UNIT 1: INTEGERS WEEK 4: Student Packet

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Name Period Date UNIT 1: INTEGERS WEEK 4: Student Packet 4.1 Integers: Multiplication and Division 1 Explore integer multiplication using a marker model.

1 Name Period Date UNIT 1: INTEGERS WEEK 4: Student Packet 4.1 Integers: Multiplication and Division 1 Explore integer multiplication using a marker model. Develop sign rules for integer multiplication and division. 4.2 Fourfold Way: Numbers, Pictures, Words, and Symbols Record observations from geometric pictures as numbers in T-tables and graphs on coordinate planes. Use inductive reasoning to find a rule from empirical data. Find inputs and outputs using a rule. 4.3 Integers: Multiplication and Division 2 Multiply and divide integers. Review math concepts from prior lessons. Demonstrate competency in whole number multiplication and properties of multiplication (highlighted review) Week 4 SP

2 FOCUS ON VOCABULARY 4 Select the word from the word bank that best completes the sentence. 1. Pictures, numbers,, and words are four different approaches to a math problem. 2. The result of multiplying numbers is called the of the numbers. 3. The numbers 2, -3, 0, -5, 7 are examples of. 4. In 28 = 4, the dividend is 28; the divisor is 7; and the is The refers to approaching a math problem in different ways. 6. Here is a sequence of numbers: 2, 4, 6, 8, 10. A rule for this sequence is get each term by adding 2 to the preceding term. 7. Below is a table of values of a function. The rule for this function is y = 3x + 1. Input x Output x + 1 Word Bank quotient fourfold way product explicit recursive symbols integers Week 4 SP0

3 4.1 Integers: Multiplication and Division 1 INTEGERS: MULTIPLICATION AND DIVISION 1 Ready (Summary) We will use patterns and a model for multiplying and dividing integers. Set (Goals) Explore integer multiplication using a marker model. Develop sign rules for integer multiplication and division. Go (Warmup) Complete the pattern started below by filling numbers in the blank spaces How would you describe the pattern formed by the numbers? Week 4 SP1

4 4.1 Integers: Multiplication and Division 1 INTEGER MULTIPLICATION Compute each product. Record your work below using positive (+) and negative (-) symbols. 1. (3) (5) = (2) (-4) = 3. (2) (3) = (place 3 groups of 5) (place 2 groups of -4) 4. (4) (-1) = 5. (3) (-6) = 6. (5) (-2) = 7. The product of a positive number and a positive number is a number. 8. The product of a positive number and a negative number is a number. Week 4 SP2

5 4.1 Integers: Multiplication and Division 1 INTEGER MULTIPLICATION (continued) Compute each product. Record your work below using positive (+) and negative (-) symbols. 9. (-3) (5) = (-2) (-4) = 11. (-3) (-3) = (take away 3 groups of 5) (take away 2 groups of -4) 12. (-2) (5) = 13. (-4) (3) = 14. (-1) (-6) = 15. The product of a negative number and a positive number is a number. 16. The product of a negative number and a negative number is a number. Do not write below this line. Week 4 SP3

6 4.1 Integers: Multiplication and Division 1 USING A MULTIPLICATION TABLE 1. Refer to the warmup for this lesson. Cut out the two strips at the bottom of the page. Then glue them appropriately to cover the zeroes in the warmup table. Notice that the grid now serves as a multiplication table. Use it to verify these products: 4(3) = located in region 3(-1) = located in region -2(5) = located in region -4(-2) = located in region -4(-3) = located in region 5(-2) = located in region Patterns for Multiplying Integers: 2. The product of a positive number and a positive number is. That is, ( + ) ( + ) = ( ) 3. The product of a positive number and a negative number is. That is, ( + ) ( - ) = ( ) 4. The product of a negative number and a positive number is. That is, ( - ) ( + ) = ( ) 5. The product of a negative number and a negative number is. That is, ( - ) ( - ) = ( ) Cut out the two strips below on the thick lines Week 4 SP4

