1. 3rd – 6th Grade Multiplication Progression

2. Overview
Multiplication in the current Idaho State Standards and the Common Core Standards
Create 3-6 Multiplication Progression
Discuss progression
Add context to the progression
Connections to future mathematics and MTI course
The goals of today’s webinar are to:
Multiplication in the current Idaho State Standards and the Common Core Standards
Create 3-6 Multiplication Progression
Discuss progression
Add context to the progression
Connections to future mathematics and MTI course
Thank you again for joining us and we hope that this is a beneficial process for everyone.
The MTI project is sponsored by the Idaho Legislature and the State Department of Education

3. Standards Names/Definitions
The standards that have been in place for the past several years and are currently being assessed on ISAT will be referred to as the current Idaho State Standards.
The new standards (adopted in spring 2011 for implementation in fall 2013) will be referred to as the Common Core State Standards
For the sake of clarity the following descriptions and names will be used to refer to the two sets of standards under discussion.
Read the two definitions
It is important to keep these two standards distinct for the present discussion.
The MTI project is sponsored by the Idaho Legislature and the State Department of Education

4. Timeline for implementation of the Common Core State Standards & Smarter Balance Assessment
The following timeline is outlined on the Idaho state department of education’s website
As you can see from the chart, the CCSS are to be implemented in the school year.
The new assessment of the CCSS is being designed by the Smarter Balance Assessment consortium. This assessment will be fully implemented in the school year. For more information on the potential design of the SBAC assessment you can go to Smarter balance assessment consortium website with the link provided on this slide.
The MTI project is sponsored by the Idaho Legislature and the State Department of Education

5. Domain Progressions in the CCCSS
In order to begin a conversation about interpreting and implementing the CCSS it is important to understand how they were constructed. The chart on this page identifies the domains found in the CCSS.
For our webinar today we will be focusing on the number and operations in base 10 domain in grades 3-5 (along with one standards from the 6th grade Number System domain) and looking at how it transitions to the ratio and proportions domain in 6th and 7th grade.
The MTI project is sponsored by the Idaho Legislature and the State Department of Education

6. Current Idaho SS
Models
Common Core SS
 3rd grade
3.M Multiply whole numbers through 10 x 10.  
Problem:
6 x 8
Model:
3.OA.3 – Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays and measurement quantities
Problem: 6 x 8
Potential Models for instruction:
Array, ratio table,
3.OA.7 - Fluently multiply within By the end of Grade 3, know from memory all products of two one-digit numbers.
 4th grade
4.M Multiply up to two-digit by two-digit whole numbers and divide whole numbers by one-digit divisors.
26 x 14
4.NBT.5 - Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations
Problem: 26 x 14
area model, ratio table, partial products
 5th grade
5.M Multiply and divide whole numbers  
5.NBT.5 - Fluently multiply multi-digit whole numbers using the standard algorithm.
Potential Models for Instruction:
traditional algorithm
Let’s start by making a comparison between the multiplication standards in the current Idaho State Standards and the Common Core State Standards.
In 3rd grade our Current Idaho Standards have students multiplying whole numbers through 10x10. We usually see this done as a set of memorized facts. The CCSS has the same standard of fluency within 100, but it comes after several other standards including the creation of arrays, area models, and equal groups.
Our current Idaho 4th grade standard is to multiply double digit numbers which would require the use of the standard algorithm.
Multiplying double digit number is also called for in the CCSS, but with strategies based on place value and properties of operation – this would include models like the area model, ratio table and partial products.
In 5th grade we see the same standard from both sets of standards documents – with both requiring the same model (the standard algorithm), however in 5th grade in the CCSS we also begin to look at multiplication of decimals which we will address in the creation of the progression.
5th grade also includes decimals which will be addressed in the webinar.

7. Creating the progression
Vertically lay out standards for 3rd-6th grade with 3rd on the bottom and 6th on the top.
Now that we have seen some differences in the standards it’s time to create the progression. We will break this process into three steps.
The first step is to lay out the multiplication standards for 3rd – 6th grade so we can see the progression of models and numbers sets as we advance through the grade levels.
Click to show example of what it might look like
The MTI project is sponsored by the Idaho Legislature and the State Department of Education

8. 6th Grade:
6.NS.3 - Fluently multiply multi-digit decimals using the standard algorithm
5th Grade:
5.NBT.5 - Fluently multiply multi-digit whole numbers using the standard algorithm.
5.NBT.7 – multiply decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations; relate the strategy to a written method and explain the reasoning used
4th Grade:
4.NBT.5 - Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations
3rd Grade:
3.OA.3 – Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays and measurement quantities . . .
3.OA.5 – Apply properties of operations as strategies to multiply and divide.
3.OA.7 - Fluently multiply within By the end of Grade 3, know from memory all products of two one-digit numbers.

