6 Math Unit... · Web viewCounting can be done during whole group instruction, mini-lesson, small group instruction, etc. ... Giving quantitative measures of center (median and/or

Grade 6 Unit 1 revised 6.30.17 Page 1Student Strategies Formative Assessment Activities and Resources

Unit Title: Measurement, Quantities, and Relationships: The Foundation of MathematicsGrade Level: 6

Timeframe: Marking Period 1Unit Focus & Essential Questions

Unit Focus: (1) Understanding that the entire 6th grade mathematics course revolves around the study of relationships between things we count and measure: quantities.(2) Understanding of the nine quantities and units associated with each of the quantities: Discrete units (or occurrences), Distance, Area, Volume, Weight, Temperature (Energy), Time, Monetary Value, and Angle of Rotation(3) Understanding that linear relationships are built by varying one of the quantities.(4) Understand how to represent linear relationships with tables/progressions that generalize to expressions/equations/functions and graphs.(5) Understand the types of questions we ask and answer about linear relationships*.(6) Continue to build fluency with the numbers (fractions, decimals, whole numbers, and percents) and calculations that arise within the linear relationships.(7) Begin to compare multiple linear relationships.

Rubric for Learning Foci for CFU’s, rubrics, differentiation, feedback Object, quantities and units of measurement Relationship of two quantities Related pairs of values Initial Value Change over an interval

Essential Questions: (1) Can we become confident in our knowledge of quantities, units, and values of measuring that are used in relationships?(2) Can we become effective and efficient at representing relationships in all four traditional representations—tables generalized to symbols, graphs, and narratives?(3) Can we become effective and efficient at asking and answering typical mathematical questions? (4) Can we become effective and efficient at talking about relationships given in any representation?(5) Can we become effective and efficient at reading and understanding others’ relationships and questions given in any representation?

CAR © 2009 Trenton Public Schools Grade 6 Mathematics 2017 Page 1 of 83


*Students should be able to ask and answer additional A- and B- type questions about the relationships. Please note that the “A-type” and “B-type” references are non-standard identifiers which are used throughout this document to maintain simplicity and clarity. As often as possible, encourage the students to create their own A- and B- type questions from the relationships and measurements that they explore. Consider posting student work around the room, then having other students visit the work (Gallery Walk) and posting their questions and answers on sticky notes as they explore.

Please note the following uses and clarifications:

For an A-type question: Given the value of the quantity you varied, find the related value of the calculated quantity. This may be described as providing the “INPUT” value with the students calculating the “OUTPUT” value.

Later, students can expect to see the same skill phrased as: Evaluate the given expression for this value of x Given the value of x, find the related value of y. Given a value for the independent variable, find the related value of the dependent variable. Evaluate function f(x) when x = …

For a B-type question: Given the value of the “calculated quantity”, find the related value of the quantity you varied.

Later, students can expect to see the same skill phrased as: Given this “OUTPUT” value, find the “INPUT” value. Given the value of y solve for x. Given a value for the dependent variable, find the related value of the independent variable. In 8th grade: What is x when the value of function f(x) is given [inverse functions]

For both types of questions, consider the following designations:Easy: the answer is in the representation.Medium: the answer could (reasonably) be in the representation.Hard: the representation needs to be extended or generalized in order to find the answer.

Questions with a twist could include…Comparative: A question that says: “Shamar measured 3 more than, twice as many as, a quantity already given.Units: A question that gives the information in one unit, but either gives additional information or asks for the answer in a different unit.Percentage: Malik has a percentage increase/decrease of a quantity already given.Find the “other value”: A jar of 50 marbles contains only red and blue marbles. If 30 of them are red, what percentage are blue?



New Jersey Student Learning Standards6.RP.3: 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.

6.RP.3a: 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.

6.EE.6: Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

6.NS.5. Apply and extend previous understandings of numbers to the system of rational numbers. Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

6.RP.3 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?

6.EE. 5: Reason about and solve one-variable equations and inequalities. Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.

6.NS.1: A. Apply and extend previous understandings of multiplication and division to divide fractions by fractions. Interpret and compute quotients of fractions, and solve word problems involving division of fractions by fractions, e.g., by using visual fraction models and equations to represent the problem. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because 3/4 of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc). How much chocolate will each person get if 3 people share 1/2 lb of chocolate equally? How many 3/4-cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length 3/4 mi and area 1/2 square mi?

6.EE. 1: Apply and extend previous understandings of arithmetic to algebraic expressions. Write and evaluate numerical expressions involving whole-number exponents.

6.NS.4: 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).



Instructional Plan

Unit 1 Instructional Pre-assessment Choose a single object and have the students, working in pairs since it is the beginning of the year, think about all of the things that could be measured about that single object. Only once the students have considered all the quantities they can, ask them to approximate the measurement of each of the quantities they considered for their object. Then create as many relationships between the quantities as they can, making tables to represent their relationships. (eg: The length of one object, the length of two objects laid end to end, the length of three objects, etc.) As they complete their work, have them record their tables on chart (or quarter chart) paper to hang on the walls. If students finish early, differentiate by asking them to create expressions and equations to represent their relationships, and even graphs.

After the assessment, discuss as a whole class, all the different relationships that came up, publicly creating the expressions and equations that were not created by individual pairs and publicly recording all the expressions and equations on a Symbolic Representation Chart and all the quantities on the Quantities Word Wall. The public recording of Quantities and Symbolic Representations is the first step to coherence.

The following day, try another, more interesting object, and repeat the pre-assessment activity. As each pair completes the activity, compare their pre-assessment with today’s class work, pointing out anything they improved upon as “learning”. If they make comments about it being so easy, respond by saying it’s no easier than yesterday, in fact, it’s exactly the same, you are simply smarter today. They learned. And you look forward to a whole year of learning. Acknowledging learning with individuals, or in this case pairs, is the first activity of building positive academic relationships with students. As a whole class, gather and publicly record the Quantities and Symbolic Relationships of this second object.



Standards & Objectives SWBAT6.RP.3: 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.

6.RP.3 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.

6.EE. 6: Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

6.NS.5. Apply and extend previous understandings of numbers to the system of rational numbers. Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.

SWBAT…CREATE & REPRESENT Create relationships from given objects Create relationships from objects of their choice Represent relationships in tables Generalize relationships to symbolic representations—expressions/equations/functions Represent relationships graphically

FLUENTLY TALK ABOUT Discuss all aspects in the Learning Focus Compare all aspects in the Learning Focus

FLUENTLY READ Interpret relationships provided in textbook problems Communicate those relationships through other representations



FLUENCY ACTIVITYUse numbers you anticipate students are likely to encounter or actually do encounter in each day’s measurements to use these numbers during the counting activity.

Counting can be done during whole group instruction, mini-lesson, small group instruction, etc.

Consider varying the activity (as students become comfortable) to include doubling, tripling, halving, and “thirding” numbers.

CREATE & REPRESENT Allow students to spend most of their time working in groups as they create the relationships in this first marking period in order to observe them working and learn their strengths.

Have the group create the table together including the creation of the three symbolic representations (Quantity 1, Quantity 2, expression/equation to show the relationship), have the person who draws best represent the relationship graphically; the person who is best with words represent the relationship narratively; and the person with the strongest personality to make sure the group stays on task and gets the job complete.

Consider stations for each different representation.

Provide students with scenarios that either decrease past zero and increase from below zero. For example:

The temperature outside at 6pm (hour zero) is 12 degrees Celsius. The temperature drops 4 degrees per hour, 5 degrees per hour.

A penguin dives from an ice cliff 30 feet high at 12 feet per second, lands in the water, and continues to dive at the same rate.

A submarine starts 100 ft below the surface of the water and rises at 8 feet per second.

Joe borrowed $20 from his parents to start a lemonade stand. He earned 50 cents for each cup sold.

FLUENTLY TALK ABOUTLet them record themselves so they can hear themselves and use the rubric on their own work

FLUENTLY READHave students read a scenario in one representation and then move through the stations creating other representations.

RUBRIC FOR LEARNING FOCI Object, quantities and units of measurement Relationship of two quantities Related pairs of values Initial Value Change over an interval

CREATE & REPRESENT As the students work, listen and look for places where individual students/pairs/groups are struggling with any Learning Foci. Record these so you can discuss them with the whole group Move toward students using the rubric on each other’s and their own work

FLUENTLY TALK ABOUTListen for Learning Foci as students are talking about their relationships.Move toward students using the rubric on each other’s and their own work

FLUENTLY READListen for Learning Foci as students are interpreting scenarios provided in textbook problems.Move toward students using the rubric on each other’s and their own work

DIFFERENTIATIONStudents who are struggling with these relationships may need to build a tape diagram or double-line model.

Providing feedback, according to a rubric of the bullets listed above, to the students on their seatwork before allowing them to make posters of it for public display will allow them to show off their best work. Attaching their seatwork to their public display will show their best learning; something they can be proud of.

As you see each student become able to do what you taught them, celebrate the learning of that individual student, eye-to-eye establishing that they can learn, in this class, from you.Grouping students who struggle talking about each of the mathematical foci listed as well as above those not struggling with anything.

FLUENCY ACTIVITYCount around the room. Make combinations

CREATE & REPRESENT Create and represent relationships

Choose 2 quantities of an object to measure, (ex: number of bottles of water and volume of water)

Vary one quantity to create a relationship. Represent as a table, generalizing to an expression and

equation Represent as a graph

FLUENTLY TALK ABOUTAs students talk about the representations, let them take turns talking about their own representation and in each other’s representation—both within their own group and each other’s groups, comparing relationships using the same and different representations of each. Then revise the representations based upon the feedback received for public display

Make public displays of revised work Revise and display the representations on chart paper

based upon the feedback received Record quantities on the Quantities Word Wall Record the symbolic representations on Symbolic

Representation Chart

FLUENTLY READHave students read scenarios from sample problems and create multiple representations of the relationships in the scenario. If students do not have something on the wall to compare the textbook problems to we didn’t measure enough

CLOSING ACTIVITYSum up each relationship discussing how the focus of the unit showed up in that particular problem/activity. Reworking the problem usually includes particular quantities and numbers, whereas, summing up usually includes the words like “quantities” and “values”

Ask: Did everyone put their quantities and units on the wall?Ask: Were there any new units today?Ask: Did everyone record their symbolic representations on the symbolic representation chart?



Reflections



Standards & Objectives SWBAT

6.RP.3b: 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?

6.EE.5: Reason about and solve one-variable equations and inequalities. Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.


SWBAT: CREATE AND REPRESENT

Represent situations involving: unit rates Represent situations and scenarios involving inequalities. Represent situations involving multiplication and division FLUENTLY TALK ABOUT discuss all aspects in the Learning Focus compare all aspects in the Learning FocusFLUENTLY READ interpret relationships provided in textbook problems communicate those relationships through other representations



FLUENCY ACTIVITYContinue to use numbers that show up in relationships, especially fractions and decimals, counting up and down, extending past zero into the negatives.

CREATE AND REPRESENTHave the group create the table together including the creation of the three symbolic representations, have the person who draws best represent the relationship graphically; the person who is best with words represent the relationship narratively; and the person with the strongest personality to make sure the group stays on task and gets the job complete.

Be sure to include progressions that contain fractions, decimals and numbers into the negatives.Discrete units (or occurrences), Distance, Area, Volume, Weight, Temperature (Energy), Time, Monetary Value, and Angle of Rotation:

Other “real world” examples can be created by choosing any two of the nine-quantities:For instance,

number and time—A student can do 75 jumping jacks in one minute;

area and number—it takes 160 square inches of paper to cover a book.

Temperature and time— A container of water at 70 degrees Fahrenheit is placed in the freezer. It cools at a rate of 1 degree every 5 minutes.

Make sure to include all 9 quantities and move away from using number as one of the quantities in every problem.

FLUENTLY TALK ABOUTStudents should be able to talk about both the scenarios when there are quantities given.

FLUENTLY READIf students do not have something on the wall to compare the textbook problems to we didn’t make enough relationships.

