Exploring the Distributive Property

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My fifth grade crew has recently been exploring the distributive property.  What’s interesting is that most of the students have used the property before, but it just wasn’t labeled.  Students mentioned this as I introduced the concept earlier last week.  Although most of the students in the class had background knowledge of how to distribute numbers, the level of that understanding differs depending on the student. The majority of students have mastered many concepts related to number sense, but pre-algebra concepts are fairly new to them. That is one of the reasons that I chose to beef up a lesson related to the distributive property. I have a few specific resources related to teaching the distributive property that I thought might be helpful for this lesson.

The substantive mathematical idea of this month-long instructional unit is to have students experience algebra and use it with geometry/measurement ideas with algebra notation. Later on in the unit students explore the distributive property, apply order of operations, simplify expressions, solve equations, utilize the Pythagorean theorem and use size-change factors. 

The lesson began with an agenda.  The mastery objective for the day was “students will be able to identify and use the distributive property to simply expressions.” I briefly explained and drew an example of the distributive property on the whiteboard. At this point I wanted students to get a quick overview of the distributive property in action. This quick overview seemed to help introduce the concept to students that haven’t seen it before.

Students were then placed in groups to complete an initial distributive property activity. A scenario was given where students were asked to purchase three gifts for three different grandkids. Each grandchild would receive the same items. Students were asked to supply number sentences.  Feel free to download the sheet here.

The groups presented their findings and number sentences. During this time I was able to showcase how the distributive property can be utilized in this scenario. Based on the responses, students were still having trouble identifying and using the distributive property. Also, I was finding that students were adding each individual number instead of using a more efficient distributive property. Seeing that students needed more practice opportunities, I decided to move on to a rectangle method activity.

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Students were then asked to find the area of the above rectangle using two different numbers sentences. I chose this particular assignment because of the math connection opportunities. Students were recently studying measurement concepts during the last unit and it’s still fresh in their minds. So, students were given rectangles that were split into segments and they were asked to show different number sentences find the area of the shaded portion.  I placed the page on the document camera and the class reviewed it together. Students were given time to reflect, make connections and ask questions during this time. I also gave students an opportunity to preview the next few lessons and see how understanding the distributive property will help them as they simplify expressions later in the week.

The distributive property activity contributes to the students’ developmental conceptual understanding of the mathematical idea. Students are asked to create a rectangle, divide it, and then use two different number sentences to showcase the shaded area. Students are using factoring strategies to group numbers in order to find the area. In doing this, students are acknowledging that the distributive property is evident in the combination process.

Screen Shot 2017-04-30 at 4.52.51 PMScreen Shot 2017-04-30 at 4.52.59 PMScreen Shot 2017-04-30 at 4.53.08 PMI believe there were challenges evident when I presented these mathematical ideas to the class. Students often come into class with preconceived notions that parentheses are only used during problems involving the order of operations. I believe that the students’ understanding of the distributive property was strengthened through the use of the rectangle area activity. Although their understanding seemed to improve, some students need to be guided through the activity. They were unsure of how to start the problem and some needed prompts.

I believe that the student work I collected suggests that the next step in my instruction is to expand on being able to use the distributive property and combine it with translating equations into expressions.  The next sequential step is to use equations to solve problems involving integers. Although students have used integers in the past, it may be beneficial to review how negative integers impact the distributive process. Also, as I gave students feedback, I wondered if they would’ve been able to complete the same number sentences, but distribute the numbers from both sides of the parentheses. For example, could they connect that 5(11 – 10) is the same as (11-10)5 ? They’ve only encountered the first example, so this may be something worth investigating for the next time I plan this lesson sequence.  Having practice with these types of problems will benefit students, as they need to have experiences with using signed numbers with expressions.

Modeling Integer Computation

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My third grade class explored integers this week.  Over the past few days students have started to become more comfortable in being able to compare and locate integers on vertical/horizontal numbers lines.  The next sequence is integer computation.  I find this to be more of a challenge for students.  Specifically, some students find the concept of subtracting a negative integer to be confusing.  Most students have encountered computation at this stage as either addition, subtraction, multiplication, or division.  The idea of subtracting a negative isn’t something that they’ve experienced and can cause students to question their own understanding.

This topic was discussed at #msmathchat last Monday night.  The consensus was that students need to experience different models to gain a better understanding of how to put together and take apart integers.

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Manipulatives, such as counters and the such are always important.  I believe most teachers use some type of manipulative to showcase integer computation.  Sometimes they’re taken away too early.

