## Grade Nine and Ten Math Integrated English Language Development: Math Explanation

### Introductory Slides (00:00–03:26)

**Narrator**: Welcome to the California Department of Education. Integrated and Designated English Language Development Transitional Kindergarten Through Grade Twelve Video Series. Mathematics with Integrated English Language Development in Grades Nine and Ten. In this unit, the students have been analyzing graphs of quadratic functions and determining how they are used to model and solve situations involving maximizing profits and area to projectile motion. In this lesson, the students will rewrite a quadratic function in vertex form to find the highest point of the graph without having to actually graph the function. At the end of the lesson, each student will contribute to a group discussion to explain the steps of completing the square describe the new formula and justify why the formula works.

**Narrator**: The California Common Core State Standards for Mathematics Driving the Lesson. The mathematics standard is Higher Mathematics Standards, Functions: Interpreting Functions, Standard 8a: Analyze functions using different representations, where students write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. Students use the process of factoring and completing the square in a quadratic function to show the zeroes, extreme values, and symmetry of the graph and interpret these in terms of a context. Watch for how this California standard is addressed throughout the lesson.

**Narrator**: The Supporting California English Language Development Standards Used in Tandem with the Mathematics Standards. The English Language Development Standards at the Bridging Level are: Grades 9–10, Part 1, Standard 1, Exchanging Information and Ideas, where students contribute to class, group, and partner discussions sustaining conversations on a variety of age and grade-appropriate academic topics by following turn-taking rules, and asking and answering relevant, on-topic questions, affirming others and providing coherent and well-articulated comments and additional information. And Grades 9–10, Part 2, Standard 3, Using Verbs and Verb Phrases, where students use a variety of verbs in different tenses and mood appropriate to the text type and discipline to create a variety of texts that describe concrete and abstract ideas, explain procedures and sequences, summarize text and ideas, and present and critique points of view. Watch how students move from early levels of proficiency toward the Bridging levels of these English language development standards throughout the lesson.

**Narrator**: Watch how the teacher leads the students toward accurate expression of their math content knowledge by providing opportunities for them to listen to the language, speak with their peers in triads with assigned roles using learned language structures and vocabulary, read and analyze academic text, then write a mathematical explanation using the appropriate structure and language features to explain the mathematical concepts learned.

### Teacher Introduces the Lesson and Students Work in Pairs (03:27–06:24)

**Teacher**: So again, just to repeat, our learning objective is going to be able to explain the process of completing the square and try to find an efficient way of doing it using math. Okay.

**Student 1**: [In Spanish] Uno por 24 es 24 obviamente.

**Student 2**: Ahh... [inaudible]

[Students writing on their whiteboard]

**Student 1**: [In Spanish] ¿Sabes cómo hacer la 2 verdad?

**Student 2**: [In Spanish] Creo.

**Student 1**: [In Spanish] En la 2, lo que yo hago es como quitar 2x. ¿Sabes cómo hacer esto?

**Student 2**: [In Spanish] ¿Aver cómo?

**Student 1**: [In Spanish] Seis más 6.

**Student 2**: [In Spanish] Ah, no se.

**Student 1**: [In Spanish] Bueno ves que hay un 2x. Entonces necesitamos 2x's para enseñar esto. Entonces necesitamos uno que se multiplique por 36 pero cuando se sumen, de 12. Okay ¿me entiendes?... Entonces lo que hago es que sumo por 36... y que nos dé… y cuando se sume nos dé 12. Como… ¿como qué número se multiplique que nos da 36 y cuando se sumen los dos números nos da 12?

**Student 2**: [In Spanish] Umm… número que nos da….

**Student 1**: [In Spanish] Lo que yo hago, mira ...

**Student 2**: [In Spanish] 16. Mmm no. 18.

**Student 1**: [In Spanish] Hay que acomodar los múltiples de 36. Hay que acomodar yo los divido por 2. Y como el 2 no funciona porque nos va salir un número decimal. Entonces hice el 3, 4, o el 5 o 6. Entonces yo lo que hice aquí utilicé el 6. Seis por 6 son 36. Pero cuando lo sumas son 12.

**Student 2**: [In Spanish] Ahhh....

**Student 1**: [In Spanish] Osea mira. Pones aquí el 6 y el 6. Ya necesitamos ver si son todos positivos. Como aquí todos son positivos, entonces estos dos van a estar así, positivos. Entonces vamos a multiplicar 6 por 6 son 36. Como dijimos. Seis por x son 6x. X por 6x son 6x y x por x es x a la segunda. Entonces todo lo sumas y nos va dar el mismo resultado.