One math textbook weighs 3 lbs. What is the greatest number of books that can be boxed so that the box weighs 25 lbs or less?

If apples cost $2 per pound, how many pounds of apples can I buy with $15?

RUBRIC FOR LEARNING FOCI Object, quantities and units of measurement Relationship of two quantities Related pairs of values Initial Value Change over an interval

CREATE & REPRESENTAs the students work, listen and look for places where individual students/pairs/groups are struggling with any Learning Foci. Record these so you can discuss them with the whole group. Move toward students using the rubric on each other’s and their own work

FLUENTLY TALK ABOUTListen for Learning Foci as students are talking about their relationships. Move toward students using the rubric on each other’s and their own work.

FLUENTLY READListen for Learning Foci as students are interpreting scenarios provided in textbook problems. Move toward students using the rubric on each other’s and their own work

DIFFERENTIATIONProviding feedback, according to a rubric of the bullets listed to the left, to the students on their seatwork before allowing them to make posters of it for public display will allow them to show off their best work. Attaching their seatwork to their public display will show their best learning; something they can be proud of.

When students struggle with calculations for any types of numbers, give them a calculator to work on it, and take note of the areas of struggle and use these to create counting around the room and combinations.

As you see each student become able to do what you taught them, celebrate the learning of that individual student, eye-to-eye establishing that they can learn, in this class, from you.Grouping students who struggle talking about each of the mathematical foci listed to the left as well as those not struggling with anything.

SpEd Modifications: Consider providing partially completed progression tables. Support with manipulatives.

G/T Modifications: Create and solve “Hard” problems with twists.

FLUENCY ACTIVITYCount around the room. Remember to including counting up and down, as well as by unit fractions and “unit” decimals (one tenth, one hundredth, etc.). Pause to “represent” the various numbers as necessary.

Make combinations and use combinations to perform operations unit by unit.

CREATE & REPRESENT Continue all activities from previous section, but now focus on the unit rate and creating scenarios and situations which require fractions and mixed numbers

Start with whole number rates, and progress to fractions and decimals as listed above.

FLUENTLY TALK ABOUT MATH As students talk about the learning foci in their representations, let them take turns talking about their own representation and in each other’s representation—both within their own group and each other’s groups, comparing relationships using the same and different representations of each. Then revise the representations based upon the feedback received.Make public displays of revised work Record quantities on the quantities word wall Record the symbolic representations on symbolic

representation chart Ask and answer A- and B-type questions Determine each of the representations from any one of the

other 4 representations

FLUENTLY READHave students read scenarios from sample problems and create multiple representations of the relationships in the scenario. If students do not have something on the wall to compare the textbook problems to we didn’t measure enough.

CLOSING ACTIVITYSum up each relationship discussing how the focus of the unit showed up in that particular problem/activity. Reworking the problem usually includes particular quantities and numbers, whereas, summing up usually includes the words like “quantities” and “values”Ask: Did everyone put their quantities and units on the wall?Ask: Were there any new units today?Ask: Did everyone record their symbolic representations on the symbolic representation chart?



Reflections:



Standards & Objectives SWBAT6.RP.3 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?

6.EE. 5: Reason about and solve one-variable equations and inequalities. Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.


SWBAT: CREATE AND REPRESENT Ask and answer problems about: unit rates, unit pricing, constant speed Ask and answer questions that use vocabulary and concepts of inequalities to scenarios Ask and answer questions that incorporate multiplication and division as inverse operations Solve problems requiring multiplication and division of fractions





FLUENCYWhen students struggle with calculations for any types of numbers, give them a calculator to work on it, and take note of the areas of struggle and use these to create counting around the room and combinations.

CREATE AND REPRESENTHave students each create a representation at the highest level they are able, using unusual quantities and a variety of different types of numbers. Then have a gallery walk so that they can ask questions of each other.

Then have them answer all of the questions asked of their relationship.

Be sure to include questions and answers that have answers that are decimals, fractions, and negatives

Show that division problems can be expressed as a large measurement being divided into smaller measurements using as many quantities as possible—

If I have a garden with an area of 4 12

square feet and each section

is 1 12

feet wide, how many one-foot long sections can I have?

You have 5 ½ cups of flour. You have a recipe for brownies that calls for 1 ¾ cups of flour. How many batches can you make?

FLUENTLY TALK ABOUT Students should be able to talk about the scenarios when there are quantities given

FLUENTLY READ

If students do not have something on the wall to compare the textbook problems to we didn’t make enough relationships.

RUBRIC FOR LEARNING FOCI Object, quantities and units of measurement A quantity varied to create a linear relationship Related pairs of values Initial Value Change over an interval

CREATE & REPRESENTUse Rubric for Learning Foci to check for each of the bulleted items as students created questions and answers. Move towards students using the rubric on each other’s questions and answers.

FLUENTLY TALK ABOUTUse Rubric for Learning Foci to check for each of the bulleted items as students talk about their questions and answers. Move towards students using the rubric on each other’s talking.

FLUENTLY READUse Rubric for Learning Foci to check for each of the bulleted items as student’s read others’ questions and answers. Move towards students using the rubric on each other’s reading.

DIFFERENTIATIONProviding feedback, according to a Rubric for Learning Foci, to the students on their seatwork before allowing them to make posters of it for public display will allow them to show off their best work. Attaching their seatwork to their public display will show their best learning; something they can be proud of.

As you see each student become able to do what you taught them, celebrate the learning of that individual student, eye-to-eye establishing that they can learn, in this class, from you.

Grouping students who struggle talking about each of the mathematical foci listed or fluency above as well as those not struggling with anything.

SpEd Modifications: Consider providing partially completed progression tables. Support with manipulatives.


FLUENCY ACTIVITYCount around the room, combinations, and operations performed unit by unit.

CREATE & REPRESENTUse the scenarios depicted around the room from previous student work.

Ask and answer type A and B questions see * (page 2)Add twists to the lists of questions.

FLUENTLY TALK ABOUTTalk about their own questions and answers

Talk about the questions and answers created by their classmates.

FLUENTLY READHave students read scenarios from sample problems and ask and answer questions about the scenarios

Show students the questions asked about the scenarios they worked with to see if they asked the same questions as the textbooks.

CLOSING ACTIVITYSum up each questions and answering activity discussing how the focus of the unit showed up in that particular problem/activity. Reworking the problem usually includes particular quantities and numbers, whereas, summing up usually includes the words like “quantities” and “values”



Reflections:



Standards & Objectives SWBAT6.EE. 1: Apply and extend previous understandings of arithmetic to algebraic expressions. Write and evaluate numerical expressions involving whole-number exponents. (squares)

SWBAT: CREATE AND REPRESENT Represent areas and volumes with exponential notation Evaluate expressions for area and volume represented in exponential notation





FLUENCYCount around the room: Remember to encourage both improper fractions (one half, two halves, three halves, four halves, etc.) and mixed numbers (one half, one, one and one half, two, two and one half, three, etc.) by periodically switching format. This will help to allow the students to build fluency. Consider the following (note the vocabulary carefully):

You are counting by quarters. Your last student finished with “23 quarters” or “23 fourths”. You say: “Wow. That’s a lot of quarters. What is another way I could say this?”

CREATE & REPRESENTHave students select a fixed width (perhaps one of the dimensions of one of their rooms). Have them count the number of square units in a rectangle of length 1 (if their room was only one unit long), then increase the length one unit at a time. Have students record this in a relationship table titled “Square units in a rectangle of width ___”.

Have students begin with a 1x1 square for their room and increase the side one unit at a timeHave students begin with a rectangular base of a room and increase the height one unit at a time.

Students should continue to increase the increments by 1 unit until they get to at least 5 (more may be necessary).

How many squares will we need to make 10 rows? 100 rows? x rows?

How many cubes will we need to make 10 layers? 100 layers? x layers

Ask and answer A-type and B-type questions.

Differentiate by providing students with unit squares (or graph paper) to physically represent their progression.

One student should draw the rectangle, one record the length, one calculate the total number of squares.

Once students make their work public, use a gallery walk and Post-Its for students to evaluate postings with the Learning rubric:

ASK AND ANSWERCreate and represent expressionsQuestions involving expressions.

RUBRIC FOR LEARNING FOCI Object, quantities and units of measurement Varying a quantity to create a linear relationship Related pairs of values Initial Value Change over an interval

CREATE & REPRESENT As the students work, listen and look for places where individual students/pairs/groups are struggling with any Learning Foci. Record these so you can discuss them with the whole group. Move toward students using the rubric on each other’s and their own work.


FLUENTLY READListen for Learning Foci as students are interpreting scenarios provided in textbook problems. Move toward students using the rubric on each other’s and their own work.



Grouping students who struggle talking about each of the mathematical foci listed to the left as well as those not struggling with anything.

SpEd Modifications: Consider providing partially completed progression tables.


FLUENCY ACTIVITYCount around the room. Remember to include counting up and down, as by whole number units (ones, tens, hundreds, etc.) well as by unit fractions (one half, one quarter, one eighth, one third, etc.) and “unit” decimals (one tenth, one hundredth, etc.).

Pause to “represent” the various numbers as necessary.

Make combinations that lead to operations students are struggling with

Vary the activity to include squaring and cubing.

CREATE & REPRESENT Create and represent relationships in a progression with area. Make sure to include squares so the representations can include exponents. Then extend each of these areas to 3-d objects to measure volumes. Make sure to include cubes so that the representations can include exponents. Create and represent expressions

ASK AND ANSWERAsk and answer A- and B-type questions involving squares. Be sure to include squares of fractional sizes (ex: How many ¼ inch squares are needed to build a square 2 ½ inches on each side? Be sure to include expressions.

FLUENTLY TALK ABOUTAs students talk about their representations, let them take turns talking about their own representation and in each other’s representation—both within their own group and each other’s groups, comparing relationships using the same and different representations of each. Make public displays of revised work.

Revise and display the representations on chart paper based upon the feedback received

Record quantities on the Quantities Word Wall Record the symbolic representations on Symbolic Representation Chart

FLUENTLY READHave students read scenarios from sample problems and create multiple representations of the relationships in the scenario. If students do not have something on the wall to compare the textbook problems to we didn’t measure enough

CLOSING ACTIVITYSum up each relationship discussing how the focus of the unit showed up in that particular problem/activity. Reworking the problem usually includes particular quantities and numbers, whereas, summing up usually includes the words like “quantities” and “values”Ask: Did everyone put their quantities and units on the wall?Ask: Were there any new units today?Ask: Did everyone record their symbolic representations on the symbolic



representation chart? Reflections:



Unit Title: Relationships Between Quantities: Ratios and RatesGrade Level: 6


Unit Focus: (1) Understanding that the entire 6th grade mathematics course revolves around the study of relationships between things we count and measure, quantities.(2) Understanding of the nine quantities and units associated with each of the quantities: Discrete units (or occurrences), Distance, Area, Volume, Weight, Temperature (Energy), Time, Monetary Value, and Angle of Rotation(3) Understanding that linear relationships are built by varying one of the quantities.(4) Understand how to represent linear relationships with tables/progressions that generalize to expressions/equations/functions and graphs.(5) Understand the types of questions we ask and answer about linear relationships.(6) Continue to build fluency with the numbers and calculations that arise within the linear relationships.(7) Begin to compare multiple linear relationships.


Essential Questions: (1) Can we become confident in our knowledge of quantities, units, and values of measuring that are used in relationships?(2) Can we become effective and efficient at representing relationships in all four traditional representations—tables that lead to symbols, graphs, and narratives?(3) Can we become effective and efficient at asking and answering typical mathematical questions? (4) Can we become effective and efficient at talking about relationships given in any representation?(5) Can we become effective and efficient at reading and understanding others’ relationships and questions given in any representation?



*Students should be able to ask and answer additional A- and B- type questions about the relationships. Please note that the “A-type” and “B-type” references are non-standard identifiers which are used throughout this document to maintain simplicity and clarity. As often as possible, encourage the students to create their own A- and B- type questions from the relationships and measurements that they explore. Consider posting student work around the room, then having other students visit the work (Gallery Walk) and posting their questions and answers on sticky notes as they explore.










