>> SPEAKER: Welcome.
Welcome to our second web site, to our second Webinar of the alternate eligible content series, the session is entitled increasing academic expectations with the alternate eligible content closer look
at math.
If you would like to it access previously recorded webinars from our winter and spring series you can find them on the alternate assessment partner page located on the PaTTAN web site. Each Webinar is
designed to build on information, from each webinar. We encourage you to access the other Webinars as needed and look forward to today's presentation where we take a closer look at math.
When you received the link for today's presentation, we also provided 3 handouts which are PaTTAN publications, we'll be referencing them in this presentation. If you didn't get a chance to down load
them, all of the information that is contained within our Webinar you'll find within some of these publications.
If you have questions during the Webinar, for content ?? you can contact the Alternateassessment@pattan.net please reference today's date. If you have questions following the Webinar you can send them
to this web address, please reference the date.
For tech support you can contact us at support@pattan.net. I'm going to give you a few seconds to write them down in case you may need them.
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Over to your left, the green number 5 arrow, points to your volume and pausing and play recording.
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URL.
Which is support@pattan.net.
Today's learner outcomes ?? we will take some time and review the basic components of effective instruction.
We talked about these in length at our January webinar. We will also spend some time looking at the content we discussed in February in regard to reducing vocabulary complexity, and communication. We
will identify and talk about a methodology of math instruction, that looks at presenting mathematics from a concrete representational and abstract perspective.
We will also take a look at number sense and the connections to the alternate eligible content.
Finally, we're going to describe some strategies to teach addition, subtraction and multiplication using models, visual representations and area maps.
Before we start, we want to let you know that we do read your answers to the survey questions.
And we do take seriously the learning that you are, as a participant, getting throughout this series.
So when we see something that we want to make sure we Clarify, we're going to make sure to bring it up at the beginning of our webinars. In this particular case, we want to make sure that we were
very, very clear on eligibility and the eligible content. We thought we would take a look at it through a lens of the students who are able to take the alternate assessment and students who take the
PSSA and the keystone exams.
Students who take the alternate eligible content are students with disabilities who are identified as having significant cognitive disabilities.
But that's not the only requirement. IEP teams determine the eligibility or the assessment. The student is determined by their IEP team to be eligible for alternate assessment then their upper edge
and boundary of learning of learning targets is the alternate eligible content.
If students with disabilities, IEP teams determine they will be taking the PSSA and keystone exams, the eligible content for the PSSA and the keystone exams is the upper boundary of learning targets
for all of these students including students with disabilities who are not eligible for the PASA.
The alternate eligible content should not be considered an entry point or target for students eligible for PSSA. If you think about it, after the eligible content we start with the upper edge we
taught you through the Webinars how to essentialize back to determine a target of learning, based on a student's instructional level.
Regardless of the student, we must always be pointing towards the target that has been determined by an IEP team. But the target that's also determined to be the upper edge of learning.
Because that is the target for the student.
And then, we can essentialize back to instructional levels and build that student back, if needed and then build that student back towards that upper boundary of learning.
So I hope that's very clear for you.
Because I'm going to tell you that question is going to be in your survey for those who watch it live.
So building upon our previous learning, back in January we talked about effective instruction.
And when we think about math instruction today, we cannot think about it outside of the perimeters of what we have already learned within ?? in regard to effective instruction.
So based on Danielson's model for instruction, and thinking about the framework Pennsylvania is using for educator effectiveness, on this slide they're listed four considerations for effective
instruction.
We know good effective instruction is good effective instruction, regardless of the cognitive ability of a student. Students can be gifted or have significant cognitive disabilities. But we can still
use the same strategies to illicit learning.
Targets and instructional sequences can be reduced in depth and breath for students with cognitive disabilities. We'll spend time practicing that skill today we're going to look at pieces of math and
thinking about effective instruction, in regard to math.
Looking at these four areas, we think about communicating with students, you know how is each student receiving the information? We did spend a lot of time in February talking about that.
And really examining each student, and what they can do in regard to communication. How does each student show what they know?