9. Creating the progression
Vertically lay out standards for 3rd-6th grade with 3rd on the bottom and 6th on the top.
Sort the cards by grade level paying attention to the models and number sets.
Our next step will be to sort the model cards. Pay attention to the number sets and models as you decide which grade level the card best fits with. Don’t worry about creating an order to the cards in each grade level – we will do this in the next step
The MTI project is sponsored by the Idaho Legislature and the State Department of Education

10. Creating the progression
Vertically lay out standards for 3rd-6th grade with 3rd on the bottom and 6th on the top.
Sort the cards by grade level paying attention to the models and number sets.
Thinking of moving from informal to formal strategies try to order the models within each grade level.
You may see more than one branch in the progression at each grade level as different informal strategies/models may lead to a different way of thinking and therefore bridge to different formal models/strategies
Now we will try to sort the models within each grade level. Thinking of moving from informal to formal strategies try to order the models within each grade level. You may see more than one branch in the progression at each grade level as different informal strategies/models may lead to a different way of thinking and therefore bridge to different formal models/strategies. In other words the models may not fit in one line within the grade levels. Think of when students are working through the new number set and where they would start and the most logical place to progress next. I will pause for a few minutes to let you order your models. (3-5 minute pause)
The MTI project is sponsored by the Idaho Legislature and the State Department of Education

11. Models that fit standards will go here
6th Grade:
6.NS.3 - Fluently multiply multi-digit decimals using the standard algorithm
5th Grade:
5.NBT.5 - Fluently multiply multi-digit whole numbers using the standard algorithm.
5.NBT.7 – multiply decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations; relate the strategy to a written method and explain the reasoning used
Models that fit standards will go here
4th Grade:
4.NBT.5 - Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations
We may see two branches within a grade level
Informal 
Informal 
Here is a visual of what that might look like
3rd Grade:
3.OA.3 – Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays and measurement quantities . . .
3.OA.5 – Apply properties of operations as strategies to multiply and divide.
3.OA.7 - Fluently multiply within By the end of Grade 3, know from memory all products of two one-digit numbers.

12. Models that fit standards will go here
6th Grade:
6.NS.3 - Fluently multiply multi-digit decimals using the standard algorithm
5th Grade:
5.NBT.5 - Fluently multiply multi-digit whole numbers using the standard algorithm.
5.NBT.7 – multiply decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations; relate the strategy to a written method and explain the reasoning used
Models that fit standards will go here
4th Grade:
4.NBT.5 - Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations
We may see two branches within a grade level
Informal 
Informal 
Now that you have had a few minutes to think about how these models could potentially be ordered, I want to pause you for a moment to share an idea. When I was looking at informal models in 3rd grade, I had two that stood out to me as where students would start in the progression – the 6x8 array and repeated addition. With the idea that these might be happening around the same time, but both leading to different formal models I put them on different branches within 3rd grade. However, within this idea of these two models happening at the same time and potentially leading different places we can still see connections between the two – like when we add the rows of 6 eight time.
3rd Grade:
3.OA.3 – Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays and measurement quantities . . .
3.OA.5 – Apply properties of operations as strategies to multiply and divide.
3.OA.7 - Fluently multiply within By the end of Grade 3, know from memory all products of two one-digit numbers.

13. Looking at the progression
We will look at each grade level to discuss how the progression of models within the grade level meet the standard(s).
Let’s take a look at the progression. We will look at each grade level to discuss how the progression of models within the grade level meet the standard or standards listed for the particular grade level. On each slide I have the way I thought about arranging the models as students are first working with the concepts and number sets, but as with any progression we may see differences based on student thinking, past experience or even the number sets.
The MTI project is sponsored by the Idaho Legislature and the State Department of Education

14. 3rd grade
3rd Grade:
3.OA.3 – Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays and measurement quantities . . .
3.OA.5 – Apply properties of operations as strategies to multiply and divide.
3.OA.7 - Fluently multiply within By the end of Grade 3, know from memory all products of two one-digit numbers.
Third grade is where we formally introduce students to multiplication. The CCSS for this grade level has many multiplication standards, some like 3.OA.3 ask specifically for models involving equal groups, arrays and measurements. CLICK We see this in our progression with our informal models like repeated addition (equal groups) and the 6 x 8 array – In 2nd grade students will have had experience with repeated addition and creating arrays so this is a natural place for 3rd graders to begin.
The repeated addition model could then be transitioned to a model like the ratio table. CLICK The ratio table can represent the same thinking as repeated addition, but can also help with the transition to multiplicative thinking like the one we see in the slide.
The array can be transitioned to the open array (open area model)CLICK as students begin to not need every square to find the product and can make larger groups of 8. In our example the student is decomposing the 8x6 array into friendly facts of 8x5 and 8x1 and then adding the products together.
All of these models lead to a deeper understanding and connections within the 10x10 multiplication facts which in turn provides fluency and meets our last 3rd grade standard on the list. CLICK We see this modeled at the end of the 3rd grade portion of the progression.
The MTI project is sponsored by the Idaho Legislature and the State Department of Education

