The Sieve of Eratosthenes: Prime Numbers, Multiples & Inverse
Lesson 10 of 21
Objective: SWBAT identify prime numbers.
To rev up our fluency to get ready for finding multiples, I like to use an app called Timed Test. This app allows me to differentiate by setting the range, time and the amount of facts a student can work on.
My students worked on this app for about 10 minutes. If they get 100%, they come and show me their work. I reset the test to a higher level by adjusting either the time, the operation (s) or the range so it caters to their growth.
I like how this works for students having difficulty with fluency because I can allow more time or graduate them by using the range of the facts. The fluency increases by them seeing and building on their success. I normally allow about 4 minutes for 80 problems. The students who master multiplication have their test either adjusted combing in division or division alone.
I set up my whiteboard with the words "prime" and " multiple" just to have on hand when the terms arise in our learning experience today.
To get kids warmed up in thinking about finding prime numbers in a different way, we are using the Sieve of Eratosthenes. (This website gives you some background and instruction). Click on any of the bolded words for background or other links to add to their knowledge as you teach. I had my students use their iPads and watch along with my direct instruction on the SB. Saying Eratosthenes is a task! Keep this up on the SB so you can use the interactive chart.
The sieve is a fun tool that helps kids see which numbers are prime. The Sieve of Erosthenes Rubric is designed so students can take responsibility for their learning after they have met with their partners to critique each other's thinking. This supports their learning and gives me a quick peek at questions they still may have about understanding prime numbers. Hopefully they will start to notice divisibility patterns and what factors divide into each number on the Sieve. It also strengthens fluency in multiplication, helping them realize the relationship between prime numbers and having only one factor pair.
What is a sieve? I illustrated it by bringing up an image of a conical shaped cooking sieve like I use for making apple sauce. We cook the apples, we press them through the conical sieve and the product winds up in the bowl. The sieve helps us get what we want...the end product is the smooth sauce in the bowl and the stuff in the sieve is thrown out or not used. I told them that we would be using the same sort of process, but using a mathematical sieve to find other ways of finding significant numbers. I explained that a sieve works like a filter. I reviewed this analogy with one student; What is one purpose of the Sieve?
Review Word: PRIME: I asked my students to define the word. They remembered what it was as I linked their past learning to their array charts that are posted about my classroom. I wrote their definition next to the word on the white board.
Notebook work and set up for practice: I asked my below grade level achieving students to list numbers to 30 in their notebooks. My at and above grade level achieving students wrote numbers all the way to 50. Suggestion: Students who have difficulty with writing numbers neatly can use grid paper. Using the website, I scrolled down to the list of numbers to 100 and used this interactive chart to draw on as I started to explicitly instruct.
We started by crossing out the number 1 since it is neither prime or composite. Then we move to 2 and draw a square around it to acknowledge that it is prime and cross out all the numbers that can be divided by 2, thinking about divisibility and patterns of even numbers. Then, we repeat the process with 3 . At this point, I wanted them to start thinking about how multiplication and division are related, so I asked them to think about what they notice as they work.
*We stopped at this point to define the word "multiple" in their own words so that students understood the purpose of what we were looking for on the sieve. Two students worked together to explain what a multiple was by discussion and example. We finally got " a multiple is the product of a number that we decide to use and another number. It's the product, but we are thinking about one factor all of the time." I do this because allowing them to come up with definitions based on their understanding and in their own words helps them synthesize their understanding. It truly becomes their word.
They did notice that 6, 12,18 were already crossed off. So, I asked them if 2 and 3 can have the same multiples sometimes. I wanted to lead their understanding that 2 and 3 have common multiples. This sets up their thinking for future understanding of common denominators and equivalent fractions and helps them relate it back to divisors. '
I asked them to really look at their chart so far and predict what we would do next. Asking them to predict reinforces reasoning and problem solving as expected in MP 1 & 2.
One student predicted that we would move to the five, but noticed that on his chart, the only multiple of 5 that was left was 25. This was an "a ha!" moment for many. I heard lots of "ohh!!! ". Some students who had number sets to 50 didn't know that 3 x 15 is 45 and so their 45 needed to be crossed off. I had them list 3x15= 5 x 9 on their notebook to help them understand those two factor pairs equal 45, assuming they know what 5x9 is.
