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Objectives – MSA now • MA.600.60.10 Read, write, and represent integers (-100 to 100) * • MA.700.60.30 Add, subtract, multiply and divide integers using one operation and integers (-100 to 100) * • MA.800.60.15 Add, subtract, multiply and divide integers using one operation (-1000 to 1000) *
Objectives – CCSS • 6.NS.5 Understand that positive and negative numbers are used together to describe quantities having opposite directions or values (e.g., temperature above/below zero, elevation above/below sea level, credits/debits, positive/negative electric charge); use positive and negative numbers to represent quantities in real-world contexts, explaining the meaning of 0 in each situation.
Objectives – CCSS • 7.NS.1c Understand subtraction of rational numbers as adding the additive inverse, p – q = p + (–q). Show that the distance between two rational numbers on the number line is the absolute value of their difference, and apply this principle in real-world contexts. • 7.NS.1d Apply properties of operations as strategies to add and subtract rational numbers.
Objectives – CCSS • 7. NS.2. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. • a. Understand that multiplication is extended from fractions to rational numbers by requiring that operations continue to satisfy the properties of operations, particularly the distributive property, leading to products such as (–1)(–1) = 1 and the rules for multiplying signed numbers. Interpret products of rational numbers by describing real-world contexts. • b. Understand that integers can be divided, provided that the divisor is not zero, and every quotient of integers (with non-zero divisor) is a rational number. If p and q are integers, then –(p/q) = (–p)/q = p/(–q). Interpret quotients of rational numbers by describing real world contexts. • c. Apply properties of operations as strategies to multiply and divide rational numbers. • 7.NS.3. Solve real-world and mathematical problems involving the four operations with rational numbers.
Algebra Tiles: BASICS • Algebra tiles can be used to model operations involving integers. • Let the small yellow square represent +1 and the small red square (the flip-side) represent -1. • The yellow and red squares are additive inverses of each other.
Algebra Tiles: Modeling integers • Using your Algebra tile mat, model each of the following integers: • A gain of 4 yards • The temperature went down 3 degrees. • A loss of 2 pounds • The stock went up 6 points
Zero Pairs • Called zero pairs because they are additive inverses of each other. • When put together, they cancel each other out to model zero.
Addition of Integers • Addition can be viewed as “combining”. • Combining involves the forming and removing of all zero pairs. • For each of the given examples, use algebra tiles to model the addition. • Draw pictorial diagrams which show the modeling.
Addition of Integers (+3) + (+1) = (-2) + (-1) =
Addition of Integers (+3) + (-1) = (+4) + (-4) =
Addition of Integers (+2) + (-3) = (+4) + (-2) = • After students have seen many examples of addition, have them formulate rules.
Game of one • Each player has 11 tiles • A player tosses the 11 tiles on the mat, records the result as an addition expression, and finds the sum on the chart. • A game continues for 10 rounds. • At the completion of the game, each player finds the sum of the 10 rounds. • The player whose sum is closest to one wins.
Subtraction of Integers • Subtraction can be interpreted as “take-away.” • Subtraction can also be thought of as “adding the opposite.” • For each of the given examples, use algebra tiles to model the subtraction. • Draw pictorial diagrams which show the modeling process.
Subtraction of Integers (+5) – (+2) = (-4) – (-3) =
Subtracting Integers (+3) – (-5) (-4) – (+1)
Subtracting Integers (+3) – (-3) • After students have seen many examples, have them formulate rules for integer subtraction.
Least is Best • Each player starts with 3 yellow tiles on a work mat. • A player spins the spinner and then subtracts the number from +3 and performs the operation using tiles if necessary. • The player records the result as a subtraction expression, and writes the difference on the chart. • The difference for Round 1 is the starting number for Round 2. This continues for 10 rounds. • At the completion of the 10 rounds, each player finds the sum of the 10 rounds. • The player with the lowest sum wins.
Multiplication of Integers • Integer multiplication builds on whole number multiplication. • Use concept that the multiplier serves as the “counter” of sets needed. • For the given examples, use the algebra tiles to model the multiplication. Identify the multiplier or counter. • Draw pictorial diagrams which model the multiplication process.
Multiplication of Integers • The counter indicates how many rows to make. It has this meaning if it is positive. (+2)(+3) = (+3)(-4) =
Multiplication of Integers • If the counter is negative it will mean “take the opposite of.” (flip-over) (-2)(+3) (-3)(-1) • After students have seen many examples, have them formulate rules.
Division of Integers • Like multiplication, division relies on the concept of a counter. • Divisor serves as counter since it indicates the number of rows to create. • For the given examples, use algebra tiles to model the division. Identify the divisor or counter. Draw pictorial diagrams which model the process.
Division of Integers (+6)/(+2) = (-8)/(+2) =
Division of Integers • A negative divisor will mean “take the opposite of.” (flip-over) (+10)/(-2) =
Division of Integers (-12)/(-3) = • After students have seen many examples, have them formulate rules.
Algebra Tiles and Integers Questions?????