# Encouraging Students To Make Deeper Mathematical Connections.

Because of all of the math talk my students do in class every day, they are very comfortable (and flexible) in sharing multiple strategies and solution paths. They can explain others’ strategies in their own words and agree and disagree with one another beautifully, however when asked to make connections between two representations (numerical or visual), I feel like I get very “surface” connections. I will read things in their journals like “They are the same because they have the same numbers” or “They are different because we double and halved the other numbers instead” or “We both used an area model” Something like this…

After going through their journals the other day with Faith, we were thinking and questioning one another about how we, as teachers, can have students dig deeper into the connections. We obviously would like to them to notice them but if not put in a position to make connections, will they on their own? Is there a way to frame a task or question that would push them to think a little deeper about how and why the two representations are alike/different yet still arrive at the same answer? How do we encourage students to make deeper, more meaningful connections when we know they can, but just may not be sure of how to get there?

Instead of a number talk they other day, I did a math routine I named “Where is _____ in _____?” I was hoping the prompt would have them go beyond just looking “at” the representations and look “into” the meaning of each representation. On the board, I posted some examples of their representations from the day before in which they had done a surface job of connecting. I had them work independently for a few minutes and then talk as a table before the group share. I was much happier with the conversation and felt like asking them to look “into” the problems really got them thinking about what the representation was showing.

This an example of two area models in which students the day before had simply said, “We both used the area model” without thinking about how they were related. I love the (.3 x .2) in each of the quadrants of the first grid.

This student had a different take on how the two area models were alike, which led to such an interesting discussion! She also did some lovely work with showing how the two distributive properties were within one another through factoring.

This student showed the distributive property and double/halving in a wonderful way…

This one was the only student who connected the area model of .6 x .4 to the strategy of .6 x .5 – .06. He showed where the .6 x .5 would be in the model and then scratched out where the extra .06 were coming off to arrive at the answer.

Having students make connections in math is so incredibly important and so difficult to do, especially with so many variations in strategies and representations. I would love to hear other ways to encourage these connections!

-Kristin

# Creating Contexts for Decimal Operations

Sometimes I have students engaging in math within a context, however at other times, we just explore some beautiful patterns we see as we play around with numbers. I see a value and need for students to experience both. This week was one of those “number weeks” and it was so much fun!

Over the past few weeks, we have been working on decimal multiplication. If you want to see the student experiences prior to this lesson, they are all over my recent blog posts….it is has been decimal overload lately:) After sharing strategies and connecting representations in this lesson, I was curious how students thought about this problem in a context because up to this point, I had not given them one for thinking about a decimal less than one times a decimal less than one.After they wrote their problem, I asked them to tell me what they were thinking about as they were deciding on the context.

I anticipated that many would refer back to what they know about taking a fraction less than 1 of a fraction less than 1, like in this example…

I love how this one said she knew she “had to start with .4” That shows the order of the numbers in the problem create a context for her. It mattered to her, taking .6 of the .4.

This student went with two different contexts and again saying that he started with the .4. This must be something we have chatted about quite a bit about because it showed up multiple times. I loved how this student said he thought about an area model in creation of his problems.

This student was great in listing all of things he was thinking about as he thought about a context..

I had students who attempted to create a “groups of” context. I don’t know if I ever realized how difficult this and how much I, as an adult, need to be able to create a visual in my mind of what is happening in a problem to make sense of it. Here is one example (not the sweetest context but she thought the Mary HAD a little lamb was clever…) She worked a bit yesterday to show what the representation would be, but kept running into problems with cutting into “.6 pieces.”

And then I have these two that had my brain reeling for a bit, for many reasons. First, does the context work with this problem? Secondly, I knew it sounded like it should work, but when I tried to make sense of it, I couldn’t create a visual. Also, as I read them, I thought I knew where it was going and the question I would pose, but it wasn’t the way they saw it ending. I asked them to create an Educreations about their problem so I could check out their thinking around the context.

Yes, Rick Astly. But the question at the end, compared to the total time Never Gonna Give You Up, threw me a bit, not where I was going with it….

