TABLE OF CONTENTS
Concepts & Skills to Maintain................................................................7
Selected Terms and Symbols .............................................7
In this unit students will:
• Represent repeated multiplication with exponents
• Evaluate expressions containing exponents to solve mathematical and real world problems
• Translate verbal phrases and situations into algebraic expressions
• Identify the parts of a given expression
• Use the properties to identify equivalent expressions
• Use the properties and mathematical models to generate equivalent expressions
Students working with expressions and equations containing variables allows for them to form generalizations. Students should think of variables as quantities that vary instead of as letters that represent set values. When students can work with expressions involving variables without the focus on a specific number or numbers that the variable may represent they can focus on the patterns that occur. It is these patterns that lead to generalizations that lay the foundation for their future work in algebra
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 especially with respect to fluency.
Apply and extend previous understandings of arithmetic to algebraic expressions.
MCC6.EE.1 Write and evaluate expressions involving whole-number exponents.
MCC6.EE.2 Write, read, and evaluate expressions in which letters stand for numbers.
MCC6.EE.2a Write expressions that record operations with numbers and with letters standing for numbers. For example, express the calculation “Subtract y from 5” as 5-y.
MCC6.EE.2b Identify parts of an expression using mathematical terms (sum, term, product, factor, quotient, coefficient); view one or more parts of an expression as a single entity. For example, describe the expression 2(8 + 7) as a product of two factors; view (8 + 7) as both a single entity and a sum of two terms.
MCC6.EE.2c Evaluate expressions at specific values of their variables. Include expressions that arise from formulas used in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (Order of Operations). For example, use the formulas V = s 3 and A = 6s2 to find the volume and surface area of a cube with sides of length s = 1 .
MCC6.EE.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.
MCC6.EE.4 Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them.) For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.
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 mathematics. 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 multi‐digit numbers and perform all operations with multi‐digit decimals. Students informally begin to make connections between covariance, rates, and representations showing the relationships between quantities.
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. For example, create a story context for (2/3) ÷ (3/4) and use a visual fraction model to show the quotient; use the relationship between multiplication and division to explain that (2/3) ÷ (3/4) = 8/9 because ¾ of 8/9 is 2/3. (In general, (a/b) ÷ (c/d) = ad/bc.) How much chocolate will each person get if 3 people share ½ lb. of chocolate equally? How many ¾ - cup servings are in 2/3 of a cup of yogurt? How wide is a rectangular strip of land with length ¾ mi and area ½ square miles?
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. For example, express 36 + 8 as 4(9 + 2).
• Variables can be used as unique unknown values or as quantities that vary.
• Exponential notation is a way to express repeated products of the same number.
• Algebraic expressions may be used to represent and generalize mathematical problems and real life situations
• Properties of numbers can be used to simplify and evaluate expressions.
• Algebraic properties can be used to create equivalent expressions
• Two equivalent expressions form an equation.
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 – Different sources use different definitions. Please preview any website for alignment to the definitions given in the frameworks.
This web site has activities to help students more fully understand and retain new vocabulary (i.e. the definition page for dice actually generates rolls of the dice and gives students an opportunity to add them).
Definitions and activities for these and other terms can be found on the Intermath website.
• Algebraic expression: A mathematical phrase involving at least one variable and sometimes numbers and operation symbols.
• Associative Property of Addition: The sum of a set of numbers is the same no matter how the numbers are grouped.
• Associative Property of Multiplication: The product of a set of numbers is the same no matter how the numbers are grouped.
• Coefficient: A number multiplied by a variable in an algebraic expression.
• Commutative Property of Addition: The sum of a group of numbers is the same regardless of the order in which the numbers are arranged
• Commutative Property of Multiplication: The product of a group of numbers is the same regardless of the order in which the numbers are arranged.
• Constant: A quantity that does not change its value.
• Distributive Property: The sum of two addends multiplied by a number is the sum of the product of each addend and the number.
• Exponent: The number of times a number or expression (called base) is used as a factor of repeated multiplication. Also called the power.
• Like Terms: Terms in an algebraic expression that have the same variable raised to the same power. Only the coefficients of like terms are different.
• Order of Operations: The rules to be followed when simplifying expressions.
• 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.