Some times don't have talk, some student he is don't have can point ?? it all will depend upon the student and in your classroom, you can only imagine you probably have students who communicate in
many different ways.
How are you eliciting. And how your providing feedback and appropriate reinforcement in regard to communication ?? all make up, that whole piece, of effective construction and communication.
When we think about using direct instruction, we think about am I providing opportunities to shape and teach new learning?
As opposed to students engaging only in repetitive and practice activities during instructional periods of time.
How much of your lesson is spent modeling? Versus guided practice, versus independent practice.
What other strategies could you include based on the student's level, to increase more independent thinking.
Often we use lots of repetition.
But there are many strategies out there, that we can help to shape learning, using direct instruction.
Engaging students in learning.
are all of the students in the classroom, active participants with the learning? Am I eliciting responses?
And how am we eliciting responses? And is it the same for all students, or am I differentiating? I would bet that many of you, are differentiating because our students, are so vast and, so different,
within that 1%.
Am I considering earlist learning? One of our teacher reviewer suggested a new program new to you through Google unique that is providing her opportunities to use Earlist learning and teaching and
engaging you don't want to use in learning.
And am I differentiating targets and expectations?
Within large and small group instruction?
Am I differentiating the type of instruction I'm doing? Or is it all one?to?one? Or is it small group? Am I building the student's ability, to learn in the variety of environments.
Am I using assessment in instruction.
How long ?? how am I using assessment to grade my instruction and am I making decisions about what I'm teaching while I'm teaching it? And then after I'm teaching it based upon what my students have
done and what their performance has been. Am I checking in with each learner?
And how are my data collection systems designed?
What kind of information am I collecting? And is it usable for me?
So those are all pieces of effective instruction that we had discussed back in January I believe. So if you would like more information on that, I would suggest you may want to go back and review it,
or view it for the first time.
There were 3 resources we provided today. And that information in these resources, has been interspersed throughout the presentation teacher desk references they're available through the PaTTAN
publications on the web site. You can order them, you can get them as they appear on the screen.
In color. Or you can also down load the copies that we have made available in a variety of places for you.
They were there with your handouts today.
You can find them on the alternate assessment partner page at the PaTTAN.net. You can also find them on the educational initiatives page with students with significant cognitive disabilities. So there
are four different ways, that you can find these resources.
The one to the left is called effective instruction.
Our next few slides, where we review effective instruction and what that looks like for any instruction regardless of what you're teaching and lesson design is included within this particular handout.
To the right you see using instructional time effectively.
Gives you some good ideas how to get the most out of those minutes that you have with your students in your classroom.
And thirdly, in the center, is essential practices for effective math instruction.
And there are many pieces within here that talk about the idea of getting to essential conceptual understanding and using modeling, and visual representation to ensure students that understand
mathematical concepts as they move through math curriculum and through the alternate eligible content, throughout the years that they're in school.
So effective instruction, implementing lessons, that's included in that first teacher desk reference that I talked to you about.
So some things that we want to remember, some things that we want to consider in regard to that, is before the lesson, of course, what alternate eligible content am I addressing?
Thinking about the grade level or levels, represented in your classroom in the content that you're going to be teaching.
What skills can I teach to help the students acquire this knowledge?
What skills can I teach to help them practice and review the knowledge? We're going to really talk about that a lot today. Because when we think about essentializing, particularly in the math content,
we may need to be essentializing back to some more basic skills. But building back up towards those grade level alternate eligible content, is a place that we will explore in the slides to come.
During the lesson, tell the students what they're learning. We're not going to hide it. We're not going to bury it in an activity but tell them exactly what they're learning because our goal is to
have them take that skill across content and across activities and generalize it. Use direct instruction, if I'm going to be teaching the student to demonstrate understanding of addition of small
sets, or demonstrate addition, demonstrate understanding what it means to put together, teach that.
And teach it to mastery.
Provide opportunities for practice. Provide feedback to the students.
As they are going through, that instruction. As the lesson comes to a close, summarize it. Restate, today we learned about putting together. Today we learned about addition.
Whatever the language is, and the vocabulary that will be familiar to students, conclude the lesson.
