Bridging the Gap: Grades 6-9
Sequence of Sessions
Overarching Objectives of this November 2013 Network Team Institute
· Participants will develop a deeper understanding of the sequence of mathematical concepts within the specified modules and will be able to articulate how these modules contribute to the accomplishment of the major work of the grade.
· Participants will be able to articulate and model the instructional approaches that support implementation of specified modules (both as classroom teachers and school leaders), including an understanding of how this instruction exemplifies the shifts called for by the CCLS.
· Participants will be able to articulate connections between the content of the specified module and content of grades above and below, understanding how the mathematical concepts that develop in the modules reflect the connections outlined in the progressions documents.
· Participants will be able to articulate critical aspects of instruction that prepare students to express reasoning and/or conduct modeling required on the mid-module assessment and end-of-module assessment.
High-Level Purpose of this Session
● Implementation: Participants will be able to articulate and model the instructional approaches to teaching the content of the first half of the lessons.
● Standards alignment and focus: Participants will be able to articulate how the topics and lessons promote mastery of the focus standards and how the module addresses the major work of the grade.
● Coherence: Participants will be able to articulate connections from the content of previous grade levels to the content of this module.
Related Learning Experiences
● This session is part of a sequence of Module Focus sessions examining the bridging the gap between grades 6-9.
Session Outcomes
What do we want participants to be able to do as a result of this session? / How will we know that they are able to do this?· Participants will develop a deeper understanding of the sequence of mathematical concepts within the specified modules and will be able to articulate how these modules contribute to the accomplishment of the major work of the grade.
· Participants will be able to articulate and model the instructional approaches that support implementation of specified modules (both as classroom teachers and school leaders), including an understanding of how this instruction exemplifies the shifts called for by the CCLS.
· Participants will be able to articulate connections between the content of the specified module and content of grades above and below, understanding how the mathematical concepts that develop in the modules reflect the connections outlined in the progressions documents.
· Participants will be able to articulate critical aspects of instruction that prepare students to express reasoning and/or conduct modeling required on the mid-module assessment and end-of-module assessment.
Session Overview
Section / Time / Overview / Prepared Resources / Facilitator PreparationWhole powerpoint / N/A / Clearly for a multitude of reasons, many students entering our classrooms do not arrive with all of the prerequisite understanding and skills needed to engage in the content of their grade level. This session is designed to prepare you to bridge the gap – to give your students a leg to stand on by giving them what they need in the most efficient and effective ways that I know of.
• Learn and experience efficient ways to assess and remediate prerequisite knowledge:
• Assessing Conceptual Understanding
• Remediating Conceptual Understanding Gaps
• Assessing Fluency
• Remediating Fluency Gaps / N/A / o Powerpoint
o Projector
o Participant Binder
Session Roadmap
Section: Bridging the Gap / Time: N/A• [minutes] In this section, you will…
• Learn and experience efficient ways to assess and remediate prerequisite knowledge:
• Assessing Conceptual Understanding
• Remediating Conceptual Understanding Gaps
• Assessing Fluency
• Remediating Fluency Gaps / Materials used include:
Time / Slide # / Slide #/ Pic of Slide / Script/ Activity directions / GROUP
1 /
2 / / You may hear it said… if you teach 6th grade, you teach K through 6, if you teach 9th grade, you teach K-9th.
Clearly for a multitude of reasons, many students entering our classrooms do not arrive with all of the prerequisite understanding and skills needed to engage in the content of their grade level. This session is designed to prepare you to bridge the gap – to give your students a leg to stand on by giving them what they need in the most efficient and effective ways that I know of.
We’ll talk about: (read off the content of the four bullets).
3 / / Outlined on this slide are some topics of conceptual understanding that are foundational to 6-9 on the whole. Some of these concepts are just being introduced in grades 6-8 (dividing fractions by fractions – grade 6, operations with negative numbers – grade 7, and systems of linear equations – grade 8); all of them are expected to have been covered before the Algebra I course.
We’re deliberately addressing conceptual understanding gaps before fluency gaps, because it is critical that students establish conceptual understanding of a concept before trying to achieve fluency with the related skills.
For the most part I will model my delivery as I see it being done in the classroom; I find this the most efficient means for explaining what I’m recommending is by demonstrating, but of course I will also add commentary as we go along.
Let’s get started with assessing understanding of the 4 basic operations and their models.
4 / / What does addition mean? Allow for participant response
Addition means putting together, putting together like objects or like quantities.
Most all of your students are able to compute addition problems of whole numbers, they may or may not be fluent adding with decimals, fractions, or measurements in unlike units. What we are assessing here is their conceptual understanding – do they recognize addition situations when they come across them. Students demonstrate their deepest understanding when they not only recognize those situations when described to them, but when they can create / conceive of a variety of addition situations on their own.
The first and most basic model of addition that students are exposed to is the part-part whole model where they are asked to find the whole, given two parts.
So, I would assess their understanding of addition at this most basic level by asking: (read first example). The most basic version of this question is simply, “Write me a word problem in which you need to add 5 + 3 to solve the problem.” You’ll want to use variations of each question that will assess the student’s understanding of this model and this operation without seeming overly simplex. In grade 7 or 8 you might ask this next example (read second example) or in grade 9 you might ask the third example (read the third example).
