see if kids can apply the strategy of Valu Pak to find errors.

Continuing our unit on length, we show two sets of objects, and ask *What do you notice? and What do you wonder?* The same green pencil is used in both comparisons as that length to measure. Students talk about the pencils being different lengths and that they’re lined up by endpoint, but they have lots of wonders about those objects. We want to hear statements like “The purple pencil is longer than the green pencil, but the blue pencil is shorter than the green pencil.” or “The blue pencil is shorter than the purple pencil.” These types of statements that can compare to a third object are really important.

Then start to compare objects that we might not be able to measure or physically put next to each other, but we can use with yarn or string to help us compare these large objects. For example, we figure out the length and width of the dry erase board we use on the show, and ask *which is longer?* By using the string to measure one of the sides, we can compare it to the other to see which is shorter or longer based on the comparison piece, which is the string.

This idea can apply on a larger scale too. We talk about different ways that kids get to school – bus, riding in a car, walking, etc. So, using this idea of a third object to compare, by looking at the map, we try to decide if Clare or Maya’s house is closer to school. By using the same string to help us measure the distances, we can figure it out.

We do a lot with the extension activity, where students can compare objects using yarn or string to help them to figure out how to create some of those comparison statements that we talked about in the show.

### Second Grade

**Focus: **311: Halves, Thirds, and Fourths / 312: Fractions with a Whole

**“I Can” statement:** I can partition circles and rectangles into halves, thirds and fourths. / I can make halves, thirds, and fourths different ways, and discover a whole.

**Extension Activity: **Split the Shape / Split the Shape, Version 2

In episode 311 for second grade, we also do a Mystery Math Mistake, bringing in an upside down, all turned around D.C. He’s really struggling with decomposing numbers to make a friendly number, and so we want to see if students can look at this in an inquiry-based way and discover the error. Xanda and Kelly are the students on this show that help us to see where that error might be.

This show begins our look at fractions! As we’ve done in different episodes before, students will look at four different images and decide which one doesn’t belong. The important part to notice here is that the shapes are partitioned in different ways, but they’re not always in equal ways. As we introduce fractions, we want kids to understand the idea of equal parts. In the show, we take a rectangle and fold it into two equal parts. We take that same rectangle and show how we could fold it into thirds. Then, we also do a non-example of thirds, which I think is really important for students see. As you’re folding something, especially into thirds, students’ spatial awareness skills are really a struggle, and so even if they’re folding something like a letter, it might not be into equal parts. Then we talk about folding something into fourths.

The second grade standards want kids to know that halves represent two equal parts, thirds represent three equal parts, and fourths represent four equal parts (which we can call quarters, another vocab word to work on!).

Then, we take this idea into circles and talk about which circles are not examples of halves. Students will see three different pictures of circles that are fractions, and students have to decide which ones are not examples of halves, fourths, or thirds. Kids must be able to look at the way things are divided and decide if they are equal. In the show, Clare and Diego are asked to divide a circle into three parts, and we have to evaluate whether their circles are partitioned equally.

Split the Shape is the extension activity for students to play with a friend. They’ll have a spinner and rectangles, squares and circles that must be partitioned into halves, thirds, and fourths, depending on what the player rolls.

As we move on to 312, you guessed it – another Mystery Math Mistake! Our friend D.C. has gone wrong in his arithmetic, and Xanda and Kelly have to find his problem!

The show begins with two rectangles and the request for them to be partitioned into quarters, or fourths. Then we ask students to shade in 1/4. The girls in the show are thinking of partitioning the rectangles in two different ways, so we talk about the different ways that you could partition a rectangle that would still be equal, but would be able to allow 1/4 to be shaded. We go through the same process with squares – looking at how to partition and shade one half of a square in different ways. The goal is to get kids to really understand why equal parts are important, especially as we look at fair shares.

We try to give scenarios to help students understand that, even though the size of the piece looks different, it still could be equivalent or equal to the fraction that we’re talking about. In our scenario, Diego’s dad makes two square pans of cornbread and slices it up for the family. Diego’s little brother feels that his piece is smaller than his brothers’ – his brother has a perfect square, where Diego has a long strip. It’s still divided into fourths, but the size of the piece looks a little bit different.

Of course, you can’t really talk about fractions and dividing things equally without dividing up a pizza! If you have a pizza, and you have friends over, how many pieces is each person going to get? Obviously, we have to know how many friends there are. If we were to have a pizza and share it with three friends, with each person getting a slice, they would get ⅓ of the pizza or the pizza would be cut into three equal parts. We would do the same thing if we had four friends coming over. We might cut the pizza into four slices and each friend would get 1/4.

Pies are another classic fractions visual, and in this episode, we match word problems to pies. Some of the pies are full and cut, some parts of the pie are empty where somebody has eaten part of it. If Noah ate most of the pie, but he left a quarter of the pie for Diego, can we find the pie which represents the problem to apply what we’re learning in the show?

For the extension activity, we play Split the Shape, Version 2! In this version, students are going to partition circles, rectangles and squares into different parts, based on the spinner that they’re using, and see who can get their parts shaded first.