The problem that sometimes comes up with this, is that students want to move towards only following a rule/process to find the solution.   This “answer getting” mentality can lead to a lack of understanding and isn’t beneficial long-term.  Wording also plays a role with integers.  Getting caught up with “add” and “subtract” can limit what students perceive.  How about find the “difference” between x and y?

Changing the wording and using a number line can make a huge difference and can empower students to rely on their own understanding of computation and integers.

I kept this chat in mind as my third grade crew finished up a lesson on integer computation.  Near the end of one lessons I gave each student a blank number line and asked them to find the difference between two integers.  The instructions are below.

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Students were given dice and headed to work.  Students ended up rolling the dice and then created their number lines.  They were required to show a number model, the number line and any type of work that was used to find the solution.  The number line was initially blank and they had to fill it in with the numbers related to their problem. There were initial questions, but it seemed as though the multiple models/strategies were beneficial.

I believe students are making progress in better understanding how to put together and take apart integers.  There’s more work ahead of us, but I’m excited about the growth so far.  Next week, the third grade class is scheduled to use a number line to show multiplication and division.  I’m thinking of using a similar model for those lessons.

Improving How Students Analyze Their Work

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One of my goals this year was to have students analyze their own work, make observations and improve. These observations have improved this year by a light margin. For example, students get back their graded paper and look over how they did. Most students look at the top for their points or some type of feedback. Some look for where something was marked incorrectly, while others look for a place in their binder to place the paper. The good news is that students are looking at their graded papers with a more critical eye. That’s a win in my book. Students are starting to observe where they needed to elaborate or change a procedure. That’s good, but the time spent looking at what to change is still minimal.

This year I introduced the NY/M model. Students were a bit hesitant at first, but I’m finding some pockets of success. Those pockets are not just related to the new model, but also a whole range of opportunities that have been put in place for students to understand where a mistake might’ve occurred. Ideally, I’d like to have students identify how the mistake or error happened and to curb that action in the future. Don’t get me wrong, I’m all for making mistakes in order to learn, but some errors impact an entire answer and I’d like students to be able to identify where that’s happening. Being able to self-reflect in order to improve is a beneficial skill.

In an attempt to provide multiple opportunities for error analysis, I’ve intentionally planned for students to identify their own math misconceptions. This has taken many different forms. I believe that students that can identify math misconceptions may be better able to proceed without making them in the future. Three tools/strategies that have been helpful in this endeavor are found below.

  • Nearpod has been a useful too this year. Specifically, having students show their work using the draw tool has helped other students identify misconceptions within their own understanding. Displaying the work on the whiteboard without a name has been especially helpful, as a student might not be embarrassed, yet the class can still learn from that particular person. I’ve used this as an opportunity to look at positive elements of student work and also look for areas that need some bolstering.
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What do you notice?

 

  • Lately I’ve been giving feedback on student papers and incorporating that into my agendas. Before passing back the papers I review the misconception list and answer questions then. I then pass out the papers and students complete the NY/M process. Generally, students make very similar errors and I attempt to address this while reviewing the agenda. This has decreased the amount of questions that students ask related to why/how to improve their answer to receive a M.

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  • On the paper I’m making a renewed effort to write feedback on homework and projects. The feedback takes many different forms and isn’t necessarily in a narrative form. Sometimes I ask question and other times I might circle/underline a specific portion that needs strengthening. This method often elicits student questions as it’s not as clear-cut as other methods. Regardless, it’s another way for students to analyze their work, make changes and turn it back in a second time.

Why is this important to me? Well, I believe that students should be provided additional opportunities to showcase their understanding. At times, I feel as though there’s a gap between what math work they show and what they’re capable of showing. Giving feedback, along with another opportunity to improve, tends to help my students show a real-time understanding of a particular concept. Ideally, this would seamlessly work and all students would move from an NYàM. It’s not all roses though. I’d say at least 50% of the students improve on their second attempt, but I’d like to see more. I believe we’re making progress and have more to go, but I believe we’re on the right track. I’m encouraged to see that this model is slowly and slightly changing the review, redo and improve cycle. This has me thinking of how to expand on it for next year. Stay tuned!

Proportions and Action Figures

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My fifth graders are in the middle of a unit on ratios and proportions.  Two weeks ago the fifth grade kids worked through a detective mystery.  It was a good start and a decent opportunity for students to explore proportions.  This week my students came back from spring break.  It took a while to get back to our regular schedule after a week off. We had to complete some review on Monday and started a new project today.  I was meaning to start it before break but we ran out of time.  I researched a few different sites (1,2,3) and decided to modified my original project.