Just like this one does, some of these next slides will show a progression of possible problems to give to students based on what grade level they are entering and/or where you feel you can most appropriately access your specific student population.
5 / / Just as watching someone else lift weights does nothing for our own muscles, listening to a teacher explain something does very little for student understanding. Pimsleur language programs are based on the notion that the brain activity that occurs when a student is asked something is starkly different and more effective towards learning than merely listening to the language. So by assessing in the manor I am illustrating now… by merely asking the question, you have already begun to remediate the missing understanding. Our general strategy is (click to advance animation) assess through questioning, discuss, and repeat perhaps again immediately perhaps the next day or next week.
6 / / The comparison model of addition (and subtraction) represents students earliest exposure to algebraic thinking. Let’s take a look at some different access points to assessing this model of addition. In the most basic form (read bullet 1). A slightly more advanced activity (read bullet 2). Or hopefully our middle and high school students are ready to engage with this problem. (read bullet 3). This last form of the problem if it were done using whole numbers is actually a 1st or 2nd grade level problem in the CCSM – why would I say this problem is exposing students to algebraic thinking? Because you can’t get the answer by merely adding the two numbers, subtracting the two numbers, multiplying them or dividing them. Let’s have a look at how we can make this last problem resonate with students through the tape diagram.
7 / / Max has 1.7 more feet of string than Joe. Who has more string Max or Joe? OK, let’s draw something to model this. I’m going to draw a bar to show Max’s string and a bar for Joe’s… (click to advance animation) whose bar should be longer? (click to advance animation) Do I know ho much string Max has? Do I know how much string Joe has? What can I label in my drawing? (click to advance animation) What else do you see? … (click to advance animation) So we know that this portion of Max’s bar is 1.7 feet. What do we know about this portion? Is it shorter or longer than Joe‘s bar? It is the same. So what do we know about these two parts combined? These 2 parts = 7.6 feet. So 1 of these parts = ?
8 / / So let’s update our strategy with a bit more detail. We assess and by default begin to remediate by posing questions that will not feel like too easy of a question. Then we discuss and/or model to scaffold students to increased understanding. Again, when it comes to modeling you need to meet your students where they are on the progression from concrete to pictorial and moving towards the ultimate goal of a ready ability to visualize and to demonstrate through a picture or concrete materials if called upon to do so.
9 / / What does subtraction mean? Subtraction means taking apart or taking away.
The first model of subtraction is the part part whole model, where students are given a part and the whole and are finding the missing part.
Again, we can ask them to come up with a word problem that represents a subtraction problem, and we choose an appropriate level at which to ask the question. The most basic, write me a word problem in which you need to find 5-3 to solve the problem. If you were asking it in the third form, students would need more than just that one expression or even more than one equation to solve the problem… require them to give you a complete solvable problem… So at this level we are tapping into their understanding of word problems involving systems of two equations with two unknowns, while simultaneously assessing their fundamental understanding of subtraction.
10 / / And here is the comparison model of subtraction. Comparisons using less than and fewer than are generally speaking less familiar to students simply because we use ‘more than’ comparisons much more commonly in our daily language with children from the time they are little. “Who has more?” and “How many more?” is much more common than “Who has fewer?” and “How many fewer?”
Again once you have established that they can do a problem like this third one, assess their full understanding by asking them to write their own word problem using the comparison model of subtraction (you can just say, write me a problem with a similar set up.)
11 / / (Use this slide to walk through modeling the problem with a tape diagram.)
12 / / What does multiplication mean?
Multiplication means putting together equal groups. (Some may say repeated addition and this is ok, but in a more contextual way it is addition (putting together) and this time it is putting together equal groups not just any sized groups.
The most basic model is the equal groups model then.
13 / / So let’s go ahead and talk about one of the properties of operations… Is it true that 5 x 3 will have the same value as 3 x 5. How can I prove this is true for any two numbers I pick? Why should it be obvious that if I have 5 groups of 3 dots put together that I will get the same number of dots as when I have 3 groups of 5 dots put together? How can I prove this?
The array model of multiplication makes this obvious and asking this question of students will reveal whether they have been exposed to this model of multiplication. If I set up my groups in an array then one can clear see that 5 groups of 3 is the same number as 3 groups of 5.
14 / / The final model of multiplication that we’ll assess is the area model.
What does area mean? (allow for participants to share)
Again, asking students to articulate this will help them refine and make permanent in their mind what area means.
How do I find area?
Write me a word problem where I am trying to find the area of something.
15 / / We also have a comparison model of multiplication. Why might a problem like this be difficult for students? (Because the total number of Starbursts is not a multiple of 5; because again, you cannot find the answer just by adding, subtracting, multiplying or dividing). Again the tape diagram is helpful here and also using actual starbursts is helpful.
Go through the tape diagram with participants.
It is not always obvious to students what the process is for setting up a situation where one has 5 times as many as the other. Deal out 24 starbursts and go through the process of, “If Meg has 1, Amy gets 5, now if I give Meg another one, I give Amy 5 more, etc.”