Performance Task Lesson - GCF

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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, we will follow a two-step reading protocol.

  1. I read the Balloon Task out loud for students as listeners. 
  2. 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. 

After reading and annotating the task description, I will ask my students the questions in bold below (each is followed by teaching moves and/or possible student responses).

What information does the problem give us?  How did you annotate?

I annotate my copy under the doc camera as students share.  This is the first opportunity of the school year to reinforce annotation expectations. Then, I circulate quickly to give real-time feedback.

What is the question asking us to find?

"The problem is asking us to find the largest number of balloon arrangements Jasmine can make."

What information do we not know?

"We don’t know how many groups Jasmine will make and we don’t know how many of each color balloon will go into her arrangements." 

Are there any words in this problem that you don’t know? 

"Arrangements." 

If we don’t figure out the meaning of a word, would you still be able to solve this problem? 

"Yes." 

What do you think this word means?

Allow 1-2 students to define the word, and then add to it, if necessary.

What do you think a reasonable prediction would be for the number of arrangements Jasmine can make?

Allow 1-2 students to share their prediction, and be sure they explain why they think theirs is a reasonable prediction. 

What would you say if someone predicted Jasmine could make 50 arrangements?

"That prediction isn’t reasonable because Jasmine doesn’t have 50 of any of the colors of balloons.  It isn't possible for her to make 50 of the same arrangements, given the balloons she has."  

Exploration

15 minutes

For this exploration, I plan to allow my students 4 minutes to work on their own, and then 8 minutes to work in partners.  

As students are working on this problem, I expect them to:

  • Work to understand the problem using appropriate annotations
  • Represent the known and unknown information precisely
  • 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 pairs are working, I circulate and engage students in conversation.  I'm asking:

  • Tell me about your strategy.
  • Why is this strategy helpful?  Why is this strategy not helpful?
  • What is it that you’re trying to do?
  • What are those numbers (if students used list of factors)?

During this work time, I am looking for student work samples to show on the board during the presentation section.  Before I teach the lesson, I also create my own samples of Student Work. This helps me to know what to look for.  It also gives me a sample to display for our discussion if students get stuck.

Presentations

15 minutes

This is the first presentation of the school year, in the very first lesson, and I anticipate students will need much guidance, practice, praise, correction, etc. I explain the importance and role of presentations – a chance to practice public speaking, a chance to share thinking, an opportunity to ask peer questions about his/her strategy, etc.  I share that I will ask questions of both presenter and the rest of the class.  

The mathematical work during this section of the lesson is to make a list of factors for each balloon color, and identifying the common factors, and then greatest common factor.  Although we'll talk about multiple student strategies during this section, I want the class to have a clear idea that a list of factors is the one efficient strategy for finding the GCF (and one that we'll focus on in the next lesson).

I will ask for a student volunteer to be the very first presenter of the school year.  Based on my observations during the lesson, I will pick a student who used  a visual strategy.  The student should present his/her thinking. I help them along the way by asking probing questions:

  • Why? 
  • How do you know?
  • Can you be more specific?

I will also suggest sentence frames that would help him/her make his/her thinking clearer. For example, rather than "I drew a picture," I help the student say "I represented the white balloons with w’s.  I know that there are 24 white balloons, so I used 24 w’s and split them into groups by…".  My goal is to help students be specific and detailed as they present. I'll ask the rest of the class if they also tried the presented strategy.  Then, I'll ask students who used the strategy to add to what’s been presented.  Finally, I ask if there are questions about the strategy.

I'll go through the same presentation process with a piece of student work that used a list of factors as a strategy.  I have a copy of this strategy ready to go, in case students don't get there themselves.   

Discussions

10 minutes

To bring this lesson to closure, I plan to guide the class through a conversation reflecting back on the problem solving strategies that were presented today.  The questions I plan to ask, and possible student responses, are below:

How are the strategies similar? Different? 

  • Both strategies involved taking the balloons and dividing them into equal groups.
  • Both strategies got to the same final answer. 
  • With the picture strategy, student had to guess and check and try many times to get to the right number of arrangements.  
  • The list strategy was faster.
  • It was easier to see where all the balloons were going with the picture strategy.
  • It was easier to see all of the possible groups with the list strategy.

Which is more efficient? Why?

  • Teaching Move: Turn and talk followed by vote and discussion (students may need help with the word ‘efficient’). 
  • Students should say that the list was more efficient because there was no need to go back and adjust and try again, like they’d need to with the picture. 

Which strategy would be better if Jasmine also had 12 purple balloons and 40 yellow balloons?  Why? 

  • Teaching Move: Turn and talk followed by vote and discussion.
  • Students may say either – some visual learners might find the model to be easier, and these numbers aren’t much different from the original problem.  If this happens, ask if 8 groups would still be the answer.  Why not?  (8 is not a factor of 12)  Most students should say that the list would be better because there are 5 different colors to keep track of.

What if Jasmine had 145 red balloons and 220 purple balloons – which strategy would be better in this situation?  Why? 

  • The second strategy would be better, because the numbers are much larger.  It wouldn’t be a good use of time to draw all of those balloons!

What if the problem just said Jasmine wanted to make equal groups?  Are there other numbers of groups Jasmine could make, if we removed the constraint of needing thelargest group? Talk with your partner for 3 minutes about this. 

  • Teaching Move: After 3 minutes, have students share out.
  • Jasmine could also have made 1, 2, or 4 groups of balloons. 

1,2, 4, and 8 are the common factors of 32, 24, and 16.  Why is 8 the correct answer to the problem we were initially asked? 

  • 8 is the right answer because it is the greatest common factor, and the problem asked us to find the largest number of groups Jasmine could make. 
  • 8 is the GCF of 32, 24, and 16 because it is the largest number that will divide evenly into all three numbers. 

I wrap up the discussion naming the key points: 

  1. There is always a greatest number that divides evenly into a set of whole numbers, and that number is called the Greatest Common Factor (GCF) of the set.  
  2. Sometimes that number is 1.  
  3. Using an organized list of factors for each number is an efficient way to identify the GCF.  

My students will have the opportunity to practice finding the GCF of a set of numbers in the next two lessons in this unit.