A Story of Eureka Math Grade, Module 4 Student File_B Contains Sprint and Fluency,, and Assessment Materials Published by the non-profit Great Minds. Copyright 2015 Great Minds. No part of this work may be reproduced, sold, or commercialized, in whole or in part, without written permission from Great Minds. Non-commercial use is licensed pursuant to a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 license; for more information, go to http://greatminds.net/maps/math/copyright. Great Minds and Eureka Math are registered trademarks of Great Minds. Printed in the U.S.A. This book may be purchased from the publisher at eureka-math.org 10 9 8 7 6 5 4 3 2 1
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Lesson 1 Lesson 1: Writing Equations Using Symbols Write each of the following statements using symbolic language. 1. When you square five times a number, you get three more than the number. 2. Monica had some cookies. She gave seven to her sister. Then, she divided the remainder into two halves, and she still had five cookies left. Lesson 1: Writing Equations Using Symbols 1
Lesson 2 Lesson 2: Linear and Nonlinear Expressions in Write each of the following statements as a mathematical expression. State whether the expression is a linear or nonlinear expression in. 1. Seven subtracted from five times a number, and then the difference added to nine times a number 2. Three times a number subtracted from the product of fifteen and the reciprocal of a number 3. Half of the sum of two and a number multiplied by itself three times Lesson 2: Linear and Nonlinear Expressions in 2
Lesson 3 Lesson 3: Linear Equations in 1. Is 8 a solution to + 9 = 13? Explain. 2. Write three different equations that have = 5 as a solution. 3. Is 3 a solution to the equation 3 5 = 4 + 2? Explain. Lesson 3: Linear Equations in 3
: Solving a Linear Equation 1. Guess a number for that would make the equation true. Check your solution. 5 2 = 8 2. Use the properties of equality to solve the equation 7 4 + = 12. State which property justifies your first step and why you chose it. Check your solution. 3. Use the properties of equality to solve the equation 3 + 2 = 11 + 9. State which property justifies your first step and why you chose it. Check your solution. : Solving a Linear Equation 4
Lesson 5 Lesson 5: Writing and Solving Linear Equations For each of the following problems, write an equation and solve. 1. Given a right triangle, find the measures of all the angles, in degrees, if one angle is a right angle and the measure of the second angle is six less than seven times the measure of the third angle. 2. In a triangle, the measure of the first angle is six times a number. The measure of the second angle is nine less than the first angle. The measure of the third angle is three times the number more than the measure of the first angle. Determine the measure of each angle in degrees. Lesson 5: Writing and Solving Linear Equations 5
Lesson 6 Lesson 6: Solutions of a Linear Equation Transform the equation if necessary, and then solve to find the value of that makes the equation true. 1. 5 ( + 3) = 1 (9 + 18) 5 3 2. 5(3 + 9) 2 = 15 2( 5) Lesson 6: Solutions of a Linear Equation 6
Lesson 7 Lesson 7: Classification of Solutions Give a brief explanation as to what kind of solution(s) you expect the following linear equations to have. Transform the equations into a simpler form if necessary. 1. 3(6 + 8) = 24 + 18 2. 12( + 8) = 11 5 3. 5 8 = 11 7 + 12 Lesson 7: Classification of Solutions 7
Lesson 8 Lesson 8: Linear Equations in Disguise Solve the following equations for. 1. = 2. = 3. = Lesson 8: Linear Equations in Disguise 8
Lesson 9 Lesson 9: An Application of Linear Equations 1. Rewrite the equation that would represent the sum in the fifth step of the Facebook problem: = 7 + 7 5 + 7 5 + 7 5 + 7 5. 2. The sum of four consecutive integers is 74. Write an equation, and solve to find the numbers. Lesson 9: An Application of Linear Equations 9
Lesson 10 Lesson 10: A Critical Look at Proportional Relationships Alex skateboards at a constant speed from his house to school 3.8 miles away. It takes him 18 minutes. a. What fraction represents his constant speed,? b. After school, Alex skateboards at the same constant speed to his friend s house. It takes him 10 minutes. Write the fraction that represents constant speed,, if he travels a distance of. c. Write the fractions from parts (a) and (b) as a proportion, and solve to find out how many miles Alex s friend s house is from school. Round your answer to the tenths place. Lesson 10: A Critical Look at Proportional Relationships 10
