N7.1.d Explain the result of dividing a quantity of zero by a non-zero quantity.
N7.1.e Explain (by generalizing patterns, analogies, and mathematical reasoning) why division of non-zero quantities by zero is not defined.
N7.2 Expand and demonstrate understanding of the addition, subtraction, multiplication, and division of decimals to greater numbers of decimal places, and the order of operations.
N7.2.a Provide a justification for the placement of a decimal in a sum or difference of decimals up to thousandths (e.g., for 4.5 + 0.73 + 256.458, think 4 + 256 so the sum is greater than 260; thus, the decimal will be placed so that the sum is in the hundreds).
N7.2.b Provide a justification for the placement of a decimal in a product (e.g., for $12.33 × 2.4, think $12 × 2, so the product is greater than $24; thus, the decimal in the final product would be placed so that the answer is in the tens).
N7.2.c Provide a justification for the placement of a decimal in a quotient (e.g., for 51.50 m ÷ 2.1, think 50 m ÷ 2 so the quotient is approximately 25 m; thus, the final answer will be in the tens). (Note: If the divisor has more than one digit, students should be allowed to use technology to determine the final answer.)
N7.3 Demonstrate an understanding of the relationships between positive decimals, positive fractions (including mixed numbers, proper fractions and improper fractions), and whole numbers.
N7.3.a Predict the decimal representation of a fraction based upon patterns and justify the reasoning (e.g., knowing the decimal equivalent of 1/8 and 2/8, predict and verify the decimal representation of 7/8).
N7.5 Develop and demonstrate an understanding of adding and subtracting positive fractions and mixed numbers, with like and unlike denominators, concretely, pictorially, and symbolically (limited to positive sums and differences).
N7.5.a Estimate the sum or difference of positive fractions and/or mixed numbers and explain the reasoning.
N7.6.b Explain, using concrete materials such as integer tiles and diagrams, that the sum of opposite integers is zero (e.g., a move in one direction followed by an equivalent move in the opposite direction results in no net change in position).
P7.3 Demonstrate an understanding of one-and two-step linear equations of the form ax/b + c = d (where a, b, c, and d are whole numbers, c less than or equal to d and b does not equal 0) by modeling the solution of the equations concretely, pictorially, physically, and symbolically and explaining the solution in terms of the preservation of equality.
P7.3.a Model the preservation of equality for each of the four operations using concrete materials or using pictorial representations, explain the process orally and record it symbolically.
P7.4 Demonstrate an understanding of linear equations of the form x + a = b (where a and b are integers) by modeling problems as a linear equation and solving the problems concretely, pictorially, and symbolically.
P7.4.a Represent a problem with a linear equation of the form x + a = b where a and b are integers and solve the equation using concrete models (e.g., counters, integer tiles) and record the process symbolically.