Performance Task Lesson - Decimal Operations

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Objective

SWBAT develop strategic competence, adaptive reasoning, and problem solving skills.

Big Idea

Problem solving is important, takes significant time and repeated efforts, and requires reflection.

Understand

5 minutes

To begin this lesson I read today's Task out loud, for students as listeners.  Then, there is a second read of problem, for annotation time (this can be out loud or independent).  The goal of this section is to help students make sense of the problem.

The questions in bold below are what I say to students, followed by possible student responses.

What information does the problem give us?  The problem tells us how many chicken strips are in one pan at each restaurant, and how much one pan of chicken costs at each restaurant.

How did you annotate? (annotate your copy under the doc camera as students share)

What is the question asking us to find? We have to decide which restaurant Ms. Martin should buy the chicken from.

What information do we not know? We don’t know which restaurant is the least expensive.

Are there any words in this problem that you don’t know?  If we don’t figure out the meaning of that word, would you still be able to solve this problem?  What do you think this word means? (It’s not likely that there are words in this problem that are unfamiliar to students, but this questioning sequence is a good one to instill in students that sometimes there are words that students might not know.  Defining those word are not always necessary in order to solve the problem)

Take a few seconds and write down a prediction about which restaurant Ms. Martin should use.  Have 1-2 students share out and justify their predictions.  As students share out, have them include their justifications.  You can also have students show if they agree or disagree with the shared predictions.

Exploration

12 minutes

Students have 4 minutes to work on their own, and then 8 minutes to work in partners.

In general, as students are working on more novel problems, I expect them to:

• Understand the problem using appropriate annotations
• Represent the known and unknown information
• Seek a strategy to solve the problem by applying understanding of mathematical connections and rules
• Determine a solution and check the reasonableness of the answer using alternate methods and re-contextualizing the solution

As students are working, I circulate around the room.  I'm asking:

• Why is this strategy helpful?  Why is this strategy not helpful?
• What information are you hoping the strategy will provide you?
• What is it that you’re trying to do?
• How will what you’re doing help you to decide which restaurant Ms. Martin should order from?
• Does the fact that you can’t purchase partial pans matter? Why?

There may not necessarily be content misconceptions that come up in this particular lesson – this is the last lesson of the unit, and students have the chance to apply what they've learned about decimals over the last few lessons.  Students may make mistakes during the sense-making phase, which can be addressed through the questions above.

After 10 minutes, ask students to raise a hand if they are ready to make a decision for Ms. Martin.  During the partner exploration time, determine the students who will present their strategies to the class.  Look for a student who used unit cost, a student who used LCM (but have your own version ready, in case the strategy was not used, and a student who used an approach that was not leading a student to a correct solution).

Presentation and Discussion

23 minutes

In this section, students have the opportunity to present their Work to the class.  The students should be engaged in meaningful conversation about the strategies they've tried.  I've included two strategies here.

• Strategy 1 involves students finding the unit cost of one chicken strip, to determine which one is cheapest.  This is the strategy that students will likely use.
• Strategy 2 involves finding the LCM for the number of chicken strips in each pan, and then finding and comparing that cost.  Students are not likely to come up with this method, but it does tie in to the work earlier in this unit and provides opportunity for conversation about solution paths.

I plan to display the work of one student who found the unit cost of one piece of chicken (student’s won’t necessarily have the language of ‘unit cost,’ which they’ll learn in the Ratios unit).  The following is a script of what I say to students.

Take a look at (name’s) work and decide what (s)he did here.  Raise your hand to get us started.  Have one student explain what they see as the strategy on the board.

Does anyone want to add to the observations about this strategy?  Raise your hand if you also tried this strategy.  Anything to add?  To the owner of the work:  (name), share with us why you decided to approach the problem in this way.

Now, when I was preparing for our class today, I also worked to decide which restaurant Ms. Martin should order from.  I came up with a different way to solve this problem.  It took me awhile, but I stuck with it, and I also decided that Ms. Martin should order from Magic Chicken.  Check out what I did!  Display your work sample, using LCM and finding the cost of buying 630 pieces of chicken

Let’s make sense of my strategy.  What do you see that I did? Take hands, and have students work together to articulate that this strategy involves finding the LCM, and then comparing the cost of buying 630 chicken strips at each restaurant.  (or, if a student did use LCM, have the student present his/her work)

Why did I (or the student) use 630 as my number of chicken strips to compare? This is the LCM of 14, 18, and 10  What does this mean, within the context of this problem?  630 is the smallest number of chicken strips that you could buy at all three restaurants.