New Jersey Student Learning Standards6.RP.A.1: Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities6.RP.A.2: Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship .6.RP.3 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.

6.EE.A.3: Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y6.EE.A.4: Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for6.NS.2 : Fluently divide multi-digit numbers using the standard algorithm.

6.NS.3: Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

6.NS.4: 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).



Instructional PlanUnit 2 Instructional Pre-assessmentChoose a simple recipe for brownies that students can begin looking at. Students will determine which quantities can be counted and measured in the recipe, while determining the proportional relationships if the recipe is doubled/ halved so as to increase or decrease the number of servings. Students will create tables showing the proportional relationships between original ingredients and the recipe that is doubled or halved. Students will be asked to count around the room using halves, quarters, and eighths. Using the recipe, students will determine if there are any relationships between cups, ounces and pounds. Tables will be used to create relationships between various units of measurements such as ounces, pounds, cups, quarts. Students will be asked to write numerical expressions for the recipe, as it pertains to the number of servings and as it relates to the number of batches. A table of such expressions will be created as well as an expression chart. All quantities will be recorded on the quantities wall by students. The numerical expressions will later be used to create scenarios, algebraic expressions and equations. Once these numerical expressions are made, it shall be discussed that several relationships can be created from the recipe. Choosing two ingredients, a chart can be created to record the changes in the volume or number of ingredients as they change. This information, will be used create graphs. Given 10% of brownies were sold, students will be asked to determine the number sold, then create a table of percentages

After the assessment, discuss as a whole class, all the different relationships that came up, publicly creating the expressions and equations that were not created by individual pairs and publicly recording all the expressions d equations on a Symbolic Representation Chart and all the quantities on the Quantities Word Wall. The public recording of Quantities and Symbolic Representations is the first step to coherence.

The following day, try another, more interesting object, and repeat the pre-assessment activity. As each pair completes the activity, compare their pre-assessment with today’s class work, pointing out anything they improved upon as “learning”. If they make comments about it being so easy, respond by saying it’s no easier than yesterday, in fact, it’s exactly the same, you are simply smarter today. They learned. And you look forward to a whole year of learning. Acknowledging learning with individuals, or in this case pairs, is the first activity of building positive academic relationships with students. As a whole class, gather and publicly record the Quantities and Symbolic Relationships of this second object.



Standards & Objectives SWBAT6.RP.A.1: Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. Make tables comparing ratios using whole numbers, and fractions.6.RP.A.2: Understand the concept of a unit rate a/b associated with a ratio a:b with b ≠ 0, and use rate language in the context of a ratio relationship6.NS.2: Fluently divide multi-digit numbers using the standard algorithm.

6.NS.3: Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

SWBAT…CREATE & REPRESENT

Create relationships from given objects Create relationships from objects of their choice Represent relationships in tables Generalize relationships to symbolic representations—expressions/equations/functions Represent relationships graphically





CREATE & REPRESENT Allow students to spend most of their time working in groups as they create the relationships in this marking period in order to observe them working and learn their strengths.

Have the group create the table together including the creation of the symbolic representations. Have the person who draws best represent the relationship graphically;


CREATE & REPRESENT As the students work, listen and look for places where individual students/pairs/groups are struggling with any Learning Foci. Record these so you can discuss them with the whole group. Move toward students using the rubric on each other’s and their own work


FLUENCY ACTIVITYCount around the room. Represent the counting on the board whenever students need to see the work to make sense of it.

Make combinations of whole numbers, decimals and fractions. Represent the combinations in writing, sometimes on the board; sometimes students can do the writing on their own paper.

Combinations should lead to performing the operations unit by unit.

CREATE & REPRESENT Create and represent relationships: Choose 2 quantities of an object to measure Vary one quantity to create a relationship. Represent as a table, generalizing to an expression and equation Represent as a graph

FLUENTLY TALK ABOUT



the person who is best with words represent the relationship narratively; and the person with the strongest personality to make sure the group stays on task and gets the job complete. Consider stations for each different representation.

Emphasize the concept that quantities can be compared as part, part, and whole. Often one is not stated explicitly.

Begin with giving unit ratios. Then move to non-unit ratios.

Emphasize that we can find a unit ratio for each ratio that is given to us. AND doubling and halving, tripling and “thirding” to find unit ratios

A BEST VALUE Store should be opened up for students to constantly work on finding the unit rate for real world problems and comparing to find the BEST VALUE

FLUENTLY TALK ABOUT

FLUENTLY READIf students do not have something on the wall to compare the textbook problems to we didn’t measure enough.


DIFFERENTIATIONStudents who are struggling with these relationships may need to build a tape diagram or double-line model, use base ten blocks or use a more tangible real world example to make connection.

When students struggle with a computation, hand them a calculator to complete the computation. Record the computation that they are struggling with and address it at a later time (mini-lesson, small group instruction, etc.)

Providing feedback, according to the learning foci, to the students on their seatwork before allowing them to make posters of it for public display will allow them to show off their best work. Attaching their seatwork to their public display will show their best learning; something they can be proud of.


SpEd Modifications: Consider helping students measure their objects, rounding to the nearest whole number when necessary and/or providing partially completed progression tables.


As students talk about the learning foci in their representations, let them take turns talking about their own representation and in each other’s representation—both within their own group and each other’s groups, comparing relationships using the same and different representations of each. Then revise the representations based upon the feedback received.

Make public displays of revised work Revise and display the representations on chart paper based upon the

feedback received Record quantities on the Quantities Word Wall Record the symbolic representations on Symbolic Representation Chart

FLUENTLY READHave students read scenarios from sample problems and create multiple representations of the relationships in the scenario.


Ask: Did everyone put their quantities and units on the Quantities Word wall?Ask: Were there any new units today?Ask: Did everyone record their symbolic representations on the Symbolic Representation Chart?



Reflections:



Standards & Objectives SWBAT6.RP.3 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.



Create relationships from given objects Create relationships from objects of their choice Represent relationships in tables Generalize relationships to symbolic representations—expressions/equations/functions Represent relationships graphically







CREATE & REPRESENT

Have the group create the table together including the creation of the symbolic representations. Have the person who draws best represent the relationship graphically; the person who is best with words represent the relationship narratively; and the person with the strongest personality to make sure the group stays on task and gets the job complete.

Emphasize the concept that different units do not change the measurements, only the representation.

FLUENTLY TALK ABOUT

FLUENTLY READ

If students do not have something on the wall to compare the textbook problems to we didn’t measure enough.

.

RUBRIC FOR LEARNING FOCI Object, quantities and units of measurement Varying a unit to create a conversion relationship Related pairs of values Initial Value Change over an interval

CREATE & REPRESENT As the students work, listen and look for places where individual students/pairs/groups are struggling with any Learning Foci. Record these so you can discuss them with the whole group. Move toward students using the rubric on each other’s and their own work







SpEd Modifications: Consider providing partially completed progression tables.G/T Modifications: Create and solve “Hard” problems with twists.


Make combinations of units used in conversions. Represent the combinations in writing, sometimes on the board; sometimes students can do the writing on their own paper.

CREATE & REPRESENT Create and represent relationships: Choose 2 units of the same quantity Vary one unit to create a relationship. Represent as a table, generalizing to an expression and equation

FLUENTLY TALK ABOUTAs students talk about the learning foci in their representations, let them take turns talking about their own representation and in each other’s representation—both within their own group and each other’s groups, comparing relationships using the same and different representations of each.

Make public displays of revised work Revise and DISPLAY the representations of conversions on chart

paper based upon the feedback received Record UNITS on the Quantities Word Wall Record the symbolic representations on Symbolic

Representation Chart

FLUENTLY READHave students read scenarios from sample problems that include conversions to complete.





Reflections:





SWBAT: CREATE AND REPRESENT Ask and answer questions about relationships expressed as expressions that involve multiplication including whole number and fractional values. Ask and answer questions about relationships expressed as equations that involve division including whole number and fractional values. Ask and answer questions about relationships expressed as inequalities that involve division including whole number and fractional values.







CREATE AND REPRESENTHave students each create a representation at the highest level they are able, using unusual quantities and a variety of different types of numbers. Then have a gallery walk so that they can ask questions of each other.

Then have them answer all of the questions asked of their relationship.

Be sure to include questions and answers that have answers that are decimals, fractions, and negatives

Show that division problems can be expressed as a large measurement being divided into smaller measurements using as many quantities as possible—

If I have a garden with an area of 4 12

square

feet and each section is 1 12

feet wide, how

many one-foot long can I have?

You have 5 ½ cups of flour. You have a recipe for brownies that calls for 1 ¾ cups of flour. How many batches can you make?

FLUENTLY TALK ABOUT:

FLUENTLY READ: If students do not have something on the wall to compare the textbook problems to we didn’t make enough relationships.


CREATE & REPRESENTAs the students work, listen and look for places where individual students/pairs/groups are struggling with any Learning Foci. Record these so you can discuss them with the whole group. Move toward students using the rubric on each other’s and their own work.








Make combinations of whole numbers, decimals and fractions. Represent the combinations in writing, sometimes on the board; sometimes students can do the writing on their own paper.

Combinations should lead to performing the operations unit by unit.

CREATE & REPRESENTUse the scenarios depicted around the room from previous student work.

ASK AND ANSWERAsk and answer type A and B questions see * (page 2)Add twists to the lists of questions.

As students answer questions, have them represent their answers in at least two different units to practice conversions.

FLUENTLY TALK ABOUTTalk about their own questions and answersTalk about the questions and answers created by their classmates.

FLUENTLY READHave students read scenarios from sample problems and ask and answer questions about the scenarios.








CREATE AND REPRESENTEnsure that students understand that every linear relationship is a proportional relationship with an initial value added. Graphically this creates a parallel line.

FLUENTLY TALK ABOUTHave students talk about and compare the relationships

with and without the initial values.

FLUENTLY READIf students do not have something on the wall to compare the textbook problems to we didn’t make enough relationships.

LEARNING FOCI Object, quantities and units of measurement Varying a quantity to create a linear relationship Related pairs of values Initial Value Change over an interval

CREATE & REPRESENTAs the students work, listen and look for places where individual students/pairs/groups are struggling with any Learning Foci. Record these so you can discuss them with the whole group. Move toward students using the rubric on each other’s and their own work

FLUENTLY TALK ABOUTListen for Learning Foci as students are talking about their relationships.Move toward students using the rubric on each other’s and their own work.






FLUENCY Count around the room beginning with the initial value and counting by the change over an interval.

CREATE & REPRESENTExtend a few of the relationships on the walls by adding an initial value to them. Add columns to the tables already posted. Add the graph to the graph already posted. Add the new expressions and equations to the Symbolic Representations Chart.

ASK AND ANSWERAsk and answer type A and B questions see * (page 2)Add twists to the lists of questions.

As students answer questions, have them represent their answers in at least two different units to practice conversions.

FLUENTLY TALK ABOUTTalk about their own questions and answersTalk about the questions and answers created by their classmates.

FLUENTLY READHave students read scenarios from sample problems and ask and answer questions about the scenarios.





Reflections:





SWBAT: CREATE AND REPRESENT Ask and answer questions about relationships expressed as expressions that involve multiplication including whole number and fractional values. Ask and answer questions about relationships expressed as equations that involve division including whole number and fractional values. Ask and answer questions about relationships with percentage twists







Spend time talking about hundreds, multiples of hundreds, and parts of hundreds to support the work of percents.

CREATE & REPRESENTTo ensure students understand a percent is one out of every hundred so knowing how many hundreds is crucial to their understanding. To ensure students understand that multiple percents are multiples of one percent is crucial to their understanding.Show that 5%=4% + 1% AND 5% = 10% x ½

FLUENTLY READStudents will be introduced to surveys, and polls. Real World Concepts 20% chance of snow 40 % of the time she walks home 60% on a math test 30% discount NJ Sales tax 7% 40% of the students like pepperoni pizza Using scenarios, discuss mark up as an additional

percentage cost added on to the original cost when seeking to make a profit. Percentage discount is taken away from the original price once the percent is found. Continue to make tables finding ½’s, 1/3’s, ¼’s of the original. Through counting note the

number of 18

’s in ¼ , number of ¼’s in ½, the number

of ¼ in 34

. Find 13

’s of whole numbers.

