Number System Standards Posters
I have found it helpful to hang these posters at the beginning of each unit. It allows me to have the standards at my fingertips as I plan. Eventually I will be more familiar with them, but until then it is a good reminder. I also find that it helps the students to see where they are in the scheme of things. I color coded each of the sub-standards to keep them separate. These posters are written in student friendly terms.
Number System Standards Posters | |
File Size: | 11 kb |
File Type: | docx |
Pre-Test 6.NS.1-6.NS.4
It is helpful for me to give a pre-test which addresses each of the standards the students are expected to learn during the unit. I use the information to form my groups for intervention and to determine the pace at which we work through the material.
Download the Pre-Test HERE from my TPT store.
Lesson 1: Review of Divisibility Patterns (Practice Standard #8)
As our students adjust to the rigor of Common Core, I have found that they have a wide range of abilities when they begin a topic. I felt it was important to make sure students are comfortable with the divisibility patterns as we move into factors and multiples, even thought this is technically not a sixth grade standard. Many of the students are working far too hard to find the answer to multiplication and division problems because they aren't able to identify patterns which would help them to find the answer mentally. Identifying patterns is Mathematical Practice Standard #8, and is one of the most helpful when solving problems using mental math.
I started with this foldable by Joy Hall at Joy of Teaching. Part of identifying patterns is that students need to struggle with the raw data to find the pattern, rather than having me provide it for them. Therefore, I only copied the foldable using the hundreds charts page. I did NOT give them the rules right away, and preferred to have them write them on their own.
I reviewed the ideas of factor and multiple with the whole class to makes sure they understood the vocabulary. Then I had them work in small groups to color in the multiples of each factor. After they finished coloring them, I asked the students to discuss with their groups any patterns they saw. They wrote these observations in their notebook.
One of the more difficult parts of describing the patterns, was to describe the patterns within the numbers. Many students began by describing the visual patterns they saw on the hundreds charts. For example, "I colored every other one." or "It's a diagonal line." I asked these students if knowing that the pattern is a diagonal is helpful when I am checking a larger number. I also directed them to start with the pattern for 2 because it is more easily identified than some of the others.
As students worked through the patterns, many of them were able to identify that 2, 4, 6, and 8 had even numbers as their multiples. The follow-up question to this discovery is to find out why 4, 6, and 8 are not factors of every even number the way that 2 is. The patterns for 4 and 8 are hard to notice unless they look at multiples past 100. I encouraged those students who were ready to extend their hundreds chart to look for these patterns. Allowing the students to struggle with these patterns, made it more interesting for them to check their hypotheses to the patterns.
The following day, I shared the Divisibility Pattern Powerpoint which I created listing the patterns. Most of the rules are pretty straight forward. The rules for 3 and 9 are more like "tricks" until you prove them. I watched this Khan Academy video to learn how it worked. Then I included a "proof" in the Powerpoint using Cuisenaire rods. It is a brief look into the Distributive Property. The rule for 7, ends up being pretty convoluted for most students. They may as well just check it by dividing by 7 in my opinion. I included the rule, but told them that I would suggest just dividing it out.
Then I had them play a modified version of this Bounce Around game. Players all begin in the middle at the START square. They roll one di to see how many spaces they will travel. The person next to them rolls one or two dice (their choice). The player moves their game piece to the space, and then if the number is divisible by the second number rolled (by the other player), the person playing earns a point. If it is not divisible by that number, the second player receives a point. The first player to five points wins the round, and they start over again. The game is called Bounce Around, because a player may move any direction per turn. If they run into the end of the board, they bounce to the other side of the board in the same row or column to continue their move. The red stars ask the player to list all factors for the number they landed on during their previous turn. The multi-colored stars ask the player to list four multiples for the number which they landed on during their previous turn.
I started with this foldable by Joy Hall at Joy of Teaching. Part of identifying patterns is that students need to struggle with the raw data to find the pattern, rather than having me provide it for them. Therefore, I only copied the foldable using the hundreds charts page. I did NOT give them the rules right away, and preferred to have them write them on their own.
I reviewed the ideas of factor and multiple with the whole class to makes sure they understood the vocabulary. Then I had them work in small groups to color in the multiples of each factor. After they finished coloring them, I asked the students to discuss with their groups any patterns they saw. They wrote these observations in their notebook.
One of the more difficult parts of describing the patterns, was to describe the patterns within the numbers. Many students began by describing the visual patterns they saw on the hundreds charts. For example, "I colored every other one." or "It's a diagonal line." I asked these students if knowing that the pattern is a diagonal is helpful when I am checking a larger number. I also directed them to start with the pattern for 2 because it is more easily identified than some of the others.
As students worked through the patterns, many of them were able to identify that 2, 4, 6, and 8 had even numbers as their multiples. The follow-up question to this discovery is to find out why 4, 6, and 8 are not factors of every even number the way that 2 is. The patterns for 4 and 8 are hard to notice unless they look at multiples past 100. I encouraged those students who were ready to extend their hundreds chart to look for these patterns. Allowing the students to struggle with these patterns, made it more interesting for them to check their hypotheses to the patterns.