His Explanation: https://www.educreations.com/lesson/embed/31398809/?ref=embed

The second one tried it out, and wasn’t so sure of his question after messing with it. The wording “.6 as small” was making me think. I was trying to make sense of that wording, do we ever say six tenths times as small? Then does his question referring back to the .4 make sense?

His Explanation:https://www.educreations.com/lesson/embed/31402039/?ref=embed

Definitely a lot for me to think about this week too! I have some amazing work with them connecting representations to write up later…they are just such great thinkers!

-Kristin

# Multiplying Decimals Less Than 1 Whole

Apologize, not much time to write, but today was so cool I had to share!Â  I am in the midst of using this work to better plan for tomorrow.

Short version: After our predictions yesterday, I posed 0.4 x 0.2 (I changed the problem to 0.6 x 0.4 for my second class) and asked the students to individually jot down what they thought the answer is. I was looking to see how they intuitively thought about the problem.Â  As expected, I saw 0.8 and 0.08 (2.4 and .24) as I walked around. I wrote both answers on the board, asked them to write their reasoning in their journals and then we shared as a class. No telling which was right or wrong, just sharing and listening.

Some great thinking and critiquing of each others’ reasoning ensued and then I sent them off to come to a consensus as a table and create a poster of how they thought about it!

Â  Â Â

Now, where to go with this work? They could just look at strategies, but I want them to think deeper about the meaning of the problem.Â  After chatting with my colleague Faith tonight, who is coming to observe tomorrow, we are going to have the students walk around to the other group posters and talk about what they saw on the other posters that changed the way they thought about the problem.

From there, I really wanted them to think about a context for this problem and Faith suggested also thinking about what happens are you begin to adjust the numbers and why….really thinking about the reasonableness of answers. What happens when one factor increases? What happens when one factor goes over a whole? What happens if the factors go into the hundredths? Does the product increase or decrease? Why?

So many fun convos to be had tomorrow!

-Kristin

# Making Decimal Predictions

Over the past weeks, I have done a lot of blogging about our work with decimal multiplication. All of this work has been focused around contexts that involve multiplication of a whole number by decimals both greater than and less than one. The students have very flexibly moved into using whole number strategies in order to multiply decimals during our number talks. Today I asked them to think about how whole numbers multiplication is similar or different from multiplication involving decimals. I was hoping to hear the relationship between the factors and the product and they did not disappoint. These are the findings from my two math classes…

I asked them to prove that a decimal greater than 1 times a whole number will have a product that is greater than both factors OR if a whole number, less than one, times a whole number will have a product that is less than one factor but greater than the other.

We shared out and ended the class predicting what they think would happen when we multiply two numbers that are less than one. This is where I saw an interesting difference in the way students thought about the problem. Some focused on the numbers and what it means in an “of” sense, while others connected to what happens with the multiplication process.

This makes for such an interesting conversation tomorrow! Excited to see the fractions come out and for students to revisit their predictions! This is the work tomorrow from last year’s experience: https://mathmindsblog.wordpress.com/2014/07/25/unanticipated-student-work-always-a-fun-reflection/

-Kristin

# Connecting Whole Number Operations to Decimal Operations

I planned today’s number talk to draw out a variety of strategies for students to reflect on as they worked through their decimal work today. I used this series of problems:

4 x 18, 12 x 18, 39 x 18

After a variety of strategies such as partial products, area model, double/half, triple/third, friendly numbers, and adjusting a factor and product I paused when I posed the final problem and asked them to estimate. Thumbs went up right away and I go predominately two answers, 800 (from 40 x 20)Â  and 720 (from 4 x 18 x 10). We discussed if it was going to be more or less than the actual answer and then we finished with a student subtracting 18 from 720 and arriving at 702.

While they were still on the carpet, I told them to be thinking of all of these strategies as they were going through their work today because we would be reflecting on them later. I posed the following problem and asked them to think about an equation and answer, “Bob is running 7 miles each day for two days, how many miles did he run.” I got 7 x 2 = 14 and then 2 x 7 = 14 because “it is two groups of 7 miles.” It was nice when a student said the commutative property makes that not matter for the answer. So I followed with, “What would it look like if he ran 0.7 mile each day? More or less than 14?” They said less because it is much smaller and we wrote 2 x 0.7 = 1.4.