Is 50% off + an additional 25% off the same as 75% off?

Look at nutritional facts from food packages

RUBRIC FOR LEARNING FOCI Object, quantities and units of measurement Varying a quantity to create a linear relationship Related pairs of values with percent change Initial value & change over an interval Change created by the percent increase (markup, raise) or

decrease (discount)

CREATE & REPRESENTAs the students work, listen and look for places where individual students/pairs/groups are struggling with any Learning Foci. Record these so you can discuss them with the whole group. Move toward students using the rubric on each other’s and their own work.



DIFFERENTIATIONStudents who are struggling with these percents may need to go back to talking about how many hundreds are in a number.





Make combinations of whole numbers, decimals and fractions. Represent the combinations in writing, sometimes on the board; sometimes students can do the writing on their own paper. Combinations should lead to performing the operations unit by unit.

Practice finding how many hundreds are in a number, beginning with 200, 300, 400, then 1200, 2500, 5600, then 350, 450, 550, then 470, 590, 870, then 543, 749, 395.

CREATE & REPRESENTHave students find 1% of a number by picking 1 from each hundred there is in that number.

Then create progressions of 1, 2, 3, 4 % and then 10% and continue the progressions to 10, 20, 30, 40 %.

Ask and answer questions about 12, 32, and 47% using combinations of 10 and 1 %.

Return to the progressions posted around the room and add percent twists to the questions and answer those questions.

FLUENTLY TALK ABOUTAs students talk about the learning foci in their representations, let them take turns talking about their own representation and in each other’s representation—both within their own group and each other’s groups, comparing relationships using the same and different representations of each.

FLUENTLY READHave students read scenarios from sample problems and find the percents in each problem, along with




Reflection:



Standards & Objectives SWBAT6.EE.A.3: Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y6.EE.A.4: Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for6.NS.4: 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).

SWBAT: CREATE AND REPRESENT Represent all of our concrete representations as abstract representations

FLUENTLY TALK ABOUT Discuss all aspects in the Learning Focus of abstract representations as well as concrete ones Compare all aspects in the Learning Focus of abstract representations as well as concrete ones

FLUENTLY READ Interpret relationships provided in textbook problems with abstract representations as well as concrete ones Communicate those relationships through other representations with abstract representations as well as concrete ones



FLUENCYAs the students continue to create, ask, and answer questions about relationship, gather the numbers students are creating and use them to count around the room.

CREATE AND REPRESENTThis is the most crucial activity for helping students to understand that algebra is a generalization of the relationships they created throughout arithmetic

Play around with combining different measurements and using multiples of expressions over many months or many people in order to talk about combining like terms and distributing.

FLUENTLY TALK ABOUTStudents should be able to talk about asking and answering questions both with the problems when there are quantities given and when there are none (just x and y given). If they struggle with the abstract one, let them pick quantities they have worked with previously to use as an example to make sense of it.

Play around with combining different measurements and using multiples of expressions over many months or many people in order to talk about combining like terms and distributing.

FLUENTLY READIf students do not have something on the wall to compare the textbook problems to we didn’t make enough relationships

RUBRIC FOR LEARNING Object, quantities and units of measurement A quantity varied to create a linear relationship Related pairs of values Initial Value Change over an interval

CREATE & REPRESENTUse Rubric for Learning to check for each of the bulleted items as students created progressions and graphs of the relationship. Move towards students using the rubric on each other’s representations.

FLUENTLY TALK ABOUTUse Rubric for Learning to check for each of the bulleted items as students talk about the representations of the relationships of the object. Move towards students using the rubric on each other’s talking.

FLUENTLY READUse Rubric for Learning to check for each of the bulleted items as student’s read/work the representations of the relationships of the object. Move towards students using the rubric on each other’s reading

DIFFERENTIATIONProviding feedback, according to a rubric for learning, to the students on their seatwork before allowing them to make posters of it for public display will allow them to show off their best work. Attaching their seatwork to their public display will show their best learning; something they can be proud of.


SpEd Modifications: Consider providing partially completed progression tables.

G/T Modifications: Create and solve “Hard” problems with twists. Create scenarios from textbook “x and y” problems.

FLUENCY ACTIVITYCount around the room. Make combinations that lead to performing operations students are struggling with unit by unit.

CREATE & REPRESENTHave a few pairs of students choose a progression on the wall and on a new piece of paper rewrite the progression replacing the quantities with x and y.

Have a few pairs of students choose a graph on the wall and on a new piece of paper redraw the graph, labeling the axes x and y.

Have a few students go to the collected chart of symbolic representations and rewrite the symbolic representation chart with x and y values.

Put all of these new pages/posters in one area in your classroom.

ASK AND ANSWERAsk and answer type A and B questions* from the original work and then ask the same questions of the abstract x and y version. Create different scenarios from the new x and y version.

FLUENTLY TALK ABOUTTalk about their own and their classmates questions and answers, in both concrete and abstract representations by constantly comparing them

FLUENTLY READHave students read abstract representation problems, make sense of them and work them.




Reflection:




Touchpoints; 6.RP.1 & 2, NS 2 & 3Scaffolded Touchpoints 1Touchpoints 6.RP.3 Scaffolded Touchpoints 2Touchpoints 6.EE.3 & 4 and 6.NS.4 Scaffolded Touchpoints 3

Create a presentation of one Touchpoint problem—presented either face-to-face, podcasted, or video recorded as practice for summative performance assessment

Homework: Provide weekly homework which involves students measuring with their families—Find 3 products in your home that are measured in weight, volume, and numberFind 3 things in the room in which you sleep that you would measure in inches, in feet, in yards.Find 3 things in the room in which you sleep that you would measure in square inches, in square feet, in square yards.Have 3 members of your family (not including yourself) approximate how many tablespoons it takes to fill a cup? Then fill it to see who is closest. Have 3 members of your family (not including yourself) approximate how many cups it takes to fill a gallon? Then fill it to see who is closest.

For each of these write down the 3 measurements/approximations and any comments folks made when giving their choices. Talk about the process of measuring, anything interesting that happened or that you learned. Bring to class by Thursday so we can talk about it all on Friday. Hang the homework on the door as they come in so that by Friday you can have a conversation and in hopes that it stirs up conversation any time you line up for anything. On Friday, get quantities on Quantities Word Wall, Create relationships between a few of them to review the week’s work with or to introduce less used quantities you have not been able to cover yet.

Summative Written Assessments

Quarterly Assessment 1 & Presentation of the measuring of others of one problem from the Quarterly assessment, either face-to-face after the assessment or podcasted/video recorded as part of the assessment.

Summative Performance Assessment

In pairs, choose a physical object, (1) represent the object visually, narratively, (2) measure everything you can about it and represent the measurements visually, narratively, and with symbols, (3) Get feedback from your classmates and/or teacher using a rubric, (4) Revise your representations, using the feedback you receive, on chart paper (4) Create a presentation of your work, face-to-face, podcasting, or videotaping.



Homework: Provide weekly homework which involves students measuring with their families—Find 3 products in your home that are measured in weight, volume, and numberFind 3 things in the room in which you sleep that you would measure in inches, in feet, in yards.Find 3 things in the room in which you sleep that you would measure in square inches, in square feet, in square yards.Have 3 members of your family (not including yourself) approximate how many tablespoons it takes to fill a cup? Then fill it to see who is closest. Have 3 members of your family (not including yourself) approximate how many cups it takes to fill a gallon? Then fill it to see who is closest.

For each of these write down the 3 measurements/approximations and any comments folks made when giving their choices. Talk about the process of measuring, anything interesting that happened or that you learned. Bring to class by Thursday so we can talk about it all on Friday. Hang the homework on the door as they come in so that by Friday you can have a conversation and in hopes that it stirs up conversation any time you line up for anything. On Friday, get quantities on Quantities Word Wall. Create relationships between a few of them to review the week’s work with or to introduce less used quantities you have not been able to cover yet.

Touchpoints; 6.NS.4 Scaffolded Touchpoints 1Touchpoints 6.EE.1 Scaffolded Touchpoints 2Touchpoints 6.RP.3, 6.EE.5, and 6.EE.6 Scaffolded Touchpoints 3



Quarterly Assessment 2—Presentation of the measuring of others with one problem from the Quarterly assessment, either face-to-face after the assessment or podcasted/video recorded as part of the assessment.


In pairs, choose a physical object, (1) represent the object visually, narratively, (2) measure everything you can about it and represent the measurements visually, narratively, and with symbols, (3) Get feedback from your classmates and/or teacher using a rubric, (4) Revise your representations, using the feedback you receive, on chart paper (4) Create a presentation of your work, face-to-face, podcasting, or videotaping.


Unit Title: Expanding Our Understanding Of NumberGrade Level: 6


Unit Focus: (1) Reinforcing that the entire 6th grade mathematics course revolves around the study of relationships between things we count and measure, quantities.(2) Focus on the quantities of temperature, elevation (linear distance), and debt (monetary value).(3) Understand how to represent the relationship of the quantities in both tables (that generalize into expressions and equations) and graphs. (4) Extend the relationships to more than two quantities.(5) Extend the types of questions we ASK AND ANSWER about the relationships*. (6) Build fluency with the values (integers)

Rubric for Learning Foci for CFU’s, rubrics, differentiation, feedback Object, quantities and units of measurement Extending to relationships of more than two quantities Related pairs of values Initial Value Change over an interval

Essential Questions: (1) Can we become confident in our knowledge of quantities, units, and values (extending to negative values) of measuring that are used in relationships?(2) Can we become effective and efficient at extending our representations of relationships to more than two quantities in all four traditional representations—tables, generalized to

symbols, graphs, and narratives?(3) Can we become effective and efficient at asking and answering typical mathematical questions, extending to inequalities? (4) Can we become effective and efficient at talking about relationships given in any representation?(5) Can we become effective and efficient at reading and understanding others’ relationships and questions given in any representation?



*Students should be able to ask and answer additional A- and B- type questions about the relationships.

Please note that the “A-type” and “B-type” references are non-standard identifiers which are used throughout this document to maintain simplicity and clarity. As often as possible, encourage the students to create their own A- and B- type questions from the relationships and measurements that they explore. Consider posting student work around the room, then having other students visit the work (Gallery Walk) and posting their questions and answers on sticky notes as they explore.










New Jersey Student Learning StandardsReason about and solve one-variable equations and inequalities.

6.EE.7. Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.

6.EE.8. Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.

Represent and analyze quantitative relationships between dependent and independent variables.

6.EE.9. Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time.

Apply and extend previous understandings of numbers to the system of rational numbers

6.NS.6. Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.

a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., –(–3) = 3, and that 0 is its own opposite.

b. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.

c. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.



6.NS.7. Understand ordering and absolute value of rational numbers.

a. Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret –3 > –7 as a statement that –3 is located to the right of –7 on a number line oriented from left to right.

b. Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write –3 oC > –7 oC to express the fact that

–3 oC is warmer than –7 oC.

c. Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world situation. For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars.

d. Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars.

6.NS.8. Solve real-world and mathematical problems by graphing points in all four quadrants of the coordinate plane. Include use of coordinates and absolute value to find distances between points with the same first coordinate or the same second coordinate



Instructional PlanUnit 3 Pre-assessment Part 1: The temperature in Toronto, Ontario (Canada) is 12 degrees Celsius on January 5. The temperature drops 4 degrees per day throughout the Winter. Represent this relationship in as many ways as you can.



