Foundation of NmbrsOperations- Test 1 Review

Nominal counting

Knowing the names of the numbers.

Ordinal counting

Knowing the number words in order and being able to recite them.

Cardinality

Understanding that the last number word used when counting a set tells how many objects are in the set.

Rational counter

A child who follows counting principles to count objects meaningfully, rather than only reciting number words.

Counting on

Starting with a known amount and counting forward to add more.

Counting on from the largest

Starting with the larger addend and counting forward by the smaller addend.

Counting back

Counting backward to subtract.

Counting off

Counting the starting number as the first number removed. For 9 − 3, count off 9, 8, 7

Counting down

Starting one below the initial number when counting the numbers removed. For 9 − 3, count down 8, 7, 6

Perceptual counter

A child who needs to see the objects being counted.

Figurative counter

A child who can count objects that are hidden by imagining them.

Motor item counter

A child who uses physical actions, such as finger movements, to represent items while counting.

Verbal unit item counter

A child who uses spoken number words as countable units and can count their counts.

Abstract unit item counter

A child who can treat numbers as abstract units without relying on objects or physical actions.

Initial Number Sequence (INS)

An early numerical level at which a child can unitize, treating a collection as one countable amount.

Unitize

Treating a group of items as a single unit.

INS+

A level beyond Initial Number Sequence that includes using ideas such as the commutative property.

Strategic Additive Reasoning (SAR)

Using number relationships and strategies, such as part-part-whole, making ten, and near doubles, to add or subtract.

Part-part-whole reasoning

Thinking of a number as a whole made from two or more parts.

Making ten

Decomposing a number so one part completes a group of ten, such as 8 + 5 = (8 + 2) + 3.

Near doubles

Using a known doubles fact to solve a nearby fact, such as 4 + 5 = 4 + 4 + 1.

Algebratize a strategy

Write a series of equations that shows why a strategy works and identify the properties used.

Number sense

A gradual and flexible understanding of numbers and their relationships.

More and less relationships

Understanding how quantities compare, including which has more, which has less, and how much more or less.

Strategic Multiplicative Reasoning (SMR)

Using number relationships and properties, especially the distributive property, to reason about multiplication and division.

Counting learning trajectory

A sequence describing how children’s counting understanding and strategies develop over time.

Join problem

An addition or subtraction story in which an amount is added to an initial amount and the result is the whole.

Join: result unknown

The initial amount and amount added are known. Example: Sandra has 8 pennies and gets 4 more. How many does she have now?

Join: initial unknown

The amount added and final amount are known. Example: Sandra gets 4 pennies and then has 12. How many did she start with?

Join: change unknown

The initial and final amounts are known. Example: Sandra starts with 8 pennies and ends with 12. How many did she get?

Separate problem

An addition or subtraction story in which an amount is taken away. The initial amount is the whole and is the largest amount.

Separate: result unknown

The initial amount and amount removed are known. Example: Sandra has 12 pennies and gives away 4. How many remain?

Separate: change unknown

The initial and remaining amounts are known. Example: Sandra starts with 12 pennies and has 8 left. How many did she give away?

Separate: initial unknown

The amount removed and remaining amount are known. Example: Sandra gives away 4 pennies and has 8 left. How many did she start with?

Part-part-whole problem

Two parts make one whole, with no action or change taking place.

Part-part-whole: whole unknown

Both parts are known. Example: George has 4 pennies and 8 nickels. How many coins does he have?

Part-part-whole: part unknown

The whole and one part are known. Example: George has 12 coins, including 8 pennies. How many are nickels?

Compare problem

A problem that compares two quantities.

Compare: difference unknown

Both quantities are known. Example: George has 12 pennies and Sandra has 8. How many more does George have?

Compare: larger unknown

The smaller quantity and difference are known. Example: George has 4 more pennies than Sandra, who has 8. How many does George have?

Compare: smaller unknown

The larger quantity and difference are known. Example: George has 12 pennies, which is 4 more than Sandra. How many does Sandra have?

Equal groups problem

A multiplication or division problem involving groups of the same size.

Equal groups: product unknown

The number of groups and size of each group are known. Example: 4 bags with 6 apples each contain 4 × 6 = 24 apples.

Equal groups: group size unknown

The total and number of groups are known. This is partitive division. Example: Share 24 apples among 4 friends

Equal groups: number of groups unknown

The total and group size are known. This is measurement division. Example: Put 24 apples into bags of 6

Multiplicative comparison

A problem comparing two quantities by stating that one is a certain number of times the other.