Look at the work under Chicken to Go.  I took 630 and divided by 14.  What does that 14 represent?  The 14 is the number of chicken strips in one pan at Chicken to Go.  So, why did I divide 630 by 14? Have students turn and talk and then have one share.  Dividing 630 by 14 will let us know how many pans Ms. Martin needs to order at Chicken to Go.

How many pans did Ms. Martin need to order at Magic Chicken?  35 How do we get to that number? Divide 630 by the number of chicken strips in one pan (18).  So, once I figured out that she’d need 45 pans at Chicken to Go, 35 pans at Magic Chicken, and 63 pans at Chicken and More, what did I figure out next, and how did I figure it out?  Cold call on a student after a few minutes of think time.  Next, you figured out the cost of the pans of chicken at each restaurant.  To find the cost, you multiplied the cost of one pan at each restaurant by the number of pans Ms. Martin needed at each restaurant.

Which is more efficient, what (student) did, or what I did? Why? Have students turn and talk, and then share out. Students should say that the student work was much more efficient.  You found the right answer, but went through more work to get there.

Now, have student with erroneous strategy present.  (name), you tried something different.  Tell us about the decisions you made.  Model here giving the student praise for sharing something that didn't work and also point out things that you like about the work (organization, annotation, something that is mathematically correct, etc – you want to set the tone that making mistakes is a totally normal part of math, and that we can all learn from mistakes, AND that presenting mistakes will be a regular, no-big-deal sort of thing)

Possible questions for presenter:  Where did you get stuck?  How did you know you were stuck?  What did you try next (or,what would you have tried next)?

Wrap up the discussion by naming the key point:  There are multiple ways to solve real-world problems, which is one of my favorite things about math!  Some strategies are more efficient than others.

Extension

10 minutes

Students will now complete an Extension to the original problem.  Before students start to work, I help them to make sense of the problem.

Now, you’ll have the chance to try an extension of this chicken problem.  This is a second part of the task we’ve just worked on and talked about.   First, though, read through the problem and look up when you’re finished.

What is this problem asking you to determine? We have to decide if Ms. Martin should still use Magic Chicken if she needs to have 70 chicken strips.  What information will you need to use from the first part of this problem, in order to work out this part of the problem?  Turn and talk with your neighbor.  Have students share out.  They should identify that they’ll need the cost information from each restaurant for this problem.  They also need to identify that Ms. Martin cannot buy partial pans.

Let’s make a prediction about the outcome of this problem. Have students vote on which restaurant they think will be the cheapest for 70 chicken strips.

Can we just say ‘hey, Magic Chicken was cheapest before, so that’s the restaurant she should buy from!’?  Why not? Because Ms. Martin can’t buy partial pans, we have to figure out how many whole pans she needs, and then find the cost.

Students complete extension problem in pairs.  As they work, I circulate around the room and check in with each pair.  I'm asking:

• What was your prediction?  Why?  How are you going to prove or disprove?
• What strategy are you using for this part?
• Why is this strategy helpful?  Why is this strategy not helpful?
• What is it that you’re trying to do?
• How will what you’re doing help you to decide which restaurant Ms. Martin should order from?
• Why not just use the cost of one piece of chicken and multiply by 70?
• Does the fact that you can’t purchase partial pans matter? Why?
• Is there anything from the work you did in the first problem that will help with the second problem?
• Can you use/Are you using the same strategy that you used in the first part?  Why or why not?
• Which restaurant should she order from? (towards the end of the 10 minutes)

The Student Work shows possible strategies students might use on the Extension problem.  In the first strategy, students may create and extend a table for each restaurant to find out the cost of at least 70 pieces of chicken.  In the second strategy, students determine how many full pans of chicken Ms. Martin needs and then find the cost of those pans.  In this instance, both of these strategies are strong.  Strategy 2 is a bit more sophisticated.

Evaluation

10 minutes

I have students raise hand for each restaurant, to vote for which restaurant Ms. Martin should order from.  I then pick one student to present her/his strategy for this extension problem.  I have the class to ask 2-3 clarifying questions of the presenter, if needed.

Students turn to their partners and summarize the strategy that was presented.  I cold call one student to share the summary of the shared strategy

Why was the strategy presented particularly helpful in the context of this problem? Have students turn and talk, and then have 1-2 students share out. (the student response here depends on what strategy was presented).

I'll wrap up the class with the key point:  there are multiple ways to approach real-world problems.  You were able to work on this problem because of all of the hard work you’ve shown in this first unit.