Standards & Objectives SWBATApply and extend previous understandings of numbers to the system of rational numbers

6.NS.6. Understand a rational number as a point on the number line. Extend number line diagrams and coordinate axes familiar from previous grades to represent points on the line and in the plane with negative number coordinates.a. Recognize opposite signs of numbers as indicating locations on opposite sides of 0 on the number line; recognize that the opposite of the opposite of a number is the number itself, e.g., –(–3)

= 3, and that 0 is its own opposite.b. Understand signs of numbers in ordered pairs as indicating locations in quadrants of the coordinate plane; recognize that when two ordered pairs differ only by signs, the locations of the

points are related by reflections across one or both axes.c. Find and position integers and other rational numbers on a horizontal or vertical number line diagram; find and position pairs of integers and other rational numbers on a coordinate plane.

6.NS.7. Understand ordering and absolute value of rational numbers.a. Interpret statements of inequality as statements about the relative position of two numbers on a number line diagram. For example, interpret –3 > –7 as a statement that –3 is located to the

right of –7 on a number line oriented from left to right.b. Write, interpret, and explain statements of order for rational numbers in real-world contexts. For example, write –3 oC > –7 oC to express the fact that

–3 oC is warmer than –7 oC. c. Understand the absolute value of a rational number as its distance from 0 on the number line; interpret absolute value as magnitude for a positive or negative quantity in a real-world

situation. For example, for an account balance of –30 dollars, write |–30| = 30 to describe the size of the debt in dollars.d. Distinguish comparisons of absolute value from statements about order. For example, recognize that an account balance less than –30 dollars represents a debt greater than 30 dollars



create relationships around temperature, debt, and elevation represent relationships in tables generalize relationships to symbolic representations—expressions/equations/functions represent relationships graphically (number lines and coordinates)

FLUENTLY TALK ABOUT discuss all aspects in the Learning Focus compare all aspects in the Learning Focus Which integer is greater and or equal to another Meaning of absolute value

FLUENTLY READ interpret relationships provided in textbook problems communicate those relationships through other representations Able to answer questions about equations in text book problems. communicate those applications of definitions through other representations



FLUENCYCount by whole numbers, then count by integers starting with 10 and counting down. Start at whole numbers count by ½, ¼, ¾, even down to 1/8. Start at whole numbers and count by dollars and cents (that are 50 cent intervals), begin at – whole numbers and count up by whole numbers and then count up by decimals to the tenths. Do these groups also as integers counting down from 1 whole and 1 dollar.

CREATE & REPRESENT

Have students create a number line with values from previous scenarios. Make the values positive and negatives integers, positive and negative fractions, and positive and negative decimals.

Use numbers from previous scenarios that represent real world situations to work with student to re-order the numbers to make sense. Use examples like temperature and sea level.

Using numbers from previous scenarios, talk with students about comparing numbers and how it applies to real world:

Ex: owing 30 dollars is better than owing 100 dollars-5⁰ is warmer than -20⁰.

Represent these scenarios on number lines.

Students will create vertical number lines with integers. Extend to horizontal number lines after students demonstrate proficiency with vertical lines.

When extending to Cartesian Coordinates, graph the quantities on separate number lines, then rotate one of them 90 degrees.


Record these representations so you can both refer back to them and discuss them with the whole group.

CREATE & REPRESENTUse Rubric for Learning Foci to check for each of the bulleted items as students create the representations of the integers

FLUENCYCount around the room and combinations.

CREATE & REPRESENTDesignate a point on the wall that will act as zero. Then place designations above and below it. Have students use sticky notes to give those places values, leave room so students can place fraction and decimals in between the whole numbers.

Have students create drawings of ocean scene including object like surfers, fish and even helicopters above. Once they have identified their relative magnitude and then have them draw a scale on the side of the drawling to give their positions. Teachers should reinforce concept that sea level represents zero. The magnitude refers to the distance from zero (also called the absolute value) and the “sign” indicates direction (above or below).

Student should be given data sets that exist in real life, like temperatures and order them on a number line.

Teachers should use a game like “Battleship” to introduce a coordinate grid. Teacher should introduce the origin. Student should be able to then graph integer pairs.

Students should be given scenarios, like earning, spending, losing, and owing money; temperatures; and elevations to graph these as points on a number line. Move towards creating a line graph showing changes over time.

Teacher will have student count and talk about distances from zero and the fact it’s always positive despite direction. Teacher will introduce symbol for absolute value.



Teacher should model using a number line to show the negative symbol means “opposite” and how the symbol direction and the value in the number of units in that direct and how the use of the symbol changes that direction. Be sure to use scenarios where the debt, temperature, or location requires the use of negative fractions and decimal values.

Students will graph integer pairs on a coordinate plane

Student will be able to write a sentences and expression that show magnitude.

Student will be able to show that absolute value is the distance a value is from zero regardless of direction, this is being evidence from class discussion, from questions like how many leagues under the sea is the shark (answer would be positive number)

Students will be able to compare integers.

FLUENTLY TALK ABOUTStudents can create scenarios for another student. Students should create groups of data that other can order on number line and compare certain data points and express with is greater.

FLUENTLY READStudent should be able to answer questions in textbook, identifying which questions and data points are statistical.

FLUENTLY TALK ABOUTUse Rubric for Learning Foci to check for each of the bulleted items as students talk about the representations of the integers comparing relationships using the same and different representations of each.

Move towards students using the rubric on each other’s representations, talking, and reading.

FLUENTLY READUse Rubric for Learning Foci to check for each of the bulleted items as student’s read/work the representations of integers

DIFFERENTIATIONStruggling students can only use whole numbers for an extended period of time. Student can exclusively work with money or temperature to ground the concept.

Providing feedback, according to a rubric of the bullets listed to the left, to the students on their seatwork before allowing them to make posters of it for public display will allow them to show off their best work. Attaching their seatwork to their public display will show their best learning; something they can be proud of.As you see each student become able to do what you taught them, celebrate the learning of that individual student, eye-to-eye establishing that they can learn, in this class, from you.Grouping students who struggle talking about each of the mathematical foci listed or fluency above as well as those not struggling with anything.

SpEd Modifications: Continue to use graphs to support placement of integers and fractions on a number line. Consider the use of a calculator to show relationships between fractions and decimal.

G/T Modifications: Have students develop their own rules for addition of integers. Begin subtracting of integers by graphing on a number line and have students develop their own rules.

Using the concept of absolute value, students can find the distance between two points. They simply start at zero, count or measure the individual distance of each point from zero and add them. (In 6th grade the points will share an x coordinate or y coordinate.)

Teacher will provide scenarios (like debt verse credit) to compare integers, and the how smaller values of debt if more favorable, and that it is true for all numbers. Negative numbers are greater the less distance they are from zero.

FLUENTLY TALK ABOUTHave student present their work and compare integers and ordering them by magnitude.

FLUENTLY READHave students read scenarios from sample problems and identify which ones are statistical.

CLOSING ACTIVITYStudent will be able to order integer by magnitude, compare them, explain their absolute value, and graphically represent them.



Reflection:



Reason about and solve one-variable equations and inequalities.

6.EE.7. Solve real-world and mathematical problems by writing and solving equations of the form x + p = q and px = q for cases in which p, q and x are all nonnegative rational numbers.

6.EE.8. Write an inequality of the form x > c or x < c to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form x > c or x < c have infinitely many solutions; represent solutions of such inequalities on number line diagrams.

Represent and analyze quantitative relationships between dependent and independent variables.

6.EE.9. Use variables to represent two quantities in a real-world problem that change in relationship to one another; write an equation to express one quantity, thought of as the dependent variable, in terms of the other quantity, thought of as the independent variable. Analyze the relationship between the dependent and independent variables using graphs and tables, and relate these to the equation. For example, in a problem involving motion at constant speed, list and graph ordered pairs of distances and times, and write the equation d = 65t to represent the relationship between distance and time

Apply and extend previous understandings of numbers to the system of rational numbers



relationships of symbolic representations—expressions/equations/inequalities – using algebraic notation relationships graphically using Cartesian coordinates Relationships using more than two related quantities

FLUENTLY TALK ABOUT discuss all aspects in the Learning Focus compare all aspects in the Learning Focus

FLUENTLY READ interpret relationships provided in textbook problems communicate those relationships through other representations Able to answer questions about equations in text book problems. communicate those applications of definitions through other representations



Composition:Using the relationships and progressions from around the room, develop additional relationships (and add columns to the progressions) from the composition of these relationships.

For example: A pallet has 1000 bricks. A mason can lay one brick in 2 minutes. If the mason charges $50 per hour, how much will it cost to lay 5 pallets of bricks?

Add: It takes … bricks to create a retaining wall….

A gallon of paint costs $30.50. One gallon can cover 200 square feet of wall. The typical apartment in Trenton has an area of 720 ft2. If you have $400, how many apartments can you paint completely?

Suppose you pay a painter $15 to paint 100 ft2, how much would it cost to paint those apartments?

Each bag of flour weighs 5 lbs. 1 pound of all-purpose flour has 3 1/3 cups in it. One bread recipe calls for 2 cups of flour. Each recipe makes 4 loaves of bread. How many loaves of bread can you make with one bag of flour? You must ONLY make full recipes.

If you can sell each loaf of bread for $1.50…

RUBRIC FOR LEARNING FOCI FOR CFU’S Object, quantities and units of measurement Relationship of two quantities Related pairs of values Initial value Change over an interval

CREATE & REPRESENTUse Rubric for Learning Foci to check for each of the bulleted items as students created equations and graphic representation or counted and measured data. Be sure to record these so you can discuss them with the whole group.

FLUENTLY TALK ABOUTUse Rubric for Learning Foci to check for each of the bulleted items as students talk about equations and graphic representation or counted and measured data. Move towards students using the rubric on each other’s representations, talking, and reading. Teacher should listen for strategies and celebrate student that talk about breaking it into step and being aware that information for the first set of calculations normally comes at the end of the problem.

ASK AND ANSWERUse Rubric for Learning Foci to check for each of the bulleted items as students create and answer A- and B-type problems based on the relationships between quantities.

FLUENTLY READUse Rubric for Learning Foci to check for each of the bulleted items as student’s read/work equations and graphic representation or counted and measured data

Move towards students using the rubric on each other’s representations, talking, and reading, comparing relationships using the same and different representations of each.

FLUENCYCount around the room and make combinations.

ASK AND ANSWERStudents should now be able to create type A and type B question for scenarios that they have also created. Once they have created and solved them, have other student create “follow-up” questions, (Example: create cost or time requirements for the scenarios, or use the item bought to create something else). Monitor the students to see if they can extend their scenarios to another set of calculations.

Have student create scenarios where there are two calculations needed to achieve the desired information. Look for them creating one relationship and then creating a second. Have them write about the process and look for them explaining that they need to break it into steps.

Have students create more of these scenarios and post them around the room. Student can perform a gallery walk

FLUENTLY TALK ABOUT Students should talk with each other about strategies to solve these composite problems during the gallery walk. Students can perform think, pair, share with their partners to solve these problems as they perform the gallery walk.

ASK AND ANSWERStudents will create and answer A- and B-type problems based on the relationships between quantities posted around the room (Gallery Walk). Students can perform think, pair, share with their partners to solve these problems as they perform the gallery walk.

FLUENTLY READ

Students should be able solve composite function word problems in the PARCC released items, using the strategies they have obtained.



DIFFERENTIATIONProviding feedback, according to a rubric of the bullets above, to the students on their seatwork before allowing them to make posters of it for public display will allow them to show off their best work. Attaching their seatwork to their public display will show their best learning; something they can be proud of.

As you see each student become able to do what you taught them, celebrate the learning of that individual student, eye-to-eye establishing that they can learn, in this class, from you.Grouping students who struggle talking about each of the mathematical foci listed or fluency above as well as those who are not struggling with anything.

SpEd Modifications: Use Algebra tiles or other manipulatives to represent quantities.

G/T Modifications:Use variables to represent relationships and extend progressions to additional quantities.

CLOSING ACTIVITYStudents will see multi-step world problems as a multiple one step problems to be solved in order. Once they break down the problem in this fashion they will be able to answer it correctly.