Provide an exit assessment. Make sure you have good data of not only what you taught, but how the students responded and what their level of performance was. Did they reach mastery as determined, when
you were planning the lesson.
If not, what do I need to teach and that is where your reflection comes in.
What does the data tell me about what I taught today? What ?? when I taught math today, and some of the mathematical concepts we were multiplying using modeling and using manipulatives and not
necessarily using digits.
What was it that the students were able to do? What could the students show me that they knew? And what is it that I need to reteach or what is it we then move onto as the next step?
What went well? What would I change?
Where will I start at the next lesson?
This is such critical information when we're teaching students and in particular when we're teaching students with the most significant cognitive disabilities.
So as a recap, when I'm making my connections to math instruction and I'm planning to use the alternate eligible content with math instruction, here are some things, some questions that might be good
check points for you.
Number one, have I reviewed the grade level math alternate eligible content? Are my students able to use that as a target based upon the data and their current instructional levels?
Or do I need to a essentialize, if I have, have I essentialized it to meaningful targets for my student?
Have I checked the math glossary if I need to? And have I reduced the complexity of the vocabulary?
Am I fluent with and confident of the expressive and receptive communicate means of each student in my classroom that I'm teaching?
Have I determined the lesson structure?
Have I incorporated pieces of direct instruction followed by guided activities?
Is there a good match or am I doing all activity based?
Am I putting those pieces in place where I'm teaching new learning?
Have I determined what assessments I'm going to use?
Both formative, how I'm checking in throughout the instruction and then something at the end, something Summative.
When I think about vocabulary, have I identified the key vocabulary? Am I considering the high frequency common words most likely located in that student's vocabulary?
On their device? Am I using those?
My goal is to ensure the students understand and meet the expectations of the alternate eligible content.
Let's look at math vocabulary examples. We did a few of these in our last webinar. If you have not seen it or you need a review I would encourage you to go back and rewatch those sections. Or,
practice or pull out some alternate eligible content and just pull out some vocabulary. Just practice reducing the complexity, thinking about your students.
The two pieces that I picked out today, I picked out a high school piece.
Where the high school alternate eligible content talks about a linear situation using graphs or numbers. And predicting or interpreting the effect of a rate of change.
In the variable or other feature. One of the big vocabulary words is understanding rate.
The student is going to need to predict or interpret an effective rate. Can they see a change? So looking at the vocabulary in the vocabulary resources for math, we looked up what rate was. And the
definition is on the screen. It talks about a ratio, it talks about different measurements. And it really gets down to a rate as a measure of change.
I also selected a 6th grade example where we identify factors for numbers 5, 10, 25 and 100 and list multiples for 10 or less.
So looking at what does factor mean? That may not mean much to my students. And it is a whole number that can divide another whole number.
With no remainder.
They even gave an example there.
So let's take a look at some possible considerations for a reduction of complexity when we think about rate.
We could use maybe some words like change between. What change is bigger?
We need to find what change may be smaller. Which change is less?
What happens when one goes up?
What happens when one goes down?
So those are ways to, kind of get to the same piece of information, that we want students to know and think about when they see this in front of them. And make a decision and interpretation or a
prediction.
We could even make it go lower and useless words.
It's all going to depend upon your students.
Let's take a look at factor. If we reduce the complexity of factor. Well we could use depending upon our student they may in, what multiply is. We could say numbers that you can multiply together to
make a number.
Maybe we could use something like specific numbers or special numbers that make up a number.
Numbers that are part of a number.
Numbers that make a number when you put together.
Numbers that go together, to make a new number.
Put together ?? make new numbers.
You could reduce that lots of different ways to help a student understand.
And again you can also reduce the number that they're working with.
Remember, we're going for conceptual understanding. And not necessarily manipulating the largest numbers that are out.
So let's review essentializing content. We looked at the December webinar when we examined IEP goals and objectives. We took a deeper look in January when we thought about planning for instruction and
thinking about, how we take that alternate eligible content. And based upon a student's present levels of performance and the data that tells me what they know and can do in math, I look at the
content and say, is that an appropriate target or do I need to reduce the complexity of that target?
