Arithmetic Check List

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Math Check List for Secondary I

Note the emphasis on the thinking part of mathematics, the intellectual or physical component of the operations.

Whole Numbers, Primes and Composites
(Number Theory)

Do you know how to use the equal sign properly? Can you recognize abuse?

Is zero a whole number? Is it a natural number? Which is larger, the set of whole numbers or the set of natural numbers?

Do you know column methods for adding, subtracting, and multiplying whole numbers with 5 or fewer digits in their decimal representation? Can you do these operation in a repeatable and reproducible way? What is the effect of an error in one step.

If you and another get different results for the same arithmetic problem, what can you say for sure?

If you and another get identical results for the same arithmetic problem, what can you say for sure?

Decimal Polynomials (coined term?) Do you know how to express a whole number as a polynomial in powers of ten with single digit coefficients from the set 0 to 9?

Do you how to do long division for a whole number n divided by a whole number d to get a quotient q and remainder r ? Do you know how to check your answer by comparing qd+r with n.

Do you know the difference between a prime or composite numbers?

How do you use the decimal representation of whole numbers to recognize multiples of 2, 3, 5 and 9.

What is the prime decomposition (also called factorization) of a whole number?

How to use tree diagrams to obtain the prime factorization of whole numbers?

Empirical Question: How is the prime factorization of a whole number unique? If you and another independent find a prime factorization of a whole number, how will your results agree?

How is the tree diagram for prime factorization and its construction is shorthand for a sequence of many equalities?

For the factorization of a whole number, Do you know how to use the list of primes whose square is less than the whole number to obtain the prime factorization efficiently with the aid of exact arithmetic with a calculator?

What is the least common multiple of a pair of (small) whole numbers? Do you know how to find the least common multiple of a pair of whole numbers by listing the multiples of each? Can you do that efficiently?

How do you use the prime factorization of a pair of whole numbers (or several) to find their least common multiple?

How do you generate (find, produce) divisors of a whole number from its prime factorization.

What does it mean for two whole numbers to be relatively prime?

If a pair of whole numbers have a common divisor (another whole number > 1) is there a prime common divisor? Explain using logic?

How do you find the greatest common divisor (g.c.d) of a pair of whole numbers (or several) from their prime factorizations.

For the pair of whole numbers M and N (pick two or argue in general), explain why the product of these numbers equals the product of their l.c.m and g.c.d That is lcm*gcd = M*N

Explain how addition and subtraction of whole numbers can be illustrated via addition of line segments or lengths.

Addition Commutes: When you are adding lengths together, does the order matter?

Explain how multiplication of whole numbers can be connected with the area of rectangles.

When you are calculating the area of a rectangle whose sides are whole number multiples of a unit length, does it matter which one is called the width and which one is called the height?

Multiplication Commutes: When you are multiplying two whole numbers together, does the order matter? Think of areas? Take a few examples.

Without doing any arithmetic, draw rectangles to explain why
the two calculations 8( 4+ 7) and 8(4) + 8(7) should give the same result. Here is a hint of the distributive property

(More general) Without doing any arithmetic, draw rectangles to explain why the two calculations A( B+ C) and AB + AC should give the same result whenever A,B and C are whole numbers. Here is the distributive property for multiplication over addition.

What are first powers of ten. Write them in decimal and exponent notation.

Fractions (before the use of signs)

Do you know how to divide an object into parts with the same shape (congruent) or with the same value (think of money).

What is a unit fraction 1/n of an object?

What is simple fraction m/n of an object or set of like objects?

What is the difference between a proper and improper fraction?

What is a mixed number?

How do measurements with a ruler provide examples of proper and improper fractions, or of mixed numbers

How can long division be used to express an improper fraction as a whole number plus a proper fraction.

What is a unit fraction of a unit fraction? Illustrate with line segments and/or a ruler. Illustrate using a unit square. as well.

What is a simple fraction of a unit fraction? Illustrate with line segments and/or a ruler. Illustrate using unit squares as well.

What is a simple fraction of a simple fraction? IIllustrate using unit squares.

What are equivalent fractions? Illustrate with line segments and/or a ruler. Illustrate using unit squares

What is the difference between a proper and improper fraction?

Do you know how to write a number given by a decimal as a mixed number or fraction?

What does it mean to raise or lower terms in a fraction?

What is the simplest form of a fraction. How is that related to the denominator and numerator being relatively pirme?

How can prime factor decompositions be use to lower terms?

How can gcds be used to lower terms.

In simplifying an improper fraction, will expressing the latter as a mixed number lead to smaller numbers to consider? How is this operation related to long division? Can you see a calculator shortcut for it? Optional: (i) How is the foregoing operation related to the Euclid Algorithm for finding a gcd of two numbers? How is the foregoing operation or Euclid's Algorithm related to continued fractions?

Multiply the tops, Multiply the bottoms is one rule for computing the product of fractions. Explain or show how and why canceling common factors can be more efficient when the product computation requires the product to be expressed in simplest terms.

Addition of fractions can be done without and with lowest common denominators. Try it both ways. Show or explain why using a least common denominator usually leads to less work in the addition and subtraction of fractions.

We can ask how many times a whole number goes into another. The answer can be a whole number with a remainder. The answer can also be a proper or improper fraction. Explain or show how with a few examples.

We can also ask how many times one length goes into another. Give a few examples with a ruler. The answer may be a proper or improper fraction, even a whole number.

Since we can ask how many times one length goes into another, we can ask how many times one fraction goes into another? The answer may be a proper or improper fraction, even a whole number. Give examples to explain or illustrate how the answer can be obtained?

What does it mean to divide one fraction by another? Why does division by a fraction give the same result as multiplying by the reciprocal of the fraction?

What does it mean to divide a length by fraction? Is the result a length or a number?

What does it mean to divide a length by a length? Is the result a length or a number?

Decimals or Decimal Fractions (before the use of signs)

How can decimals that include a decimal point (Decimal fractions) be written as fractions where the denominator is a power of ten. What are the factors of the denominator?

Do you know how to write a fraction whose denominator is a product of 2s, 5s and/or 10s exactly as a decimal? Give a few examples in which there are more fives than twos, fewer fives than twos, and the same number of fives and twos.

Do you know how to do long division with decimals to approximate the quotient to two or several places after the decimal point? Do you know how to check your answer?

Do you know how to shift the decimal points in long division of decimal numbers so that the divisor becomes a whole number? Do you know how to justify the shift using the concept of equivalent fractions and rules for dividing fractions.

Do you know column methods for adding, subtracting, and multiplying decimals with 5 or fewer digits in their decimal representation before and after the decimal point? How can the method for multiplication be justified?

To be continued.