# 6th Grade Common Core Math

## Tips, Mathematical Practices, and Ideas about Integers

### Topic 10: Integers

Note: The original dates you set for this unit was December 6-18 (Dec.19 posttest/pretest). However, most of you will be starting this unit soon since your pacing is determined by student understanding.

 10-1 Understanding Integers 6.NS.5 M06-S1C1-04, M06-S1C1-05 10-2 Comparing and Ordering Integers 6.NS.7.a M06-S1C1-04 10-3 Rational Numbers on a Number Line 6.NS.6.c M06-S1C1-03, M06-S1C1-04 10-4 Step-Up Lesson: Adding Integers 7.NS.1.b M06-S1C2-01 10-5 Step-Up Lesson: Subtracting Integers 7.NS.1.c M06-S1C2-01 10-6 Step-Up Lesson: Multiplying Integers 7.NS.2.a 10-7 Step-Up Lesson: Dividing Integers 7.NS.2.b 10-8 Absolute Value 6.NS.7.d M06-S1C1-05 10-9 Graphing Points on a Coordinate Plane 6.NS.6.c M06-S4C3-01 K104 Missing Coordinates** (See double dose)J31 Use Reasoning** M06-S4C3-02M06-S5C2-09 10-10 Problem Solving: Use Reasoning 6.G.3 M06-S5C2-09

### Double Dose Recommendations:

• Kim Sutton math routines
• Pre-teach the following:
• Missing Coordinates, 3 days (AZ7, K104)
• Use Reasoning, 1 day (J31)

### Tip of the Week:

• Start your unit by looking at the performance task. Have students create need-to-know lists for what they need to know in order to complete the task.

• After each lesson, refer back to the need-to-know list to see which pieces they’ve learned in order to reach the final goal of completing the performance task.

### Technology Integration Resources:

Below are some Learn Zillion videos to use for each of the lessons beyond SuccessNet:

If you have iPads in your class, Educreations has a graphing grid with the coordinates to use as background paper and students can create tutorials with this app.

### Mathematical Practices:

In enVision, the “Hands-On, Minds-On” quick lesson to introduce the concept also incorporates Math Practices. When I was in the classroom, I’d use my pretest data and formative assessments to determine which students need pre-teaching. Every day I’d pre-teach a small group, and I’d spend time using these lessons to develop the concept and connect prior knowledge. When I’d start my lesson, I could quickly introduce it using the Interactive Learning, beneficial to all students.

## The Importance of Conceptual Understanding in Math

Why not just teach the steps?

Students take information from the world using all their senses. Why should learning mathematics be much different? Let’s look at how athletes learn to play sports. Imagine taking a child to watch a professional baseball player for 50 minutes a day and then be expected to win a game. It seems ridiculous because in order to play the sport, the player must feel the bat and gloves in their hands. They must run the bases to determine how long it will take. They need to imagine hitting the ball into the outfield. No matter how much you watch the pros, until you get your whole body involved, there is no way to really excel at the sport. This is not much different than a student watching a teacher for 50 minutes and then be expected to successfully complete their math tasks. Truly understanding mathematics involves more than just memorizing steps or formulas. In order to truly comprehend mathematical concepts, our teaching methods should involve the use of moving and placing manipulatives in order for students to feel and see what is really happening in a problem and therefore be able to predict a reasonable outcome.

### What do conceptual learning activities look like?

Conceptual learning activities often incorporate the mathematical practices.

• Make sense of problems and persevere in solving them : When a student asks and demonstrates the reasonableness of the answer, they are using this math practice (and arguably others as well). For example:
•  “What is 7.245 x 6.18?” A student demonstrates conceptual understanding of mathematics when she or he explains that 850.015 cannot be a reasonable product because by estimating the smallest possible response as 7 X 6, and the largest possible response as 8 X 7; therefore making the product between 42 and 56.

