U5 Patterns Inside Numbers
BTC p107 Tax Collector
These are 12 envelopes, each one with some money in it. This is your money. I am just holding it for you. But, you can have any of these envelopes whenever you want. You just have to ask for it. So, which one do you want first? [They choose the 12]
OK. There you go. This envelope is now gone.
Right [snap fingers]. I forgot to tell you that we have to pay taxes on this money. And because you took the $12, the tax collector will take the 1, 2, 3, 4, and 6 envelopes [cross out the envelopes as you say the numbers]. Why does the tax collector take these envelopes?
Right! The tax collector takes the factors of whatever envelope you took. Ok, so which envelope do you want. [They choose 11]
Ok. But I forgot to mention that the tax collector always wants some taxes. So, when you choose an envelope there must always be at least one envelope for the tax collector to take -- one factor for the tax collector to take. So, can you take the $11?
[No.] So, which envelope can you take?
Ok. You take the $10. What does the tax collector take? [$5]
Ok. What next? [Nothing].
Ok. Now the tax collector is very kind and does not want to see anything go to waste, so they will take the rest of the envelopes [cross the rest out].
While working whole-class, you got $22 in total. You can get way more money! Your job is to work with your group to get more than $22.
Factors: the numbers you multiply together to get a specific product. (e.g. 1, 2, 3, 4, and 6 are all factors of 12). A factor is a number that divides another number evenly, with no remainder.
Common factors: Factors that are shared (e.x. the numbers 6 and 12 share a factor of 2)
Greatest Common Factor (GCF): the largest shared factor (e.x. the largest common factor between the numbers 6 and 12 is 6)
Prime number: a whole number greater than 1 that has only two factors: 1 and itself (e.g., 2, 3, 5, 7).
Use factor trees and prime numbers to find the GCF
Use The Ladder Method to find the GCF. Multiply ALL the numbers on the left side of the ladder. (Don't include the bottom numbers when multiplying.)
T1
My Responsibilities Nov 12
Due 11/12
1. CYUqs (do some with factor trees and some with the ladder method):
You are all going to do some of these. But, there are three things for you to keep in mind. First, do your own. Second, you choose where to start. Third, check your work with the students around you, and if you need help, get help. And if someone needs help, give help.
Check answers to make sure you are understanding:
2. Be sure that your notes are your notes are up to date:
3. Stuck? Choose to persevere: review your thinking, try different approaches, ask for help, search
Turn to your neighbor and describe what a factor is.
After completing the Check-your-understanding Questions, which were important for everyone to do?
What would you do if you were asked to find the GCF of 3 (or more numbers)?
Can we find the greatest common factor of 60, 24, and 48
On the blank side, write your first and last name and U5T1. Then, find the GCF of:
A. 24 and 30
B. 60 and 80
Lists will always work, but can be time-consuming.
The Ladder Method can be used to find the LCM too. Multiply ALL the numbers on the outside: (the numbers on the left and the numbers on the bottom.
What is the difference between the GCF and the LCM (compare and contrast)?
T2
My Responsibilities Nov 13
Due 11/13
1. CYUqs (do some with lists and some with the ladder method):
You are all going to do some of these. But, there are three things for you to keep in mind. First, do your own. Second, you choose where to start. Third, check your work with the students around you, and if you need help, get help. And if someone needs help, give help.
Check answers to make sure you are understanding
2. Be sure that your notes are your notes are up to date:
3. Stuck? Choose to persevere: review your thinking, try different approaches, ask for help, search
Before discussing your the Check-your-understanding Questions, turn to your neighbor and describe what a multiple is.
When you both agree, take turns showing each other one problem you thought represented the basic idea of how to find a LCM.
Did you notice that sometimes the LCM was simply the product of the two numbers?
But other times, finding the LCM took making a list (or using the Ladder Method).
If you could make up a rule for when to multiply to find the LCM and when to not multiply, what would the rule be?
On the blank side, write your first and last name and U5T2.
