## Monday, September 9, 2013 - Common Factors and Multiples

__Homework__"Activity 27" and "What did Arf the dog give his master for his birthday?" (front and back of handout) is due Friday, September 13. Students numbered each set of problems for Activity 27 #1-13 and there are 8 numbered problems for the Arf birthday preset assignment. Students should have 21 problems completed by Friday showing all organized work.

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**While students worked on solving questions 4 and 12, Bellwork for week 3 and week 4 and Dividing Decimals by Decimals p.139 #2-30 (even only) were returned. For Bellwork, we first discussed what they were solving for in the problem, then shared different mathematical expressions that could or could not be used emphasizing what each number given in the problem meant. We discussed why division was the operation needed - how many groups can be taken out of a given whole.**

__Bellwork__

__Classwork__After Bellwork, we looked at work that was returned and how points were earned through the work each student completed. Just a reminder, Bellwork and Classwork in the 20% Practice category are graded on completion. 30% Progress grades are graded on accuracy.

After reviewing graded work, we began "Classifying Numbers." I demonstrated a Venn diagram (used to explore relationships between things) by exploring prime numbers less than 9 and multiples of two less than nine. The set included numbers 1-9. After finding numbers less than nine falling in either or both categories, students were shown to place the remaining numbers outside the circles within the rectangle. The example Venn diagram is shown below. Students worked with a partner to complete their own Venn diagrams where they found common factors between sets of numbers. We will continue to use Venn diagrams to find common multiples between sets of numbers tomorrow.

## Tuesday, September 10, 2013 - Classifying Numbers and Applying Factors and Multiples

__Bellwork__Students finished "Classifying Numbers" by completing a total of 4 Venn diagrams for Sets A - D and answering follow-up questions for each set. Big ideas to take away: (1) common factors or multiples exist in the overlapping/intersecting part of the two circles, (2) factors evenly divide numbers, (3) multiples are products, (4) the greatest common factor or GCF of a set of numbers will not exceed the value of the given numbers, (5) one is a common factor for all numbers and sometimes it is the GCF, (6) the least common multiple or LCM is sometimes the product of the given numbers, (7) there are an infinite number of common multiples between two or more numbers.

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__Classwork__The math department was instructed by administration to administer the AIMS Test to all students. Students who were absent today need to schedule time to take this test preferably during tutoring hours as it is a timed assessment. This test will not impact the student's class average.

After the AIMS Test, students worked in groups of 4 to solve problems A-D from "Riding Ferris Wheels." Tomorrow, we will finish discussing their solutions as a class and will continue to apply multiples and factors in other real-world situations.

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## Wednesday, September 11, 2013 - Applying Factors and Multiples

**Students found common multiples 1 to 100 and the least common multiple (LCM) for eight pairs of numbers using #1-8 under "Applications" from the "Riding Ferris**

__Bellwork__

Wheels" task.

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**We finished examining problems A-C from "Riding Ferris Wheels." In some classes, we acted out the difference in rates to demonstrate why Jeremy's rides were longer than Deborah's. We used pictures to represent how many revolutions Jeremy and Deborah took in each scenario and summarized each situation in the following way:**

__Classwork__

For A: "At the end of 60 seconds (Jeremy and Deborah were both at the bottom again), Jeremy made 1 revolution while Deborah made 3."

For B: "At the end of 150 seconds, Jeremy made 3 revolutions while Deborah made 5."

For C: "At the end of 70 seconds, Jeremy made 7 revolutions while Deborah made 10."

Big ideas. These three scenarios show three different relationships between the numbers and their least common multiple.

For 60 and 20,

**, 60.**

__the multiple is one of the two numbers__For 50 and 30,

**.**

__the multiple is a combination of common factors between 50 and 30 and their remaining uncommon factors__For 10 and 7,

**.**

__the multiple is the product of 10 and 7__Finding the product of two numbers will certainly give you a common multiple, but it may not be the smallest or the LCM.

The following task, "Looking at Cicada Cycles," was introduced to a few classes. These cicadas unique 13 and 17-year life cycles are designed to outsmart their predators. A 12-year cycle would make them more susceptible to different generations of predators because of the numerous factors 12 have. Picking prime numbers is so clever! Students will continue to examine different generations of cicadas tomorrow to see when the 13 and 17-year cicada will emerge from the ground together. What a sight that would be!

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## Thursday, September 12, 2013 - Applying Factors and Multiples

**Students completed #31-32 under "Applications" from the "Riding Ferris Wheels" task (posted above). They were given the equation 7 x 9 = 63 and was asked to write as many sentences as they could using the following terms: factor, divisor, product, multiple, and divisible by.**

__Bellwork__

The sentences varied, but were similar to these:

7 is a factor of 63.

9 is a factor of 63.

7 is a divisor of 63.

9 is a divisor of 63.

63 is a product of 7 and 9.

63 is a product of 9 and 7.

63 is a multiple of 7.

63 is a multiple of 9.

63 is divisible by 7.

63 is divisible by 9.

Which terms are synonyms of each other? Factor and divisor can be substituted for each other as well as product and multiple. Knowing which number gets divided and which number does the dividing is still being reinforced, as well as understanding the difference between a factor and a multiple.