• Model with mathematics: A student demonstrates conceptual understanding of prime numbers when she or he is asked to prove if a number is a prime or composite number, and she/he creates a model to demonstrate it is prime by building only two arrays to show the different multiplication combinations of the number, while a composite number has more than two.

ResourceWhat is Conceptual Understanding?

### Performance Tasks demonstrate Conceptual Understanding

Before beginning a new topic, study the performance based assessment. What are the students truly expected to be able to do conceptually? This should be our “GPS.” We need to have a clear understanding of the concept at hand to effectively facilitate learning.

Do you teach the steps, or do you teach the concepts?

How do you allow students to demonstrate their conceptual understanding daily?

I wanted to give a big thank you to Robyn Gonzales, Common Core Coach and Title I Teacher at SMES, for the ideas, resources, and the wording in the top and bottom portions of this post.

## Tips, Mathematical Practices, and Ideas about Dividing Fractions and Mixed Numbers — Week 2

### Topic 9- Dividing Fractions and Mixed Numbers

November 21-December 4 (Dec. 5 posttest/pretest)

#### Lesson

9-1 Understanding Division of Fractions

6.NS.1

#### AZ Standards

M06-S1C2-04, M06-S1C2-05, M06-S5C1-01

9-2 Dividing a Whole Number by a Fraction 6.NS.1 M06-S1C2-04, M06-S1C2-05, M06-S5C1-01
9-3 Dividing Fractions 6.NS.1 M06-S1C2-04
9-4 Estimating Quotients 6.NS.1 M06-S1C2-04, M06-S1C3-02
9-5 Dividing Mixed Numbers 6.NS.1 M06-S1C2-04
9-6 Solving Equations 6.EE.7 M06-S1C2-04
9-7 Problem Solving: Look for a Pattern 6.NS.6 M06-S3C1-01

### Double Dose Recommendations:

• Kim Sutton math routines.
• Pre-teach the following: Area, 5 days (#34-38)

### Tip of the Week:

The Performance Assessment has students working with fractions in recipes. Here’s an online activity that complements the performance task:

### Feeding Frenzy

In this activity, students will multiply and divide a recipe to feed groups of various sizes. Students will use unit rates or proportions and think critically about real world applications of a baking problem.

Videos:

### Mathematical Practices:

The CCSS says this about Model with Mathematics, “In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community.”
With the holidays, could you ask the students to Model with Math by asking  questions about how much food your household needs to buy to host a holiday party?
How do students demonstrate their conceptual understanding of mathematics in your classroom? What performance tasks do you have students work on?

## Tips, Mathematical Practices, and Ideas about Dividing Fractions and Mixed Numbers

### Topic 9- Dividing Fractions and Mixed Numbers

November 21-December 4 (Dec. 5 posttest/pretest)

#### Lesson

9-1 Understanding Division of Fractions

6.NS.1

#### AZ Standards

M06-S1C2-04, M06-S1C2-05, M06-S5C1-01

9-2 Dividing a Whole Number by a Fraction 6.NS.1 M06-S1C2-04, M06-S1C2-05, M06-S5C1-01
9-3 Dividing Fractions 6.NS.1 M06-S1C2-04
9-4 Estimating Quotients 6.NS.1 M06-S1C2-04, M06-S1C3-02
9-5 Dividing Mixed Numbers 6.NS.1 M06-S1C2-04
9-6 Solving Equations 6.EE.7 M06-S1C2-04
9-7 Problem Solving: Look for a Pattern 6.NS.6 M06-S3C1-01

### Double Dose Recommendations:

• Kim Sutton math routines.
• Pre-teach the following: Area, 5 days (#34-38)

### Tip of the Week:

• The Topic Opener/STEM Project on page 201 not only brings in math in the context of the real world, but it also requires informational text reading and researching.
• New York City Department of Education created a performance task, Share My Candy, for dividing fractions by fractions, which includes a rubric.