1. Find the LCM of 8 and 12
2. Ananya likes to go to Starbucks every 10 days and Anika likes to go every 12 days. Ananya and Anika both went to Starbucks today. After how many days do they go to Starbucks on the same day again?
The secret I use to multiply large numbers mentally is called the Distributive Property.
First, I decompose one of the factors in the expression.
For example, in the expression 7 ⋅ 12, I decompose the 12 into 10 + 2. Then, I multiply the 7 by 10 and the 7 by 2 and add up the products.
7 ⋅ 12 = 7 (10 ⋅ 2)
* to prove that this works, you can simplify both sides to get 84 = 84
I want you to think of the next 10 problems like puzzles.
When you figure it out, be sure to share the secret with each of your group members.
#1 notice that the number outside of the parentheses is a common factor of the numbers on the other sides of the equals sign
T3
My Responsibilities Nov 14
Due 11/14
1. CYUqs (Be sure to do at least 8 of the medium questions!)
You are all going to do some of these. But, there are three things for you to keep in mind. First, do your own. Second, you choose where to start. Third, check your work with the students around you, and if you need help, get help. And if someone needs help, give help.
Check answers to make sure you are understanding
2. Be sure that your notes are your notes are up to date:
3. Stuck? Choose to persevere: review your thinking, try different approaches, ask for help, search
Show the distributive property algorithm, starting with an expression and sharing the factor outside the parentheses with the terms inside.
Note: If you simplify both sides (working inside parentheses first), you will get the same numbers (a.k.a they are equivalent expressions)
Unit 5 Test is this Thursday 11/20
GCF via Ladder Method
LCM via Ladder Method
A
Compare and Contrast Nov 17
Due 11/17
p11: 1-10; p13: 17-25 (Volume 1, Ch1-1)
Rewrite the problem. Show your thinking (factor trees with prime numbers OR the Ladder Method). Label and circle your answer (GCF = or LCM =).
Work together. If you need help, get help. And be sure to check answers while you're working.
A ratio is tool for comparing two quantities of the SAME things
It tells you how much of one thing there is compared to another thing.
Today, we're going to relate factors and multiples to ratios.
Ratios show up in many places in real life: morning oatmeal 2 parts water to 1 part oats; cinnamon roll recipe calls for 15 cups of flour to 3 cups of milk
Scale up: want to make oatmeal for MORE than just my family
Scale down: want to make LESS, so we don't have so much left over
Now that you know how a ratio table works (how all the ratios are connected), let's look at a ratio table with missing numbers.
#1 Can't figure it out? Don't see a pattern? Add a row and scale down
#2 You can always scale down, scale up or vice versa
T4
My Responsibilities Nov 18
Due 11/18
1. CYUqs:
You are all going to do some of these. But, there are three things for you to keep in mind. First, do your own. Second, you choose where to start. Third, check your work with the students around you, and if you need help, get help. And if someone needs help, give help.
Check answers to make sure you are understanding:
2. Be sure that your notes are your notes are up to date:
3. Stuck? Choose to persevere: review your thinking, try different approaches, ask for help, search
Before discussing the GCF and LCM problems, turn to your elbow partner and tell them which one is the smaller and which one is the bigger: factors or multiples.
When you are in agreement, show each other at least one LCM and one GCF problem from yesterday that you answered correctly.
Discuss any problems that were challenging.
After completing the Check-your-understanding Questions, which were important for everyone to do?
A unit rate is a special kind of ratio that compares two DIFFERENT things, but makes the second number equal to one single unit (1).
When you're finding the unit rate, are you scaling up or down?
When you use the unit rate to find a larger number, is that scaling up or down?
The numbers you are using to scale up or down, what kind of numbers are they?
T5
My Responsibilities Nov 19
Due 11/19
1. CYUqs:
You are all going to do some of these. But, there are three things for you to keep in mind. First, do your own. Second, you choose where to start. Third, check your work with the students around you, and if you need help, get help. And if someone needs help, give help.