Lesson 11 Lesson 11: Constant Rate Vicky reads at a constant rate. She can read 5 pages in 9 minutes. We want to know how many pages,, Vicky can read after minutes. a. Write a linear equation in two variables that represents the number of pages Vicky reads in any given time interval. b. Complete the table below. Use a calculator, and round answers to the tenths place. (time in minutes) Linear Equation: (pages read) 0 20 40 60 c. About how long would it take Vicky to read 25 pages? Explain. Lesson 11: Constant Rate 11
Lesson 12 Lesson 12: Linear Equations in Two Variables 1. Is the point (1, 3) a solution to the linear equation 5 9 = 32? Explain. 2. Find three solutions for the linear equation 4 3 = 1, and plot the solutions as points on a coordinate plane. Linear Equation: 4 3 = 1 Lesson 12: Linear Equations in Two Variables 12
Lesson 13 Lesson 13: The Graph of a Linear Equation in Two Variables 1. Ethan found solutions to the linear equation 3 = 8 and graphed them. What shape is the graph of the linear equation taking? 2. Could the following points be on the graph of + 2 = 5? Lesson 13: The Graph of a Linear Equation in Two Variables 13
Lesson : The Graph of a Linear Equation Horizontal and Vertical Lines 1. Graph the linear equation + =, where = 0, = 1, and = 1.5. 2. Graph the linear equation + =, where = 1, = 0, and = 5 2. : The Graph of a Linear Equation Horizontal and Vertical Lines 14
Lesson 3. What linear equation represents the graph of the line that coincides with the -axis? 4. What linear equation represents the graph of the line that coincides with the -axis? : The Graph of a Linear Equation Horizontal and Vertical Lines 15
Lesson 15 Lesson 15: The Slope of a Non-Vertical Line 1. What is the slope of this non-vertical line? Use your transparency if needed. 2. What is the slope of this non-vertical line? Use your transparency if needed. Lesson 15: The Slope of a Non-Vertical Line 16
Lesson 16 Lesson 16: The Computation of the Slope of a Non-Vertical Line Find the rate of change of the line by completing parts (a) and (b). a. Select any two points on the line to label as and. their coordinates. b. Compute the rate of change of the line. Lesson 16: The Computation of the Slope of a Non-Vertical Line 17
Lesson 17 Lesson 17: The Line Joining Two Distinct Points of the Graph = + Has Slope 1. Solve the following equation for : 35 7 = 49. 2. What is the slope of the equation in Problem 1? 3. Show, using similar triangles, why the graph of an equation of the form = is a line with slope. Lesson 17: The Line Joining Two Distinct Points of the Graph = + Has Slope 18
Lesson 18 Lesson 18: There Is Only One Line Passing Through a Given Point with a Given Slope Mrs. Hodson said that the graphs of the equations below are incorrect. Find the student s errors, and correctly graph the equations. 1. Student graph of = 1 2 + 4: Error: Correct graph of the equation: Lesson 18: There Is Only One Line Passing Through a Given Point with a Given Slope 19
Lesson 18 2. Student graph of = 3 5 1: Error: Correct graph of the equation: Lesson 18: There Is Only One Line Passing Through a Given Point with a Given Slope 20
Lesson 19 Lesson 19: The Graph of a Linear Equation in Two Variables Is a Line 1. Graph the equation = 5 10 using the 4 -intercept point and slope. 2. Graph the equation 5 4 = 40 using intercepts. Lesson 19: The Graph of a Linear Equation in Two Variables Is a Line 21
Lesson 19 3. What can you conclude about the equations = 5 10 and 5 4 = 40? 4 Lesson 19: The Graph of a Linear Equation in Two Variables Is a Line 22
Lesson 20 Lesson 20: Every Line Is a Graph of a Linear Equation 1. Write an equation in slope-intercept form that represents the line shown. 2. Use the properties of equality to change the equation you wrote for Problem 1 from slope-intercept form, = +, to standard form, + =, where,, and are integers, and is not negative. Lesson 20: Every Line Is a Graph of a Linear Equation 23
Lesson 20 3. Write an equation in slope-intercept form that represents the line shown. 4. Use the properties of equality to change the equation you wrote for Problem 3 from slope-intercept form, = +, to standard form, + =, where,, and are integers, and is not negative. Lesson 20: Every Line Is a Graph of a Linear Equation 24