The following day, I shared the Divisibility Pattern Powerpoint which I created listing the patterns. Most of the rules are pretty straight forward. The rules for 3 and 9 are more like "tricks" until you prove them. I watched this Khan Academy video to learn how it worked. Then I included a "proof" in the Powerpoint using Cuisenaire rods. It is a brief look into the Distributive Property. The rule for 7, ends up being pretty convoluted for most students. They may as well just check it by dividing by 7 in my opinion. I included the rule, but told them that I would suggest just dividing it out.
Then I had them play a modified version of this Bounce Around game. Players all begin in the middle at the START square. They roll one di to see how many spaces they will travel. The person next to them rolls one or two dice (their choice). The player moves their game piece to the space, and then if the number is divisible by the second number rolled (by the other player), the person playing earns a point. If it is not divisible by that number, the second player receives a point. The first player to five points wins the round, and they start over again. The game is called Bounce Around, because a player may move any direction per turn. If they run into the end of the board, they bounce to the other side of the board in the same row or column to continue their move. The red stars ask the player to list all factors for the number they landed on during their previous turn. The multi-colored stars ask the player to list four multiples for the number which they landed on during their previous turn.
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Lesson 2: Greatest Common Factor (GCF) and Least Common Multiple (LCM) [6.NS.4]
Lesson 3: Prime Factorization and Review of GCF/LCM [6.NS.4]
Several of my students recognized that there may be a faster way to come up with the greatest common factor using the factor tree approach. They tried to identify the factors, but quickly found out that there were many ways a number could be decomposed into its factors. This caused unreliability for their strategy. These students were naturally on their way to the prime factorization strategy for finding the greatest common factor. I love when my students do all the work for me!
During this lesson, I introduced students to the idea of prime factorization. As a warm-up, I had student decompose several numbers in as many ways as they could think of. We compared their responses on the board and noticed that although each was different, there were some similarities between the factors. I challenged the students to see if they could continue decomposing their numbers into factors until they weren't able to be decomposed any further. Some students recognized at once that they were going to end up with all prime numbers. Others attempted to decompose prime numbers into multiples of 1 (i.e. 2 x 1). When I asked them if identifying that 1 is a factor helped them at all, they realized that this wasn't necessary to show as a factor.
Students were able to practice factoring several more numbers to their prime factors. I then showed them how to find their common factors using a Venn Diagram. The factors shared by both numbers were written in the middle of the diagram. I also had them circle these factors on each of their factor trees, so that we could track the factors we had used. The remaining factors went in their respective portion of the Venn Diagram. I asked students to identify the greatest common factor that they were able to see in their initial factor trees (because they had multiple ways of decomposing the same number, they often were able to see the GCF this way as well).
I asked the class to show how the prime factors helped them figure out the GCF in their notebook. I asked one of my students to describe for the class their understanding. For those students who were unable to find the pattern independently, they were able to clarify with a whole group discussion. I then gave one more example for all students to complete and discuss with a partner.
Finally, I showed them how they can use the same Venn Diagram to find the LCM as well. In order for a multiple to be created which is divisible by both numbers, it must be created by the factors of both original numbers. For example:
LCM(42, 70) can be found by factoring 42 into its prime factors (3, 2, 7) and 70 into its prime factors (2, 7, 5). In the Venn Diagram, the 2 and 7 are already covered. We know that 42 and 70 both are comprised by a factor of 2 and a factor of 7. However, in order for a number to be divisible by 42, it also must have a factor of 3. In order for a number to be divisible by 70, it must have a factor of 5 as well. These are the two factors which are NOT shared in the Venn Diagram by both numbers. In order to find the LCM without listing out each and every factor, you must multiply the original number by the factor which it is "missing". 42 must be multiplied by a factor 5, and 70 must be multiplied by a factor of 3. This creates a multiple (210) which can be broken into prime factors of (2, 7, 3, and 5). It is confusing to explain in words, but using the Venn Diagram, students are able to see what the missing piece will be, and they will be able to more easily find the LCM when it is not readily apparent. This also prepares them for algebraic thinking in finding the least common denominator, and factoring polynomials.
During this lesson, I introduced students to the idea of prime factorization. As a warm-up, I had student decompose several numbers in as many ways as they could think of. We compared their responses on the board and noticed that although each was different, there were some similarities between the factors. I challenged the students to see if they could continue decomposing their numbers into factors until they weren't able to be decomposed any further. Some students recognized at once that they were going to end up with all prime numbers. Others attempted to decompose prime numbers into multiples of 1 (i.e. 2 x 1). When I asked them if identifying that 1 is a factor helped them at all, they realized that this wasn't necessary to show as a factor.