15. 4th grade
4th Grade:
4.NBT.5 - Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations
Let’s move to 4th grade and see how the potential models and strategies progress as our number sets increases. Again we will see two branches which show the next logical steps to move from informal to formal models, but we will make connections to other models as we go.
CLICK We could again see students starting with repeated addition, but with our larger number sets this can become very inefficient quickly. This model can again transition to the ratio table CLICK where we are able to bridge additive thinking to multiplicative like we see in the table on the screen as we start to make larger groups and make connections to place value. CLICK Being able to be flexible with numbers and find ways to flexibly compose quantities (like finding 14 groups of 26) leads students to be able to mentally decompose and multiply or think of composing the 14 groups without the need for a ratio table like we see in the last model in this branch. Walk through last model in branch with pen.
Let’s look at another possible branch in 4th grade and see if we can make some connections between branches. CLICK We would start with the area model (possibly done on graph paper) where we have every number represented. Students could find the product by repeatedly adding rows/columns, but we need them to transition to seeing larger quantities like the 20x10 to help us transition to more efficient models. The model on the screen is decomposed by place value – which is how we eventually need students to decompose to get to models like partial products, but we could also see this done in many other ways. Can you see the parts of the ratio table and the mental math strategy in the area model. I’ll give you a minute to look. Point out where we see each on the screen (USE PEN)
Just like repeated addition, the area model that has every square shown can become very inefficient with larger number sets, but is place that students need to live for a while as they are wrapping their head around the quantities and their ability to decompose the factors. When this transition is made we can move to an open area model CLICK in which each square does not need to be represented. We still want to make the connection to area and proportionality as we use this model as we decompose the factors and find the parts of the product.
Finally we can transition the area model when done by place value to partial products CLICK. We see the same process done in both this and the area model except here we do not have the visual representation.
The MTI project is sponsored by the Idaho Legislature and the State Department of Education

16. 5th grade
5th Grade:
5.NBT.5 - Fluently multiply multi-digit whole numbers using the standard algorithm.
As we move to 5th grade we wrap up one number set and move to another. With this idea in mind, I have separated the 5th grade standards into two slides.
Our first is fluently multiplying multi-digit whole numbers with the standard algorithm. Reflecting back on what 4th grade has done to help build the foundation and understanding for this, 5th grade will now just need to take the final steps to formalize this number set by adding one last model for multiplying whole numbers – the standard algorithm.
A big focus in 5th grade is on the number sets and models we see with the second multiplication standard.
The MTI project is sponsored by the Idaho Legislature and the State Department of Education

17. 5th grade  building up to the problem
0.4 x 0.3
5th grade is the first time students will operate with decimals in the CCSS, so before we look at the models for the decimal portion of the 5th grade progression we are going to step back for a moment and look at a model with a smaller number set that could potentially build up to the models we see in our progression.
Take a moment to look at the model for the problem. Can you see how the models shows 0.3 of 0.4? I’ll pause to give you a few seconds to look. (30 second pause)
We see the 0.3 shown in the model by looking at all of the purple shading (circle). We then take 0.4 of the 0.3 by shading 4 of the 10 rows within the three columns (show with pen). This can be difficult to visualize without a model like the 10x10 grid because the 0.4 is in reference to the 0.3 so we are actually scaling down the 0.3 to 0.4 of its original size. Prior to multiplying with decimals (or fractions) students have only been exposed to multiplication as scaling up, so we need to spend time developing the idea of scaling down and connect this idea to visual models. This will also help set them up for proportional reasoning in later grades.
The MTI project is sponsored by the Idaho Legislature and the State Department of Education

