FOSS and Common Core Math Grade 8

Transcription

1 FOSS and Common Core Math Grade 8 Contents Introduction... K1 Ratios and Proportional Relationships for Grades K4 The Number System for Grades K8 Expressions and Equations for Grades K14 Functions for Grade 8... K20 Geometry for Grades K22 Statistics and Probability for Grades K24 INTRODUCTION The adoption of the Common Core State Standards for Mathematics calls for shifts in focus, coherence, and rigor. The teaching of the standards should be focused on the important content, coherent from one grade level to the next, and rigorous in requiring conceptual understanding, fluency, and application. Within this area of application, FOSS provides fertile ground for the use of mathematics. The FOSS Program integrates mathematics with science throughout the grade 8 courses. In active investigations, students apply mathematics during data gathering and analysis. Support is provided within the Investigations Guide to scaffold student understanding throughout these activities. In addition, you may want to collaborate with your students math teacher to discuss the techniques and processes students are accustomed to using in math classes to help create interdisciplinary connections that will support students thinking around mathematical content. This chapter gives an overview of how FOSS addresses the Common Core State Standards for Mathematics through science. It also points out specific instances in which students excercise those skills during science instruction. NGSS_Com_Core_MATH_Grade8 2C.indd 1 Copyright The Regents of the University of California K1 4/9/18 9:24 AM

2 FOSS and Common Core Math Grade 8 K2 Mathematical Practices Mathematical practices consist of eight processes and proficiencies that are important for all students. 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Within the context of science, students use some of these mathematical practices on a regular basis. According to Next Generation Science Standards (volume 2, appendix L, p. 138), the three CCSSM practice standards most directly relevant to science are: MP.2. Reason abstractly and quantitatively. MP.4. Model with mathematics. MP.5. Use appropriate tools strategically. When students reason abstractly and quantitatively and model with mathematics, they are using math in context. They work with symbols and their meanings and represent and solve word problems. Students choose and correctly use the available tools to collect data and solve problems. In the grade 8 courses, students engage with these three practices during the active investigations. Here are some examples. In Investigation 2 of the Heredity and Adaptations Course, students model heredity using Punnett squares to determine offspring potential. Students then predict genetic variation of offspring based on parental traits before using an online simulation to breed imaginary creatures called Larkeys, and analyze the population results to draw conclusions about population changes and trait outcomes. In Investigation 1 of the Electromagnetic Force Course, students measure forces on various objects using a spring scale and draw conclusions about how net force affects motion. They are presented with additional problems in which they must calculate net force and make claims about the motion that will result.

3 Introduction In Investigation 3 of the Gravity and Kinetic Energy Course, students analyze results of a simulated crashes to draw conclusions about how speed and mass affect energy transferred during a collision. They determine mathematical relationships based on the observed results, and conclude that when mass doubles, kinetic energy doubles, but that when speed doubles, kinetic energy increases by a factor of 4 (2²). In Investigation 2 of the Waves Course, students conduct controlled experiments with spring waves to determine how increased amplitude or frequency affect the energy transferred by the wave. After collecting data and discussing results, students are posed with the additional challenge of inferring how changing wavelength would affect wave energy. They refer to the velocity equation (introduced in Investigation 1) v = f l to make the inference that increasing frequency would necessarily decrease wavelength, which is known as an inverse relationship. In Investigation 7 of the Planetary Science Course, students use a scaling factor to determine the scaled sizes and distances of a solar system model that has a 1 cm Sun. After confirming their calculations, students construct a model including the inner planets, then have a discussion about the vast distances between solar system objects and why it is not easily feasible to include the outer planets on a classroom model of this scale. Mathematical Content The mathematical content in eighth grade is organized around five concepts. The number system Expressions and equations Functions Geometry Statistics and probability The following pages have a table that identifies the opportunities to engage students in developing these mathematical concepts. It lists the math content for eighth grade and points out relevant opportunities in the five FOSS courses for grade 8. FOSS and Common Core Math Grade 8 K3

4 FOSS and Common Core Math Grade 8 RATIOS AND PROPORTIONAL RELATIONSHIPS FOR GRADES 6 7 Standard Waves Course Gravity and Kinetic Energy Course Understand ratio concepts and use ratio reasoning to solve problems. Grade 6 K4 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 wholenumber 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. Inv 3, Part 2, Step 14, Recommendations for discussions Inv 4, Part 3, Step 5, Introduce resolution Inv 1, Part 1, Step 10, Discuss speed Inv 1, Part 2, Step 13, Define acceleration Inv 1, Part 3, Step 17, Change units to meters Inv 1, Part 3, Step 18, Compute the acceleration of gravity Inv 2, Part 1, Step 6, Predict weight and mass