When they were all finished, I asked them to really look at their chart. What numbers were left?
All students could see right away all of the prime numbers that were left. They knew they were prime numbers. I asked them why we didn't need to use 7 and find more multiples. They could explain that all the multiples of 7 were already crossed off. But, those who wrote down numbers to 50 disagreed with me and said that they needed seven because 49 was not crossed off yet. This shows reasoning skills I had hoped for! I asked if they wrote their number line to 100 if they would have to continue with more prime numbers. I wanted them to see that the longer the number line, the more they would have to check the multiples. When we were all done, we used the interactive chart on the website to check each prime number and its multiples and independently evaluate how we did.
I asked them to look at their Sieve of Erosthenes Rubric and read what the rubric was measuring. We discussed what self evaluation meant because I wanted them to understand the purpose behind it. I asked them to evaluate themselves on the first point only by looking at the SB chart and their number list. I wanted them to decide how well they picked out the prime numbers and score themselves on that only.
After they evaluated the first part of the rubric, I asked them to read the second and third sections of the rubric. I asked them if they could fill out the next section yet? They knew we needed to have a partner discussion and use the correct vocabulary words.
Video fun: I asked my students to video each other explaining the Sieve and the process we used to find all of the prime numbers. (using their iPads ) I wanted them to include anything they discovered or if they had any "aha" moments.
As I roved the room, I could see the video processes and explanation taking place. I asked students to email the videos to their partners so they could look at themselves and listen to their own reasoning. They could video themselves, but I wanted the other person to video because then the student talking would be less focused on looking at themselves and more focused on what they were saying.
*If you do not have iPads or video access, students can create a poster or chart during their explanation of their work to the other. Those could be used as a gallery walk at the end of the class.
RTI Watching students video themselves is a great way to determine if students understand fully what they had just done or the concept of why. I intervened and worked with two students who were really clueless and could not repeat the process. I helped them by questioning and working through the process. Slowly, both of them could use words correctly and explain what was going on with their sieve. I used this moment to really look at how they all were using vocabulary and explaining the process. Several needed to be guided with using. When they were done, they watched their own video and then evaluated themselves on their rubric and added up the score.
I asked them to write any questions they had on the lines under the rubric.
Showing Their Videos
"Roll em!" Three volunteers offered to show their videos on the SB using Apple TV. We watched each and critiqued if they had used the word "multiple". I asked students to critique the samples by using "Wishes and Stars". We say " I wish you would have...." and " I really liked how you..." I wanted students to acknowledge and be aware of what was missing or what was excellent. I chose just three for the sake of time. We could discuss each one, but they had already seen their own work. This experience exercises Math Practice Standard 3 as we critique the understanding. This process helped students weed out their misconceptions about confusing multiples with factors.
As we talked about the videos the word "multiple" surfaced many times and I felt they had a good grasp on understanding what a multiple meant. One student explained that the sieve was like panning for gold, but we were looking for prime numbers. I thought that was great! That was better than my applesauce analogy!
Wrap Up and Connect
The last question on the rubric is designed to introduce the idea of connecting division with multiples and prime numbers being a divisor. I demonstrated on the whiteboard using a math mountain. I drew the number 18 on the board and asked a student to tell me what prime factor could be listed on the bottom. One student said three. I wrote the three on the bottom and then put the six on the other side.
How does the prime factor become a divisor? Can we show that using other ways? I asked them if they could see where the the prime number plays the role of the divisor? I asked for thumbs up to show understanding. All thumbs were up and I pointed again to the three as the divisor saying 18 divided by the prime number 3 is 6.
I left this thought open and told them that the last question was designed to make them think and connect division to multiplication. We had a short discussion about it, I clarified the meaning of the question and I assigned this writing piece for homework. Through this, I will move to tomorrow's lesson on divisibility, finding factors and practicing more box method division. It sets up the continuation of satisfying the part of the standard that asks students to show and prove the relationship between division and multiplication.