8 Math Unit 4 P… · Web view(12) Can we become ... Unit 1 Pre-assessment. ... Have students record their quantities publicly on the Quantities Word Wall

Student Strategies Formative Assessment Activities and Resources

Unit Title: Measuring, Representing our measuring, and Reading others’ representations of measuringGrade Level: 8

Timeframe: Marking Period 4Unit Focus & Essential Questions

Unit Focus:(1) Understanding that the entire 8th 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 proportional and 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 values (fractions, decimals, large and small whole numbers) and calculations that arise in the domain and range of the linear relationships.(7) Begin to compare multiple linear relationships.

Rubric for Learning Population Quantity and the units being measured of

the population Related pair of actual values Linear model/line of best fit

Initial value Change over an interval

Related pair from the model

Rubric for Learning Real numbers

RationalIrrational

Magnitude of an irrational numberDecimal approximationNumber line approximation

Rubric for Learning

Quantity and the units in which the students are measuring.

Representations of geometric quantities

Relationships between the quantities

Essential Questions:(1) Can we become confident in our knowledge of quantities, units, and values of measuring that are used in measuring populations?(2) Can we become effective and efficient at representing bivariate data as a scatterplot or categorical data as a two way table?(3) Can we become effective and efficient at asking and answering questions about scatter plots?(4) Can we become effective and efficient at talking about relationships given in any representation of a population?(5) Can we become effective and efficient at reading and understanding others’ representations of bivariate data of a population?(6) Can we become effective and efficient at determining whether a real number is rational or irrational?(7) Can we become effective and efficient at using rational approximations of irrational numbers?(8) Can we become effective and efficient at comparing real numbers?(9) Can we become effective and efficient at identifying properties of shapes when presented in scenarios?(10) Can we become effective and efficient calculating values?(11) Can we become effective and efficient answering and answering questions about attributes of shapes?(12) Can we become effective and efficient about talking about attributes of shapes?(13) Can we become effective and efficient about reading about attributes of shapes?

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*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?

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New Jersey Student Learning Standards

CCSS.MATH.CONTENT.8.SP.A.1Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.CCSS.MATH.CONTENT.8.SP.A.2Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.CCSS.MATH.CONTENT.8.SP.A.3Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.CCSS.MATH.CONTENT.8.SP.A.4Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?CCSS.MATH.CONTENT.8.NS.A.1Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.CCSS.MATH.CONTENT.8.NS.A.2Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.CCSS.MATH.CONTENT.8.G.C.9Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

Instructional PlanUnit 1 Pre-assessment

Have the students choose a population in which they can measure 2 quantities about each member of the population and gather that data, represent this in a graph and create line of best fit and talk about the actual pair of values and the pair of values from the model.

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Standards & Objectives SWBATCCSS.MATH.CONTENT.8.SP.A.1Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

CCSS.MATH.CONTENT.8.SP.A.2Know that straight lines are widely used to model relationships between two quantitative variables. For scatter plots that suggest a linear association, informally fit a straight line, and informally assess the model fit by judging the closeness of the data points to the line.

CCSS.MATH.CONTENT.8.SP.A.3Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept. For example, in a linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additional 1.5 cm in mature plant height.

CCSS.MATH.CONTENT.8.SP.A.4Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table. Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects. Use relative frequencies calculated for rows or columns to describe possible association between the two variables. For example, collect data from students in your class on whether or not they have a curfew on school nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?

SWBAT…CREATE & REPRESENT

create scatterplots and 2 way tables (for categorical data) from things that were counted and measured between 2 quantities of each member of a population (distinguish what the population is)

represent relationships graphically describe patterns determine change over an interval (slope) determine initial value (y-intercept)

FLUENTLY TALK ABOUT discuss all aspects in the Focus for Learning

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

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FLUENCYStart at whole numbers count by 10, stop at intervals of

40.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.

CREATE & REPRESENTHave the group create/collect the data 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.

Some examples of things that can be used to measure about populations which yield different scatterplots to analyze are:

eat fewer calories weight goes down ship more items you have less money in your

account the hours I study and the score I make on the test

Some examples for collecting categorical data:

Girls/boys, how they get to school, walk, ride the bus, bike etc.

Age range and the types of movies they like, comedy, drama, romance, etc

FLUENTLY TALK ABOUT

When students talk about their or other’s work be sure they are using the language the same as they have been using throughout the year.

FLUENTLY READ

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

CREATE & REPRESENTUse Rubric for Learning to check for each of the bulleted items as students created representations of the data from the populations

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

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 population.

DIFFERENTIATION

Providing 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.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

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

CREATE & REPRESENTChoose population to find 2 things (of the 9 quantities) that can be counted and measured about that population, record them. Then graph this relationship. Create a line that best fits the data so that the linear relationship can be talked about. Determine the meaning of related pairs of the actual values and a pair from the model (from the line of best fit), identify and give meaning to initial values and the change over an interval.

Ask and answer questions type A and B questions about the representations.

Also, create/collect categorical data and collect the data on two way frequency table.

FLUENTLY TALK ABOUTStudents practice talking about the math in their own and their group members’ representations, then students talk about the math in other group’s representations using the rubric for learning.

FLUENTLY READHave students read others’ scenarios from textbooks or sample PARCC problems and create multiple representations of the relationships from them.Have students record their quantities publicly on the Quantities Word Wall. Have students record the symbolic representations on the Symbolic Representation Chart.