Touchpoints; Scaffolded Touchpoints 1Touchpoints Scaffolded Touchpoints 2Touchpoints Scaffolded Touchpoints 3

Create a presentation of one Touchpoint problem—presented either face-to-face, podcasted, or video recorded as practice for summative performance assessmentHomework: Make an equation to explain how to double or triple a recipe.Graph the total cost of buying a video game every day for a week.Create a progression and equation using something you measure the next night have other student create type a and type b questions for the progression, and the third night have the original students answer the questions.



Reflection:




Quarterly Assessment 3—Presentation of one problem from the Quarterly assessment, either face-to-face after the assessment or podcasted/video recorded as part of the assessment.


Student as class will create factory. They will design a product. Student with break down all parts of the factory and create progressions, equations, and graphs for all part of the factory to explain the production and what would happen of the factory was successful and was scaled.



Unit Title: Making Sense Of Data: An IntroductionGrade Level: 6


Note that the following skills will be studied in detail in Unit 4. The purpose for placing these in Unit 3 is exposure and exploration to the greatest extent possible given time prior to PARCC testing.

Unit Focus: (1) Understanding that the entire 6th grade mathematics course revolves around the study of relationships between things we count and measure, quantities.(2) Understanding of the nine quantities and units associated with each of the quantities.(3) Understanding that statistical properties are built by varying one of the quantities.(4) Understand how to represent the statistical properties of quantities, and how to represent them through the measures of central tendency.

(5) UNDERSTAND THE TYPES OF QUESTIONS WE ASK AND ANSWER ABOUT THE STATISTICAL PROPERTIES AND MEASURE OF CENTRAL TENDENCIES.(6) Continue to build fluency with the values (fractions and whole numbers) and calculations that arise within the proportional relationships.(7) Begin to compare multiple proportional relationships.

Rubric for Learning Foci Population Quantity and the units in which the students are measuring. Existence of variability

Central tendencies (Mean, Median Mode) Variability (Range, Mean absolute deviation) Outliers

Essential Questions: (1) Can we become effective and efficient at identifying what a statistical question is which a question about a population?(2) Can we become effective and efficient representing statistical values?(3) Can we become effective and efficient answering and answering questions about central tendencies, variability, outliers and adding or subtracting data points?(4) Can we become effective and efficient about talking statistical values?(5) Can we become effective and efficient about reading statistical values?














New Jersey Student Learning StandardsSummarize and describe distributions.

6.SP.B.4Display numerical data in plots on a number line, including dot plots, histograms, and box plots.6.SP.B.5Summarize numerical data sets in relation to their context, such as by: 6.SP.B.5.A Reporting the number of observations. 6.SP.B.5.B Describing the nature of the attribute under investigation, including how it was measured and its units of measurement. 6.SP.B.5.C

Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.

6.SP.B.5.D Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.

Develop understanding of statistical variability.

6.SP.A.1Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, "How old am I?" is not a statistical question, but "How old are the students in my school?" is a statistical question because one anticipates variability in students' ages.

6.SP.A.2Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.

6.SP.A.3Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.


http://www.corestandards.org/Math/Content/6/SP/A/3/



http://www.corestandards.org/Math/Content/6/SP/B/5/d/

http://www.corestandards.org/Math/Content/6/SP/B/5/c/

http://www.corestandards.org/Math/Content/6/SP/B/5/b/

http://www.corestandards.org/Math/Content/6/SP/B/5/a/

http://www.corestandards.org/Math/Content/6/SP/B/5/



Instructional Plan ReflectionUnit 3a: Pre-assessment. Ask an open-ended statistical question such as:

Who is the better player: LeBron James or Steph Curry? Who is the best player of all time: Wilt Chamberlain, Julius Erving, Michael Jordan, Kobe Bryant, LeBron James, etc?Which is the best movie?

Why? Can you support your reasoning? What information would you need to decide?

After the assessment, discuss as a whole class, attributes and quantities discussed and their comparisons. How do you tell which is “better” or “worse” overall?

Using the Socratic method, ask them questions that lead to types of statistics that can be collected, and used, to help answer questions. Cover the idea of variability and need for multiple data points. Discuss how this information can be used in “real world” applications, such as the amount to offer a player for a contract, the length of time of the contract, the number of jerseys to order for a sale, the price an item should sell for, offering scholarships to students, or jobs to potential employees etc.

Teacher can help students all use consistent and correct vocabulary with which to describe the data distributions.



Standards & Objectives SWBAT 6.SP.B.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots.6.SP.B.5.C Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.6.SP.B.5.D Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.

CREATE & REPRESENT Describe the data Organize the date in line plots, histograms and box and whisker plots Express measures of central tendency, mean, median and mode.

FLUENTLY TALK ABOUT Discuss what would be the best measure of central tendency of a given data set.

ASK AND ANSWER QUESTIONS ABOUT THE DATA SETS

FLUENTLY READ interpret relationships provided in textbook problems communicate those relationships through other representations






FLUENCYCount around the room.Students will be given set of data and order it least to greatest.

CREATE & REPRESENTMath Talk: Teacher will discuss way to look at data, applying what student already know. Students should describe center of the data, but with fully defined medians and the methods.

Be sure that students understand that the median value is the one where half of the values are greater and half of the values are less than the median.

Be sure that the students understand that the mode is the value that most of the population has.

The mean is the value if each item would have if the quantity were evenly distributed.

CREATE & REPRESENT (CENTRAL TENDENCY)Students will use measurements of central tendency to make box and whiskers plots with using measures of central tendency and the data the collected.

Students will use their box and whisker plots to find interquartile range.

FLUENTLY TALK ABOUTStudents will able to take data points from others and create measures of central tendency.

Teacher will introduce the concept of outlier and how it affects data

Student will describe the effects of an outlier being introduced to their data and what measures of central tendency are changed and what measures are not.

Students will look at two sets of data and describe their similarities and differences in terms of measurements of central tendency and variability.

RUBRIC FOR LEARNING FOCI Population Number in the population Quantity and the units in which the students are

measuring. Existence of variability

CREATE & REPRESENTUse Rubric for Learning Foci to check for each of the bulleted items as students created graphic representation and analysis of the data.

FLUENTLY TALK ABOUTUse Rubric for Learning Foci to check for each of the bulleted items as students talk about the graphic representation and analysis of the data, comparing relationships using the same and different representations of each. Move towards students using the rubric on each other’s representations, talking, and reading.

FLUENTLY READUse Rubric for Learning Foci to check for each of the bulleted items as student’s read/work the representations of the representation and analysis of the data. Move towards students using the rubric on each other’s representations, talking, and reading.

DIFFERENTIATIONProviding feedback, according to a rubric of the bullets listed to the left, to the students on their seatwork before allowing them to make posters of it for public display will allow them to show off their best work. Attaching their seatwork to their public display will show their best learning; something they can be proud of.As you see each student become able to do what you taught them, celebrate the learning of that individual student, eye-to-eye establishing that they can learn, in this class, from you.Grouping students who struggle talking about each of the mathematical foci listed or fluency above as well as those not struggling with anything.

Central tendencies (Mean, Median, Mode) Distribution (quartiles, median, range)

FLUENCYCount around the room and combinations. Students will be given set of data and order it least to greatest.

CREATE & REPRESENT (DATA SETS) Students will pick population and a quantity (out of the 9

quantities) and create multiple data points by counting and measuring.

Collect their data (on Post-It notes) and present it to the class on the board.

Organize the data (visually) to start to make sense out of it.

Students will create several of these and post them around the room.

FLUENTLY TALK ABOUTGo around to the data around the room: Ask about the minimum, maximum, the population (number of data points), the values, the attributes measured or counted, most common value, and “the middle”

CREATE & REPRESENT (CENTRAL TENDENCY)Select a physical object of which you have multiple (ex: Unifix

Cubes). Distribute unevenly to a group of students (ex: 5 students receive quantities of 1, 2, 3, 4, and 5 objects). Have students discuss the “equity” and distribution of the situation and ways to remedy it.

Discuss (perhaps without the vocabulary) mean, median, mode, and range.

Model each of the students’ suggestions with the objects.

Repeat with other uneven numbers (example: 16 objects, and 5 groups) to demonstrate how fractions can come into play with discrete quantities (ex: the average American family has 2.1 children or the average number of people in line for a movie is 8.5).



FLUENTLY READStudent will be given scenarios able to make graphical representation, find the measure of central tendency and talk about the data, including identifying outliers. Student should be able to answer questions in textbook.

ELL:SPED Accommodations:Gifted & Talented:

SpEd Modifications: Provide manipulatives (index cards, Unifix cubes, etc.) so students can manipulate the data physically to find “centers”. Apply a number line so they can see distributions.

G/T Modifications:Consider incorporating bivariate data, fractional values, negative values.

Choose a population of an object (such as the weight of the backpacks, aspects of students (such as their height) or books in the room, the cars that pass by the window, etc.) in whichthere is multiple, choose a quantity (such as the weight of the book) about them and measure and count for value. Have students determine the “middle”. You can also have them determine the middle of the “top half” and the middle of the “bottom half” of the data (Upper and Lower Quartiles).Students will represent the data as stem or leaf plots or histograms.

Arrange the data, graphically represent it. Analyzes it using the measurements of central data. Student work, graphical representation, calculations, and descriptions of the scenarios will be displayed around the classroom.

FLUENTLY TALK ABOUTStudents practice talking about the math in their own representations within their groups.Students will discuss with others which measures of central tendency describe the data, and which best describes the center of the data and why.Teacher will introduce “outliers” into the data set and students will create new measures of center and describe which have changed, which have stayed the same.

FLUENTLY READHave students read scenarios from sample problems and create multiple representations of the statistical analysis in the scenario.

Students should be able to analyze and discuss the data in terms of the vocabulary to answer the questions given in the text book and PARCC-released questions.

CLOSIING ACTIVITYSum up each how data points can be organized in way that we can make meaning from them. Students will able to represent data as a single number, talk about any factors affecting that number, graphically represent them, and compared it with other data sets.



Reflection:



Touchpoints; Scaffolded Touchpoints 1Touchpoints Scaffolded Touchpoints 2Touchpoints Scaffolded Touchpoints 3


Homework: Provide weekly homework which involves students measuring with their families—Create three statistical questions from things you observe in the lunch roomRecords data points bout a one quantity base on things found in your homeFind the mean median and mode, of the heights of 7 friends and familyCreate a box plot of the amount of house you slept each day this weekFind the MAD of the monthly average temp of Trenton for 2017 and compare it to the MAD of the month average temp of San Francisco. Explain your findings.




In pairs, you be an assigned an attribute and quality student will find data point and graphically represent them. They find measures of central tendency and compare them another group that had the same quantity and attribute. Measure of variability and center will be found. Students will write two paragraphs on their findings.


Grade 6 Unit 4 Page 62 of 83 revised 6.30.17Student Strategies Formative Assessment Activities and Resources

Unit Title: 4A: Introduction to StatisticsGrade Level: 6


Note that the following skills will may have been introduced in Unit 3. They should be explored in detail in unit 4.

Unit Focus: (1) Understanding that the entire 6th grade mathematics course revolves around the study of relationships between things we count and measure, quantities.(2) Understanding of the nine quantities and units associated with each of the quantities.(3) Understanding that statistical properties are built by varying one of the quantities.(4) Understand how to represent the statistical properties of quantities, and how to represent them through the measures of central tendency.(5) Understand the types of questions we ask and answer about the statistical properties and measure of central tendencies.(6) Continue to build FLUENCY with the values (fractions and whole numbers) and calculations that arise within the proportional relationships.(7) Begin to compare multiple proportional relationships.

Rubric for Learning Foci Population Quantity and the units in which the students are measuring. Existence of variability Outliers

Essential Questions: (6) Can we become effective and efficient at identifying what a statistical question is which a question about a population?(7) Can we become effective and efficient representing statistical values?(8) Can we become effective and efficient answering and answering questions about central tendencies, variability, outliers and adding or subtracting data points?(9) Can we become effective and efficient about talking statistical values?(10) Can we become effective and efficient about reading statistical values?