And we have called that essentializing the content based upon the university of Oregon and some of the work that's been done out there.
So essentializing the content is essentially, essentially saying, have I identified what the student needs to know with this piece of alternate eligible content. It is usually the noun. Often the
noun. But really look at the piece of content and think about what does it want the student to know?
Then, once we have that identified, and we have circled it, we may go next to say, what does the student need to do with that content that they need to know?
And we boxed it. It's often the verb but really look at the content. Sometimes there's one or two pieces of each of those.
So really, examine it.
And have I identified the content in which the student needs to operate with the know and do. What is the context? Once we have that figured out, then we can start reducing the complexity in all of
those different areas. And they could look different, depending upon the student. So let's practice a quick math example. We'll use that 6th grade piece of alternate eligible content.
And this is identifying the factors for number 5, 10, 25 and 100 and list multiples for numbers 10 or less. As an instructor as a teacher as a former life skills teacher, if I was going to
essentialize this, because that word and is in there you see it's in italics I would break that apart, for an essentialization process.
And, in lieu of time today we'll look at the first piece of it. Identify factors for numbers 5, 10, 25 and 100.
So if I wanted to say what is it that students need to know typically the noun if I needed to identify what they needed to know, in that top piece of information, it would be factors.
In the second piece, it would be multiples. I don't need them to find numbers I need them to find the multiples. What do I want them to do with the factors?
I want them to identify the factors.
What do I want them to do with the multiples?
Well they just need to list those.
In what context, do they need to identify factors?
They need to identify factors for numbers 5, 10, 25 and 100 and when we looked in the glossary, we know exactly what that means.
Second part, what's the context?
They need to list multiples for numbers 10 or less.
So as I was saying earlier, in lieu of time we looked at what that could look like if you were identifying some targets. And we have just used the top example, the ?? the first part of this piece of
alternate eligible content, identify factors for numbers 5, 10, 25 and 100.
So for my up are level, I may ask the student to show me, point to, and I'm going to use numbers. I'm going to use digits and sets.
That multiply together, so they will have objects there, to help support their understanding of the numbers or pictures. Um, that multiply together to make 5, 10, 25 and 100. So I took out the word
factors.
In less complexity, use a table and manipulatives, show me sets of objects paired with numbers that you can use to make 5, 10 and 25.
So I could do one set of five. Two sets of 5, to show 10. I could use five sets of 5 to show 25.
Or 1 set of 25.
Or one set of 10.
And then finally, for the less complex, point to the sets of objects that make five and 10.
One set of five, makes five. Two sets of five make ten.
I could even reduce that further. And just work with the number 5.
Show me five.
Again, depending upon your student, you need to know what your student's instructional level is, start there, get mastery and then you can move and shape the learning towards that alternate eligible
content. So we're going to teach some ?? teaching math, basic understandings. I'm going to go through, um ?? and look at some basic understand linings and thinking about math.
And thinking about what we're doing in Pennsylvania.
But really, focusing on ?? talking about modeling visual reputations and thinking about math from a perspective that doesn't always mean numbers.
Pennsylvania has been using a model, methodology called concrete representational and abstract.
What is really nice about this model and what really, I think has stood out, is that, we use the same language, across all 3 models.
So what we might be using and saying and describing, when we're using digits, and numbers, we could also do at a very concrete level using objects and visual representations and pictures.
We have also been using calculators for a lot of years to assist students to manipulate algorithms which is a big word. I will say it's a big word. It's a mathematical term that is used to describe
addition, subtraction, multiplication division.
And making some numerical application in real life situations. Well we can use a concrete and a modeling and a, um, visual representation model to teach concepts that make ?? have students make sense
of numbers. And then, the implications for applying that to the real world are just amazing. I'm so excited to be able to show this to you today.
It doesn't mean calculators still cannot be used. I don't want you to walk aware, Sharon said we don't use calculators anymore, that's not true. They're a source of technical assistance, but it's also
important that we now have an opportunity to teach students with significant cognitive disabilities underlying mathematical concepts, that can be really foundational for success as we, get our
students community ready.