### Third Grade

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**Focus: **311: Fraction Number Lines / 312: Locate Unit Fractions

**“I Can” statement:** I can learn about fractions using a number line. / I can partition number lines to locate unit fractions.

**Extension Activity: **Create and partition number lines / Partition, locate, label fractions

Episode 311 for third grade begins with a Mystery Math Mistake on a concept students have studied in a previous show. D.C. is looking at 18 x 9 and is a bit confused with his decomposing and composing as he’s making his area model with partial parts of partial products. Trevor and Marcus are going to help Mrs. Askew figure out where the mistake is and get D.C. back on his feet!

For this episode, we’ll continue our work with fractions that we began a few episodes ago. In fact, we have 16 shows on fractions for third grade because it’s such an integral part of the students’ success as they move on to fourth grade.

Our “I Can” statement for this episode is: I can learn about fractions using a number line. For the beginning of the show, we’ll have two number lines for students to look at. One of the number lines is partitioned starting at zero and then going to 10, and the other one begins at zero and goes to 1, with a tick mark in the middle. Students notice different things about these two number lines – one of the big things we want them to see is that the quantity of the number line differs based on how it’s partitioned. We look at what we think that tick mark might represent as it’s halfway in between zero and 10 and the other one is halfway between zero and one.

We then take different number lines, and find different ways that we can sort them. This just gets kids comfortable with the idea of taking a number line and separating it into fractional pieces. One student says that the number lines have tick marks at the whole numbers, so they might look at sorting those where they see a number line start at zero and end at six, and there’s no actual fractional parts in the middle but they’re actually looking at just whole numbers. The other ones are a fraction number lines that have just fractions in them at the tick marks.

A common point of confusion for third graders is being able to make a number line proportional. It goes back to the spatial awareness thing! So we spend time in the episode looking at how students can create their own number lines. We look at where half should be labeled on a number line, or how a person might think of half as a number line divided into two half parts, just like you would with fraction strips. Many of the common misconceptions are addressed while we’re creating number lines in this segment.

We spend the rest of the show really looking at number lines and figuring out how to fold them so we can make different number lines. We make a number line that’s in halves, fourths, eighths, thirds and sixths. Then we compare those to the fraction number strips that we made in the previous show.

For their extension activity, students are going to create and partition number lines. It’s really important to make sure that kids go slow here to make sure they understand the parts. Many students, if they’re trying to divide a number line into fourths, will put in four tick marks. In the show, we stress that the fourth tick mark is usually on the whole number, depending on how you’re dividing it. So, if you want to partition your line into fourths, you would only need to add three lines. This helps them make that connection that’s needed.

In show 312, we get another Mystery Math Mistake! We’re doing 15 x 3, and D.C. just cannot seem to get it right. The boys help set him straight to make sure he has all the parts in his area model to do his multiplication.

Our “I Can” statement is: I can partition number lines to locate unit fractions.

Unit fractions? What are those?? Often, math books talk about unit fractions, but students don’t really understand what they are. We know that a unit fraction always has one in the numerator, and then it has a denominator. Think of a unit fraction as a fraction that describes the pieces that we’re talking about. If I have something divided into thirds, the unit fraction we’re talking about would be ⅓. If I had something divided into six parts, the unit fraction would be 1/6.

To get students thinking about this, we start with four images to figure out which doesn’t belong. Students have to look carefully at the number lines and fraction strips to see where things are labeled and have them pay attention to the details in these. We really want to make sure that students can clearly locate and label fractions on a number line, because there are important things to include: partitions of equal parts, a dot, a label for the fraction.

To apply this concept, we have three friends that create a number line – Clare, Andre and Diego. We want to have the number line partitioned into fourths. It is really helpful when kids can look at things that maybe aren’t perfect and bring their reasoning to why they agree or disagree with the way somebody has partitioned the number line. We have some people that feel that the number lines are done exactly right. A lot of these number lines are no longer ending at one but they’re extending to two. And so we go through and kind of have an evaluative look to see how these number lines are really created.

We then start practicing partitioning number lines, locating and labeling the fractions. We label 1/3, we label 1/2, we label ⅛. Sometimes, a fractions number line might end at four, not just at one. Students should know where 1/8 is and how you’re not going to find 1/8 appearing between one and two, and two and three, because 1/8 is only between zero and one. This will help the kids to get that relationship.

Since we represent fractions three different ways – fraction strips, area models, and number lines – we want to ask students *which way do you like to look at fractions?*

Of course, the extension activity is to partition, locate and label fractions on the number line, so that students will have lots of great practice with their new skill!

I think your students are really going to like our Mystery Math Mistake! It’s a really fun spin on math that helps kids really get interested in trying to find the error. If you want to take it a step further in your classroom, have the students create their own Mystery Math Mistake! If you want to check out more on Mystery Math Mistakes, visit our SIS4Students page to see a whole week’s worth of problems that we did during the beginning parts of COVID. See if you can spot the mistake and let us know!

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