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Contents - Lessons & Practice

Math

Algebra 1

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Algebra 1

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Questions from the 144 chapters of Algebra 1. Each item asks for a term, a definition, or a claim the chapter itself states.

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Algebra 1

144 lessons

One lesson for each chapter of Algebra 1, in the book’s order. The lesson teaches what that chapter says you need to know, then checks it.

  1. 1Students Start HereWe're genuinely excited for you to join our community and embark on a transformative learning journey!
  2. 2Preparing for SuccessUse the information in this section to support and elevate your academic journey.
  3. 3Unit 1 Overview and ReadinessA solution of an equation is a value of a variable that makes a true statement when substituted into the equation.
  4. 41.1 Exploring Expressions and EquationsThe big thing you learned is that an equation is a statement that an expression has the same value as another expression.
  5. 51.2 Writing Equations to Model Relationships, Part 1You will be asked to evaluate the percentage of a number mentally.
  6. 61.3 Writing Equations to Model Relationships, Part 2The two quantities, x and y , are related
  7. 71.4 Equations and Their SolutionsFor a value to be a solution to an equation, it must make the equation true
  8. 81.5 Equations and Their GraphsA is the only graph that intersects the vertical axis at 0 (or that starts at the origin)
  9. 91.6 Equivalent EquationsTell whether two expressions are equivalent and explain why or why not
  10. 101.7 Explaining Steps for Rewriting EquationsIllustrate that dividing by a variable is not used in solving equations because it can lead to equations that have fewer solutions than the original equation
  11. 111.8 Choosing the Correct Variable to Solve For, Part 1You also wrote an equation in the most useful form that would be needed to find specific information
  12. 121.9 Choosing the Correct Variable to Solve For, Part 2You then rearranged the equation to find the number of cars and number of trucks that could be in the shipment
  13. 131.10 Connecting Equations to Graphs, Part 1You then worked on solving the equations for y and recognized where the slope and y -intercept are located in the new equations
  14. 141.11 Connecting Equations to Graphs, Part 2Take an equation of the form a x + b y = c and rearrange it into the equivalent form y = m x + b
  15. 151.12 Writing the Equation of a LineYou also worked to find the slope and y -intercept from a graph to then write the equation of the line in slope-intercept form
  16. 161.13 Lines from Tables and GraphsFor questions 1 - 3, write the equation of
  17. 171.14 Writing Equations of Parallel and Perpendicular LinesUsing the slope and a point, you wrote equations in point-slope form and slope-intercept form
  18. 181.15 Direct VariationSet up and solve direct variation problems
  19. 19Project 1: Slopes and InterceptsIn this culminating project, you will integrate ideas from the unit.
  20. 20Unit 2 Overview and ReadinessWe connect the points with a straight line to get the graph of the equation.
  21. 212.1 Writing and Graphing Systems of EquationsRefer to two equations as a system of equations
  22. 222.2 Writing Systems of EquationsThis summer, the Farmer’s Almanac is calling for very high temperatures in central Texas.
  23. 232.3 Solving Systems by SubstitutionYou determined whether a given graph could represent a specific system of equations using your knowledge of horizontal and vertical lines and the slopes and intercepts of those lines
  24. 242.4 Solving Systems by Elimination, Part 1Demonstrate that adding or subtracting equations in a system creates a new equation, where one of the solutions to this equation is the solution to the system
  25. 252.5 Solving Systems by Elimination, Part 2By determining that the equation was still true, you discovered that adding equivalent values to each side of an equation maintains its truth
  26. 262.6 Solving Systems by Elimination, Part 3Then you created new equations by multiplying any of the original equations by the same number.
  27. 272.7 Systems of Linear Equations and Their SolutionsWhen solving by graphing, the graphs of the equations are the same line.
  28. 282.8 Representing Situations with InequalitiesYou also explored the value that is at the boundary of an inequality and consider whether it is or isn't a solution to an inequality
  29. 292.9 Solutions to InequalitiesYou know that an open circle is used to represent < and > , while a closed circle represents ≤ and ≥
  30. 302.10 Writing and Solving Inequalities in One VariableHe plans to buy some prepared dishes from a supermarket
  31. 312.11 Graphing Linear Inequalities in Two VariablesThe boundary between the two regions is the graph of an equation that is related to the inequality
  32. 322.12 Using Linear Inequalities as ConstraintsThis prepared you to write linear inequalities that represent constraints in situations and graphing the solution regions
  33. 332.13 Solving Problems with Inequalities in Two VariablesConnect the different representations and interpret them in terms of the situation, when given inequalities, graphs, and descriptions that represent the constraints in a situation
