Today's Number Talk
For a detailed description of the Number Talk procedure, please refer to the Number Talk Explanation. For this Number Talk, I am encouraging students to represent their thinking using an array model.
Task 1: 12/4
After I wrote the first task on the board, students immediately said the answer was three. I simply said: Prove it to me! Some students drew a rectangle with 12 in the center, placed 4 on one side, and placed a variable (often the first letter of their names) on the other side: 12:4. Other students decomposed the 12: 12:4 = 8:4 + 4:4.
Task 2: 24/4
During the next task, I was happy to see students doubling the 4 x 3 array equalling 12 to get a 4 x 6 array equalling 24: 24:4..
Task 3: 240/4
Task 4: 480/4
Throughout every number talk, I continually model student thinking on the board to inspire other students. This also requires students to use math words to explain their thinking instead of relying on a model to represent the math. As students solved each task, I wrote the answers on the board to encourage students to use prior tasks to solve the more complex tasks: Tasks on Board.
Explanation Behind the Venn Diagram
When reading through the Common Core Standards, I noticed that Finding the Greatest Common Factor is not taught until 6th grade (6.NS.4). However, when looking at the Bloom's Taxonomy Pyramid, I wanted my students to go beyond the first three tiers of learning. By asking students to analyze and compare the factors for two numbers and to create a Venn Diagram to show their thinking, students will be involved in higher level thinking processes than simply asking students to identify factors.
To begin today's lesson, I introduced the goal: I can identify the common factors between two numbers. I asked: What do you think common means? Turn and Talk with your math partner: What are five things you and your partner have in common? We came back together and discussed common further. I then explained: When we are looking for common factors, we will be looking for shared factors.... or factors that two numbers share.
For math, I always have a stack of multi-colored papers placed inside sheet protectors to help provide students with a larger workspace and to help them organize their thinking. At this point, I asked each student to get two colored papers of like colors (two green or two blue, etc.). I also passed out a Venn Diagram (found at eduplace.com) to each pair of students. Pairs of students organized their desks like this: Arrangement of Papers on Desks..
Then, I explained: A Venn Daigram is used to compare two topics. For example, if we were comparing two classmates, we would write their names on the topic lines above each circle. Then, we would place any common (or shared) characteristics inside the area that their circles overlap. We would place any unshared characteristics in the outer portion of their circles where there is no overlapping. Today, we are going to use this same idea with math.
Factor Tree for 12
Once students were ready, I asked each partner to label one page, "Factor Tree," and the other page "Factor Pairs." I then did the same on either side of a Venn Diagram on the board. I began by placing 12 on one side of the Venn Diagram and 18 on the other. I explained: Let's have one partner create a factor tree for 12 and the other partner can find a factor tree for 18. However, for this task, I want you to work alongside of me, step-by-step.
I continued: Let's start with the number 12. Here's a picture of the teacher model: Factor Tree & Factor Pairs for 12.
1. Write the target number, 12, at the top your board.
2. Ask: What two factors, when multiplied together, equal 12? Because this method is called PRIME factorization, the goal is to find all the prime factors for 12. I always ask: Does 2 go into 12 evenly? Students responded, "Yes! 2 x 6!" I modeled how to write 2 x 6 as part of our factor tree.
3. Whenever you are using the prime factorization method, I'd like you to circle all of the prime numbers. Does anyone see a prime number in our factor tree so far? (Two!) How do you know 2 is prime? I pointed to the I'm Prime Chant and prompted students to sing the Prime Song.
What about the 6? Is 6 prime or composite? (Composite!) How do you know? (Because it has more than two factors)
4. Whenever you have a composite number left, ask, "What goes into this number evenly?" So... what goes into 6 evenly? (Two... 2 x 3!) What should we circle? (The 2! And the 3!) Why? (The 2 and 3 are prime!) What does prime mean again? I began singing and students joined in: I'm prime! P-R-I-M-E. The only factors are one and me! Numbers that divide evenly!