5 Ratios and Proportional Relationships for Grades 6 7 Planetary Science Course Heredity and Adaptation Course Electromagnetic Force Course Extending Inv 1, Apply math to a map Inv 6, Part 1, Step 8, Introduce astronomical unit and light-year Inv 3, Part 2, Step 2, Engage with walking-stick activity Inv 3, Part 2, Step 8, Debrief online activity Inv 2, Part 3, Step 17, Have a sensemaking discussion Inv 2, Part 3, Step 20, Test the model further FOSS and Common Core Math Grade 8 K5

6 FOSS and Common Core Math Grade 8 RATIOS AND PROPORTIONAL RELATIONSHIPS FOR GRADES 6 7 (continued) Standard Waves Course Gravity and Kinetic Energy Course Analyze proportional relationships and use them to solve real-world and mathematical problems. 1. Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units. For example, if a person walks 1/2 mile in each 1/4 hour, compute the unit rate as the complex fraction 1/2 / 1/4 miles per hour, equivalently 2 miles per hour. Inv 1, Part 1, Step 9, Define frequency Grade 7 K6 2. Recognize and represent proportional relationships between quantities. a. Decide whether two quantities are in a proportional relationship, e.g., by testing for equivalent ratios in a table or graphing on a coordinate plane and observing whether the graph is a straight line through the origin. b. Identify the constant of proportionality (unit rate) in tables, graphs, equations, diagrams, and verbal descriptions of proportional relationships. c. Represent proportional relationships by equations. For example, if total cost t is proportional to the number n of items purchased at a constant price p, the relationship between the total cost and the number of items can be expressed as t = pn. d. Explain what a point (x, y) on the graph of a proportional relationship means in terms of the situation, with special attention to the points (0, 0) and (1, r) where r is the unit rate. Inv 1, Part 2, Step 18, Discuss the equation Inv 2, Part 1, Step 17, Introduce inverse relationship Inv 2, Part 2, Step 8, Have a sense-making discussion Inv 2, Part 2, Step 10, Continue discussion with weight on other planets Inv 2, Part 2, Step 13, Discuss jumps, force, and mass Inv 2, Part 2, Step 14, Review other planetary data Inv 2, Part 2, Step 15, Introduce Newton s second law Inv 2, Part 2, Step 17, Practice with other examples Inv 3, Part 2, Step 8, Analyze results for mass Inv 3, Part 2, Step 10, Confirm speed relationship Inv 3, Part 2, Step 12, Review relationships Inv 3, Part 2, Step 13, Practice applying knowledge (optional) Inv 4, Part 1, Step 3, Introduce video: Understanding Car Crashes

7 Ratios and Proportional Relationships for Grades 6 7 Planetary Science Course Heredity and Adaptation Course Electromagnetic Force Course Inv 5, Part 2, Step 7, Determine impactor size Inv 9, Part 1, Step 12, Discuss findings Inv 9, Part 2, Step 16, Search for Earthlike planets FOSS and Common Core Math Grade 8 Inv 2, Part 3, Step 17, Have a sensemaking discussion Inv 2, Part 3, Step 20, Test the model further K7

8 FOSS and Common Core Math Grade 8 THE NUMBER SYSTEM FOR GRADES 6 7 Standard Waves Course Gravity and Kinetic Energy Course Compute fluently with multi-digit numbers and find common factors and multiples. 2. Fluently divide multi-digit numbers using the standard algorithm. 3. Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation. Grade 6 K8 Apply and extend previous understandings of numbers to the system of rational numbers. 5. 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.

9 The Number System for Grades 6 7 Planetary Science Course Heredity and Adaptation Course Electromagnetic Force Course Inv 2, Part 2, Step 11, Show slide show: Heredity Inv 1, Part 1, Step 10, Introduce altitude and elevation FOSS and Common Core Math Grade 8 K9

10 FOSS and Common Core Math Grade 8 THE NUMBER SYSTEM FOR GRADES 6 7 (continued) Grade 6 K10 Standard 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. Waves Course Inv 4, Part 2, Step 11, Create digital waves Gravity and Kinetic Energy Course

11 The Number System for Grades 6 7 Planetary Science Course Heredity and Adaptation Course Inv 1, Part 1, Step 13, Construct a model time line Inv 1, Part 2, Step 10, Compare the organisms Electromagnetic Force Course FOSS and Common Core Math Grade 8 K11

12 FOSS and Common Core Math Grade 8 THE NUMBER SYSTEM FOR GRADES 6 7 (continued) Grade 7 K12 Standard Apply and extend previous understandings of operations with fractions to add, subtract, multiply, and divide rational numbers. 1. Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram. a. Describe situations in which opposite quantities combine to make 0. For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged. b. Understand p + q as the number located a distance q from p, in the positive or negative direction depending on whether q is positive or negative. Show that a number and its opposite have a sum of 0 (are additive inverses). Interpret sums of rational numbers by describing realworld contexts. c. Understand subtraction of rational numbers as adding the additive inverse, p q = p + ( q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. d. Apply properties of operations as strategies to add and subtract rational numbers. 3. Solve real-world and mathematical problems involving the four operations with rational numbers. Waves Course Gravity and Kinetic Energy Course Inv 2, Part 2, Step 12, Measure jump heights