SUMMARY

Sum 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”

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Rubric for Learning Population Quantity and the units being measured of

the population Related pair of actual values Linear model/line of best fito Initial valueo Change over an interval

Related pair from the model


struggling with anything.

Reflection:

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Standards & Objectives SWBAT

CCSS.MATH.CONTENT.8.NS.A.1Know that numbers that are not rational are called irrational. Understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.CCSS.MATH.CONTENT.8.NS.A.2Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π2). For example, by truncating the decimal expansion of √2, show that √2 is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.SWBAT…

CREATE & REPRESENT use a table to list real numbers and determine if they are rational or irrational based on if the numbers can be written as a ratio use a number line to approximate locations of irrational numbers

FLUENTLY TALK ABOUT discuss the rational and irrational number comparisons discuss using decimal and number line approximations estimate the value of rational numbers discuss all aspects in the Focus for Learning

FLUENTLY READ interpret that there are numbers that are not rational approximate irrational numbers by rational numbers

FLUENCYCount around the room and combinations (real numbers)

CREATE AND REPRESENTHave students use a table to list real numbers and determine if

they are rational or irrational based on if the numbers can be written as a ratio

use a number line to approximate locations of irrational numbers

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.


CREATE & REPRESENTCreate and represent:

tables to determine is a list real numbers and determine if they are rational or irrational based on if the numbers can be written as a ratio

a number lines to approximate locations of irrational numbers Be sure to include progressions that contain real numbers (rational and

irrational). Ask type A and B questions. See *

FLUENTLY TALK ABOUTUse the Rubric of Learning to fluently talk about:

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Rubric for Learning Real numbers

RationalIrrational

Magnitude of an irrational numberDecimal approximationNumber line approximation


When planning, be sure to consider:

irrational numbers and their decimal and number line approximations

rational and irrational number comparisonsBe sure to include progressions that contain real number set (rational and irrational).

FLUENTLY TALK ABOUT

Students should be able to talk about scenarios:-rational and irrational comparisons- magnitude of irrational numbers- Decimal and number line approximations of irrational numbers

FLUENTLY READStudent should be able to answer questions in textbook and Touchpoints.

ELL:SPED Accommodations:Gifted and Talented:

CREATE & REPRESENTUse Rubric for Learning to check for each of the bulleted items as students created progressions and graphs of the relationship.

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 representations, talking, and reading.

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.

DIFFERENTIATION

Providing 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.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.

-rational and irrational comparisons- magnitude of irrational numbers- Decimal and number line approximations of irrational numbers

FLUENTLY READ

Student should be able to: answer textbook and Touchpoint questions : determine whether a real number is rational or irrational Know that there are numbers that are not rational, and approximate them

by rational numbers

SUMMARYSum 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”

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Reflection:

Standards & Objectives SWBATCCSS.MATH.CONTENT.8.G.C.9Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

CREATE & REPRESENT create and represent volumes of cones, cylinders and spheres create questions about lengths when given volumes, or volumes when given lengths create abstract representations of the measuring you are doing of the cones, cylinders and spheres

FLUENTLY TALK ABOUT discuss how to find the various types of measurements for cones, cylinders and spheres discuss how to find a volume when given values or find values when given a volume

FLUENTLY READ Other’s representations of cylinders, cones and spheres using the focus for learning.

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FLUENCY1/3, start 0, count by 1/3Start at whole numbers and tenths and take half.Activities with combinations of units and operations performed unit by unitAs the students begin to create their progressions of items in the classrooms, teachers need to gather the numbers students are creating and use them to count around the room.

CREATE AND REPRESENTHave students find the area of a circle and begin a progression by varying height of the cylinder.When finding volume of cylinder talk about it as the height of one layer of the area of the circular base and how it is filling it up, make a measurement of the circle (may be a generalization at first before you introduce the formula as a way to get the exact area of a circle.Use cylinders, spheres and cones with the same diameter and height and fill with rice to find the relationship between the various shapes for determining the formulas (There are plastic shapes in the Glenco kits that work well for this activity).

FLUENTLY TALK ABOUT

Students should be able to talk about both the scenarios when there are quantities given and when there are none (just x and y given, concrete and abstract representations)

FLUENTLY READ

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

CREATE & REPRESENTUse Rubric for Learning to check for each of the bulleted items as students created progressions and graphs of the relationship.

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 representations, talking, and reading.

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.

DIFFERENTIATION

Providing 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.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.


CREATE & REPRESENTFill volumes of cylinders, cones and spheres.Have the students record the representations of measurements, generalize the ones you are able to into a formula (cylinders)

FLUENTLY TALK ABOUTUse the rubric for learning to fluently talk about their own relationshipUse the rubric for learning to fluently talk about relationships created by classs mates

As students talk about the representations, let them take turns talking about the math in 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 READ

Have students read scenarios from sample problems and create multiple representations of the relationships in the scenario..Have students record their quantities publicly on the Quantities Word Wall. Have students record the symbolic representations on the Symbolic Representation Chart.

SUMMARYSum 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”

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Rubric for Learning Quantity and the units in which the students

are measuring. Representations of geometric quantities Relationships between the quantities


Reflection:

Touchpoints: 8.SP.1, 8.SP.2, 8.SP.3, 8.SP.4, 8.NS.1, 8.NS.2, 8.G.9

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 guys 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 4—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.

Summative Performance Assessment

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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.

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8 Math Unit 4 P… · Web view(12) Can we become ... Unit 1 Pre-assessment. ... Have students record their quantities publicly on the Quantities Word Wall