  34. 342.14 Solutions to Systems of Linear Inequalities in Two VariablesYou recalled that a solution to a linear equation in two variables is any pair of numbers that makes the equation true, and that a solution to a system of two equations in two variables is a pair of numbers that makes…
  35. 352.15 Solving Problems with Systems of Linear Inequalities in Two VariablesMaking comparisons prompted you to think about the solutions to the equations, inequalities, or systems that were represented.
  36. 36Project 2: Modeling with Systems of Inequalities in Two VariablesStep 1 - Examine the following nutrition information for some trail mix ingredients
  37. 37Unit 3 Overview and ReadinessA scatter plot is a graph of numerical data with two variables.
  38. 383.1 Linear ModelsDraw a linear model that fits the data well and use the linear model to estimate values
  39. 393.2 Fitting LinesThe solid line is the better fit for the data since it goes through the middle of the data and follows the negative trend of the data with very few points a far distance from the line of best fit
  40. 403.3 ResidualsA residual is the difference between the actual y -value for a point and the expected y -value for the point on the linear model with the same associated x -value
  41. 413.4 The Correlation CoefficientA doesn’t belong because it is the only scatter plot with data that fit a line well and are increasing
  42. 423.5 Using the Correlation CoefficientYou thought about whether there would be a strong correlation or not as well as whether the relationship was a positive or negative correlation
  43. 433.6 Causal RelationshipsYou began to recognize that some variables may be related, but one does not always cause the other to change.
  44. 44Project 3: Two-Variable StatisticsCollect data, create a linear model to fit the data, determine if the linear model is a good fit, and use the information from my linear model to answer questions
  45. 45Inquiry Project: Defining FunctionsAccess the interactive simulator to begin this project
  46. 46Unit 4 Overview and ReadinessThe slope of a line is the ratio of the change in y and the change in x .
  47. 474.1 Describing and Graphing SituationsA customer at a bagel shop is buying 13 bagels.
  48. 484.2 Function NotationEach graph represents the distance of a dog from a post as a function of time since the dog owner left to purchase something from a store.
  49. 494.3 Interpreting Using Function NotationThe output value at f(4) is greater than the output value at f(0) because at 4 seconds the drone is 4 meters off the ground while at 0 seconds, the drone is on the ground (0 meters)
  50. 504.4 Using Function Notation to Describe Rules, Part 1You also compared output values when the same input was entered into different equations
  51. 514.5 Using Function Notation to Describe Rules, Part 2Be able to identify if a graph represents a function through the vertical line test
  52. 524.6 Features of GraphsYou completed mathematical statements and found the value of variables by reading the information from a graph
  53. 534.7 Finding SlopeRecall that the slope of a line is the rate of change of the linear function.
  54. 544.8 Using Graphs to Find Average Rate of ChangeEstimate or calculate the average rate of change between two points from the graph of a function
  55. 554.9 Interpreting and Creating GraphsVocabulary words from prior lessons, such as “maximum,” “minimum,” “increasing,“ and “decreasing,” were used to review and to describe the graphs
  56. 564.10 Comparing GraphsMake sense of an equation of the form f ( x ) = g ( x ) in terms of a situation and a graph, and know how to find the solutions
  57. 574.11 Graphing a Function Using TransformationsThe constant term is different in each equation
  58. 584.12 Domain and Range, Part 1You noticed the reasonableness of an input depends on the situation’s context
  59. 594.13 Domain and Range, Part 2You then determined the domain and range when you were given more information about the graphs
  60. 604.14 SequencesYou then thought of another rule for the sequence and gave a different list of the next 3 terms
  61. 614.15 Introducing Geometric SequencesThe first two sequences are increasing, and the second two are decreasing
  62. 624.16 Introducing Arithmetic SequencesConsider the function f given by f ( n ) = 3 n - 7 .
  63. 634.17 Representing SequencesAn arithmetic sequence is a sequence in which each term is the previous term plus a constant
  64. 644.18 The nᵗʰ Term of a SequenceIt is possible to use them interchangeably
  65. 65Project 4: Using Functions to Model Battery PowerFor questions 1 - 2 , use the following prompt and image
  66. 66Unit 5 Overview and ReadinessYou can also do the reverse—identify the slope and y -intercept and use them to find the equation of the line
  67. 675.1 Properties of ExponentsTo simplify, the exponential terms are added and the same base is kept.
  68. 685.2 Rational ExponentsNotice that the root and exponents are the same and simplify to leave the integer only
  69. 695.3 Patterns of GrowthBegin making connections between linear and exponential growth and the real world (especially financially)
