Number Sense

Number Sense- Whole Numbers and Decimals




(taken from “Big Ideas by Dr. Small”):


1. The place value system we use is built on patterns to make our work with numbers more efficient.
2. Students gain a sense of the size of numbers by comparing them to meaningful benchmark numbers.
3. Decimals are an alternative representation to fractions, but one that allows for modeling, comparisons, 
and calculations that are consistent with whole numbers; because decimals extend the pattern of the 
base ten place value system.
4. A decimal can be read and interpreted in different ways; sometimes one representation is more useful 
than another in interpreting or comparing decimals or for performing and explaining a computation.

GOAL: I can demonstrate and explain equivalent representations using powers of ten
GOAL: I can identify and explain patterns within our place value system (including decimals)
GOAL: I can use these patterns to represent whole and decimals numbers in standard form, expanded form, in pictures, and in words
GOAL: I can compare and order whole and decimals numbers and plot them on a number line
GOAL: I can round whole and decimals numbers to meaningful benchmarks

  •  represent, compare, and order whole numbers and decimal numbers from 0.001 to 1 000 000, using number lines with appropriate increments, base ten materials for decimals);
  •  demonstrate an understanding of place value in whole numbers and decimal numbers from 0.001 to 1 000 000, using base ten materials to represent the relationship between 1, 0.1, 0.01, and 0.001) (Sample problem: How many thousands cubes would be needed to make a base ten block for 1 000 000?);
  • read and print in words whole numbers to one hundred thousand, using meaningful contexts
  • solve problems that arise from real-life situations and that relate to the magnitude of whole numbers up to 1 000 000 (Sample problem: How would you determine if a person could live to be 1 000 000 hours old?);
  • identify composite numbers and prime numbers, and explain the relationship between them (i.e., any composite number can be factored into prime factors) (e.g., 42 = 2 x 3 x 7).

ADDING AND SUBTRACTING

BIG IDEAS:

(taken from “Big Ideas by Dr. Small”):
  1. A personal “invented” algorithm is often more meaningful and sometimes equally efficient as a conventional algorithm.
  2. Decimals are an alternative representation to fractions, but one that allows for calculations that are consistent with whole numbers.

STUDENT LEARNING GOALS:

GOAL: I can use strategies to estimate sums and differences of decimal numbers.
GOAL: I can add whole and decimal numbers.
GOAL: I can subtract whole and decimal numbers.

CURRICULUM EXPECTATIONS:

  • add and subtract decimal numbers to thousandths, using concrete materials, estimation, algorithms, and calculators
  • use estimation when solving problems involving the addition and subtraction of whole numbers and decimals, to help judge the reasonableness of a solution

MULTIPLYING AND DIVIDING

BIG IDEAS:

(taken from “Big Ideas by Dr. Small”):
  1. A personal “invented” algorithm is often more meaningful and sometimes equally efficient as a conventional algorithm.
  2. Thinking of numbers as factors or multiples of other numbers provides alternative representations of those numbers.
  3. Just as multiplication and division are intrinsically related, so are factors and multiples.

STUDENT LEARNING GOALS:

GOAL: I can use strategies to estimate products and quotients.
GOAL: I can multiply two-digit numbers.
GOAL: I can divide three-digit numbers by one-digit numbers.

CURRICULUM EXPECTATIONS:

  • multiply and divide decimal numbers to tenths by whole numbers, using concrete materials, estimation, algorithms, and calculators (e.g., calculate 4 x 1.4 using base ten materials; calculate 5.6 ÷ 4 using base ten materials)
  •  multiply whole numbers by 0.1, 0.01, and 0.001 using mental strategies (e.g., use a calculator to look for patterns and generalize to develop a rule); 
  • multiply and divide decimal numbers by 10, 100, 1000, and 10 000 using mental strategies (e.g.,“To convert 0.6 m2 to square centimetres, I calculated in my head 0.6 x 10 000 and got 6000 cm2.”)
  • explain the need for a standard order for performing operations, by investigating the impact that changing the order has when performing a series of operations

FRACTIONS, RATIOS AND RATES

BIG IDEAS:

(taken from “Big Ideas by Dr. Small”):
  1. Fractions can represent parts of regions, parts of sets, parts of measures, division, or ratios. These meanings are equivalent.
  2. A fraction is not meaningful without knowing what the whole is.
  3. Renaming fractions is often the key to comparing them or computing with them. Every fraction can be renamed in an infinite number of ways.
  4. Ratio and rates, just like fractions and decimals, are comparisons of quantities.
    • A ratio compares quantities with the same unit
    • A rate compares quantities with different units

STUDENT LEARNING GOALS:

GOAL: I can represent fractions and their equivalents.
GOAL: I can relate fractions to decimals.
GOAL: I can order fractions and mixed numbers with like denominators.
GOAL: I can identify and solve problems using ratios and rates.

CURRICULUM EXPECTATIONS:

  • represent, compare, and order fractional amounts with unlike denominators, including proper and improper fractions and mixed numbers, using a variety of tools (e.g., fraction circles, Cuisenaire rods, drawings, number lines, calculators) and using standard fractional notation (Sample problem: Use fraction strips to show that 1 is greater than .);
  • estimate quantities using benchmarks of 10%, 25%, 50%, 75%, and 100% (e.g., the container is about 75% full; approximately 50% of our students walk to school)
  • represent ratios found in real-life contexts, using concrete materials, drawings, and standard fractional notation (Sample problem: In a classroom of 28 students, 12 are female.What is the ratio of male students to female students?);
  • determine and explain, through investigation using concrete materials, drawings, and calculators, the relationships among fractions (i.e., with denominators of 2, 4, 5, 10, 20, 25, 50, and 100), decimal numbers, and percents
  • represent relationships using unit rates (Sample problem: If 5 batteries cost $4.75, what is the cost of 1 battery?).

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