**Student 2**: [In Spanish] Eres muy científico. [inaudible]

**Student 1**: [In Spanish] Trata de ser la segunda.

**Teacher**: You guys are doing great. I love how you're helping each other.

### Whole Class Debrief (06:25–07:11)

**Teacher**: All right class, you guys did a great job so come on back to me. So, let's share some of our answers. I'm just going to call on a few of you if you don't mind. So, looking at number one. We have a rectangle here with an area of 24. Dylan, what did you put for possible side lengths?

**Student 2**: I got 6 and 4.

**Teacher**: Okay. So, we could have one side that's six and another side that's four. How did you know that these would work?

**Student 2**: Um…

**Student 1**: [whispers] Multiply.

**Student 2**: I multiplied them and got this. [pointing to his whiteboard]

**Teacher**: Okay good. So, they're going to multiply and they're going to give you that area, okay. As we know, the length times the width is going to give us our area.

### Setting up Expectations of Roles for Group Work (07:12–10:15)

**Teacher**: Okay. Now before you start choosing roles, I just want to go over these, right here. Now it says, "please keep these goals in mind while working with your group." You want to explore different ways to think about a pattern. You can use drawings, diagrams, charts, whatever you want. Discuss your actions with your teammates. So, as you're going through the exercise you want to talk and share what you're thinking. You also want to try to understand what your, the other teammate is thinking as well. So, if they're doing something that you're not sure of, ask them a question. Could you explain this for us a little bit better? Make sure you're sharing all of your ideas and steps. Use mathematical language to discuss your process. So that's kind of using algebra to try to discuss it. And that's going to be towards the end. We're going to try to put an equation or a formula to the process.

**Teacher**: So, we have a few different roles. Currently, we are in groups of three which is perfect. So, the time keeper, you can just knock that out. Don't worry about that. We're gonna have a facilitator, a scribe, and a speaker. So, this just gives you kind of an outline of what's expected of you. I'm going to pass out this sheet as well and this kind of gives you an idea of how you should communicate in that role. So, questions you could ask your team depending on the role you have chosen. So, the facilitator encourages group to understand each other's way of thinking. Make sure everyone is on task and doing their best. And takes on roles that are unfilled. So, you're gonna kind of watch the time and make sure we're up to date with everything. The scribe does all the writing on the assignment. And you have to use everybody's input. Okay. So, you want to be the final say on how we want to word our answers. And then the speaker is going to share what the group is thinking if, if called on. Okay. And also, the speaker is the only one who can ask me a question. So, if someone in your group has a question, you need to pass it along to the speaker. And the speaker can ask me the question and call me over. So, I'm just gonna go ahead and give you about a minute. So, go ahead and choose your role and write your name down next to it.

**Teacher**: So, think about your roles. Think about the sentence starters here especially the facilitator. So, the facilitator can kind of start the discussion and ask each member what they think here. My question is, just kind of loosely answer the question, is there a specific way that you have to split the x's? Okay. So why don't you guys go ahead. I want the facilitator to make sure that both of the other two answer that question. So just take a moment to air that out and make sure we say it out loud.

### Students Work in Triads (10:16–10:44)

**Student 1**: Actually, you just have to count this and this and you multiply it by itself.

**Student 3**: 10. 25. Like that.

**Student 1**: Okay, so why like that. There's 5 over here, right. There's 5 over here. So, the best way to just multiply it and this is gonna be the same.

**Student 3**: Uh-hmm.

**Student 1**: So, you multiply this. 4 times 4 is gonna be 16. 5 times 5 is gonna be 25. So, it's more easier.

### Teacher Set up the Next Activity Whole Group (10:45–12:38)

**Teacher**: So, what we're going to be doing on the next activity is very similar. Okay. But we're going to be using the box method to try to help us figure out the bottom. And again, keep in mind we want to try to make a formula. Maybe some of us already I heard some people saying, "Oh I think I have it already." That's great. Let's work towards that and share with our— the people at our table. And try to make sure that we all can understand what you're beginning to see. Okay, so here comes the next one.

[Teacher hands out paper]

**Teacher**: Now the first couple questions on here you need to answer with a complete sentence. So, I want to make sure we're discussing together and kind of putting the same or similar answer here. And again, keep in mind your roles and the sentence starters for your conversations.

