In this unit, students develop an understanding of integers and rational numbers. They begin to transition from informal and intuitive awareness to more formal models. They also develop and apply algorithms for operations on positive and negative numbers, and explore the properties that guide these algorithms.
PA.CCSS.Math.Content.CC.2.1.7.E.1 Apply and extend previous understandings of operations with fractions to operations with rational numbers. (7.NS.1, 7.NS.2, 7.NS.3)
PA.CCSS.Math.Conetnt.CC.2.2.7.B.3 Model and solve real-world and mathematical problems by using and connecting numerical, algebraic, and/or graphical representations. (7.EE.3)
#1 Make sense of problems and persevere in solving them.
Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends.
#3 Construct viable arguments and critique the reasoning of others.
Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.
#4 Model with mathematics.
Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.
#5 Use appropriate tools strategically.
Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, or a calculator. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.
#6 Attend to precision.
Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context.
#7 Look for and make use of structure.
Mathematically proficient students look closely to discern a pattern or structure. Mid-level students will see 7 × 8 equals the well-remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property.
In Grade 6, instructional time focused on four critical areas: (1) connecting ratio and rate to whole number multiplication and division and using concepts of ratio and rate to solve problems; (2) completing understanding of division of fractions and extending the notion of number to the system of rational numbers, which includes negative numbers; (3) writing, interpreting, and using expressions and equations; and (4) developing understanding of statistical thinking. In this unit, the second area is emphasized and extended, with thorough analysis and application of algorithms and properties.
In the Grade 6 book, Bits and Pieces II, students develop an understanding of whole and rational numbers, how to use models to develop an understanding of mathematical concepts and understand arithmetic operations with rational numbers. These concepts are extended in Accentuate the Negative when students are asked to define and develop understanding of, explore relationships between, and develop understanding of arithmetic operations with positive and negative integers (e.g. interpreting positive numbers as a gain and negative numbers as a loss).
In Grade 6 when studying Bits and Pieces II, students learned to use a number line to develop equivalence and operations of fractions and decimals, and also to develop an understanding of the Commutative Property using whole and rational numbers. When students previously worked through the books Data About Us, Covering and Surrounding, Variables and Patterns, and Stretching and Shrinking they were introduced to using a coordinate grid with positive coordinates using the order of operations to solve problems in a context. In this book, Accentuate the Negative, students learn to extend the number line and coordinate grid to include negative coordinates. They also develop an understanding of the Commutative Property, the Distributive Property, and the order of operations (Accentuate the Negative, pg 7).
Accentuate the Negative give students experiences with positive and negative numbers, ordering, and informal operations in a variety of contexts so that subsequent formal work can be based on “what makes sense.” Positive and negative numbers in the form of integers, fractions, and decimals are also represented on a number line. Students formulate algorithms for addition and subtraction of positive and negative numbers, think about the meaning of operations from several perspectives, and use different representation models (number line and chip board).
The number line and fact families as well as the contexts of time, distance and speed are used to develop students’ understanding of multiplication and division of positive and negative numbers. Since students have had few informal exposures to multiplication and division of integers, a useful context for questions leading up to these skills (motion at different rates of speed) is explained before asking students to formulate algorithms for multiplication and division. Chip board models are also useful, but they have limitations when trying to model the product or quotient of two negative numbers.
Finally, students compare algebraic properties of the operations on positive and negative numbers to those of the number system of only positive numbers (Accentuate the Negative, pg.3).
Major misconceptions by and struggles for students in this unit include:
- Incorrectly transferring algorithms for addition and subtraction of integers to situations that require multiplication and division and vice versa
- Belief that integers (for example) such as -1 have less value than the integer -32
- Incorrect placement of fraction and decimal representations of rational numbers on a number line
- Adding or subtracting the absolute value of integers without regard to the signs and in the absence of absolute value symbols
- Finding the difference between integers without number lines and without being able to count the spaces
- Finding the halfway point between two opposite numbers without the use of a number line
- Estimating the value of points on a number line when all of the whole integer tick marks are not labeled
- Remembering that the vocabulary words “opposite” and “additive inverse” are synonymous
- Writing equations from number line/motion models
- Solving multi-step addition/subtraction situational problems involving integers
- Understanding why “zero pairs” must be added to chip models to subtract an integer by a negative number
- Decontextualizing the Commutative Property of addition and multiplication in given situations to write algebraic algorithms
- Rewriting subtraction of integer problems as addition (recognizing that the subtraction sign becomes the negative sign of the following number)
- Rewriting “subtracting by a negative” as “adding a positive”
- Explaining why subtraction and division are not commutative
- Confusing the quadrant numbers on a coordinate plane
- Confusing the x- and y-coordinates in an ordered pair
- Knowing what sign to give an answer when multiplying or dividing more than two integers
- Remembering the order of operations
- Writing two equivalent expressions from area models
- Distributing the factor outside of the parentheses to all integers inside of the parentheses during the Distributive Property