They went back to their tables and I asked them to think about how we could represent these two equations on a number line. My thought was that it would give them a visual of the size (magnitude) of the jump and help in determining reasonableness. Eh, mistake on my part…I forced that number line on them and, while it was fairly easy for most, some really struggled. The upside was, it was a great formative assessment for me to see how students approach number lines (many like putting the 1/2 in the middle, yet had 14 on the end and were confused). We used number lines a lot in our fraction unit and definitely went past a whole on them, however I guess I did not really make that connection to fraction multiplication on a number line. Mental note for future work:)

After we had our number lines up on board and talked about determining reasonableness based on the factors. I posed this, “Let’s say Bob ran 2.8 miles a day for 8 days. What would be a reasonable estimate for his total miles.” They talked at tables, came back with 24, 20-24, and 17. We talked about the actual being more or less than each estimate.
Fabulous, now I want you to show how you could find the actual answer as many ways as you can.”

They went around to the different tables and talked about which strategies they had that were the same, one they maybe had not thought about and then which one they connected most with. After they finished walking around, I had them reflect on any of those questions in their journals…

Aww….”a bad number you can round to a happy number.” ðŸ™‚

I love this student picked this strategy up from another group!

“The distribution property”:)

Can you tell they did not take to that number line at all? Not one number line. I also anticipated some fraction work, but they were really working with the decimal in connecting with whole number multiplication. It was a really fun day of math!

-Kristin

# Decimal Multiplication: Whole # x Decimal

Through numerous Decimal Number Talks, Investigations on tenths, hundredths, and thousandths grids, and many findings about decimal operations, we are approaching our last couple lessons in our decimal unit. Not that the work with decimals ever ends, but our unit ends with decimal times decimal and the generalization of a “rule” for multiplying decimals. I have many thoughts about the new Investigations unit on multiplication of decimals but I am very excited about the connections my students have made between whole number and decimal operations. I do attribute a lot of their flexibility to our Number Talks though:)

I wanted to assess where they were before we moved into a decimal times decimal work because I think there is a lot of reasoning to do there before we come to a generalization!Â  I was really excited to see the use of multiple strategies!

First, I had students who were still treating the decimal operations like whole number operations and reasoning about where the decimal point “makes sense.” I do love this because it is heavy in estimation and sense making about what is reasonable. It is obviously not the most efficient strategy, but I what I truly learned from this, is that I need to do more whole number multiplication work with this student to build efficiency…

I have students that love partial products….(and I cannot get some students to stop saying the “box method”….:)

I loved this area model because of the size of the .4 side. She was very particular about that!

Some friendly number work…I especially loved her estimation first….yeah!

I had some who multiplied the decimal by 10 and then divided their product by 10…

Saw some halving and doubling…

I had a student think about the decimal as a fraction. It starts at the top and then he jumps to the bottom of the page.He said he multiplied 9 x 12 to find out how many “rows” he would have, 108. Then he divided it by ten because there were 10 rows in each grid.Â  It was interesting!

So tomorrow we start decimal by decimal multiplication…I feel great about our start and I look forward to having them reason about decimals less than a whole times less than a whole.

-Kristin

# Decimal Multiplication

This is a quick lunchtime post, so not much time to reflect or analyze, but wanted to throw it out there..

We did a Decimal number talk today and ended with the problem 5 x 4.6

I had students double half, use partial products, and use friendly numbers. (Incorrect answers were also written in case you are wondering about the 21.6 and 35…we looked for errors also)

Then one student said that she bumped the 5 up to 6 and did 6 x 4 because she knew that faster, however she couldn’t figure out how to adjust her answer based on what she did. We had already had someone explain how they adjusted from 5 x 5, but this was not the same. She knew that she needed to subtract 1 (because we had already established 23 was correct) but where was that 1 coming from?

I sent them back to their groups to talk about it.