### Technology Integration Resources:

Videos:

Online & interactive practice, that progressively gets more difficult from IXL:

### Mathematical Practices:

MP6- Attend to Precision

• Communicate precisely to others
• Use clear definitions in discussion with others and in their own reasoning
• State the meaning of symbols they choose (equal sign)
• Specific with units of measure, labeling, and quantities
• Calculate accurately and efficiently

Question/Practice Stems

• Why do you believe that to be true?
• How did you reach your conclusion?
• What math vocabulary did you use today?
• How did you use it?
• What does that symbol mean?
• What would happen if your removed the symbol?
• How would the problem change?
• Does my work need labeling? If so, what? Why?

Are students applying the math practices daily?  How do you know?  What do the math practices sound like and look like in your classroom?

## Going Deeper

I’ve noticed that our 6th grade math teachers are going deeper with the math, which requires more time. Therefore, half of our students are still working on Topic 6, while a quarter of our students are learning through Topic 7, and the other quarter of our students are working on Topic 8.

Here are links to those posts:

Dig deeper into math concepts, beyond the computational skills

Photo Credit: Jeff Mikels via Compfight

If your students are currently learning through Topic 8, I’d like to encourage you to try the Performance Task, some of the DOK ideas, or PBL ideas to go deeper. It might take a little longer than the Essential Map (pacing) that you created, but that’s okay because it will mean students are learning it on a deeper level.

What opportunities are you providing your students to deeply explore math concepts beyond the computational math skills?

## Tips, Mathematical Practices, and Ideas about Multiplying Fractions and Mixed Numbers

### Topic 8- Multiplying Fractions and Mixed Numbers

November 12-November 19 (Nov. 20 posttest/pretest)

#### Lesson

8-1 Multiplying a Fraction and a Whole Number

6.NS.1*

#### AZ AIMS Standards

M06-S1C2-04, M06-S1C2-05,  M06-S5C1-01

8-2 Estimating Products 6.NS.1* M06-S1C2-04, M06-S1C3-01
8-3 Multiplying Fractions 6.NS.1* M06-S1C2-04, M06-S1C3-01
8-4 Multiplying Mixed Numbers 6.NS.1* M06-S1C2-04, M06-S1C3-02,  M06-S5C1-01
8-5 Problem Solving: Multiple-Step Problems 6.G.1 M06-S1C2-04, M06-S5C2-01, M06-S5C2-02

### Double Dose Recommendations:

• Preteach Perimeter: 3 days (#31-33)
• Kim Sutton Math Routines

### Tip of the Week: PBL

How do we move up Bloom’s Taxonomy and go deeper with Depth of Knowledge (DOK)? What beginning PBL is available to help with DOK and Bloom’s?

#### Description

The Parchitecture Project

Using Google Sketch Up, students design a student play-area for the new High Tech High K-8 school in Chula Vista. Students create design firms, conduct student surveys for input on their design, determine budget costs, and propose their ideas to a panel of adults and peers. Students have complete control of the design of the space with the stipulation that it must be safe and accessible to all students.
Prime Putt – Putt
• 6.G.1
• 7.G.1
• 7.G.4
• 7.G.6
• 7.RP.2
Prime Putt – Putt golf is looking to refurbish their miniature golf course. Mrs. Math, the owner, has outlined the desired repairs. Students are to submit a scale drawing of existing golf holes along with a list of materials necessary to give Prime Putt – Putt the makeover it desperately needs.
Step Right Up for a Good Cause
• 6.NS.1
• 5.NF.1
• 5.NF.1
• 6.NS.4
• 7.SP.5
• 7.SP.7
A local charity needs your help!  You have been asked to plan a Family Fun Night in order to raise money for the charity.  You will develop games and find theoretical and experimental probability of each game.   You will plan a concession stand menu with combo choices, shop for game prizes and concession needs, and propose a layout for the event.  In the end, you will present a proposal to a panel of charity officials that includes the projected cost and profit for the event.
Let’s Party!