CAR © 2009











CAR © 2009


New Jersey Student Learning StandardsSummarize and describe distributions.

6.SP.B.4 Display numerical data in plots on a number line, including dot plots, histograms, and box plots.6.SP.B.5 Summarize numerical data sets in relation to their context, such as by:

6.SP.B.5.A Reporting the number of observations. 6.SP.B.5.B Describing the nature of the attribute under investigation, including how it was measured and its units of measurement.

6.SP.B.5.C Giving quantitative measures of center (median and/or mean) and variability (interquartile range and/or mean absolute deviation), as well as describing any overall pattern and any striking deviations from the overall pattern with reference to the context in which the data were gathered.

6.SP.B.5.D Relating the choice of measures of center and variability to the shape of the data distribution and the context in which the data were gathered.

Develop understanding of statistical variability.6.SP.A.1 Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, "How old am I?" is not a statistical question, but "How old are the students in my school?" is a statistical question because one anticipates variability in students' ages.6.SP.A.2 Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape.6.SP.A.3 Recognize that a measure of center for a numerical data set summarizes all of its values with a single number, while a measure of variation describes how its values vary with a single number.

Standards & Objectives SWBATCREATE & REPRESENT

Describe the data Determine the size of the sample. Organize the data in line plots, histograms and box-and-whisker plots Express measures of central tendency, mean, median and mode. Express measurements of variability Use words to describe the relationship of data to measures of the central tendencies

FLUENTLY TALK ABOUT Discuss what would be the best measure of central tendency of a given data set Discuss whether or not a question is statistical in nature. Discuss the population and the quantities being measured Compare two data sets using variability measurements Ask and answer questions about the data sets


CAR © 2009






http://www.corestandards.org/Math/Content/6/SP/B/5/b/

http://www.corestandards.org/Math/Content/6/SP/B/5/a/




Instructional PlanUnit 4a Instructional Pre-assessment:

Ask an open-ended statistical question such as:Who is the better player: LeBron James or Steph Curry? Who is the best player of all time: Wilt Chamberlain, Julius Erving, Michael Jordan, Kobe Bryant, LeBron James, etc?Which is the best movie?

Why? Can you support your reasoning? What information would you need to decide?

CAR © 2009


FLUENCY Count around the room and combinations. Student will be given set of data and order it least to

greatest.

CREATE & REPRESENT (DATA SETS)Example: Population: Bookbags in the room. Quantity: weight of the bookbag.

When presenting data, arrange it in terms of histograms, frequency tables, box plots, stem-and-leaf plots, data tables, etc. Show them the data presented in multiple formats.

Math Talk: Teacher will discuss way to look at data, applying what student already know. Students should describe center of the data, but with fully defined medians and the methods.

Be sure that students understand that the median value is the one where half of the values are greater and half of the values are less than the median.

Be sure that the students understand that the mode is the value that most of the population has.

The mean is DEFINED as the value if each item would have if the quantity were evenly distributed. It is FOUND by dividing the sum by the number of elements. [Please be sure the students understand the difference between the “definition” and the “formula”]

CREATE & REPRESENT (CENTRAL TENDENCY) Students will use measurements of central tendency

to make box and whiskers plots with using measures of central tendency and the data the collected.

Students will use their box and whisker plots to find interquartile range.

Students will calculate Mean absolute deviation of their data.

RUBRIC FOR LEARNING FOCI Population Quantity and the units in which the

students are measuring. Existence of variability Outliers

Record these so you can discuss them with the whole group.

CREATE & REPRESENTUse Rubric for Learning Foci to check for each of the bulleted items as students created graphic representation and analysis of the data.

FLUENCY Count around the room and combinations. Student will be given set of data and order it least to greatest.

CREATE & REPRESENT (DATA SETS) Students will pick population and a quantity (out of the 9 quantities)

and create multiple data points by counting and measuring. Collect their data (on Post-It notes) and present it to the class on the

board. Organize the data (visually) to start to make sense out of it.

Students will create several of these and post them around the room.

FLUENTLY TALK ABOUTGo around to the data around the room: Ask about the minimum, maximum, the population (number of data points), the values, the attributes measured or counted, most common value, and “the middle”

The various arrangements of data into histograms, frequency tables, box plots, stem-and-leaf plots, data tables, etc. Discuss the data presented in multiple formats.

CREATE & REPRESENT (CENTRAL TENDENCY)

Select a physical object of which you have multiple (ex: Unifix Cubes). Distribute unevenly to a group of students (ex: 5 students receive quantities of 1, 2, 3, 4, and 5 objects).Have students discuss the “equity” and distribution of the situation and ways to remedy it.Discuss (perhaps without the vocabulary) mean, median, mode, and range.Model each of the students’ suggestions with the objects.

Repeat with other uneven numbers (example: 16 objects, and 5 groups) to demonstrate how fractions can come into play with discrete quantities (ex: the average American family has 2.1 children or the average number of people in line for a movie is 8.5).

Choose a population of an object (such as the weight of the backpacks, aspects of students (such as their height) or books in the room, the cars that pass by the window, etc.) in which there are multiple, choose a quantity (such as the weight of the book) about them and measure and count for value. Have students determine the “middle”. You can also have

CAR © 2009


them determine the middle of the “top half” and the middle of the “bottom half” of the data (Upper and Lower Quartiles).

Students will represent the data as line plots, stem and leaf plots or histograms.

Arrange the data, graphically represent it. Analyze it using the measurements of central data. Student work, graphical representation, calculations, and descriptions of the scenarios will be displayed around the classroom.

FLUENTLY TALK ABOUT (CENTRAL TENDENCY)Students practice talking about the math in their own representations within their groups.

Students will discuss with others which measures of central tendency describe the data, and which best describes the center of the data and why.

Teacher will introduce “outliers” into the data set and students will create new measures of center and describe which have changed, which have stayed the same.

Teacher will describe a situation. Students will then create a data set at each step to meet the criteria in the situation.

Ex: 1. George took 5 tests, and has an average score of 90.

a. No two scores were the same.b. No score was equal to the mean.c. The median score was above the mean.d. Etc…

Create others along those lines…include outliers, calculating “What would he need to achieve a 91 average after his next test?”

CAR © 2009


FLUENTLY TALK ABOUTStudents will able to take data points from others and create measures of central tendency.

Teacher will introduce the concept of outlier and how it affects data

Student will describe the effects of an outlier being introduced to their data and what measures of central tendency are changed and what measures are not.

Students will look at two sets of data and describe their similarities and differences in terms of measurements of central tendency and variability.

FLUENTLY READStudent will be given scenarios able to make graphical representation, find the measure of central tendency and talk about the data, including identifying outliers

Student should be able to answer questions in textbook,

FLUENTLY TALK ABOUTUse Rubric for Learning Foci to check for each of the bulleted items as students talk about the graphic representation and analysis of the data. Move towards students using the rubric on each other’s representations, talking, and reading.

FLUENTLY READUse Rubric for Learning Foci to check for each of the bulleted items as student’s read/work the representations of the representation and analysis of the data comparing relationships using the same and different representations of each.

Central tendencies (Mean, Median Mode) Variability (Range, Mean absolute deviation) Outliers Outliers organization of the values of quantities




SpEd Modifications: Grouping students who struggle talking about each of the mathematical foci listed or FLUENCY above as well as those not struggling with anything.

G/T Modifications:How do outliers affect each of the presentations of data?

CREATE & REPRESENT (VARIABILITY)Teacher will give students different data sets representing

the same attributes (ex: weight of student backpacks). Have students compare the data points and distributions.

Discuss variations from center, including range, MAD and IQR. Using MAD and IQR and describe the data set. Example, two data sets of temperature of cities: if a person wants a consistent temperature, in what city should they live?

FLUENTLY READHave students read scenarios from sample problems and create multiple representations of the statistical analysis in the scenario. Be sure that students are able to analyze and discuss the data in terms of the vocabulary to answer the questions given in the text book and PARCC questions.

CLOSING ACTIVITY Sum up each how data points can be organized in way that we can make meaning from them. Students will able to represent data as a single number, talk about any factors affecting that number, graphically represent them, and compared it with other data sets.

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Unit Title: 4b: Introduction to Statistical VariabilityGrade Level: 6

Timeframe: Marking Period 4bUnit Focus & Essential Questions

Develop understanding of statistical variability.6.SP.A.1 Recognize a statistical question as one that anticipates variability in the data related to the question and accounts for it in the answers. For example, "How old am I?" is not a statistical question, but "How old are the students in my school?" is a statistical question because one anticipates variability in students' ages.6.SP.A.2 Understand that a set of data collected to answer a statistical question has a distribution which can be described by its center, spread, and overall shape..


Groups of statistical data points/questions. Categorize the data and their questions as statistical or non-statistical. Create definitions of center and shape. Use student created definitions to identify a data set that can answer a statistical question. Describe center, spread and shape of data.

FLUENTLY TALK ABOUT working definitions of center, spread and over shapes of data. What made a question statistical and how to identify it.

FLUENTLY READ Able to identify statistical questions in text book problems. Communicate those applications of definitions through other representations

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Unit 4b: Pre-assessment. Present students with different sets of data with scenarios: questions include examples that are only one data point (such as: “Do you like vanilla ice cream?” or “How many days did you work this week?”) and ones that are don’t not contain variability (such as “How many days are there this month?” or “How many people in this classroom have brown hair?”). Have students organize the data into categories of their choosing. Ask them to write how they defined their groups and why each data set and question falls into that category.

After the assessment, discuss as a whole class, all their categories and choices. Record the choices and talk about the choices they made. Using the Socratic method, ask them questions that lead to the definition of a statistical question. Cover the idea of variability and need for multiple data points. (Comparison of a data set with a population vs. one data point and comparing data sets with variability vs. one with no change)

Continuing that day or following day, giving students more sets of data. Have students, categorize them, ask them to check them with their partner. Discuss with the whole class. Teachers should see from their reasoning, if they are correctly categorizing and point out when they are using the working definition to make the decisions. Teacher should also note uses of the words and concepts of: sample, population, representative, change, variable, variability, average, deviation. When students use the concept without the vocabulary, reinforce the vocabulary. Ensure that students are using the vocabulary correctly. Students will then be given set of data and describe the shape using words. Teacher can help students all use consistent and correct vocabulary with which to describe the data distributions.

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FLUENCY ½, start at whole numbers count by ½ , ¼ , ¾ , even

down to 1/8 Start at whole numbers and count by dollars and cents (that are 50 cent intervals), begin at – whole numbers and count up by whole numbers and then count up by decimals to the tenths. Counting/combinations of values (fractions percent’s and decimals) used in Touchpoints Student will be given ranges of number to order from least to greatest.

Student will be given set of data and order it least to greatest.

CREATE & REPRESENTAllow students to spend most of their time working with the data set and relation questions and make their own decisions about how to organize them. Be sure to include scenarios involving negative numbers, fractions, and decimals. Also include data sets and questions that are not statistical such as one data point or data that is not variable.When expressing the data, ensure they are expressing magnitude of fractions and decimals correctly.

Use Socratic method to have student to define what makes a question statistical question.

FLUENTLY TALK ABOUTStudents will use working definitions to identify the when data sets and questions are statistical and then talk about the shape and centers of the data using words.

When students talk about their work or others’ work: have them describe why another group made the decision about whether the question is statistical and how the data is described.

FLUENTLY READStudent should be able to answer questions in textbook, identifying which questions and data points are statistical.

Note: Grade 6 uses univariate data exclusively (such as the average weight of the backpacks in the room). You may wish to avoid bivariate data questions (such as the number of rainy days in March over a 10 year period).

RUBRIC FOR LEARNING Population, quantity, and the units in which the students are

measuring. Population Quantity and the units in which the students are measuring. Existence of variability

CREATE & REPRESENTUse Rubric for Learning to check for each of the bulleted items as students created identifications of statistical data

FLUENTLY TALK ABOUTUse Rubric for Learning to check for each of the bulleted items as students talk about the representations of the statistical data comparing relationships using the same and different representations of each.


