Connecting and Extending Math Strategies

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This post has been marinating for a few weeks. I started thinking about it before NCTM and afterwards found it just as valuable.  In Late April my third grade students were exploring fractions.  This third grade group had experiences putting together and taking apart fractions earlier in the year.  They also used number lines to multiply a unit fraction by a whole number.  Students have slowly and steadily improved their confidence in being able to use fractions when completing application problems.  That’s when I came across a problem that seemed to stretch the fraction strategies that they’ve been using.

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I displayed this problem on the whiteboard.  I gave the students some time to digest the task and asked them what their first step would be.  A few students wanted to dive right in and use repeated addition to find the solution.  They thought this might be the most efficient method and what’s available in their strategy toolbox.   Other students thought that there might be a better strategy.  One students had an idea to possibly use an area model for this problem.  Curious looks came across the classroom as this particular student spoke up a bit more.  She mentioned that using a partial-products strategy could be helpful in a situation like this.  Some students agreed, while others were shaking their heads in disagreement.  For the most part, the class is very familiar with partial-products so the area model wouldn’t be a stretch.  Students also had an understanding that multiplication might be one way to tackle this problem.

I gave students time to discuss in their groups the different ways to solve the problem.  There was a lot of disagreement happening during this time.  I wanted to stop the whole class because of the noise, but the math conversations were actually really good. Students were vocal about using an area model, while some students said that they could find the answer by using a traditional method.  More students seemed confused, as this isn’t really a method that students have used with decimals.  I gave the class time to independently try out their strategy and find a solution.  Students worked on this for around five minutes.  Pictures were drawn, number lines constructed, and repeated addition was in full swing.  Some students had a correct solution, but I didn’t give an affirmation (which is crazy tough not to do).  I asked students to show a number model and most generated (5 * $1.13) + $1.39 = ?  I displayed the different strategies on the document camera.  I then asked the class if it’d be possible to use fractions to find a solution for this?  Just then, a group of students raised their hands.  They said that the 1.13 could be represented by 1 13/100 while 1.39 is equivalent for 1 39/100.  A collective whispering sound came across the classroom.  Then a bunch of hands came up from all over the room.  The class went through this strategy to find a solution.  Our number model changed a bit.

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The class then took the number model and used an area model.

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The students then added the 1  39/100 to the 5 65/100 to get 6  104/100.    Students then said that the answer is 7 and 4 hundredths.  Another students chimed in that the answer can’t just be written as a decimal and that it’s $7.04.  Students were satisfied with this method and asked if they could use it in the future.  I asked them why not?  A couple students said that they didn’t see fractions in the problem.  They saw decimals and that automatically changed the approach to a more “decimal-friendly” computation strategy. A few students replied that they didn’t think they could use it because it wasn’t taught.   Ouch.

After the class I pondered that response and thought about what is explicitly taught and what we expect students to explore and construct on their own.  Do we truly give students opportunities to discover mathematics in a way in which they can bridge strategies?  Not one student verbally called out to using a decimal to fraction conversion for this problem.  This has me wondering how explicit teachers are in empowering students to bridge strategies and concepts.  Are they given that freedom?  For example, are fraction strategies only used when students see fractions?  Is the substitution method only used when students see 1-step equations?   I believe it depends on the situation and background information also plays a role here. Having a deeper conceptual understanding of fractions and decimals might have led students to look at how the last strategy might be helpful.  Also, while an area model might be helpful now, the strategy will most likely change as a student’s math journey progresses.  Multiple strategies are beneficial for any learner, but this lesson has me asking more questions.  It was a terrific lesson and I know students were making connections.  That’s an #EduWin in my book. Next week the third grade class will be investigating angles.

Side note:  I finally submitted by components for board certification this week.  It’s a huge relief to send the files off, but now it’s a waiting game as I won’t get the scores back until December.

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