Unit 5: Patterns, Factors, and Multiples

Hi there!  I hope that everyone had a wonderful Thanksgiving break. I look forward to seeing you all tomorrow and getting back into learning mode!  We will spend Monday reviewing Unit 5: Patterns, Factors, and Multiples.  Students will take the Unit 5 Test on Tuesday.  Students will get to bring home their review and their math notebooks to study and look over on Monday night.  Below is a recap of Unit 5.  I forgot to take pictures of some of the posters that we used in class before leaving for break, so I am sharing some from the internet, but the charts in our class look just like them!

We began the unit by learning about patterns.  Students explored patterns and came up with their own patterns.  They also had to be able to come up with a rule when given a pattern.  For example, if given the following pattern: 4, 9, 7, 12, 10, 15, 13, students would have to determine that the rule is add 5 then subtract 2.

Next students learned about factors and multiples. Below you will find the definitions and examples for each of these!


A factor is a number that is multiplied by another number to get a product. In the example below, 6 and 3 are factors.
6 x 3 =18

Below is an example of a poster that looks just like the one we have posted in the classroom to help students remember what a factor is!

A multiple is a number that is the product of given number and some other number.  In the example below, 18 is a multiple.
6 x 3 = 18

Below is an example of a the poster that looks just like the one we have posted to help students remember what a multiple is!

Next students learned how to find factors and common factors. Students were taught to use a T chart like the one below to help them organize their thinking when finding factors & common factors of two numbers.
We began with the number 36. First, we organized our work in a t-chart, like the one shown below.  Students write the number they are finding factors for (36) at the top of the chart.  Next, we go through each single digit and decide if it's a factor or not.  It sounds something like this:
Is 1 a factor?  Yes.  1 and what?  1 and 36.
Is 2 a factor?  Yes.  How do you know?  Because it's even.  2 and what?  2 and 18 (they divide if they don't know).
Is 3 a factor?  Yes.  3 and what?  3 and 12.
Is 4 a factor?  Yes.  4 and what?  4 and 9.
Is 5 a factor?  No.  How do you know?  Because 36 doesn't have a 0 or 5 in the 1s place.
Is 6 a factor?  Yes.  6 and what?  6 and 6.
Is 7 a factor?  No.  How do you know?  7 x 4 is 28 and 7 x 5 is 35.
Is 8 a factor?  No.  How do you know?  8 x 4 is 32 and 8 x 5 is 40.
Is 9 a factor?  Yes, but we already have it on our list.

The t chart is also helpful for finding common factors of 2 different numbers. As you can see in the picture above, we used a Tchart to find all of the common factors of 12 and 24. First we found the factors of 12 and then the factors of 24.  Then we circle all of the factors they have in common.

I also set up common factor workstations in our classroom.  Students went around the room in teams with clip boards and found common factors of different numbers using the strategies that they learned in class.  Below you will find pictures of the mathematicians at work! 

Students were also taught to use a venn diagram to help them find common multiples of a number. Below is an example that we did together using a venn diagram to find common multiples of 3 and 6..we didn't get a chance to finish, but as you can see we could have continued!

Students also learned about divisibility.  A number is divisible by another number if the quotient is a multiple or counting number and the remainder is 0.  For example, 12 is divisible by 6, because 6 can divide evenly into 12 without a remainder.  However 13 is not divisible by 6, because 13 can not be divided evenly by 6 without a remainder.  Students explored divisibility with many different problem solving situations.  As students solved problems, and we noticed that a number was divisible by another number, we added to our divisibility chart on the board.  Students later used the patterns that they noticed on the chart to make a divisibility rule foldable in their notebook.  For example, students noticed that if number had a 0 or 5 in the ones place then it was divisible by 5. Another example, students noticed that if a number had a 0, 2, 4, 6, or 8 in the ones place that the number was divisible by 2. Below you can find a picture of the divisibility chart on the board.  It was so powerful for students to notice the patterns and come up with the rule on their own, rather than me just telling them! Students will be bringing home the divisibility rules chart in their notebook on Monday!  You will find an example of students using the divisibility posters to find divisibility rules to add to their foldable!

Examples of divisibility rules foldable!

Next, students learned about prime and composite numbers! Rather than giving students the definition of prime and composite numbers, I gave them a problem and let them develop their own definition of prime and composite numbers!  To begin the lesson, students solved the following problems in their notebooks!

Students used square tiles and grid paper to determine that you could only arrange 13 tables in one way to make a rectangle, which is 1 row of 13.  However, 12 tables could be arranged different ways! ( 1 row of 12, 2 rows of 6, 6 rows of 2, 3 rows of 4,and 4 rows of 3).  From there, we discussed the difference between the 2 numbers.  I told them that 12 is composite number and 13 is prime.  Based on their observations from the problem of the day, they were able to determine their own definitions of prime and composite numbers!  So valuable!  See below....

Division Strategies!

This week, we're wrapping up Unit 4: Strategies for Dividing by 1-Digit Divisors.  We will review for the test on Tuesday. Students will take the Unit 4 test on Wednesday.  For homework on Tuesday, students will bring home an additional review.  I would suggest going over the questions with them!

I wanted to take the time to re-cap the division strategies that your child has learned throughout this unit.

We began the unit by having students write on a post it "What Do I Already Know About Division". Some students were able to tell me that when we divide we work with equal groups.  They were also able to tell me what the quotient, dividend, and divisor were.  Some of them even knew that sometimes there are remainders when we are dividing!  I was impressed!