FLUENTLY READUse Rubric for Learning to check for each of the bulleted items as student’s read/work the representations of the statistical data

DIFFERENTIATION

Providing feedback, according to a rubric of the bullets listed above, to the students on their seatwork before allowing them to make posters of it for public display will allow them to show off their best work. Attaching their seatwork to their public display will show their best learning; something they can be proud of.

As you see each student become able to do what you taught them, celebrate the learning of that individual student, eye-to-eye establishing that they can learn, in this class, from you.Grouping students who struggle talking about each of the mathematical foci listed or fluency above as well as those not struggling with anything.

SpEd Modifications: Grouping students who struggle talking about each of the mathematical foci listed or FLUENCY above as well as those not struggling with anything.

G/T Modifications:How do outliers affect each of the presentations of data?

FLUENCYCount around the room and combinations.

CREATE & REPRESENT Organize data points/ questions Define what makes a statistical question Use word to describe data sets Measurements of data variation (spread): range,

IQR, MAD

FLUENTLY TALK ABOUT Students should be able to identify others thinking,

why they chose that a data set/question was statistical.

Critique how the student describes the data and if it was appropriate.

Discuss the variation of the data and how it impacts the use of that data.

FLUENTLY READHave students read scenarios from sample problems and identify which ones are statistical.

SUMMARYStudents will be able to describe data sets and their accompanying data sets. They can identify if the data set and questions are statistical. They will also be able to describe the shape, spread and center of the data set.

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Reflection:

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Unit Title: 4c: Geometric RepresentationsGrade Level: 6

Timeframe: Marking Period 4cUnit Focus & Essential Questions

Unit Focus: (1) Understanding that the entire 6th grade mathematics course revolves around the study of relationships between things we count and measure, quantities.(2) Understanding of the nine quantities and units associated with each of the quantities.(3) Understanding that geometry involves the exploration of physical representations of four of the nine quantities, specifically: length, area, volume, and angle of rotation (4) Understand how the quantities of area and volume are related to length.(5) Understand the types of questions we ask and answer about the physical properties of shapes.

(6) CONTINUE TO BUILD FLUENCY WITH THE VALUES (FRACTIONS AND WHOLE NUMBERS) AND CALCULATIONS THAT ARISE WITHIN THE GEOMETRIC RELATIONSHIPS.

Rubric for Learning Foci Quantity and the units in which the students are measuring. Representations of geometric quantities Relationships between the quantities

Essential Questions: (1) Can we become effective and efficient at composing and decomposing shapes?(2) Can we become effective and efficient at identifying properties of shapes when presented in scenarios?(3) Can we become effective and efficient calculating values?(4) Can we become effective and efficient answering and answering questions about attributes of shapes?(5) Can we become effective and efficient about talking about attributes of shapes?(6) Can we become effective and efficient about reading about attributes of shapes?

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Instructional PlanUnit 4c Instructional Pre-assessment: Present the students with an empty box (from cereal, granola bars, crackers, etc.) and both a unit square and a unit cube. Ask them to estimate the number of unit squares needed to cover the bottom of the box, the entire box, and the number of cubes necessary to fill the box. Ask them why someone (the manufacturer of the stuff originally packaged in the box) might need to know that information.

Solve real-world and mathematical problems involving area, surface area, and volume.

6.G.1 Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.6.G.3. Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side joining points with the same first coordinate or the same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems.

STANDARDS & OBJECTIVES SWBAT …

CREATE & REPRESENT A parallelogram given a rectangle and making non-perpendicular cut through it Triangles of varying shapes by cutting different parallelograms on a diagonal Polygon shapes graphed on coordinate grid Area using squares or grids Compositional 2-d shapes using rectangles and triangles

FLUENTLY TALK ABOUT Area of rectangles, parallelograms, trapezoids and triangles Decomposing complex shapes into rectangles, parallelograms, trapezoids, and triangles Characteristics of special quadrilaterals and composite figures (made from rectangles, triangles, and other shapes)

FLUENTLY READ Other people’s representations of compositions of shapes Textbook problems involving rectangles, quadrilaterals, triangles, and composite shapes

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FLUENCY Count around the room. Be sure to “double” and “half”, as these numbers come

into play with the area of triangles and trapezoids. Make combinations, being sure to use fractions and

decimals (negatives do not work well in this unit, but you may want to practice them, anyway).

CREATE & REPRESENT Provide students with unit cubes and boxes of different

shapes and dimensions to measure. If available, provide them with the same box, but different sized cubes (inches, centimeters, etc.)

Have students draw representations of their figures on a coordinate grid, both using the origin as a vertex and NOT using the origin as a vertex.

FLUENTLY TALK ABOUT Students should be able to discuss how they laid out their

unit squares (ex: 3 rows of 5 squares) to answer area of rectangles.

Students should be able to discuss the “transformations” needed to translate a rectangle into a parallelogram, triangle, and trapezoid.

Students should be able to recognize that two congruent right triangles form a rectangle, and two congruent non-right triangles and trapezoids create a parallelogram, etc.

FLUENTLY READStudent should be able to answer questions from assessments, released items, and textbook.

RUBRIC FOR LEARNING FOCI Quantity and the units in which the students are

measuring. Representations of geometric quantities Relationships between the quantities

CREATE & REPRESENTUse Rubric for Learning Foci to check for each of the bulleted items as students determine the size of their box, in terms of length, width, height, area, and volume.

With the parallelogram, they should measure both the side lengths and the height (suggest this, if they struggle).

If they get stuck finding the area with the parallelogram, provide scissors. Instruct them to cut a perpendicular line across the parallelogram and move one piece to create a rectangle.

With the triangles and trapezoids, be prepared with congruent duplicate shapes, so that the students can build a parallelogram by rotating the copy 180 degrees to create a parallelogram

FLUENTLY TALK ABOUTUse Rubric for Learning Foci to check for each of the bulleted items as student’s read/work the representations process of finding area

FLUENTLY READUse Rubric for Learning Foci to check for each of the bulleted items as student’s read/work the representations of the statistical data

FLUENCY Count around the room. Make combinations, being sure to use fractions and


CREATE & REPRESENT Give students a rectangle to measure. They should find

the number of squares “long” and “wide” their box is. When they do this, be sure they know that they are measuring in “square” lengths… not square units.

Ask them to figure out how many squares they would need to completely cover the box.

Repeat this with a parallelogram.

Repeat this with a right triangle Repeat this with a non-right triangle Repeat this with a trapezoid.

FLUENTLY TALK ABOUTThe number of squares long and wide, perimeter, and area of each of the shapes.Students should be able to clearly articulate the area of a rectangle is base times height (rows and columns of unit squares).Students should also be able to “derive” the formulas of the area of a triangle (½ base times height), and trapezoid (average of the bases times height or ½ (sum of the bases) x height).

FLUENTLY READUse Rubric for Learning Foci to check for each of the bulleted items as student’s read/work the process of finding area of different shapes.

DIFFERENTIATION STRATEGIES

SpEd Modifications: Provide students with combinations of rectangles,

ASK AND ANSWERBoth A- and B- type questions about faces, dimensions, and area of rectangles, triangles, parallelograms, and trapezoids from textbooks and assessments.

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rectangles and triangles, and missing side lengths to compute area and perimeter. Show students how to draw in “missing” sides to see the rectangles more clearly.

G/T Modifications:Provide students images of figures with parts missing (such as a rectangle with a triangle cut out) to find the areas of the shaded regions.

CLOSING ACTIVITYCreate a design with pattern blocks. Determine the total area covered.

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Solve real-world and mathematical problems involving area, surface area, and volume.

6.G.2 Find the volume of a right rectangular prism with fractional edge lengths by packing it with unit cubes of the appropriate unit fraction edge lengths, and show that the volume is the same as would be found by multiplying the edge lengths of the prism. Apply the formulas V = l w h and V = B h to find volumes of right rectangular prisms with fractional edge lengths in the context of solving real-world and mathematical problems.6.G.4 Represent three-dimensional figures using nets made up of rectangles and triangles, and use the nets to find the surface area of these figures. Apply these techniques in the context of solving real-world and mathematical problems.


Volume of rectangular prisms in terms of rows, columns, and layers of unit cubes. Nets that can be folded into rectangular and prisms Surface area of triangular and rectangular prisms area using squares or grids compositional 2-D shapes using rectangles and triangles.

FLUENTLY TALK ABOUT Volume of rectangular prisms in terms of rows, columns, and layers of unit cubes. Surface area composed as the total area of all surfaces Each individual surface as a two-dimensional shape.

FLUENTLY READ Other people’s representations of prisms Textbook problems involving prisms

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FLUENCY Count around the room. Make combinations, being sure to use fractions and


RUBRIC FOR LEARNING FOCI Quantity and the units in which the students are

measuring. Representations of geometric quantities Relationships between the quantities

FLUENCYCount around the room.Make combinations, being sure to use fractions and decimals.

CREATE & REPRESENTGive students a box to measure. They should find the number of

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CREATE & REPRESENTPrepare boxes of different shapes, sizes, and dimensions, as well as squares and cubes of various sizes.

Have students draw representations of their rectangles on a coordinate grid, both using the origin as a vertex and NOT using the origin as a vertex.

Be sure that students understand that volume refers to the amount of space within (number of cubes that would fit inside) a 3-D shape and surface area refers to the number of squares it takes to cover it.

FLUENTLY TALK ABOUT The number of cubes long, wide, and high, area of each of

the faces, and volume of the box. The total lengths of each of the sides in terms of

measurements as well as numbers of cube lengths The volume as the number of “layers” of cubes from the

bottom row.

CREATE & REPRESENTUse Rubric for Learning Foci to check for each of the bulleted items as students determine the size of their box, in terms of length, width, height, area, and volume.

FLUENTLY TALK ABOUTUse Rubric for Learning Foci to check for each of the bulleted items as student’s read/work the representations process of finding area

FLUENTLY READUse Rubric for Learning Foci to check for each of the bulleted items as student’s read/work the representations of the statistical data

cubes “long”, “wide”, and “high” their box is. When they do this, be sure they know that they are measuring in “cube lengths” not cubic units.

Ask them to figure out how many cubes they would need to fill the box one row deep.

Be sure that they understand that this number is the volume of one layer. It is equal to the number of unit squares it takes to cover the bottom of the box, and represents a certain number of “cube lengths” long and “cube lengths” wide.

Ask them to figure out how many cubes needed to fill the box. Ask them to determine the number of squares needed to cover the prism.Ask them to determine the effect on the box if another level were added.Vary the “cubes” so that they are one unit, two units, ½ unit, ¼ unit on each side. Include “cubes” that are other sizes (such as 2 ¼ units).

FLUENTLY TALK ABOUTThe number of cubes long, wide, and high, area of each of the faces, and volume of the box.The total lengths of each of the sides in terms of measurements as well as numbers of cube lengthsThe volume as the number of “layers” of cubes from the bottom row.

ASK AND ANSWERBoth A- and B- type questions about faces, dimensions, and area of rectangles, triangles, parallelograms, and trapezoids from textbooks and assessments.

CLOSING ACTIVITYGiven a commercially available box, relate the different aspects studied in this unit to features of the box.

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Reflection

Touchpoints; Scaffolded Touchpoints 1Touchpoints Scaffolded Touchpoints 2

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Touchpoints Scaffolded Touchpoints 3


Homework: Make an equation to explain how to double or triple a recipe.Graph the total cost of buying a video game every day for a week.Create a progression and equation using something you measure the next night have other student create type a and type b questions for the progression, and the third night have the original students answer the questions.




Student as class will create factory. They will design a product. Student with break down all parts of the factory and create progressions, equations, and graphs for all part of the factory to explain the production and what would happen of the factory was successful and was scaled.

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6 Math Unit... · Web viewCounting can be done during whole group instruction, mini-lesson, small group instruction, etc. ... Giving quantitative measures of center (median and/or

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