This CRA is not a program. It's not a curriculum. It's just one way of thinking about teaching math and teaching math for students to understand it from a very conceptual viewpoint. The alternate
eligible content when you get right down to it and you take the time and you look at it, I give great Kudos to our teachers, many of you who may be sitting out there listening to this or listening to
it, from a recorded version, but, the thoughtfulness that went into getting it to a conceptual meaningful level for students was really quite, quite applaudable and I think that, as you see that, we
can teach mathematical concepts we don't have to use large numbers, we can use smaller digits, to teach the same concepts that will provide a great foundation for our students as they go out into the
world.
And become problem solvers and use numbers in ways that are meaning full and increase independence and increase the ability to be successful post secondary. And post school in their outcomes.
So just a bottom line to know that this is not just a great idea that everyone is jumping on the band wagon this really has a research base behind it. The institute of educational science it's a
research arm, of the U.S. Department of Education but it has no political influence in it, whatsoever. It just looks at, good solid research that supports what we do.
The first statement talks about intervention materials, should include opportunity its for students to work with visual representations ?? to help them understand the mathematical ?? instruction
should be systematic we know the research supports that. That includes using models, to be proficient, problem solvers to help kids think and to provide, guided practice and lots of feed back to
ensure the students learn these mathematical concepts.
We also know that students, there's research about the students who struggle in math.
You know, that we, um, know that accurate and fluent retrieval of math facts, the reduced memory deficits, and get to the conceptual understanding we reduce the complexity. We connect to really
meaningful situations, and teach math.
In that, from that perspective.
And just to situate it, this particular slide, I want to talk about the fact that you know we teach, when we think about the language of numbers and, I'm bringing language into this, because we know
language is a huge component of learning for our students with the most significant disabilities.
And the language is really a place that ?? and language of numbers is a place that I think we get hung up on when our ?? we're teaching math with our students. And just looking at this slide we have
?? you know, letters, kids learn letters there's a name sound and word. They put them together to make words. We have nine digits in math.
And 10 symbols all together 0?9 and we teach kids, we can teach them up through 10, we're really going to have some productive math learning using models for some higher level math concepts.
Math, we teach names for the digits.
You know, in ?? our letters and our language, there's just, particular ways that, certain, letters make sounds like Carla, piece chocolate, when you pick a number like 7, when you think of math, we
end up having lots of, um, different ways 7 comes, 7 is 17, it's a different word, 7 is 207, 71, negative 7, 0.7, 1/7, so we have a lot of different ways we are representing this, which can become
confusing and our kids get bogged down in language. When we think about a modeling perspective and in Pennsylvania, using a framework that looks at the concrete, where we make sense of numbers,
removing objects, and things, we can make sense by drawing and we can make sense with symbols which are our digits. We use consistent language we can teach the same thing.
Often and I, myself as a teacher went right to the digits right to the numbers.
So ?? it's very important, um, that we look at math from a different sequence of instruction, though, however many of our students, may always be at the concrete level but that's ok.
They can still be learning conceptually, align to what other kids can be learning. It gives a fluency across the levels for a wide varied of the learners within 1% we know we have students it at the
very concrete level and kids that are manipulating numbers and that's ok.
We can teach it, across those ?? those particular methodologies.
So this just, this slide is really telling us that using a modeling methodology really helps us reach a variety of students. Be able to teach some things we may not have been able to teach before.
And that's what makes it really exciting.
So let's take a look at what this potentially be for our population of students when we're teaching math.
In our third grade and fourth grade what you see up on the screen are alternate eligible content, students by third grade are being asked to look at two digit numbers and round up to the nearest 10
which is taking in someplace value and understanding the different places of where numbers live in the number system.
And then, again, at fourth grade they're asked to compare values and determine which is greater than, less than or equal.
So we typically, in early Numeracy, in teaching quantity, has an effective place. We teach symbols, 0?9.