  70. 705.4 Representing Exponential GrowthYou looked at patterns in exponents to determine the value of zero as an exponent and applied the basic rules of exponents with zero
  71. 715.5 Representing Exponential DecayKnow the meanings of “exponential growth” and “exponential decay.”
  72. 725.6 Negative Exponents and Scientific NotationYou used the Product Property of Exponents to simplify the expressions
  73. 735.7 Analyzing GraphsThe numbers were written in both decimal and fraction form, and you may have noticed different patterns in each format
  74. 745.8 Exponential Situations as FunctionsInterpreting the graph in terms of the context helped prepare you for your work with functions in the rest of the unit
  75. 755.9 Interpreting Exponential FunctionsLin and Diego are discussing two expressions: x 2 and 2 x
  76. 765.10 Looking at Rates of ChangeIn particular, the focus for this activity was on how exponential functions have different rates of change for different input intervals, which is not the case for linear functions
  77. 775.11 Modeling Exponential BehaviorYou became more comfortable with the fact that the data are not exactly exponential and with choosing different ways for deciding on an appropriate exponential decay factor
  78. 785.12 Reasoning about Exponential Graphs, Part 1She continues spending a third of what is left each week thereafter
  79. 795.13 Reasoning about Exponential Graphs, Part 2F is the only one that doubles when n increases by 1
  80. 805.14 Which One Changes Faster?You also had to justify your response and decide what information would make it easier to determine if the graph was linear or exponential
  81. 815.15 Changes Over Equal IntervalsFor each given expression, write an equivalent expression with as few terms as possible
  82. 82Project 5: Introduction to Exponential FunctionsUse the following prompt and table to answer questions 1 - 7
  83. 83Unit 6 Overview and ReadinessThe exponent 4 means there are 4 factors of y
  84. 84Unit 6 Inquiry Project: Area Model MultiplicationAccess the interactive simulator to begin this project
  85. 856.1 Add and Subtract PolynomialsMonomials, binomials, and trinomials are all different types of polynomials.
  86. 866.2 Multiplying PolynomialsMultiplying a polynomial by a monomial is done by applying the Distributive Property of Multiplication over Addition
  87. 876.3 Dividing PolynomialsDivide polynomials using synthetic division
  88. 886.4 Greatest Common Factor and Factor by GroupingThis is an important skill used in factoring polynomials later in the lesson
  89. 896.5 Factor TrinomialsYou also learned some tricks to help determine the factors of a trinomial, such as when the middle term is negative and the last term is positive, the signs in the binomial factors must both be negative.
  90. 906.6 Factor Special ProductsIdentify which of the polynomials is a perfect square trinomial
  91. 916.7 General Strategy for Factoring PolynomialsYou created sample polynomials that would fit into each strategy
  92. 92Project 6: Polynomials and RectanglesComplete the table with the length, width, and area of each rectangle.
  93. 93Unit 7 Overview and ReadinessThe first term in the result is the product of the first terms in each binomial
  94. 947.1 Patterns of ChangeThe x -values are 1 , 2 , 3 , 4 , and 5 in all three tables
  95. 957.2 Introduction to Quadratic RelationshipsTell whether a pattern is growing linearly, exponentially, or quadratically
  96. 967.3 Determining if a Function is QuadraticKnow that, in a pattern of shapes, the step number is the input and the number of squares is the output
  97. 977.4 Comparing Quadratic and Exponential FunctionsIn each pattern, the number of small squares is a function of the step number, n
  98. 987.5 Building Quadratic Functions to Describe Situations, Part 1The big thing you learned is that the discovered patterns help determine the function rule
  99. 997.6 Building Quadratic Functions to Describe Situations, Part 2The big lesson was that adding a quadratic term to a linear function has an effect of “bending” the graph, as the output values are no longer changing at a constant rate
  100. 1007.7 Domain, Range, Vertex, and Zeros of Quadratic FunctionsChoose a domain that makes sense in a revenue situation
  101. 1017.8 Equivalent Quadratic ExpressionsRewrite quadratic expressions in different forms by using an area diagram or the distributive property
  102. 1027.9 Standard Form and Factored FormThe sum of the partial products are x 2 + - x + 1 or x 2 - 2 x + 1
  103. 1037.10 Graphs of Functions in Standard and Factored FormsYou then made observations about how the graphs and forms are related in helping identify intercepts
  104. 1047.11 Graphing from the Factored FormYou used this function to complete a table and determine the x -intercepts and vertex of the graph of the function
  105. 1057.12 Graphing the Standard Form, Part 1Proactive people don’t accept the world as it is or wait for direction from others to initiate change.
  106. 1067.13 Graphing the Standard Form, Part 2In the right column, one of the factors is just a variable whose coefficient is 1 or -1, and the other factor is either a sum or a difference