The class began by talking about proportions and scale models.  The discussion lasted around five minutes and then we reviewed the concept and vocabulary with a Kahoot  The majority of kids were able to answer the questions using estimation, but many were challenging, which was good because we were able to stop and use different strategies.  Most students wanted to cross-multiply for everything, but by the end of the activity students were starting to see the value in diversifying their strategies.  I felt like spending time on this was worthwhile.  This experience reminded me of a Tweet from #msmathchat from last night.

Afterwards, I introduced the action figure project to the students.  Students measured the dimensions of the action figures.  They measured the figure in millimeters, converted it to centimeters and eventually to inches.

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They then measured their own dimensions with a partner and compared them to the action figure.

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Many of the students were able to use proportions as a tool to find a solution.  Some students had a bit of trouble tackling the issue of converting the units.  Overall though, students are becoming better and using different strategies to solve proportion problems – an #eduwin in my book.  You can access the files that I used for this project here.

 

Math BreakoutEdu

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Today is the last day before my school’s Spring Break. Generally, my classes end up finishing up a particular unit before a large break. This time is different. Both my fourth and fifth grade classes are in the middle of a unit.   I’m also finding that both classes are due for some review. Foundational pieces involving place value and order of operations are tripping up some of the students. While looking around for resources, I came across a BreakoutEdu that corresponds with March Madness. I need to give a huge shoutout to Rita for creating this game. I’ve used Breakout in my class before, but am still in the rookie stage. I printed out the files and started to compile them last week. I figured out which locks where needed and started to compile a few different ideas on how it would work.

What’s great is that my school’s media specialist, the fantastic @mrsread, has a teacher BreakoutEdu box that’s available for checkout. I was able to checkout the box and fiddle with the locks earlier this week. I was able to get most of the locks figured out and reset to the codes needed for the activity.  I say most and not all because the multi-lock is still giving me issues. After checking on the forums it seems like this tends to happen more frequently than I originally thought.

After becoming a bit more confident in how to use the set in my own classroom, I decided to use the Breakout with a fourth grade math class this past Thursday. Since I couldn’t use the multi-lock, I decided to use a combination lock that I had at home. I put together a small Google form that coordinated with that particular lock. The next day I spent my planning period organizing the materials. I decided to go with manila envelopes to store the papers and deviously hid them around the classroom. I introduced the game with the slide show in the file at the bottom of this post.

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From the March Madness Breakout PowerPoint

The kids were excited as they already completed a Breakout a few months ago. I told the students that four manila folders were in the room and they had to find them to locate the clues to open the box. I then started the timer and they were off.

Observations

The class of 21 split themselves fairly evenly and started working on the tasks. It just so happened the Google form was completed quickly and one of the locks was open in less than five minutes. That wasn’t my intention.  I was hoping it would be a bit more challenging.   The other tasks, especially the order of operations, took more than 20 minutes to complete. I noticed that around 4-5 students would be working on the sheet while others congregated and tried to find more clues. Some of the kids were making simple errors with the order of operations. The bracket challenge was also tricky, as some students didn’t understand how a bracket worked. Students would complete the bracket and not understand that the larger number would move on to the next section. I could tell that students were getting frustrated as time ticked away.   I didn’t interject although I wanted to help. Eventually, students had to use a hint card, but they prevailed. We had a great conversation afterwards using the Breakout reflection cards. This was also great for me to hear, as students gave feedback about which particular tasks were the most difficult and how they contributed to their team.

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The Breakout worked so well that I decided to use it with my fifth graders the next day. I changed up the Google form piece and made it more aligned to what we’ve been learning over the past few months. I even added a question where students had to translate a problem from German to English. I may or may not have had a ton of fun helping create questions for the game.

Overall, the game went just as well with the fifth grade group, although they had more trouble with the locks. They were a bit confused with the combination lock. Once they figured out at that skill the class opened up the final locks with about 15 minutes or so to spare. The class didn’t have time to review the reflection cards. I’m hoping we can take those out after spring break is over.

Rita’s files for this Breakout can be accessed here. Feel free to use the Google forms (1) (2) that can be copied and used as well.

Experiencing Proportions

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My fifth grade class just started a unit on ratios and proportions.  This is fairly new vocabulary to them, although students have used proportions before.  They’ve used them before to convert measurements and fractions, as well as other items.  They weren’t called proportions at that time.  Instead, I remember discussing them as conversions or creating equivalent fractions.

