One Step Equations and Inequalities
TABLE OF CONTENTS
Key Standards ......................................................................................................................4
Concepts & Skills to Maintain.............................................................................................8
Selected Terms and Symbols
In this unit students will:
• Determine if an equation or inequality is appropriate for a given situation
• Represent and solve mathematical and real world problems with equations and inequalities
• Interpret the solutions to equations and inequalities
• Represent the solutions to inequalities on a number line
• Analyze the relationship between dependent and independent variables through the use of tables, equations and graphs
Beginning experiences in solving equations will require students to understand the meaning of the equation as well as the question being asked. The use of illustrations, drawings, and balance models to represent and solve equations and inequalities will help students to develop this understanding. Solving equations and inequalities will also require students to develop effective strategies such as fact families, and inverse operations. As effective strategies are developed students will revisit rate and proportional reasoning problems and solve them using strategies developed in solving similar one-step equations.
Students will represent, model and solve equations and inequalities that are based on mathematical and real world problems. Presented with these situations, students must determine if a single value is required as a solution or if the situation allows for multiple solutions will be included. This creates the need for both equations (single solution for the situation) and inequalities (multiple solutions for the situation). When working with inequalities, students will work with situations in which the solution is not limited to the set of positive whole numbers but includes positive rational numbers. As an extension to this concept, certain situations may require a solution between two numbers. Therefore, the exploration with students as to what this would look like both on a number line and symbolically will be explored.
The process of translating between mathematical phrases and symbolic notation is essential in the writing of equations/inequalities for a situation. This is a two-way process and students will be able to write a mathematical phrase for an equation.
The goal is to help students connect the pieces. This is done by having students use multiple representations for mathematical relationships. Students will translate freely among the story, words (mathematical phrases), models, tables, graphs and equations/inequalities. Given any one of these representations students should be able to develop the others.
Although the units in this instructional framework emphasize key standards and big ideas at specific times of the year, routine topics such as estimation, mental computation, and basic computation facts should be addressed on an ongoing basis. Ideas related to the eight practice standards should be addressed constantly as well. To assure that this unit is taught with the appropriate emphasis, depth, and rigor, it is important that the tasks listed under "Evidence of
Learning" be reviewed early in the planning process. A variety of resources should be utilized to supplement this unit. This unit provides much needed content information, but excellent learning activities as well. The tasks in this unit illustrate the types of learning activities that should be utilized from a variety of sources.
STANDARDS ADDRESSED IN THIS UNIT
Mathematical standards are interwoven and should be addressed throughout the year in as many different units and activities as possible in order to emphasize the natural connections that exist among mathematical topics.
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
MCC6.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.
MCC.6.EE.7 Solve real-world and mathematical problems by writing and solving equations of the form and for cases in which p, q and x are all nonnegative rational numbers. qpx=+qpx=
MCC.6.EE.8 Write an inequality of the form or to represent a constraint or condition in a real-world or mathematical problem. Recognize that inequalities of the form or have infinitely many solutions; represent solutions of such inequalities on number line diagrams. cx< cx>cx< cx<
Represent and analyze quantitative relationships between dependent and
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.
Understand ratio concepts and use ratio reasoning to solve problems.
MCC.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.
MCC.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.
MCC.6.RP.3b Solve unit rate problems including those involving unit pricing and constant speed.
MCC.6.RP.3c 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.
MCC.6.RP.3d Use ratio reasoning to convert measurement units; manipulate and transform units appropriately when multiplying or dividing quantities.
Standards for Mathematical Practice:
The Standards for Mathematical Practice describe varieties of expertise that mathematics educators at all levels should seek to develop in their students. These practices rest on important "processes and proficiencies" with longstanding importance in mathematics education. The first of these are the NCTM process standards of problem solving, reasoning and proof, communication, representation, and connections. The second are the strands of mathematical proficiency specified in the National Research Council’s report Adding It Up: adaptive reasoning , strategic competence, conceptual understanding (comprehension of mathematical concepts, operations and relations), procedural fluency (skill in carrying out procedures flexibly, accurately, efficiently and appropriately) and productive disposition (habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy).
1. Make sense of problems and persevere in solving them. In grade 6, students solve problems involving ratios and rates and discuss how they solved them. Students solve real world problems through the application of algebraic and geometric concepts. Students seek the meaning of a problem and look for efficient ways to represent and solve it. They may check their thinking by asking themselves, "What is the most efficient way to solve the problem?", "Does this make sense?", and "Can I solve the problem in a different way?"
2. Reason abstractly and quantitatively.
In grade 6, students represent a wide variety of
real world contexts through the use of real numbers and variables in mathematical
expressions, equations, and inequalities. Students contextualize to understand the meaning of the number or variable as related to the problem and decontextualize to manipulate symbolic representations by applying properties of operations.
3. Construct viable arguments and critique the reasoning of others. In grade 6, students construct arguments using verbal or written explanations accompanied by expressions, equations, inequalities, models, and graphs, tables, and other data displays (i.e. box plots, dot plots, histograms, etc.). They further refine their mathematical communication skills through mathematical discussions in which they critically evaluate their own thinking and the thinking of other students. They pose questions like "How did you get that?", "Why is that true?" "Does that always work?" They explain their thinking to others and respond to others’ thinking.
4. Model with mathematic contextually s. In grade 6, students model problem situations symbolically, graphically, tabularly, and contextually. Students form expressions, equations, or inequalities from real world contexts and connect symbolic and graphical representations. Students begin to explore covariance and represent two quantities simultaneously. Students use number lines to compare numbers and represent inequalities. They use measures of center and variability and data displays (i.e. box plots and histograms) to draw inferences about and make comparisons between data sets. Students need many opportunities to connect and explain the connections between the different representations. They should be able to use all of these representations as appropriate to a problem context.