**Teacher**: So, B for number 1 here asks us, "What are the dimensions of the square formed by the one tiles?" So, we had talked about what the dimensions or side lengths are for the whole square, the big square. But if you notice, we also have a square down here with our constant term, right? This makes a square as well. So, I'm asking you, what are the dimensions? So, what are the side lengths here? Okay and the side lengths here. And try to see if there's a relationship between all of them that you've built, okay. And see if we can put that into words. Does that help a bit? It could get a bit messy folks, but that's okay. So, let's just try to apply the same ideas that we've noticed, the same pattern that you've noticed and try to apply it when you get to the back side.

### Students Work in Triads (12:39–17:54)

**Student 1**: It wouldn't work.

**Student 3**: I don't understand this right here.

**Student 1**: What don't you understand? I've never really used the box method so I don't know what this is.

**Student 1**: Well, let's try to add your way. How do you do it?

**Student 3**: I am just guessing.

**Student 1**: Well, actually you… this will be like x plus one and x plus one. But then it wouldn't give you 3x. Look, try it. Try that. Uh, it wouldn't, it wouldn't work. So, we need... So, I was, what I was thinking was putting negative numbers, you know. Negative numbers.

**Student1**: [In Spanish] Pruébalo aver poner los negativos números acá porque no se puede dividir en dos. Porque si lo dividimos en dos, sería como 1... 1.25x. Entonces necesitaríamos un número negativo.

**Student 4**: [In Spanish] Podríamos sumar.

**Student 1**: [In Spanish] No creo que podamos.

**Student 4**: [In Spanish] Si dijo el maestro que podemos.

**Student 1**: [In Spanish] Ahora solo pon el punto en medio y .25.

**Student 1**: It says how do we use b to find c? How do we use b?

**Student 3**: Hmm.

**Student 1**: We divide it first right.

**Student 3**: Two.

**Student 1**: Two. If you divide it and then we'll…

**Student 3**: Multiply by 2.

**Student 1**: There you go.

**Student 1**: [In Spanish] ¿Qué hicimos con el 7?

**Student 2**: [In Spanish] Lo dividimos.

**Student 1**: [In Spanish] Lo dividimos. Y como se, ¿cómo qué hicimos con el 10.5?

**Student 2**: [In Spanish] Lo multiplicamos por 2.

**Student 1**: [In Spanish] ¿Por dos?

[Teacher talking]

**Teacher**: I think about some of us are just about getting done. So I am gonna give you about five more minutes on this and then we will bring it back to discuss what we found.

**Student 1**: [In Spanish] Lo multiplicamos por el mismo. Ahora ponlo acá en inglés.

[Students writing]

**Student 1**: We divide. We… Ponle, we divide... We divide... by 2.

**Student 2**: [In Spanish] ¿Qué se va ser hacer aquí?

**Student 5**: Facilitator, what do you think the goal is?

**Student 6**: Well basically under like two, when he was writing on the whiteboard, we're trying to find the ones.

**Student 7**: That's right.

**Student 6**: We'll do what with the ones?

**Student 6**: Find it to put it in the formula.

**Student 5**: Ok so...

**Student 7**: That will fit in the...

**Student 5**: How many ones do we need to make it a perfect square?

**Student 7**: Oh okay.

**Student 6**: Yeah, we're trying to find the factor of 4.

**Student 5**: See how many ones we need to make it a perfect square and turn it into a factored form. Okay. You happy?

[Students writing]

**Student 7**: See how many… ones are needed… are needed to be able to make the perfect square period. There.

**Student 5**: You happy?

**Student 6**: We gotta find a reason for dividing it. Oh, because you have to put the x's on, on the side of the x squared. Um, parallel.

**Student 5**: Parallel. Why do you divide the linear coefficient? The linear coefficient is that… Uh, why do we divide it the way we do? Because we're trying to make a perfect square.

**Student 6**: No, but that's our final solution.

**Student 7**: Yeah.

**Student 6**: Giving a final solution doesn't mean why we divide it.

**Student 5**: No, for b it says why did you divide the linear coefficient the way you do?

**Student 6**: I don't know like...

**Student 5**: Because we need to make a perfect square.

**Student 6**: Our formula was x, x divided by 2 squared.

**Student 5**: Yeah.

**Student 6**: They're asking why we divide it? Like what happens to when we get it?

**Student 5**: Where do you see that? Where do you see that? Where do you see that question?

**Student 7**: Isn't it this one.

**Student 5**: Read that. Read that question for me?

**Student 6**: Why?

**Student 5**: Read that question. Read that.

**Student 6**: Why did you divide the linear coefficient?

**Student 5**: So where are getting this formula stuff from?

**Student 6**: Because it's asking what you do.

**Student 5**: It doesn't say anything about the formula. It's just asking why do we split the... why do you split the... x's in half?