So, on Tuesday the class was formally introduced to proportions.  I started off by using a Brainpop video that helped introduce the concept.  We watched the video twice and answered the quiz as a class.  Through this process it seemed as though students were starting to become more familiar with proportions.

The class then picked up their math reflection journals.  The class completed a few proportion examples.  They were able to use a few different strategies to complete the proportions and seemed most comfortable by using a cross-multiply and solve for x strategy.  Maybe so, because that’s what’s fresh in their minds from a past pre-algebra unit.  We continued to work on a few different math journal pages.  The majority of students were starting to pick a strategy to solve the proportions.  Although most were feeling confident about proportions, I had a group of students that were having trouble.  I decided to reach out to a few different people about proportions and found the Tweet below.

After reviewing the documents, I decided to try out the Illuminations activity with my kids.  I brought all the kids to the front of the classroom and explained that they will be solving a mystery.  The kids were stoked.  I had to go over the directions multiple times, but after around 1o minutes I believe they were all on board.  I put the students in teams.  I then told them that not all students would be catching the same culprit.  Students were confused about this, but I thought it added a wrinkle to the activity.  I passed out the sheets to the students and they were off to working on their own.

Almost all of the teams had questions about how to proceed.  I had the teams tackle the problems on their own for the first ten minutes.  Fortunately, teams started to show some perseverance and solved the first problem based on the clue that they were provided. Photo Mar 11, 6 25 34 PM The teams used the strategies discussed earlier in their journal pages.  After around 20 minutes I had teams starting to come up to me with their final answer.  I gave them a thumbs up or thumbs down.  If they didn’t have the right culprit I asked them to redo a specific question that would move them in the right direction.

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I feel like this was an activity that helped students become more aware of proportions and how to solve them.  The overall goal was successful, although I need to reflect on some of the team dynamics that played out.  Not all the teams worked well together.  Some students were more confident than others, and some students wanted to let others do the majority of the work.  I think this tends to happen in varying degrees during group work.  Although this happened, I still feel that the student conversations added to activity.  The ideas and strategies that were being discussed seemed to benefit all involved.

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Illuminations warned me from the start  🙂

 

Next week I’ll be using the questions to have the students reflect on this activity.

 

Tinkering with Second Attempts

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I believe teachers are always tinkering to see what works best.  I’m in the midst of doing just that with my NY/M strategy.  Near the beginning of the year I noticed that students would take quizzes and focus on the top third of the page.  Near the top, often right next to the assignment title would be a score.  That score would mean the world to the kid. Students would take a look at the point value and immediately make an evaluation based on that that score alone.   The internalization and analysis that I’d hope for wouldn’t happen.  So this year I decided to move a bit closer to a standards-based model with assignments.  Students would be allowed to redo an assignment for credit.

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At first this strategy seemed to make a huge difference.  Students took more risks and it seemed to curb some of their anxiety.  In the most positive cases, I’d write specific feedback on the students’ papers and place a NY at the top of the paper.  Students would redo the sheet and put it back into the turn in bin.  The less-than-perfect cases would involve me putting  a NY at the top of the page without additional markings.  In all honestly, sometimes the errors were careless mistakes and didn’t require much feedback on my part.  Regardless, students were turning back in the NYs and around 60-70% would receive Ms in return.  This was good news.  Although I was glad that this strategy seemed to be working, I started noticing a trend.

Some students would put less effort in completing the assignment the first time knowing that they’d have another opportunity.  Also, students would redo an assignment and not truly analyze what they did wrong in the first place, so they ended up netting zero.  This started to discourage me and that’s when I started tinkering again.   I asked a few different people on Twitter about the logistics behind retakes in their math class.  One person stated that her students fill out a form before retaking the assignment, while another mentioned that the student would be required to get extra help before the retake. I don’t believe there’s  a right or wrong answer to this.  I believe that there needs to be some type of reflection/feedback that occurs before the retake.  That could help students become more aware of what happened during the first attempt and prepare them a bit better for the second.  Being aware of where a hiccup happened is usually the first step in the reflection process.  So starting this week my students will be filling out the sheet below before retaking a quiz.

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I’ll probably continue to tinker with the wording, but it’s a start.  I want students to be able to analyze their first attempt and find the reason behind the retake.  Sometimes, the reason is because of a simple mistake, but I’d like kids to move beyond that as most reasons for the second attempt are mathematical.  I’m looking forward to seeing how this works during the last third of the school year.