Lesson 21 Lesson 21: Some Facts About Graphs of Linear Equations in Two Variables 1. Write the equation for the line shown in the figure below. 2. A line goes through the point (5, 7) and has slope = 3. Write the equation that represents the line. Lesson 21: Some Facts About Graphs of Linear Equations in Two Variables 25
Lesson 22 Lesson 22: Constant Rates Revisited 1. Water flows out of Pipe A at a constant rate. Pipe A can fill 3 buckets of the same size in 14 minutes. Write a linear equation that represents the situation. 2. The figure below represents the rate at which Pipe B can fill the same-sized buckets. Which pipe fills buckets faster? Explain. Lesson 22: Constant Rates Revisited 26
Lesson 23 Lesson 23: The Defining Equation of a Line 1. Do the graphs of the equations 16 + 12 = 33 and 4 + 3 = 8 graph as the same line? Why or why not? 2. Given the equation 3 = 11, write another equation that will have the same graph. Explain why. Lesson 23: The Defining Equation of a Line 27
Darnell and Hector ride their bikes at constant speeds. Darnell leaves Hector s house to bike home. He can bike the 8 miles in 32 minutes. Five minutes after Darnell leaves, Hector realizes that Darnell left his phone. Hector rides to catch up. He can ride to Darnell s house in 24 minutes. Assuming they bike the same path, will Hector catch up to Darnell before he gets home? a. Write the linear equation that represents Darnell s constant speed. b. Write the linear equation that represents Hector s constant speed. Make sure to take into account that Hector left after Darnell. c. Write the system of linear equations that represents this situation. : Introduction to Simultaneous Equations 28
d. Sketch the graphs of the two equations. e. Will Hector catch up to Darnell before he gets home? If so, approximately when? f. At approximately what point do the graphs of the lines intersect? : Introduction to Simultaneous Equations 29
Lesson 25 Lesson 25: Geometric Interpretation of the Solutions of a Linear System 2 = 1 Sketch the graphs of the linear system on a coordinate plane:. = 5 5 a. the ordered pair where the graphs of the two linear equations intersect. b. Verify that the ordered pair named in part (a) is a solution to 2 = 1. c. Verify that the ordered pair named in part (a) is a solution to = 5 5. Lesson 25: Geometric Interpretation of the Solutions of a Linear System 30
Lesson 26 Lesson 26: Characterization of Parallel Lines Does each system of linear equations have a solution? Explain your answer. 1. = 5 3 4 + 2 = 5 4 2. = 2 5 3 4 8 = 11 1 + = 8 3. 3 + 3 = 12 Lesson 26: Characterization of Parallel Lines 31
Lesson 27 Lesson 27: Nature of Solutions of a System of Linear Equations Determine the nature of the solution to each system of linear equations. If the system has a solution, then find it without graphing. 1. = 1 2 + 5 2 2 = 7 2. = 2 3 + 4 2 + 1 2 = 2 = 3 2 3. 3 + = 2 Lesson 27: Nature of Solutions of a System of Linear Equations 32
Lesson 28 Lesson 28: Another Computational Method of Solving a Linear System Determine the solution, if it exists, for each system of linear equations. Verify your solution on the coordinate plane. = 3 5 1. = 3 + 7 = 4 + 6 2. 2 = 11 Lesson 28: Another Computational Method of Solving a Linear System 33
Lesson 29 Lesson 29: Word Problems 1. Small boxes contain DVDs, and large boxes contain one gaming machine. Three boxes of gaming machines and a box of DVDs weigh 48 pounds. Three boxes of gaming machines and five boxes of DVDs weigh 72 pounds. How much does each box weigh? 2. A language arts test is worth 100 points. There is a total of 26 questions. There are spelling word questions that are worth 2 points each and vocabulary word questions worth 5 points each. How many of each type of question are there? Lesson 29: Word Problems 34
Lesson 30 Lesson 30: Conversion Between Celsius and Fahrenheit Use the equation developed in class to answer the following questions: 1. How many degrees Fahrenheit is 11 C? 2. How many degrees Fahrenheit is 3 C? 3. Graph the equation developed in class, and use it to confirm your results from Problems 1 and 2. Lesson 30: Conversion Between Celsius and Fahrenheit 35
Lesson 31 Lesson 31: System of Equations Leading to Pythagorean Triples Use a calculator to complete Problems 1 3. 1. Is 7, 20, 21 a Pythagorean triple? Is 1,, a Pythagorean triple? Explain. 2. Identify two Pythagorean triples using the known triple 9, 40, 41. 3. Use the system + = = to find Pythagorean triples for the given values of = 2 and = 3. Recall that the solution in the form of, is the triple,,. Verify your results. Lesson 31: System of Equations Leading to Pythagorean Triples 36