Students were able to practice factoring several more numbers to their prime factors. I then showed them how to find their common factors using a Venn Diagram. The factors shared by both numbers were written in the middle of the diagram. I also had them circle these factors on each of their factor trees, so that we could track the factors we had used. The remaining factors went in their respective portion of the Venn Diagram. I asked students to identify the greatest common factor that they were able to see in their initial factor trees (because they had multiple ways of decomposing the same number, they often were able to see the GCF this way as well).
I asked the class to show how the prime factors helped them figure out the GCF in their notebook. I asked one of my students to describe for the class their understanding. For those students who were unable to find the pattern independently, they were able to clarify with a whole group discussion. I then gave one more example for all students to complete and discuss with a partner.
Finally, I showed them how they can use the same Venn Diagram to find the LCM as well. In order for a multiple to be created which is divisible by both numbers, it must be created by the factors of both original numbers. For example:
LCM(42, 70) can be found by factoring 42 into its prime factors (3, 2, 7) and 70 into its prime factors (2, 7, 5). In the Venn Diagram, the 2 and 7 are already covered. We know that 42 and 70 both are comprised by a factor of 2 and a factor of 7. However, in order for a number to be divisible by 42, it also must have a factor of 3. In order for a number to be divisible by 70, it must have a factor of 5 as well. These are the two factors which are NOT shared in the Venn Diagram by both numbers. In order to find the LCM without listing out each and every factor, you must multiply the original number by the factor which it is "missing". 42 must be multiplied by a factor 5, and 70 must be multiplied by a factor of 3. This creates a multiple (210) which can be broken into prime factors of (2, 7, 3, and 5). It is confusing to explain in words, but using the Venn Diagram, students are able to see what the missing piece will be, and they will be able to more easily find the LCM when it is not readily apparent. This also prepares them for algebraic thinking in finding the least common denominator, and factoring polynomials.
Prime Factorization PPT | |
File Size: | 5965 kb |
File Type: | ppt |
divisibility_patterns.ppt | |
File Size: | 611 kb |
File Type: | ppt |
Lesson 4: Introduction to Fraction Operations {6.NS.1}
The pre-test I give at the beginning of this unit only gives me so much information. Before beginning my division of fractions unit, it is helpful to me if I know what level of understanding my students have of fractions and division. When you put the two together, it gets messy, especially if they don't have a clear understanding of each of these components.
For this activity, I split students in small groups based on the results of their pre-test. Those who did well on the test became the group leaders, and those who did poorly were group members. I asked students to think of the fraction 3/4. They were to discuss and brainstorm with their group all of the different ways they could think of to represent 3/4. They thought of different scenarios and visual representations. I had them compare their individual brainstorm notes with their group and create a simple poster (one per group). They then did the same with defining or explaining the process of division.
The posters included some misconceptions about both the fractions and division. After the unit is in full swing, I ask students to revisit their posters and see if they have an additions or revisions.
My favorite part of this activity is my time spent circulating and notating interesting comments or misconceptions students may have. I carry a clipboard that has my class list with a box next to each student's name. I notate in shorthand things that help me group students for intervention and classwork groupings. I can pair students with similar difficulties or strengths depending on the activity.
Once the posters were finished, I had the groups do a "gallery walk" and leave notes for groups about things they disagreed with or had questions about. They also left compliments or additions.
For this activity, I split students in small groups based on the results of their pre-test. Those who did well on the test became the group leaders, and those who did poorly were group members. I asked students to think of the fraction 3/4. They were to discuss and brainstorm with their group all of the different ways they could think of to represent 3/4. They thought of different scenarios and visual representations. I had them compare their individual brainstorm notes with their group and create a simple poster (one per group). They then did the same with defining or explaining the process of division.
The posters included some misconceptions about both the fractions and division. After the unit is in full swing, I ask students to revisit their posters and see if they have an additions or revisions.
My favorite part of this activity is my time spent circulating and notating interesting comments or misconceptions students may have. I carry a clipboard that has my class list with a box next to each student's name. I notate in shorthand things that help me group students for intervention and classwork groupings. I can pair students with similar difficulties or strengths depending on the activity.
Once the posters were finished, I had the groups do a "gallery walk" and leave notes for groups about things they disagreed with or had questions about. They also left compliments or additions.
l3_fantastic_fractions.pptx | |
File Size: | 1184 kb |
File Type: | pptx |
l5_fantastic_fractions_continued.pptx | |
File Size: | 1520 kb |
File Type: | pptx |
division_of_fraction_by_whole_number.docx | |
File Size: | 123 kb |
File Type: | docx |
Marimba Makers PPT | |
File Size: | 1392 kb |
File Type: | pptx |
Fraction Operation Jeopardy | |
File Size: | 3724 kb |
File Type: | pptx |
fraction_warm-up_examples.pdf | |
File Size: | 156 kb |
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6ns1_test.pdf | |
File Size: | 876 kb |
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Valentine's Party Planning | |
File Size: | 169 kb |
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