5. Use appropriate tools strategically. Students consider available tools (including estimation and technology) when solving a mathematical problem and decide when certain tools might be helpful. For instance, students in grade 6 may decide to represent similar data sets using dot plots with the same scale to visually compare the center and variability of the data. Additionally, students might use physical objects or applets to construct nets and calculate the surface area of three dimensional figures.
6. Attend to precision. In grade 6, students continue to refine their mathematical communication skills by using clear and precise language in their discussions with others and in their own reasoning. Students use appropriate terminology when referring to rates, ratios, geometric figures, data displays, and components of expressions, equations or inequalities.
7. Look for and make use of structure. Students routinely seek patterns or structures to model and solve problems. For instance, students recognize patterns that exist in ratio tables recognizing both the additive and multiplicative properties. Students apply properties to generate equivalent expressions
(i.e. 6 + 2x = 3 (2 + x) by distributive property) and solve equations (i.e. 2c + 3 = 15, 2c
= 12 by subtraction property of equality), c=6 by division property of equality). Students
compose and decompose two and three dimensional figures to solve real world problems involving area and volume.
8. Look for and express regularity in repeated reasoning. In grade 6, students use repeated reasoning to understand algorithms and make generalizations about patterns.
During multiple opportunities to solve and model problems, they may notice that a/b ÷ c/d = ad/bc and construct other examples and models that confirm their generalization. Students connect place value and their prior work with operations to understand algorithms to fluently divide multidigit numbers and perform all operations with multidigit decimals. Students informally begin to make connections between covariance, rates, and representations showing the relationships between quantities.
Apply and extend previous understandings of multiplication and division to divide
fractions by fractions.
MCC6.NS.1 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.
Compute fluently with multi-digit numbers and find common factors and multiples.
MCC6.NS.2 Fluently divide multi-digit numbers using the standard algorithm.
MCC6.NS.3 Fluently add, subtract, multiply, and divide multi-digit decimals using the
standard algorithm for each operation.
MCC6.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.
•Represent, analyze, and generalize a variety of patterns with tables, graphs, words, and, when possible, symbolic rules;
•Relate and compare different forms of representation for a relationship;
•Use values from specified sets to make an equation or inequality true.
•Develop an initial conceptual understanding of different uses of variables;
•Graphs can be used to represent all of the possible solutions to a given situation.
Many problems encountered in everyday life can be solved using proportions, equations or inequalities.
CONCEPTS/SKILLS TO MAINTAIN
It is expected that students will have prior knowledge/experience related to the concepts and skills identified below. It may be necessary to pre-assess in order to determine if time needs to be spent on conceptual activities that help students develop a deeper understanding of these ideas.
• Using parentheses, brackets, or braces in numerical expressions and evaluate expressions with these symbols.
• Write and interpret numerical expressions.
• Generating two numerical patterns using two given rules.
• Interpret a fraction as division
• Operations with whole numbers, fractions, and decimals
SELECTED TERMS AND SYMBOLS
The following terms and symbols are often misunderstood. These concepts are not an inclusive list and should not be taught in isolation. However, due to evidence of frequent difficulty and misunderstanding associated with these concepts, instructors should pay particular attention to them and how their students are able to explain and apply them.
The definitions below are for teacher reference only and are not to be memorized by the students. Students should explore these concepts using models and real life examples. Students should understand the concepts involved and be able to recognize and/or demonstrate them with words, models, pictures, or numbers.
The websites below are interactive and include a math glossary suitable for middle school children. Note – At the middle school level, different sources use different definitions. Please preview any website for alignment to the definitions given in the frameworks.
[ http://www.amathsdictionaryforkids.com/ ]http://www.amathsdictionaryforkids.com/
This web site has activities to help students more fully understand and retain new vocabulary
[ http://intermath.coe.uga.edu/dictnary/homepg.asp ]http://intermath.coe.uga.edu/dictnary/homepg.asp
Definitions and activities for these and other terms can be found on the Intermath website. Intermath is geared towards middle and high school students.
Addition Property of Equality: Adding the same number to each side of an equation produces an equivalent expression.
• Constant of proportionality: The constant value of the ratio of two proportional quantities x and y; usually written y = kx, where k is the constant of proportionality. In a proportional relationship, y = kx, k is the constant of proportionality, which is the value of the ratio between y and x.
• Direct Proportion (Direct Variation): The relation between two quantities whose ratio remains constant. When one variable increases the other increases proportionally: When one variable doubles the other doubles, when one variable triples the other triples, and so on. When A changes by some factor, then B changes by the same factor: A=kB, where k is the constant of proportionality.
• Division Property of Equality: States that when both sides of an equation are divided by the same number, the remaining expressions are still equal
• Equation: A mathematical sentence that contains an equal sign
• Inequality: A mathematical sentence that contains the symbols >, <, , or .
• Inverse Operation: A mathematical process that combines two or more numbers such that its product or sum equals the identity.
• Multiplication Property of Equality: States that when both sides of an equation are multiplied by the same number, the remaining expressions are still equal.
• Proportion: An equation which states that two ratios are equal.
• Subtraction Property of Equality: States that when both sides of an equation have the same number subtracted from them, the remaining expressions are still equal.
• Term: A number, a variable, or a product of numbers and variables.
• Variable: A letter or symbol used to represent a number or quantities that vary