Multiplicative comparison: product unknown

The smaller quantity and multiplier are known. Example: Jill picks 6 apples and Mark picks 4 times as many

Multiplicative comparison: set size unknown

The larger quantity and multiplier are known. Example: Mark picks 24 apples, which is 4 times Jill’s amount

Multiplicative comparison: multiplier unknown

Both quantities are known. Example: Mark picks 24 apples and Jill picks 6

Combination problem

Counting the possible pairings between two or more sets.

Product of measures

A multiplication situation in which two measurements produce a new measurement, such as length × width = area.

Set model of multiplication

Representing multiplication with equal groups of objects.

Array model of multiplication

Representing multiplication with objects arranged in equal rows and columns.

Area model of multiplication

Representing multiplication with the area of a rectangle.

Volume model of multiplication

Representing multiplication with layers or groups of cubes in a three-dimensional figure.

Length model of multiplication

Representing multiplication with lengths, bars, or jumps on a number line.

Combinations model of multiplication

Representing multiplication by counting all possible pairings between sets.

Partitive division

Division in which the number of groups is known but the number in each group is unknown.

Measurement division

Division in which the size of each group is known but the number of groups is unknown.

Division by zero

Undefined. For example, 2 ÷ 0 has no quotient because no number multiplied by 0 equals 2.

Zero divided by a nonzero number

Equals 0. For example, 0 ÷ 2 = 0 because 0 × 2 = 0.

Zero divided by zero

Undefined because every number multiplied by 0 equals 0, so there is no single quotient.

Product

The result of multiplication.

Factor

A number multiplied by another number to produce a product.

Dividend

The quantity being divided, usually the total.

Divisor

The number by which the dividend is divided.

Quotient

The result of division.

Basic fact fluency

Knowing and using efficient strategies for basic addition, subtraction, multiplication, and division facts.

One-more-than or two-more-than fact

Using a known amount and adding one or two more.

Doubles fact

An addition fact with two equal addends, such as 4 + 4 = 8.

Addition near-doubles strategy

Using a double to solve a nearby fact: 4 + 5 = (4 + 4) + 1 = 9.

Make-ten strategy

Breaking apart an addend to complete ten: 8 + 5 = 8 + (2 + 3) = (8 + 2) + 3 = 13.

Near-squares strategy

Using a square fact to multiply: 7 × 6 = (6 + 1) × 6 = 36 + 6 = 42.

Multiplication near-doubles strategy

Splitting a factor into equal parts: 6 × 4 = 6 × (2 + 2) = 12 + 12 = 24.

Using fives facts

Using a known fact with 5: 7 × 6 = 7 × (5 + 1) = 35 + 7 = 42.

Using tens for nines

Multiplying by 10 and subtracting one group: 8 × 9 = 8 × (10 − 1) = 80 − 8 = 72.

Decompose

Break a number into parts that are easier to use in a calculation.

Substitution

Replace a number or expression with an equal value.

Equal sign

Shows that the expressions on both sides have the same value, including at every step of a strategy.

Justifying 8 + 6 by making ten

8 + 6 = 8 + (2 + 4) = (8 + 2) + 4 = 10 + 4 = 14. Decompose 6, then use the associative property.

Additive identity property

Adding 0 leaves a number unchanged: a + 0 = 0 + a = a.

Multiplicative identity property

Multiplying by 1 leaves a number unchanged: a × 1 = 1 × a = a.

Commutative property of addition

Changing the order of addends does not change the sum: a + b = b + a.

Commutative property of multiplication

Changing the order of factors does not change the product: a × b = b × a.

Associative property of addition

Changing the grouping of addends does not change the sum: (a + b) + c = a + (b + c).

Associative property of multiplication

Changing the grouping of factors does not change the product: (a × b) × c = a × (b × c).

Distributive property

Multiplying a sum by a number gives the same result as multiplying each addend and adding: a(b + c) = ab + ac.

Additive inverse

A number and its opposite add to 0: a + (−a) = 0.

Multiplicative inverse

A nonzero number and its reciprocal multiply to 1: a × (1/a) = 1.

Inverse relationship between addition and subtraction

Subtraction undoes addition: if a + b = c, then c − b = a.

Inverse relationship between multiplication and division

Division undoes multiplication when the divisor is nonzero: if a × b = c, then c ÷ b = a.

Which problem type begins with the total or largest amount?

Separate.

Which computational equation can be used to solve 3 + 7 = 10 for the first addend?

10 − 7 = __.

Which multiplication model counts pairings between two sets?

Combinations.

Which property explains a(b + c) = ab + ac?

The distributive property.