Check answers to make sure you are understanding:
2. Be sure that your notes are your notes are up to date:
3. Stuck? Choose to persevere: review your thinking, try different approaches, ask for help, search
A
Summarize Nov 19
Due 11/20
Click the link below to open the unit's graphic organizer:
Complete each box by adding notes, examples, and annotations (notes that explain what is happening in your examples). Refer to your notes, CYUqs, entry tickets, vertical collaboration, Google Search, etc. Work together. Help each other.
You may use the notes you make on this graphic organizer during the test (nothing else).
Unit 5
Take out your Chromebook and some things to write with. If needed, you may use your graphic organizer. No multiplication charts, watches, calculators, or pencil pouches.
DO NOT SIGN IN TO YOUR CHROMEBOOK. If you are signed in, sign out. Click the Apps in the bottom left corner > Pear Assessment > Login with Google (sign in with your FCUSD email @student.fcusd.org). Sign in again with @student.fcusd.org and your password > Continue
Click Start Assessment to open the Unit 5 Answer Key 2025
When finished, put your test (with your name on it) and graphic organizer in the basket in the back. You may read (Sora), draw, write, work on an assignment, or rest. Do not talk.
Minimum day
Nov 21 Goal Setting
Due 11/21
Start by opening PowerSchool. Click the overall grade.
On your Goal Sheet, check off each goal that you have a 3 or better in.
If you have a score of less than 3, leave the "Got It" blank.
Where it says "Date", that is the date you plan to retake and show me you understand!
At that the bottom of your goal sheet you see the NEXT STEPS for goals that are less than 3. Please follow those steps BEFORE the retake date: 12/4
Please raise your hand to let me know!
After reviewing your tests, I see that EVERYONE has been working hard and doing their best. I am proud of YOU.
Take a moment to reflect on the efforts you put into this unit:
During vertical collaboration, have you been working with your groups: engaging in the conversations, asking question when you're confused, really focusing on the math?
When working on the CYUqs, are you doing 8-12 problems? Are you focusing on the medium levels, trying a few spicy questions, checking answers, and asking for help if you need it?
Here's how I graded this assessment: There were 3 questions per learning goal: mild, medium, and spicy. If you correctly answered 3 questions, score is a 4; if 2 were correct, score is a 3; if 1 was correct, score is a 2; if none were correct, score is a 1
When you look at PowerSchool, Qs 1-3 are U5T1, Qs 4-6 are U5T2, Qs 7-9 are U5T3*, Qs 10-12 are U5T4, Qs 13-15 are U5T5*
Refer to the Proficiency Levels to know what your work says about where you are right now in terms of your understanding on the key learning target(s):
A
Test Corrections Nov 21
Due 11/26
The most important part of the test happens right now: reviewing your work and learning from mistakes.
1. Open Pear Assessment. On your paper test, make a smiley face by the questions you correctly answered. Make a check mark by the ones you missed.
2. Find an expert to help you with the problem you missed. Compare and contrast your work and their work. Ask questions. Think. Don't copy.
3. On your corrections worksheet, neatly rewrite the problem, show your thinking, and circle the correct answer.
4. Before moving to the next problem, write down your mistake and what you will do to not make that mistake again.
* If you are curious to know what 3's and 4's looked like, review the randomly chosen exemplars (provided in class)
Thanksgiving Break
Thanksgiving Break
Thanksgiving Break
Thanksgiving Break
Thanksgiving Break
December 12 is the last day to submit the U3T4 Knowledge Application assignment.
Why, How, and What
In order to make sense of the world (DI1), students will reason abstractly and quantitatively (SMP2) and look for and make use of structure (SMP7) while exploring changing quantities (CC2)
Grade Six Content Standards
Consider how numbers are made up, exploring factors and multiples, visually and numerically. (p33)
NS4. Find the greatest common factor of two whole numbers less than or equal to 100 and the least common multiple of two whole numbers less than or equal to 12. Use the distributive property to express a sum of two whole numbers 1–100 with a common factor as a multiple of a sum of two whole numbers with no common factor. For example, express 36 + 8 as 4 (9 + 2).