13 The Number System for Grades 6 7 Planetary Science Course Heredity and Adaptation Course Electromagnetic Force Course Inv 2, Part 3, Step 4, Calculate hours of sunlight Inv 4, Part 1, Steps 8-9, Discuss moonrise, Analyze moonrise data FOSS and Common Core Math Grade 8 Inv 1, Part 1, Step 14, Conduct the investigations Inv 1, Part 1, Step 16, Move to Pushes and Pulls B Inv 1, Part 1, Step 18, Preview Pushes and Pulls C Inv 1, Part 2, Step 6, Collect data and determine measurements Inv 1, Part 3, Step 8, Introduce the notebook sheets Inv 2, Part 3, Step 12, Conduct new experiments Inv 3, Part 3, Step 9, Analyze results K13

14 FOSS and Common Core Math Grade 8 EXPRESSIONS AND EQUATIONS FOR GRADES 6 8 Standard Waves Course Gravity and Kinetic Energy Course Apply and extend previous understandings of arithmetic to algebraic expressions. Grade 6 K14 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 3y. Represent and analyze quantitative relationships between dependent and independent variables. 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. Inv 1, Part 1, Step 17, Develop speed equation Inv 1, Part 1, Step 20, Compare seeds Inv 1, Part 2, Step 13, Define acceleration Inv 1, Part 2, Step 14, Calculate acceleration Inv 1, Part 3, Step 11, Complete calculation for first frames Inv 2, Part 2, Step 15, Introduce Newton s second law Inv 2, Part 2, Step 17, Practice with other examples

15 Expressions and Equations for Grades 6 8 Planetary Science Course Heredity and Adaptation Course Electromagnetic Force Course Inv 2, Part 3, Step 7, Calculate and graph the information Extending the Inv 2, Recreate Eratosthenes s discovery Inv 5, Part 2, Step 7, Determine impactor size FOSS and Common Core Math Grade 8 K15

16 FOSS and Common Core Math Grade 8 EXPRESSIONS AND EQUATIONS FOR GRADES 6 8 (continued) Standard Waves Course Gravity and Kinetic Energy Course Solve real-life and mathematical problems using numerical and algebraic expressions and equations. Grade 7 K16 3. Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions, and decimals), using tools strategically. Apply properties of operations to calculate with numbers in any form; convert between forms as appropriate; and assess the reasonableness of answers using mental computation and estimation strategies. For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $ If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation. Inv 2, Part 2, Step 10, Continue the discussion with weight on other planets

17 Expressions and Equations for Grades 6 8 Planetary Science Course Heredity and Adaptation Course Electromagnetic Force Course Inv 2, Part 3, Step 4, Calculate hours of sunlight Inv 4, Part 1, Steps 8-9, Discuss moonrise, Analyze moonrise data FOSS and Common Core Math Grade 8 K17

18 FOSS and Common Core Math Grade 8 EXPRESSIONS AND EQUATIONS FOR GRADES 6 8 (continued) Standard Waves Course Gravity and Kinetic Energy Course Grade 8 K18 Work with Radicals and integer exponents. 2. Use square root and cube root symbols to represent solutions to equations of the form x2 = p and x3 = p, where p is a positive rational number. Evaluate square roots of small perfect squares and cube roots of small perfect cubes. Know that 2 is irrational. 4. Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used. Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities (e.g., use millimeters per year for seafloor spreading). Interpret scientific notation that has been generated by technology. Understand the connections between proportional relationships, lines, and linear equations. 5. Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed. Inv 3, Part 2, Step 8, Analyze results for mass Inv 3, Part 2, Step 10, Confirm speed relationship Inv 3, Part 2, Step 12, Review relationships Inv 3, Part 2, Step 13, Practice applying knowledge (optional) Inv 1, Part 1, Step 15, Represent speed with graphs Inv 1, Part 1, Step 20, Compare seeds Inv 1, Part 2, Step 10, Discuss slope

19 Expressions and Equations for Grades 6 8 Planetary Science Course Heredity and Adaptation Course Electromagnetic Force Course FOSS and Common Core Math Grade 8 Inv 1, Part 1, Step 13, Construct a model time line Inv 1, Part 2, Step 10, Compare the organisms K19