And we also teach ten, if we can teach students to see numbers through quantities as opposed to what they're called, we can move them to higher levels. Instead of just focusing on names we have kids
that we teach 1 to 1 correspondence you know the students I've had them too, they will start counting loose their place they will go back. Um, and this goes ?? becomes a ?? a process where they,
continually, are trying to count from 0 or 1 and, if we can teach conceptually we can have students actually, um, through magnitude be able to see quantities like five and ten and start, from that
point to be, um, understanding numbers and be able to compose and decompose numbers which was really important when we think about operations.
We're using a lot of supports we're out in classrooms doing this, using number lines, ten frames, place value mats.
And, um, when students have difficulty in math, um, they're seeing counting as I just described, that being hung up on that 1:1 correspondence always can be very ineffective. We need to really be
thinking about how to take kids to a place where we can rethink counting.
By teaching counting quantitatively. What does one look like? What does two look like? What does three look like? And using visual cues to help the students make the connections. Once we get above the
ten, instead of introducing new language, 11, 12, 13, 14, having kids match pictures ?? of numbers to that teach them quantitatively what 11 is really one 10, 11. 12 is one 10 and 2 ones. 20 is two
tens. 22 ?? two tens. 38 is 3 tens, 8 ones.
88 is 8 tens 8 ones, students learn less language and build more conceptual knowledge of what that number is.
And I get excited thinking about transferring that learning to real life situations. Think about students knew those numbers from that quantitative perspective, the impact of imagining money when
you're out at the store making change, understanding what you have to pull out of your wallet. Understanding what you're pulling out of the MAC machine.
It ?? I think we can go on and on thinking about how we can take that in the work world and everything, to ?? to make sense for kids about numbers.
So we think about counting examples with manipulatives. We look at, here's a two ten frames, at the top is five. Can we get students to associate looking at that, every time they see that,
automatically knowing that's five.
Same thing at the bottom, automatically realizing that's ten.
We would have to shape and teach that learning. But that's possible.
We can also represent five in a different way.
If a student, better understands it through dice or through looking at it, in this model, you know, five can be thought of that way, ten can be thought of the two fives next to each other. The ?? how
you manipulate it, is up to how the student learns best. But what we want to do, most of all is make sure that the students can start quantitatively see numbers. We're out in different classes,
classrooms with kids with autism and classrooms with kids with significant cognitive disabilities and manipulating this information. So let's quickly move on and see how we can transfer that into
some content of addition and subtraction that ?? and multiplication that lines into our alternate eligible content.
We can transfer seeing those numbers in different visual formats to actually using other manipulatives. Here's a cube and a stack of cubes, they both equal 10. Teach our students, we transfer that by
teaching it and then, ten tenss equals that, what that block at the bottom. That's 100. The student could recognize that as 100 items.
So here is an example of alternate eligible content across grades 3, 4 and 5.
That examine place value what you're looking on the screen is grade 3, alternate eligible content is to the right.
But here's grade four.
And we're still looking at, um, values and modeling relationships, between multi?digit numbers.
Um, and here's in grade five, identifying place value and 3 digit numbers through the use of models, even if you're teaching students at the secondary level, 6, 7, 8, high school you may need to
essentialize your content back to ensure your students understand place value from a conceptual standpoint.
And understanding that and being able to manipulate addition, subtraction, multiplication and division through understanding place value, can really help you to that higher level math of what students
need to know and be able to do when we think about some ?? 342, can you use two digit numbers.
You can use single digit and you may, start with single digits build it up two digits, up to 3 digits. For our example wills today we'll use 3 digit numbers. So this particular example, is we see this
as 342. Which is 300s, 3, 100s, 4 tens and two ones. So what does that look like at a concrete level? What are 300s?
The student using manipulatives, that's 3, 100, 4 tens.
And we can write, the 300 in there and model that.
If the student understands it at the conceptual level, that is 3, 100s. 4, tens we layout 4, tens.
We have four tens is the same as 40.
And then we have two ones.
That's what two ones look like.
So that's decomposing that number of 342, is really, 3, 100s, plus 4 tens plus two ones.
If we did it representation tally a student is going to draw I'll show you a picture of that, they would draw 3 boxes, 3, 100s, plus four tens. Plus two ones.