RP3. Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.
a. Make tables of equivalent ratios relating quantities with whole number measurements, find missing values in the tables, and plot the pairs of values on the coordinate plane. Use tables to compare ratios.
b. Solve unit rate problems including those involving unit pricing and constant speed. For example, if it took 7 hours to mow 4 lawns, then at that rate, how many lawns could be mowed in 35 hours? At what rate were lawns being mowed?
c. Find a percent of a quantity as a rate per 100 (e.g., 30% of a quantity means 30/100 times the quantity); solve problems involving finding the whole, given a part and the percent.
d. Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.
BTC p107 Tax Collector
These are 12 envelopes, each one with some money in it. This is your money. I am just holding it for you. But, you can have any of these envelopes whenever you want. You just have to ask for it. So, which one do you want first? [They choose the 12]
OK. There you go. This envelope is now gone.
Right [snap fingers]. I forgot to tell you that we have to pay taxes on this money. And because you took the $12, the tax collector will take the 1, 2, 3, 4, and 6 envelopes [cross out the envelopes as you say the numbers]. Why does the tax collector take these envelopes?
Right! The tax collector takes the factors of whatever envelope you took. Ok, so which envelope do you want. [They choose 11]
Ok. But I forgot to mention that the tax collector always wants some taxes. So, when you choose an envelope there must always be at least one envelope for the tax collector to take -- one factor for the tax collector to take. So, can you take the $11?
[No.] So, which envelope can you take?
Ok. You take the $10. What does the tax collector take? [$5]
Ok. What next? [Nothing].
Ok. Now the tax collector is very kind and does not want to see anything go to waste, so they will take the rest of the envelopes [cross the rest out].
While working whole-class, you got $22 in total. You can get way more money! Your job is to work with your group to get more than $22.
Factors: the numbers you multiply together to get a specific product. (e.g. 1, 2, 3, 4, and 6 are all factors of 12). A factor is a number that divides another number evenly, with no remainder.
Common factors: Factors that are shared (e.x. the numbers 6 and 12 share a factor of 2)
Greatest Common Factor (GCF): the largest shared factor (e.x. the largest common factor between the numbers 6 and 12 is 6)
Prime number: a whole number greater than 1 that has only two factors: 1 and itself (e.g., 2, 3, 5, 7).
Use factor trees and prime numbers to find the GCF
Use The Ladder Method to find the GCF. Multiply ALL the numbers on the left side of the ladder. (Don't include the bottom numbers when multiplying.)
T1
My Responsibilities Nov 12
Due 11/12
1. CYUqs (do some with factor trees and some with the ladder method):
You are all going to do some of these. But, there are three things for you to keep in mind. First, do your own. Second, you choose where to start. Third, check your work with the students around you, and if you need help, get help. And if someone needs help, give help.
Check answers to make sure you are understanding:
2. Be sure that your notes are your notes are up to date:
3. Stuck? Choose to persevere: review your thinking, try different approaches, ask for help, search
Turn to your neighbor and describe what a factor is.
After completing the Check-your-understanding Questions, which were important for everyone to do?
What would you do if you were asked to find the GCF of 3 (or more numbers)?
Can we find the greatest common factor of 60, 24, and 48
On the blank side, write your first and last name and U5T1. Then, find the GCF of:
A. 24 and 30
B. 60 and 80
Lists will always work, but can be time-consuming.
The Ladder Method can be used to find the LCM too. Multiply ALL the numbers on the outside: (the numbers on the left and the numbers on the bottom.
What is the difference between the GCF and the LCM (compare and contrast)?
T2
My Responsibilities Nov 13
Due 11/13
1. CYUqs (do some with lists and some with the ladder method):
You are all going to do some of these. But, there are three things for you to keep in mind. First, do your own. Second, you choose where to start. Third, check your work with the students around you, and if you need help, get help. And if someone needs help, give help.