The first strategy that students were introduced to was using multiples and compatible numbers to help estimate when dividing.  Below is an example of how to use multiples to estimate quotients.

Students were presented with the following problem.

This student solved the problem by multiplying by 8 until they got close to 110.  As you can see from their work, there would be about 13 or 14 boxes of pumpkin muffins.  This answer is reasonable because 110 is between 104 & 112.  This student was called up under the ELMO to share with the class.  The conversations among the students were really powerful!

Then we filled out the anchor chart below as a class to match the problem and continued solving other problems that required students to use multiples to help them estimate quotients.

We also talked about using compatible numbers when using estimation to divide.  Compatible numbers are numbers that are easy to work with. For example, the following problem would require students to divide 132 by 3.

132 does not divide easily by 3 mentally.  So students may choose to divide 120 by 3, since 120 is relatively close to 132.  This is easy to do, since students already know that 12 divided by 3 equals 4, then they know 120 divided by 3 equals 40.  They could have also chosen to divide 150 by 3, since they know 15 divided by 3 equals 5, then they know 150 divided by 3 is 50.  They would then know that the actual answer is between 40 and 50.  Estimating is a great "real life" tool.  It helps students know if their actual answer is reasonable or not.

The next strategy that students used to divide was the distributive property.  This seemed to be a popular strategy among the classes!  You can use the distributive property to break apart numbers to make them easier to divide.  Students were presented with the following problem that required them to divide 69 by 3.  Many of the students were able to recognize that they could break apart 69 into 60 and 9.  Then they could divide 60 by 3 and 9 by 3 and then add those partial quotients together.

This is Hannah sharing her thinking in front of the class!

I love how Jackson checked his answer by adding up 23 3 times to see if it equalled 69.

We recorded the student's thinking on a anchor chart so that they could refer back to it during our division unit.  

The next strategy that students learned for dividing was using repeated subtraction.  The problem below shows how students used repeated subtraction to solve 72 divided by 6. Students started with 72 and subtracted groups of 6 until they got to 0.  The amount of times that they subtracted 6 (12 times) was their quotient, because that told them how many groups of 6 were in 72.

Students were then taught to divide using regrouping.  This is also know as the "traditional algorithm" for dividing. This was the way that I was taught in school.  However, I was taught just to memorize the rules/steps without understanding why I was doing each step. For example, "just bring down the number when subtracting".  I taught this strategy with students using base-ten blocks and students had to explain each step and what they were doing in order to build a conceptual understanding behind this strategy.

We also worked with dividing with remainders.  Students seemed to really enjoy this topic.  Students also learned how to interpret remainders when dividing.  This means that students had to decide how to use the remainder based on the context of the problem and what it was asking.

Here are students playing a game titled "Remainder Travel".  They got to spin and divide.  Their remainder determined how many spaces they got to move.  They really enjoyed this game!

Students learned 5 strategies for "Interpreting the Remainder" when dividing.  The strategy that they chose depended on what the question was asking.

These are the 5 strategies:

1.) Drop the remainder (Ignore it)

2.) The remainder is the answer

3.)  Use the remainder to determine how many more to the next whole

4.) Round the quotient up to the next whole

5.)  Use the remainder in fraction form as part of the whole

Students created a foldable to help them practice the strategies for interpreting remainders.  They solved each problem and glued it under the flap of the strategy used so that they had an example for each strategy.

For the problem below, students solved 743 divided by 46 and got the answer, which was 16 with a remainder of 7.  Students had to use the strategy of using the remainder to determine how many more to the next whole.  They knew that there was 7 marbles left over so to figure out how many more marbles were needed to complete the next bag, they had to think " I know that 46 marbles complete a bag and they already have 7 marbles left over so they would need 39 more marbles to complete the last bag."

For the problem below, students had to use the strategy the remainder is the answer.  It is the same problem as above, but the question is different.  Students knew from solving the problem above that there were 7 marbles left over as the remainder.  Therefore, the answer was 7 marbles left over.  

For the problem below, students had to use the strategy drop/ ignore the remainder.  The problem was simply asking how many bags they could fill.  The answer is 16, the remainder of 7 would be ignored to answer this question.  

For this problem below, students had to use the strategy of round the quotient up to the next whole.  Once solving, students knew that 16 bags could be filled with a remainder of 7 marbles.  To be able to hold all of the marbles, one more bag would be needed for the extra marbles.  Therefore, 17 bags would be needed.  You had to round the quotient up to the next whole (17) because of the 7 remaining marbles.  

For the last problem, which can be seen below, students had to use the strategy of using the remainder as a fraction.   Each person could get 4 whole cookies.  The remaining cookie could be split into fourths to be divided among the 4 friends.  Therefore, each child would get 4 whole cookies and one fourth of the other cookie.

I have a feeling that the students will do awesome on the Unit 4 test on Wednesday.  Their scores will be sent home at the end of this week.  Thank you for your continued support at home!  If you're reading, please tell your students to whisper "class blog" to Mrs. Davis tomorrow for a special treat.  This is a little strategy that we are using to make sure everyone is viewing the class blog/newsletters!  

Pumpkin Day

The kiddos had a blast at Pumpkin Day! Stations were set up for them to enjoy various activities and treats!!!!