  107. 1077.14 Graphs That Represent SituationsThe height, in inches, of a frog's jump is modeled by the equation h ( t ) = 60 t - 75 t 2 , where the time t , after the frog jumps, is measured in seconds
  108. 1087.15 Vertex FormThe expressions are in standard form, factored form, and one other form
  109. 1097.16 Graphing from the Vertex FormYou graphed the vertex and two other points to complete the graph
  110. 1107.17 Changing the VertexThe value of g is always 4 greater than the value of f , no matter which value of x we choose
  111. 111Project 7: Design a FountainIn this culminating project, you will integrate ideas from the unit by designing a fountain with given criteria.
  112. 112Unit 8 Overview and ReadinessWrite Equivalent Expressions: Mini-Lesson Review
  113. 1138.1 Finding Unknown InputsA mechanical device is used to launch a potato vertically into the air.
  114. 1148.2 When and Why Do We Write Quadratic Equations?A strategy such as this might be difficult to use when the solutions are harder to find
  115. 1158.3 Solving Quadratic Equations by ReasoningWhat 8.3 Solving Quadratic Equations by Reasoning asks you to know.
  116. 1168.4 Solving Quadratic Equations with the Zero Product PropertyDetermine which of the statements are true if g · h = 0 .
  117. 1178.5 How Many Solutions?This includes not taking the square root of a negative number and accurately determining how many solutions an equation might have
  118. 1188.6 Rewriting Quadratic Expressions in Factored Form, Part 1The top shape is a square measuring 8 in.
  119. 1198.7 Rewriting Quadratic Expressions in Factored Form, Part 2This built a foundation for understanding how to find the factored forms of different quadratic expressions
  120. 1208.8 Rewriting Quadratic Expressions in Factored Form, Part 3Rewrite a quadratic expression of the form x 2 - c into its factored form
  121. 1218.9 Solving Quadratic Equations by Using Factored FormRearrange a quadratic equation to be written as expression in factored form = 0 and find the solutions
  122. 1228.10 Rewriting Quadratic Expressions in Factored Form, Part 4Being able to recognize detailed aspects of these expressions helps provide a better understanding that is useful when solving by factoring
  123. 1238.11 Writing Quadratic Equations Given Real SolutionsWrite the quadratic function f ( x ) = x 2 - 9 x + 20 in factored form
  124. 1248.12 Using Technology to Find the Quadratic RegressionMake predictions about a real-world situation using a quadratic data set
  125. 125Project 8: Modeling Rocket FlightThe goal of each round is to write the quadratic expression as it has been given to your partner.
  126. 126Unit 9 Overview and ReadinessIf a n and b n are real numbers, and n ≥ 2 is an integer, then
  127. 1279.1 What Are Perfect Squares?One method involved taking the square root of both sides of the equation.
  128. 1289.2 Completing the Square, Part 1Explain how you know the expressions you selected are perfect squares
  129. 1299.3 Completing the Square, Part 2Completing the square can be used to solve quadratic equations
  130. 1309.4 Completing the Square, Part 3Complete the square for quadratic expressions of the form a x 2 + b x + c when a is not 1 and explain the process
  131. 1319.5 Quadratic Equations with Irrational SolutionsYou graphed the equations to find the approximate solutions using a graphing calculator
  132. 1329.6 The Quadratic FormulaEvaluate the expressions and find the two numbers
  133. 1339.7 Applying the Quadratic FormulaThe correct answers are 2 and 8: - ( - 5 ) + 9 = 5 + 3 = 8 and - ( - 5 ) - 9 = 5 - 3 = 2
  134. 1349.8 Deriving the Quadratic FormulaOne way to solve the quadratic equation x 2 + 5 x + 3 = 0 is by completing the square .
  135. 1359.9 Writing Quadratics in Different FormsA ball is thrown upward from the top of a 40 foot high building at a speed of 80 feet per second.
  136. 1369.10 Rewriting Quadratic Expressions in Vertex FormWhen given a quadratic expression in factored form, rewrite it in standard form
  137. 1379.11 Using Quadratic Expressions in Vertex Form to Solve ProblemsF ( 1 ) can be expressed in words as "the value of f when x is 1." Find or compute
  138. 138Project 9: Using Quadratic Equations to Model Situations and Solve ProblemsA Linear Function and a Quadratic Function
  139. 139Productive Struggle“Failure” gets an undeserved bad reputation in education.
  140. 140Cooperative Learning and Productive DiscourseMeaningful learning that can transfer to new contexts is more likely to occur in classrooms with rich classroom discourse.
  141. 141Motivation in the ClassroomThere is a scourge of what can be called “false growth mindsets” among educators.
  142. 142MetacognitionSpaced retrieval practice is the process of distributing many, short sessions of learning over a longer period of time, as opposed to cramming fewer, longer sessions of learning over a shorter period of time.
  143. 143Multimedia Learning PrincipleThat said, it turns out the notion of “learning styles” is not actually supported by research, and in fact a large body of work suggests that everybody learns better through a combination of words and pictures.
  144. 144Course DesignStudents are asked to consider what happens when a pattern is generated from a specific value that is repeatedly added or subtracted (linear).