Check answers to make sure you are understanding
2. Be sure that your notes are your notes are up to date:
3. Stuck? Choose to persevere: review your thinking, try different approaches, ask for help, search
Before discussing your the Check-your-understanding Questions, turn to your neighbor and describe what a multiple is.
When you both agree, take turns showing each other one problem you thought represented the basic idea of how to find a LCM.
Did you notice that sometimes the LCM was simply the product of the two numbers?
But other times, finding the LCM took making a list (or using the Ladder Method).
If you could make up a rule for when to multiply to find the LCM and when to not multiply, what would the rule be?
On the blank side, write your first and last name and U5T2.
1. Find the LCM of 8 and 12
2. Ananya likes to go to Starbucks every 10 days and Anika likes to go every 12 days. Ananya and Anika both went to Starbucks today. After how many days do they go to Starbucks on the same day again?
The secret I use to multiply large numbers mentally is called the Distributive Property.
First, I decompose one of the factors in the expression.
For example, in the expression 7 ⋅ 12, I decompose the 12 into 10 + 2. Then, I multiply the 7 by 10 and the 7 by 2 and add up the products.
7 ⋅ 12 = 7 (10 ⋅ 2)
* to prove that this works, you can simplify both sides to get 84 = 84
I want you to think of the next 10 problems like puzzles.
When you figure it out, be sure to share the secret with each of your group members.
#1 notice that the number outside of the parentheses is a common factor of the numbers on the other sides of the equals sign
T3
My Responsibilities Nov 14
Due 11/14
1. CYUqs (Be sure to do at least 8 of the medium questions!)
You are all going to do some of these. But, there are three things for you to keep in mind. First, do your own. Second, you choose where to start. Third, check your work with the students around you, and if you need help, get help. And if someone needs help, give help.
Check answers to make sure you are understanding
2. Be sure that your notes are your notes are up to date:
3. Stuck? Choose to persevere: review your thinking, try different approaches, ask for help, search
Show the distributive property algorithm, starting with an expression and sharing the factor outside the parentheses with the terms inside.
Note: If you simplify both sides (working inside parentheses first), you will get the same numbers (a.k.a they are equivalent expressions)
Unit 5 Test is this Thursday 11/20
GCF via Ladder Method
LCM via Ladder Method
A
Compare and Contrast Nov 17
Due 11/17
p11: 1-10; p13: 17-25 (Volume 1, Ch1-1)
Rewrite the problem. Show your thinking (factor trees with prime numbers OR the Ladder Method). Label and circle your answer (GCF = or LCM =).
Work together. If you need help, get help. And be sure to check answers while you're working.
A ratio is tool for comparing two quantities of the SAME things
It tells you how much of one thing there is compared to another thing.
Today, we're going to relate factors and multiples to ratios.
Ratios show up in many places in real life: morning oatmeal 2 parts water to 1 part oats; cinnamon roll recipe calls for 15 cups of flour to 3 cups of milk
Scale up: want to make oatmeal for MORE than just my family
Scale down: want to make LESS, so we don't have so much left over
Now that you know how a ratio table works (how all the ratios are connected), let's look at a ratio table with missing numbers.
#1 Can't figure it out? Don't see a pattern? Add a row and scale down
#2 You can always scale down, scale up or vice versa
T4
My Responsibilities Nov 18
Due 11/18
1. CYUqs:
You are all going to do some of these. But, there are three things for you to keep in mind. First, do your own. Second, you choose where to start. Third, check your work with the students around you, and if you need help, get help. And if someone needs help, give help.
Check answers to make sure you are understanding:
2. Be sure that your notes are your notes are up to date:
3. Stuck? Choose to persevere: review your thinking, try different approaches, ask for help, search
Before discussing the GCF and LCM problems, turn to your elbow partner and tell them which one is the smaller and which one is the bigger: factors or multiples.
When you are in agreement, show each other at least one LCM and one GCF problem from yesterday that you answered correctly.
Discuss any problems that were challenging.
After completing the Check-your-understanding Questions, which were important for everyone to do?