20 FOSS and Common Core Math Grade 8 FUNCTIONS FOR GRADE 8 Standard Waves Course Gravity and Kinetic Energy Course Use functions to model relationships between quantities. Grade 8 K20 4. Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values. 5. Describe qualitatively the functional relationship between two quantities by analyzing a graph (e.g., where the function is increasing or decreasing, linear or nonlinear). Sketch a graph that exhibits the qualitative features of a function that has been described verbally. Inv 1, Part 1, Step 14, Compare tracks Inv 1, Part 2, Step 9, Complete data tables Inv 1, Part 2, Step 12, Review student responses

21 Functions for Grade 8 Planetary Science Course Heredity and Adaptation Course Electromagnetic Force Course Inv 1, Part 1, Step 15, Discuss findings for Pushes and Pulls A Inv 5, Part 1, Step 21, Graph experimental data Inv 7, Part 4, Step 15, Discuss growth rate FOSS and Common Core Math Grade 8 K21

22 FOSS and Common Core Math Grade 8 GEOMETRY FOR GRADES 7 8 Standard Waves Course Gravity and Kinetic Energy Course Draw, construct, and describe geometrical figures and describe the relationships between them. 1. Solve problems involving scale drawings of geometric figures, including computing actual lengths and areas from a scale drawing and reproducing a scale drawing at a different scale. Grade 7 Grade 8 K22 Solve real-life and mathematical problems involving angle measure, area, surface area, and volume. 6. Solve real-world and mathematical problems involving area, volume and surface area of two- and threedimensional objects composed of triangles, quadrilaterals, polygons, cubes, and right prisms. Understand congruence and similarity using physical models, transparencies, or geometry software. 1. Verify experimentally the properties of rotations, reflections, and translations: a. Lines are taken to lines, and line segments to line segments of the same length. b. Angles are taken to angles of the same measure. c. Parallel lines are taken to parallel lines. Inv 3, Part 1, Step 13, Measure incident and reflected angles Inv 3, Part 4, Step 6, Conduct refraction test

23 Geometry for Grades 7 8 Planetary Science Course Heredity and Adaptation Course Electromagnetic Force Course Inv 3, Part 2, Step 4, Review scaling (optional) Inv 3, Part 2, Step 7, Develop a plan Inv 4, Part 1, Step 21, Demonstrate lunar eclipse Extending the Inv 5, Scale craters to a local map Inv 7, Part 1, Step 6, Calculate diameter and distance Inv 7, Part 1, Step 13, Extend the investigation with homework Inv 2, Part 2, Step 8, Introduce light as energy FOSS and Common Core Math Grade 8 K23

24 FOSS and Common Core Math Grade 8 STATISTICS AND PROBABILITY FOR GRADES 7 8 Standard Waves Course Gravity and Kinetic Energy Course Investigate chance processes and develop, use, and evaluate probability models. Grade 7 K24 6. Approximate the probability of a chance event by collecting data on the chance process that produces it and observing its long-run relative frequency, and predict the approximate relative frequency given the probability. For example, when rolling a number cube 600 times, predict that a 3 or 6 would be rolled roughly 200 times, but probably not exactly 200 times. 7. Develop a probability model and use it to find probabilities of events. Compare probabilities from a model to observed frequencies; if the agreement is not good, explain possible sources of the discrepancy. a. Develop a uniform probability model by assigning equal probability to all outcomes, and use the model to determine probabilities of events. For example, if a student is selected at random from a class, find the probability that Jane will be selected and the probability that a girl will be selected. b. Develop a probability model (which may not be uniform) by observing frequencies in data generated from a chance process. For example, find the approximate probability that a spinning penny will land heads up or that a tossed paper cup will land open-end down. Do the outcomes for the spinning penny appear to be equally likely based on the observed frequencies?

25 Statistics and Probability for Grades 7 8 Planetary Science Course Heredity and Adaptation Course Electromagnetic Force Course Inv 2, Part 3, Step 4, Breed the larkeys Inv 5, Part 2, Step 16, Draw conclusions about Earth impacts FOSS and Common Core Math Grade 8 Inv 2, Part 3, Step 7, Breed the F1 generation Inv 2, Part 4, Step 6, Practice using Punnett squares Inv 2, Part 4, Step 10, Tally the F3 generation traits Inv 2, Part 4, Step 16, Use a reading comprehension strategy K25

26 FOSS and Common Core Math Grade 8 STATISTICS AND PROBABILITY FOR GRADES 7 8 (continued) Standard Waves Course Gravity and Kinetic Energy Course Investigate patterns of association in bivariate data. Grade 8 K26 3. Use 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. Inv 1, Part 2, Step 10, Discuss slope

27 Statistics and Probability for Grades 7 8 Planetary Science Course Heredity and Adaptation Course Electromagnetic Force Course FOSS and Common Core Math Grade 8 K27

28 FOSS and Common Core Math Grade 8 K28