As caterers, students are bidding on a job to plan a birthday party for a 13-year old.  Given a budget of \$250, they will submit a party proposal for 30 guests that includes a budget spreadsheet, written description of party and events, menu, map of room, and an oral presentation.

### Mathematical Practices: Make sense of problems and persevere in solving them.

This tip comes from the ADE:

In grade 6, students solve problems involving ratios and rates and discuss how they solved them. Students solve real world problems through the application of algebraic and geometric concepts. Students seek the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinking by asking themselves, “What is the most efficient way to solve the problem?”, “Does this make sense?”, and “Can I solve the problem in a different way?”.

What opportunities are you providing students to persevere with problem solving (and PBL)?

## Tips, Mathematical Practices, and Ideas about Multiplying, Fractions, and Mixed Numbers

### Topic 8- Multiplying Fractions and Mixed Numbers

November 12-November 19 (Nov. 20 posttest/pretest)

#### Lesson

8-1 Multiplying a Fraction and a Whole Number

6.NS.1*

#### AZ AIMS Standards

M06-S1C2-04, M06-S1C2-05,  M06-S5C1-01

8-2 Estimating Products 6.NS.1* M06-S1C2-04, M06-S1C3-01
8-3 Multiplying Fractions 6.NS.1* M06-S1C2-04, M06-S1C3-01
8-4 Multiplying Mixed Numbers 6.NS.1* M06-S1C2-04, M06-S1C3-02,  M06-S5C1-01
8-5 Problem Solving: Multiple-Step Problems 6.G.1 M06-S1C2-04, M06-S5C2-01, M06-S5C2-02

### Double Dose Recommendations:

• Preteach Perimeter: 3 days (#31-33)
• Kim Sutton Math Routines

### Tip of the Week:

Bloom’s/DOK- Are we providing opportunities for our students to analyze, evaluate, create, and construct during math?

• Analyzeconstruct models, graphing information, comparing and contrasting values
• Evaluate– Prepare a list of criteria for solving, make a booklet that includes mathematical rules for solving and potential tools list

### Technology Integration Weekly Resources:

You could have students view the video, and ask them to then create their own video (or tell a partner if you do not have video equipment), to demonstrate with models and create their own “tutorial” with models.

By having them create their own tutorial, or presentation to a partner, they would then be using different multiple intelligences, and mathematical practices, to explain the “why” behind the mathematical computations.

How else can you take advantage of these videos as a learning opportunity for students?

### Mathematical Practices: Model with Mathematics

Model with mathematics includes:

• Understand this is a way to reason quantitatively and abstractly (able to decontextualize and contextualize)
• Are able to simplify a complex problem and identify important quantities to look at relationships
• Represent mathematics to describe a situation either with an equation or a diagram and interpret the results of a mathematical situation
• Ask themselves, “How can I represent this mathematically?”

In model with mathematics, students will:

• Apply prior knowledge to new problems and reflect
• Use representations to solve real life problems
• Apply formulas and equations where appropriate

Teachers will:

• Pose problems connected to previous concepts
• Provide a variety of real world contexts
• Use intentional representations
• Gives students opportunities to model with mathematics and describe the mathematical situation

Some questions to develop mathematical thinking could be:

• What number model could you construct to represent the problem?
• How would it help to create a diagram, graph, table …?
• What are some ways to visually represent …?

What opportunities are you providing students to analyze, evaluate, and create during math?