A unit rate is a special kind of ratio that compares two DIFFERENT things, but makes the second number equal to one single unit (1).
When you're finding the unit rate, are you scaling up or down?
When you use the unit rate to find a larger number, is that scaling up or down?
The numbers you are using to scale up or down, what kind of numbers are they?
T5
My Responsibilities Nov 19
Due 11/19
1. CYUqs:
You are all going to do some of these. But, there are three things for you to keep in mind. First, do your own. Second, you choose where to start. Third, check your work with the students around you, and if you need help, get help. And if someone needs help, give help.
Check answers to make sure you are understanding:
2. Be sure that your notes are your notes are up to date:
3. Stuck? Choose to persevere: review your thinking, try different approaches, ask for help, search
A
Summarize Nov 19
Due 11/20
Click the link below to open the unit's graphic organizer:
Complete each box by adding notes, examples, and annotations (notes that explain what is happening in your examples). Refer to your notes, CYUqs, entry tickets, vertical collaboration, Google Search, etc. Work together. Help each other.
You may use the notes you make on this graphic organizer during the test (nothing else).
Unit 5
Take out your Chromebook and some things to write with. If needed, you may use your graphic organizer. No multiplication charts, watches, calculators, or pencil pouches.
DO NOT SIGN IN TO YOUR CHROMEBOOK. If you are signed in, sign out. Click the Apps in the bottom left corner > Pear Assessment > Login with Google (sign in with your FCUSD email @student.fcusd.org). Sign in again with @student.fcusd.org and your password > Continue
Click Start Assessment to open the Unit 5 Answer Key 2025
When finished, put your test (with your name on it) and graphic organizer in the basket in the back. You may read (Sora), draw, write, work on an assignment, or rest. Do not talk.
Minimum day
Nov 21 Goal Setting
Due 11/21
Start by opening PowerSchool. Click the overall grade.
On your Goal Sheet, check off each goal that you have a 3 or better in.
If you have a score of less than 3, leave the "Got It" blank.
Where it says "Date", that is the date you plan to retake and show me you understand!
At that the bottom of your goal sheet you see the NEXT STEPS for goals that are less than 3. Please follow those steps BEFORE the retake date: 12/4
Please raise your hand to let me know!
After reviewing your tests, I see that EVERYONE has been working hard and doing their best. I am proud of YOU.
Take a moment to reflect on the efforts you put into this unit:
During vertical collaboration, have you been working with your groups: engaging in the conversations, asking question when you're confused, really focusing on the math?
When working on the CYUqs, are you doing 8-12 problems? Are you focusing on the medium levels, trying a few spicy questions, checking answers, and asking for help if you need it?
Here's how I graded this assessment: There were 3 questions per learning goal: mild, medium, and spicy. If you correctly answered 3 questions, score is a 4; if 2 were correct, score is a 3; if 1 was correct, score is a 2; if none were correct, score is a 1
When you look at PowerSchool, Qs 1-3 are U5T1, Qs 4-6 are U5T2, Qs 7-9 are U5T3*, Qs 10-12 are U5T4, Qs 13-15 are U5T5*
Refer to the Proficiency Levels to know what your work says about where you are right now in terms of your understanding on the key learning target(s):
A
Test Corrections Nov 21
Due 11/26
The most important part of the test happens right now: reviewing your work and learning from mistakes.
1. Open Pear Assessment. On your paper test, make a smiley face by the questions you correctly answered. Make a check mark by the ones you missed.
2. Find an expert to help you with the problem you missed. Compare and contrast your work and their work. Ask questions. Think. Don't copy.
3. On your corrections worksheet, neatly rewrite the problem, show your thinking, and circle the correct answer.
4. Before moving to the next problem, write down your mistake and what you will do to not make that mistake again.
* If you are curious to know what 3's and 4's looked like, review the randomly chosen exemplars (provided in class)
Thanksgiving Break
Thanksgiving Break
Thanksgiving Break
Thanksgiving Break
Thanksgiving Break
December 12 is the last day to submit the U3T4 Knowledge Application assignment.