18. 5th grade
5th Grade:
5.NBT.7 – multiply decimals to hundredths, using concrete models or drawings and strategies based on place value, properties of operations; relate the strategy to a written method and explain the reasoning used
Let’s take a look at how these models could potentially progress through the grade level. We will start with an by extending the example from the last slide to the 10x10 grid model that we have in our progression.
CLICK. Let’s investigate how this models our problem. Walk through model explaining parts (USE PEN) Just like our other area models in which each number (in this case hundredths) is represented this model gets inefficient quickly, but it is a place students need to start to help make sense of operating with this new number set.
From the 10x10 grids we could go down to paths. We’ll first investigate moving to the open area model. CLICK This is just like the area model we saw for our 4th grade number set as we decompose by place value and multiply to find part of the product. Just like students need an understanding of place value to multiply by 10, 100 or 1000 with whole numbers, they also need an understanding of place value to multiply by 0.1 or This is another standard we see in 5th grade in the CCSS which helps to support students’ number sense as they apply it to models like the open area model. We see this being applied as the student is able to multiply 0.3 x 0.4 or 0.4 x 2 in the model.
The area model can then transition to partial products just like we saw in 4th grade. CLICK These are decomposed the same way and give the same quantities but partial products does so without the iconic model.
Another path we could take from the 100s grid is to a bar model which can transition to a ratio table. CLICK In the bar model we start with the bar representing 2.3 and scale down to find 0.4 of the bar. Walk through solving with bar model (USE PEN) The bar model provides an iconic representation of the problem that will easily transfer to a symbolic model like the ratio table. We can see the same thinking in the ratio table. Walk through ratio table (USE PEN) Let’s take a minute and see if we can make the connection between the 100’s grid and the bar model and ratio table. Where do you see the 0.1, 0.2 and 0.4 of 2.3 in the bar model represented in the 100’s grid? I will pause for a minute to let you look. (30 second pause) Walk through connections (USE PEN)
The multiplicative thinking used in the ratio table can also transition to other ideas of scaling like we see in the last model CLICK. In this model we see the student’s mental process as they scale the 2.3 up to 4 times it’s size and then divide by 10 to find 0.4 of 2.3
Just like in the other grade levels we can see many connections among the models as we progress from informal to more formal strategies.
The MTI project is sponsored by the Idaho Legislature and the State Department of Education

19. 6th grade
6th Grade:
6.NS.3 - Fluently multiply multi-digit decimals using the standard algorithm
We finally get to 6th grade as we look the formalization of multiplication of decimals. Again, reflecting back on what 5th grade has done to build flexibility and understanding with this number set, 6th grade can formalize to the traditional algorithm. This adds one more strategy to for students to solve multiplication of decimals.
The MTI project is sponsored by the Idaho Legislature and the State Department of Education

20. Adding context to the progression
Take your context cards and place them next to the model(s) in the progression that would be driven by each context.
It’s time to take our progression one step further. We are going to add context cards to the progression to investigate how we can drive particular models in the progression at each grade level by providing different contextual problems to students. Take your 6 context cards and decide which model in the progression would be best driven by the context.
The MTI project is sponsored by the Idaho Legislature and the State Department of Education

21. How can the different contextual situations help with the progression?
Different context can lead to different ways of thinking about the mathematics, which can lead to different branches of the progression.
Now that you have the context added to the progression, reflect on the following question: How can the different contextual situations help with the progression? Small group then large group discussion
Context can be beneficial in many ways, but here is how I think about it related to the progression. CLICK
Different context can lead to different ways of thinking about the mathematics, which can lead to different branches of the progression. An example of this is when we use a problem like the donut problem to drive the ratio table in 4th grade, but give a question about laying sod (which relates to area) to drive the area model. Another example is when we see the gas problem driving the use of a bar model or ratio table as we represent 2.3 being 1 gallon and then scaling down from there to find 0.4 of a gallon.
The MTI project is sponsored by the Idaho Legislature and the State Department of Education

22. Why is each branch of the progression important?
Partial Products:
Traditional algorithm
Computation
Decompose numbers
Ratio Table:
Multiplicative thinking
Composing number
Multiplication as scaling
Ratio and proportion for Middle School
We see two branches in each grade level as we work through the progression, one that includes the ratio table and another that leads to partial products. So why is it important for students to work through both branches of the progression? Small group the large group discussion
Partial products:
Traditional – transitions well to the traditional algorithm
Computation – applying fluency, efficiency with some number sets
Decompose numbers – distributive property, decompose both numbers
Ratio Table:
Multiplicative thinking –
Composing number – composing 14 in our example
Multiplication as scaling
Ratio and proportion for middle school – scaling ratios up and down
The MTI project is sponsored by the Idaho Legislature and the State Department of Education

23. Building Mathematical Understanding
Take Students’ Ideas Seriously
Press Students
Conceptually
Focus on the Structure of the Mathematics
Building Mathematical Understanding
Finally let’s take a look at the 5 big ideas for Building Mathematical Understanding and where they came up in our progression.
Take Student’s Ideas Seriously: Listening to how students are thinking about solving multiplication problems allows us to identify where they are in the progression
Press Students Conceptually: As we move through the progression we are pushing students to more efficient models
Encourage Multiple Strategies: we see this as we look at the multiple models and benefits of each in the progression
Address Misconceptions: This wasn’t big in our webinar, but defiantly something we need to address with students as we help them create connections and meaning
Focus on the Structure of Mathematics: This was a big idea of our webinar as we look as the structure, big ideas and connections within the operations of multiplication across multiple grade levels.
Address Misconceptions
Encourage Multiple Strategies/ Models
© CDMT
January 19

24. Thank you for attending!
Questions
Contact Information
Jackie Ismail
DMT Website-
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