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## Tips, Mathematical Practices, and Ideas about Adding and Subtracting Fractions and Mixed Numbers

### Topic 7 — Adding and Subtracting Fractions and Mixed Numbers

Oct. 30-Nov. 7 (Nov. 8 posttest/pretest)

#### Lesson

7-1 Adding and Subtracting: Like Denominators

#### AZ AIMS Standards

M06-S1C2-07

7-2 Least Common Multiple 6.NS.4 M06-S1C1-02
7-3 Adding and Subtracting: UnlikeDenominators 6.NS.1* M06-S1C2-07
7-4 Estimating Sums and Differences of Mixed Numbers 6.NS.1* M06-S1C3-01
7-6 Subtracting Mixed Numbers 6.NS.1*
7-7 Problem Solving: Make a Table 6.RP.1

Double Dose Recommendations — Preteach the following:

• Relating measures, 2 days (#24‐#25). Look for science connections.
• Elapsed time, 3 days (#26‐28)

### Tip of the Week:

• Assessing/Higher Level Learning:  Are we having our students complete the Performance Based Assessments at the end of each Topic?
• Enrichment: In your teacher’s manual you will find ideas for enrichment under “Advanced/Gifted” on page 160D. There are two different ideas listed here.

### Technology Integration Weekly Highlight:

• Math in the Real World Project: The math project on page 161 in the Teacher’s Guide, connects to the real world by researching how many miles the top hiking trails are in the USA (the Appalachian Trail, the Pacific Crest Trail, and the Continental Divide Trail); then have the students pick one of the three trails and show the portion of the total trail in one state as a fraction. Then make a table that lists the state and the portion of the trail as a fraction.
• Students would use the computer for their research, and they could also use it for their reflection on the learning that took place in the above project. This would also be a great way to bring in the mathematical practice of attend to precision.
• Students could use Educreations on the iPad to narrate and show their thinking.
• On the netbooks or thin clients, students could create their table in Google Docs, then narrate their reflection by connecting Voice Comments (in Kaizena) to their document. Click here for instructions (see slide #14).

### Mathematical Practices:

The following comes from the ADE: 6.MP.6. Attend to precision. In grade 6, students continue to refine their mathematical communication skills by using clear and precise language in their discussions with others and in their own reasoning. Students use appropriate terminology when referring to rates, ratios, geometric figures, data displays, and components of expressions, equations or inequalities.

Some questions that can be asked for attending to precision includes:

• What mathematical terms apply in this situation?
• How did you know your solution was reasonable?
• What mathematical language…, definitions…, properties can you use to explain…?
• Explain how you might show that your solution answers the problem.

How do you use, and challenge students to use mathematics vocabulary precisely and consistently?

## Tips, Mathematical Practices, and Ideas about Adding and Subtracting Fractions and Mixed Numbers

### Topic 7 — Adding and Subtracting Fractions and Mixed Numbers

Oct. 30-Nov. 7 (Nov. 8 posttest/pretest)

#### Lesson

7-1 Adding and Subtracting: Like Denominators

#### AZ AIMS Standards

M06-S1C2-07

7-2 Least Common Multiple 6.NS.4 M06-S1C1-02
7-3 Adding and Subtracting: UnlikeDenominators 6.NS.1* M06-S1C2-07
7-4 Estimating Sums and Differences of Mixed Numbers 6.NS.1* M06-S1C3-01
7-6 Subtracting Mixed Numbers 6.NS.1*
7-7 Problem Solving: Make a Table 6.RP.1

### Double Dose Recommendations — Preteach the following:

• Relating measures, 2 days (#24‐#25). Look for science connections.
• Elapsed time, 3 days (#26‐28)

### Technology Integration Weekly Highlight:

Here are some websites with fractions for students to interact and engage with:

• Comparing Fractions — If you have students who need more experience with understanding fractions, this might be a place to build background. Have them use manipulatives while playing this game.
• Equivalent Fractions Game — If you have students who need more experience with understanding fractions, this might be a place to build background. Have them use manipulatives while playing this game.
• Equivalent Fractions Target Shoot — If you have students who need more experience with understanding fractions, this might be a place to build background. Have them use manipulatives while playing this game.
• LCM — This is an online worksheet. It gives immediate feedback to student about accuracy.
• LCM of three numbers —  This is an online worksheet. It gives immediate feedback to student about accuracy.
• Adding Mixed Numbers  —  This is an online worksheet. It gives immediate feedback to student about accuracy.