7 4.1 Integers: Multiplication and Division 1 RELATING DIVISION AND MULTIPLICATION Complete the patterns. Multiplication Fact A. ( 5 ) ( 6 ) = 30 B. ( 4 ) ( ) = -12 C. ( ) ( 2 ) = -16 D. ( -6 ) ( ) = 24 Corresponding Division Facts ( 30 ) (_6_) = 5 ( 30 ) ( ) = 6 ( -12 ) ( 4 ) = ( ) ( -12 ) ( ) = 4 ( -16 ) (_2_) = ( ) ( -16 ) ( ) = 2 ( 24 ) ( -6 ) = ( ) ( 24 ) ( ) = -6 Patterns for Dividing Integers: 2. The quotient of a positive number divided by a positive number is. That is, ( + ) ( + ) = ( ) 3. The quotient of a positive number divided by a negative number is a. That is, ( + ) ( - ) = ( ) 4. The quotient of a negative number divided by a positive number is. That is, ( - ) ( + ) = ( ) 5. The quotient of a negative number divided by a negative number is. That is, ( - ) ( - ) = ( ) Week 4 SP5

8 4.1 Integers: Multiplication and Division 1 HOT PIECES AND COLD NUGGETS Suppose scientists discover an amazing way to control the temperature of liquid. They ve invented hot pieces and cold nuggets that maintain their temperature. If it's a hot summer day and your beverage is too warm, place some cold nuggets in it to cool it down. Too cold now? Take some cold nuggets out. If it's a cold winter evening and your beverage is too cool, place some hot pieces in to warm it up. Too hot now? Take some hot pieces out. Think: Put in (+) Take out ( ) Hot Pieces (Hot Positive) Cold Nuggets (Cold Negative) In other words: Put in 1 hot piece + (+1) 1 degree hotter Put in 1 cold nugget + (-1) 1 degree colder Take out 1 hot piece (+1) 1 degree colder Take out 1 cold nugget (-1) 1 degree hotter 1. Suppose you put 2 groups of 3 hot pieces into a liquid. Does the liquid get hotter or colder? By how much? Write an equation to represent this. = 2. Suppose you put 2 groups of 3 cold nuggets into a liquid. Does the liquid get hotter or colder? By how much? Write an equation to represent this. = 3. Suppose you take out 2 groups of 3 hot pieces from a liquid. Does the liquid get hotter or colder? By how much? Write an equation to represent this. = 4. Suppose you take out 2 groups of 3 cold nuggets from a liquid. Does the liquid get hotter or colder? By how much? Write an equation to represent this. = Week 4 SP6

9 4.1 Integers: Multiplication and Division 1 MORE INTEGER MULTIPLICATION AND DIVISION Compute. If needed, draw positive symbols (+), negative symbols (-), and zero pairs (-3) 2. (-28) (-4) 3. (-6) (-6) 4. (-5) 9 5. (-40) (-39) (-3) 8. (-5) (-12) (-11) (-4) (-4) Week 4 SP7

10 4.2 Fourfold Way: Numbers, Pictures, Words, and Symbols FOURFOLD WAY: NUMBERS, PICTURES, WORDS, AND SYMBOLS Ready (Summary) We will extend geometric patterns. We will use tables, graphs, words, and symbols to describe the patterns. Set (Goals) Record observations from geometric pictures as numbers in T-tables and graphs on coordinate planes. Use inductive reasoning to find a rule from empirical data. Find inputs and outputs using a rule. Go (Warmup) Continue each pattern for steps 1-5. Write a rule (in words) that fits each pattern, describing what is done to the stage number to arrive at an output value. Stage # Rule (in words) Pattern = 5 = = stage # plus Pattern = 4 = 2 4 = 3 4 Pattern =1 2 = = 2 2 = 3 3 = = Week 4 SP8

11 4.2 Fourfold Way: Numbers, Pictures, Words, and Symbols NUMBERS, PICTURES, WORDS, AND SYMBOLS 1 A Geometric Pattern Table 1. Build and draw the first several steps suggested by this pattern. Step # (Input) Arithmetic # of toothpicks (Output) Step or 6 + 1(3) 9 Step or + ( ) Step 3 3 Step 4 4 Step Week 4 SP9