**Student **6: Yeah, why?

**Student 5**: Yeah, because we need to make a perfect square. What if we, if there were 8 and we did 3 and 5, it wouldn't be a perfect square. It would be a rectangle, which would ruin the whole project pretty much. But if we split them in half, then you get a perfect square.

**Student 6**: I know, but like...

**Student 5**: Yeah, yeah.

**Student 6**: It's asking why we divide it.

**Student 5**: Yeah, you'd get a perfect square.

**Teacher**: OK, I am gonna stop you folks in one minute, in one minute.

**Student 6**: Wait, we divided the coefficient.

**Student 5**: We're not talking about the coefficient. Oh, linear coefficient.

**Student 7**: Yes.

**Student 5**: Which is the axis.

**Student 6**: OK. Got it.

[Student laughs]

**Student 6**: Dude, I was so tripped out.

### Whole Class Share Out and Closing (17:55–21:21)

**Teacher**: So, I'm just going to hop around and see some of our findings from this activity. Okay we're gonna, um, we're gonna start with this group over here. So, the question was, "What was the goal?" Okay or the end product? What were we trying to do? And I need to hear from the speaker right. Oh. What was the end goal for us when we were trying to complete the square as the process?

**Student 8**: We were kinda looking like at the area of the square.

**Teacher**: Okay the area of what?

**Student 8**: The square.

**Teacher**: Good. So, our goal here... is we want to find... Say it again.

**Student 8**: The area.

[Teacher writing on the board]

**Teacher**: Okay, good. So, we want to find an area of the square, and specifically when we were completing it remember the one thing that's missing is the c value. So specifically, we're looking for this missing piece of the square. But in order to figure out that number, we want to make sure that the end result is going to be a complete square. So good. Perfect.

**Teache**r: Okay, now question number two, thank you. Why do you divide the linear coefficient the way you do? Okay. What's the purpose of that? We're going to go to this group right here. Jose is going to share it for us.

**Student 9**: To divide the coefficient into two in order to distribute equally.

**Teacher**: Wow! So, did anyone else talk about dividing it by two? If you guys wanted to show your hands real quick. Okay, good. So those of us got to it and we had mentioned that a little bit earlier. So that's beautifully said. We're going to divide it by two.

[Teacher writes on the board]

**Teacher**: To keep it... Say it again. To keep it equal on both sides. Remember we're making a square here. We want to make sure that it's equal on both sides. Question C then, "How do you find the constant term?" How do we then find the constant term and Claudier looks, oh Claudier looks eager and just so now we can go ahead.

**Student 5**: Okay so you divide the x term in half and multiply by itself or you square it.

**Teacher**: Ohh. So, you divide the x by 2.

**Student 5**: Yeah.

**Teacher**: And then you square it.

**Student 5**: Yeah.

**Teacher**: Okay. So, did uh, I think I want to go with this group here, Thalia's group.

**Student 10**: One half b squared.

**Teacher**: So, our c, our constant term could equal...

**Student 10**: Parentheses one half b.

[Teacher writes on board]

**Teacher**: So, if you take one half of b and remember we're talking about b is going to be our linear coefficient here. Okay, one half b and then what?

**Student 10**: Squared on the outside.

**Teacher**: Squared. Okay, we're just gonna wrap this up. Okay, because time snuck up on us. That struggle you guys had was so beautiful to me. I loved it. Okay. I love when you struggle because that's when learning is happening, okay. It's not because I don't want you guys to be happy. But the end goal, when we finally figure this out on our own, it is so special. It is so much better that way than me just up here telling you.

### Closing Slides (21:22–22:38)

**Narrator**: Reflection and Discussion. Reflect on the following questions: First, how did you observe the following focal content standards and supporting English language development standards being implemented in this grade nine and ten integrated English language development lesson? Interpreting Functions: Standard 8a. English Language Development: Part 1, Standard 1, Exchanging Information and Ideas; and Part 2, Standard 3, Using Verbs and Verb Phrases. Second, what features of integrated English language development did you observe in the lesson?

**Narrator**: Now pause the video and engage in a discussion with colleagues.

[pause]

**Narrator**: The California Department of Education would like to thank the administrators, teachers, and students who participated in the making of this video. This video was made possible by the California Department of Education in collaboration with WestEd and Timbre Films.

Questions:

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Last Reviewed: Tuesday, June 18, 2024