Factor Pairs for 12
Now we are ready to write the prime factorization equation! I modeled how to write the equation under the factor pairs heading: 2 x 2 x 3 = ____. I then explained a new way of showing how to find factor pairs with this equation. I circled the 2 and then the 2 x 3. I then placed 2 x 6 on the other side of the prime factorization equation.
I then rewrote the equation again (2 x 2 x 3 = ____). This time, I modeled how to circle 2 x 2 and then the 3. Then, I wrote 4 x 3 on the other side of the prime factorization equation.
Students also completed the same steps: Student's Desk for 12.
Factor Tree & Factor Pairs for 18
We then completed the same factor tree process with the number 18. As a result, the prime factorization equation was 2 x 3 x 3. Again, we circled prime factors within the equation to identify all the factor pairs for 18. Here's a picture of the teacher model: Factor Tree & Factor Pairs for 18..
Student also completed the same steps:Student's Desk for 18.
I then pointed to the overlapping portion of the Venn Diagram circles and asked: What factors do both numbers have in common? Students' hands shot up! I called on students, one at a time, to come up to the board to write a common factor. Students also whispered with each other during this time to complete their own Venn Diagrams: Student's Venn Diagram for 12 & 18.. We then moved on to unshared factors for 12 and 18. Here's a part of this class discussion as students Completed the Venn Diagram. This was the end result: Venn Diagram for 12 and 18..
To provide students with guided practice, I then asked partners to work together to find the common factors for 30 and 48: Student's Factors for 30. and Student's Factors for 48.. During this time, I conferenced with students and provided support when needed.
After some time, I asked for students to help me complete the Factor Tree & Factor Pairs for 30. and Factor Tree & Factor Pairs for 48.. Here, a student shows how to use the prime factorization equation to find 16 x 3 = 48. Another student came up and revealed some Confusion with Factor Pairs.. This was a perfect teachable moment as I knew many other students were probably experiencing the same confusion! Here, a student shows how Rearranging the Factors. can be helpful to find factors!
Finally, students completed the Venn Diagram for 30 & 48. together. I simply invited any student to come to the board to take turns completing the chart: Completing the Venn Diagram for 30 & 48. This was the end result: Venn Diagram for 30 & 48.
Students were now ready to practice finding common factors on their own!
Inspired by a 5th grade unit called Bridges in Mathematics, I created a sheet of Riddles. for students to solve. Each riddle required students to examine the factors of numbers within 100 and to identify the common factors of two numbers.
I explained: Today, you get to try solving some math riddles! I heard one student say, "I love riddles!" As I passed out the page of Riddles.to each pair of students I continued: Today, I want you to continue using the factor trees and prime factorization to identify factor pairs. In each riddle, you'll find two numbers that share a common factor. The riddle will also provide you with other clues to help you identify the number. When you think you have solved the riddle, raise your hand and I will check your work and the number. Keep your answers top secret! Ready... Set... Go!
Conferencing with Students
During this time, I conferenced with each group of students in-between checking student work and riddle answers. The room filled with excitement as students began raising their hands, "We got it! We got it!" Sometimes students would even tell me, "Even though we haven't shown our work, we know the answer to this one!"
When checking students, I would ask: What is the answer? How do you know? Can you show me your evidence? I knew that by encouraging students to construct arguments based upon evidence, students would be practicing Math Practice 3: Riddle for 60 & 100.
While conferencing with students, I checked for student understanding of the strategies: Student Finding the Factor Pairs for 24. I also provided support whenever possible: Finding Factors. After some time, students became quite proficient with identifying factors by circling: Conference with Student.
To support another student who just moved to America and is learning her multiplication facts, I showed her how to use a calculator to find factors. This helped to provide her with an access to learning: Student Using a Calculator.
As this pair of students completed each riddle, they wrote fun phrases on top of each fold: Students Folded the Riddles Over as they Finished. You can really tell that they enjoyed learning today!
As students finished, I asked them to help others without giving away the answers of course!
To bring closure to this lesson, I celebrated students who were on task and truly involved in math today. We cleaned up our materials and I encouraged students to take their math riddles home to challenge family members!