300 plus 4 ones ?? is the same as 342.
So let's take a look of alternate eligible content across four and six, that examine addition and subtraction even if you're teaching students it at the secondary level at 6, 7, 8 high school you may
need to essentialize, to ensure understanding for your students using visual models, using place value, using objects. Modeling, and ?? um, area maps to ensure that your students understand the basic
conceptual knowledge behind this, before moving onto higher order math.
So in our example here, here you see add and subtract whole numbers with sums and differences less than 1,000, Kudos to our teams of teachers that worked on this.
They did not say it to be 1,000, or 100, it is anything less than 1,000, it gives you great latitude for a variety of students in the population. In grade 6 solve problems using up to 3 digit numbers
again. Up to 3 digit numbers. In any of the operations. It could be single digits or double digits, depending upon the level of complexity, for your student, and the data of where your student is
starting you may need to start with identifying numbers.
And the place value. But ?? um I think you're getting the idea.
So let's take a look at what that could look like. 5, if we read that problem 5, 100, two tens, five ones, plus 1, 100, 3 tens, and six ones it could look like this.
525 plus 136.
So let's start with our first number. We're going to work left to right.
We'll work left to right across the top and right to left.
We're going to start with 5, 100s, what does 5, 100s look like. 5, 100s. What's next? Two tens. What does that look like?
And five ones.
And there we have 5 ones.
Now we're going to go to our bottom number. We're going to work right to left.
How many ones?
We have 6 ones.
Now if I have been teaching my students, to look at numbers and look at numbers conceptually, what can we do with the five, we have five ones and six ones do we have enough there to make ten ones?
Absolutely.
So we would ?? take ten ones and we turn it into one ten.
And you would have to practice with your students take ten ones make it 1, ten. Take ten tenss and make it 100. We would teach it and shape it. So then we show it, over here with our numbers is one.
Then of course we carry it, we have 1 one more ten here. Here's our one 10, we added over here in the left.
So you are our next, how many tens? We have 3 tens. We would add 3 tens, 3 tens plus 3 tens equals the students would say 6 tens we would add a six.
How many hundreds? One, how many hundreds do we have all together? 5 plus 1, 100, equals 6, one hundreds, our total is 6, 100, 6 tens, one, one or 661.
You can show represent it by drawing our boxes 5 boxes.
Two tens. How many ones. Five money ones.
How many ones are on the bottom, we have 6 ones.
Five ones plus 5 ones equals ?? this is the same as 110. We have 3 tens we have a total of 6 tens. Another, 1, 100, we have 6, 100s, 6 tens and 1 one.
661.
Let's look at a subtraction.
Again, we're using double digits, I mean 312 minus 261.
We have 3 digit numbers.
So we would start out with how many hundreds in that top number? We have 3, 100s, how many ten, one.
How many ones, two ones. Okay.
And then, what we need to do, is take a look on the bottom, we're going to start left to right. And see, do we have enough hundreds to take away? Because we're going to take away and again you may
have to essentialize to your students, do they understand take away? Do we have enough hundreds to take away, yes. We do have enough hundreds to take away. Do we have enough tens to take away from
the ten we have there?
Well, no we only have one ten. So ?? we would teach our students, how many tens in 100, we can convert, 1, 100 to 10, tens.
We would do that. We would show it, over here.
We took away 1, 100. And we made, 10, tens. So we have 11, tens here.
So ?? do we have enough ones to take away from the ones, absolutely, let's start our math here. We take 1, 1, from 2, 1s, equals 1, 1.
We take 6, tens from 11 tens and how many tens are left? Five tens.
We take two, 100s from 2 and we have ?? no 100s, so we have an answer of 5 tens or 1, one, or 51. So going through the representation, similar process.
And you get the similar answer.
So ?? we have been out in classrooms and this is what it looks like. Kids are actually manipulating these different pieces.
But kids have the option. Some kids, prefer using it at the concrete manipulative levels, some kids like the representational level. But the bottom line it is making sense to them.
So now let's take a quick look at multiplication ?? grades 5 and grade 6.