### Tip of the Week: Examples and Explanations

This tip comes from the ADE about finding the GCF and LCM: 6.NS.4

• What is the greatest common factor (GCF) of 24 and 36? How can you use factor lists or the prime factorizations to find the GCF?

Solution: 22 • 3 = 12. Students should be able to explain that both 24 and 36 have 2 factors of 2 and one factor of 3, thus 2 x 2 x 3 is the greatest common factor.)

• What is the least common multiple (LCM) of 12 and 8? How can you use multiple lists or the prime factorizations to find the LCM?

Solution: 23 • 3 = 24. Students should be able to explain that the least common multiple is the smallest number that is a multiple of 12 and a multiple of 8. To be a multiple of 12, a number must have 2 factors of 2 and one factor of 3 (2 x 2 x 3). To be a multiple of 8, a number must have 3 factors of 2 (2 x 2 x 2). Thus the least common multiple of 12 and 8 must have 3 factors of 2 and one factor of 3 ( 2 x 2 x 2 x 3).

• Rewrite 84 + 28 by using the distributive property. Have you divided by the largest common factor? How do you know?
• Given various pairs of addends using whole numbers from 1-100, students should be able to identify if the two numbers have a common factor. If they do, they identify the common factor and use the distributive property to rewrite the expression. They prove that they are correct by simplifying both expressions.
• 27 + 36 = 9 (3 + 4)

63 = 9 x 7

63 = 63

• 31 + 80

There are no common factors. I know that because 31 is a prime number, it only has 2 factors, 1 and 31. I know that 31 is not a factor of 80 because 2 x 31 is 62 and 3 x 31 is 93.

### Mathematical Practices:

6.MP.7. Look for and make use of structure.

Students routinely seek patterns or structures to model and solve problems. When students can rewrite 84 + 28 by using the distributive property, then answer if they divided by the largest common factor, then they are looking for and making use of structure.

Students will:

• look for, develop, and generalize relationships and patterns
• apply conjectures about patterns and properties to new situations

Teachers will:

• provide time for applying and discussing properties
• highlight different approaches for solving problems

What higher level questions will you ask during Topic 7 to help students look for and make use of structure?

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## Tips, Mathematical Practices, and Ideas about Decimals, Fractions, and Mixed Numbers

### Topic 6- Decimals, Fractions, and Mixed Numbers

October 23-28 (Oct. 29 posttest 6/pretest 7)

#### Lesson

6-1 Fractions and Division

6.NS.1*

#### AZ AIMS Standards

M06-S1C1-03

6-2 Fractions and Decimals 6.NS.1* M06-S1C1-01
6-3 Improper Fractions and Mixed Numbers 6.NS.1* M06-S1C1-01, M06-S1C1-04
6-4 Decimal Forms of Fractions and Mixed Numbers 6.NS.1* M06-S1C1-01, M06-S1C1-04
6-5 Problem Solving: Draw a Picture** 6.NS.3

** See in double dose time.

### Double Dose Recommendations:

• Cyclical Review from Quarter 1.
• Pre-teach continuing measurement, 2 days (#22-23). Look for science connections.
• 6-5 Problem Solving: Draw a Picture, 2 days

### Tip of the Week

This tip comes from the ADE:

### Mathematical Practices

This tip comes from the ADE:

Math Practice #4: Model with Mathematics

In grade 6, students model problem situations symbolically, graphically, tabularly, and contextually. Students form expressions, equations, or inequalities from real world contexts and connect symbolic and graphical representations. Students begin to explore covariance and represent two quantities simultaneously. Students use number lines to compare numbers and represent inequalities. They use measures of center and variability and data displays (i.e. box plots and histograms) to draw inferences about and make comparisons between data sets. Students need many opportunities to connect and explain the connections between the different representations. They should be able to use all of these representations as appropriate to a problem context.

If you asked students to represent a fraction in three different ways, how would that help them think about the mathematical practice of modeling with mathematics?

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