**Working in groups of four, students diligently worked to figure out how many years it would take for the 13 and 17-year cicada to emerge from the ground together. They discovered that it would take 221 years! They also discovered that in this situation the least common multiple was the product of 13 and 17. Some students did this at the start and were prompted with the question, "How do you know there isn't an earlier year?" They knew that 221 years was a possible year, but to be sure, they had to find all the multiples of 13 and 17 leading up to 221. When they found no other common multiple, they knew they found the number of years it would take before the cicadas would emerge from the ground together.**

__Classwork__

Students were then asked to examine this phenomena with 12, 14, and 16-year cicadas. When would they emerge from the ground together? They embarked on the same method of listing multiples with some finding the product of the three numbers. The product was 2688 years, which is a common multiple, but there is a smaller multiple, 336. This led to the idea that there must be a faster way to find the LCM that doesn't require listing multiples.

We examined how we can find the least common multiple using prime factorization. Many of them left class bewildered and some starting to see. In the next few days, we will work on using prime factorization and the "ladder method" to find both the greatest common factor and least common multiple. You can find a copy of the cicada task sheet posted under Wednesday, September 12th.

## Friday, September 13, 2013 - Applying Factors and Multiples, Finding the GCF and LCM Using the Ladder Method

Homework and Bellwork were collected at the beginning of class.

Homework was given on Monday involving 21 problems from the handout with "Activity 27" on one side and "What did Arf the dog give his master for his birthday?" on the other side. Students were reminded to turn in organized work and not just the handout.

Monday - two word problems involving division with decimals - #4 and #12

Tuesday - 4 Venn diagrams and 12 follow-up questions involving these diagrams

Wednesday - #1-8 using "Riding Ferris Wheels" - for the 8 pairs of numbers given, find all common multiples from 1 to 100 and the LCM

Thursday - #31 and #32 using "Riding Ferris Wheels" - create sentences for the equation 7 x 9 = 63 using the terms factor, divisor, product, multiple, and divisible by; find the missing factor for question 21 parts a and b.

Today's Bellwork will be turned in next Friday. This way, students have time to correct, finish, and/or receive extra help during tutoring before submitting their work. Class time will no longer be given to students to organize their Bellwork as it has been done for the last 5 weeks. An ongoing list of Bellwork assignments that will be due on the upcoming Friday will be posted on the board.

Applying factors and multiples , #10 - 15 using "Looking at Cicada Cycles." Please see attachment from Wednesday.

Guided practice was given to find the GCF and LCM using traditional methods. These methods were then compared to the Ladder Method using problems #1-3 from p.175 of the Holt textbook. Students were shown page 173 in case they ever wanted to see more examples of this method. Students were shown the reason why the Ladder Method helps them find the GCF and LCM. It pulls the common factors and leaves remaining factors that are unique to the make-up of each number. Since multiples are products of a given number, the multiple must include all factors of the original number. Problem #3 showed the students that the Ladder Method does not work for finding the LCM if you have more than 2 numbers. In this case, we would need to use prime factorization. Prime factorization will be used on Monday to help students find the LCM of 3 or more numbers.

Homework was given on Monday involving 21 problems from the handout with "Activity 27" on one side and "What did Arf the dog give his master for his birthday?" on the other side. Students were reminded to turn in organized work and not just the handout.

__Bellwork for this week included work from Monday through Thursday:__Monday - two word problems involving division with decimals - #4 and #12

Tuesday - 4 Venn diagrams and 12 follow-up questions involving these diagrams

Wednesday - #1-8 using "Riding Ferris Wheels" - for the 8 pairs of numbers given, find all common multiples from 1 to 100 and the LCM

Thursday - #31 and #32 using "Riding Ferris Wheels" - create sentences for the equation 7 x 9 = 63 using the terms factor, divisor, product, multiple, and divisible by; find the missing factor for question 21 parts a and b.

Today's Bellwork will be turned in next Friday. This way, students have time to correct, finish, and/or receive extra help during tutoring before submitting their work. Class time will no longer be given to students to organize their Bellwork as it has been done for the last 5 weeks. An ongoing list of Bellwork assignments that will be due on the upcoming Friday will be posted on the board.

__Bellwork that is not labeled with dates and headings and is not organized will get points deducted.__**It is the student's responsibility to keep Bellwork organized, to staple it Thursday night, and to be ready to turn in on Friday at the start of class.**__Today's Bellwork__Applying factors and multiples , #10 - 15 using "Looking at Cicada Cycles." Please see attachment from Wednesday.

__Classwork__Guided practice was given to find the GCF and LCM using traditional methods. These methods were then compared to the Ladder Method using problems #1-3 from p.175 of the Holt textbook. Students were shown page 173 in case they ever wanted to see more examples of this method. Students were shown the reason why the Ladder Method helps them find the GCF and LCM. It pulls the common factors and leaves remaining factors that are unique to the make-up of each number. Since multiples are products of a given number, the multiple must include all factors of the original number. Problem #3 showed the students that the Ladder Method does not work for finding the LCM if you have more than 2 numbers. In this case, we would need to use prime factorization. Prime factorization will be used on Monday to help students find the LCM of 3 or more numbers.

***Notebook Quiz 2 will be given next week, so please be ready. Make sure all assignments are dated, labeled, and in chronological order. Bellwork can be arranged at the front of the notebook in its own section or at the beginning of each week's classwork.***Notebook Quiz 2 will be given next week, so please be ready. Make sure all assignments are dated, labeled, and in chronological order. Bellwork can be arranged at the front of the notebook in its own section or at the beginning of each week's classwork.