Take a look at it, um, multiply single digit whole numbers in 5th, 6th grade. Solve problems up to 3 digit numbers in any of the four operations. We know multiplication comes in there.
7th grade there it is again. Solve multiplication and division problems.
Positive negative numbers. 8th grade we're looking at exponents which is multiplication, and again, high school. And the reason I'm showing those examples is because I want you to envision as you
essentialize you may be essentializing back to the basic addition and subtraction and teaching your students conceptually, what that might look like.
So one way, because it's ?? this looks like, this is algebra, in high school there are only a couple of pieces of it, many the alternate eligible consent. One way to think of algebra is think about
it, as essentializing basic skills we're talking about today.
So ?? let's take a look at multiplication.
Five ones times one one we're using an area map.
We have five ones across the top. One one across the left side. We're going to multiply and the best part of this, is when you multiply one by one, you're going to come up with a one.
One times 10 is going to be a ten. If you're using you know, if it was 25 times 1 you would have two tens, times two tens, five ones times one.
You still, only get itself, when you multiply by one. So 1, 1 times 11, is 1.
As you can see, you would do that all the way across, the student would count up or ?? they're looking at the conceptual amount that's 5 ones. You know or they may see it as five, down at the bottom.
If you did it, representationally, we use, circles for ones.
And again you do the same way. They draw in their circles ?? still comes out to five ones. What does it look like if we did it with ?? one times 10, is 10.
And ?? one times 1, of course is one.
Your student, um, either looks at it, knows those amounts and it's how many, 8 tens, four ones or 84. That would be the answer. And conceptually, they can understand what that amount looks like. If we
did it representationally, we would do it like this. Where you're again multiplying down.
And then the student adds it up, 8 tens, four ones, 84.
For more information, and training regarding modeling, visual representation and our Pennsylvania CRA model, there is information out there for you.
They are having CRA days as you can see most of them are done for this. However, there will be more next year. And, that we will, get that information to you and I do have a way for you to get that
information in the 15?16 school year if you're interested in looking at this through fractions integers and equations there are still sessions available in the State for you to be part of those
presentations and, they are hands on you'll be working and learning.
We also wanted to provide for you today, the ability and this is in your PowerPoint a wiggio, that we have through Pattan math we did today, add an alternate assessment folder in that, on that Wiggio
I would encourage you to join that and the first thing you'll find under that folder tab is the presentation from today in the PowerPoint version.
What you got in your handouts is a PDF version you did not get to see how that all manipulates and give you an opportunity to go back and play with it.
To see how that all works. And so, that is on this site on the page, please sign up. Go to the folder, under the folder tab there's a folder alternate assessment, click on the little gear that's by
the file and select the down load option. That will down load for you. I understand if you just try to click on the link it may not work as well for you.
So you want to click on the little gear by the file. And select the down load option.
Okay? So ?? please, these are resources for you, and then once you get onto this Wiggio you'll get automatic updates they will let you know when those next presentations are going to be happening in
your area.
We are also looking at other opportunities to get this information out to you.
So you can practice and learn more about the possibilities for ?? using models, visual representation and area maps, when teaching math.
I want to thank today, before we're done ?? our teacher reviewers. We had ?? quite a few of them.
Fabulous, fabulous feedback thank you Dawn, Sherri, Dawn, Shelli, Jackie and Charlie, great great feedback and great input to today's presentation. We know that kids learn lots of different styles and
we have very challenging students out there. So, so we're looking for partners in this work as you're working with this in your classroom, please contact us and let us know. We would love to get out
and see it. Love to have you come in and present with me for our webinar series I'm sure they will be continuing after this spring series.
Act 48 participants ?? reminder, to click on the link to complete your survey for today.
And for your questions and to enter today's code that you will receive in just a minute.
Or less.
And to receive your act 48 hours remember you have to attended all four sessions live.
And complete all the surveys.
Thank you very much ?? this was a lot of information, I hope you're as excited as I am about what math can look like for our students who are eligible for the alternate assessment and are using the
alternate eligible content. Thank you very much.
And all times please contact me we're looking for reviewers for the next sessions.
I wish you all a very good evening.
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