12 4.2 Fourfold Way: Numbers, Pictures, Words, and Symbols NUMBERS, PICTURES, WORDS, AND SYMBOLS 1 (continued) Transfer the data from the previous page into the table below. Table # of Step # toothpicks (Input) (Output) Graph 2. Recursive Rule: 3. Explicit Rule (in words): 4. Explicit Rule (in symbols): 5. How many toothpicks are in step #50? 6. In what step number are there exactly 87 toothpicks? Week 4 SP10

13 4.2 Fourfold Way: Numbers, Pictures, Words, and Symbols NUMBERS, PICTURES, WORDS, AND SYMBOLS 2 A Geometric Pattern Table 1. Build and draw the first several steps suggested by this pattern. Step # (Input) Arithmetic # of toothpicks (Output) Step or + ( ) Step 2 2 Step 3 3 Step 4 4 Step Week 4 SP11

14 4.2 Fourfold Way: Numbers, Pictures, Words, and Symbols NUMBERS, PICTURES, WORDS, AND SYMBOLS 2 (continued) Transfer the data from the previous page into the table below. Table # of Step # toothpicks (Input) (Output) Graph 2. Recursive Rule: 3. Explicit Rule (in words): 4. Explicit Rule (in symbols): 5. How many toothpicks are in step #50? 6. In what step number are there exactly 74 toothpicks? Week 4 SP12

15 4.2 Fourfold Way: Numbers, Pictures, Words, and Symbols NUMBERS, PICTURES, WORDS, AND SYMBOLS 3 A Geometric Pattern Table 1. Build and draw the first several steps suggested by this pattern. Step # (Input) Arithmetic # of toothpicks (Output) Step or + ( ) Step 2 2 Step 3 3 Step 4 4 Step Week 4 SP13

16 4.2 Fourfold Way: Numbers, Pictures, Words, and Symbols NUMBERS, PICTURES, WORDS, AND SYMBOLS 3 (continued) Transfer the data from the previous page into the table below. Table # of Step # toothpicks (Input) (Output) Graph 2. Recursive Rule: 3. Explicit Rule (in words): 4. Explicit Rule (in symbols): 5. How many toothpicks are in step #50? 6. In what step number are there exactly 64 toothpicks? Week 4 SP14

17 4.3 Integers: Multiplication and Division 2 SKILL BUILDER 1 Complete these statements for remembering multiplication and division rules for integers. 1. The product of two positive numbers is. 2. The product of two negative numbers is.. 3. The product of a negative number a positive number is. 4. The quotient of a positive number and a positive number is. 5. The quotient of a negative number and a positive number is. 6. The quotient of a negative number and a negative number is. Compute. 7. 3(5) 8. 2 (-5) 9. (-3) (-9) (-6) 13. Juan says: a negative and a negative make a positive. He applies this rule and writes the following. (-12) + (-3) = 15 Is Juan correct? Explain. Week 4 SP15

18 4.3 Integers: Multiplication and Division 2 SKILL BUILDER 2A Compute. 1. 8(-2) = (-6) (-4) Week 4 SP16

19 4.3 Integers: Multiplication and Division 2 SKILL BUILDER 2B Distributive Property: For all numbers a, b, and c, ab ( + c) = ab+ ac. Use the distributive property to compute = 8 ( + 4) 8 14 = (8 ) + (8 ) 8 14 = = = 9 ( + 4) 9 34 = (9 ) + (9 ) 9 34 = = Complete the table and find a rule for the number of toothpicks in the figure at step n. Step 1 Step 2 Step 3 Step # n # of toothpicks 3 5 Rule: Week 4 SP17

20 4.3 Integers: Multiplication and Division 2 SKILL BUILDER 3A Compute = (-1) 4. (-6)(-4) Which division sentence may be used to check that 8 4 = 32? Choose the best answer. A. 8 4 = 2 B = 3 C = 3 D = 8 8. Lucio says: 3 groups of 4 is the same as 4 groups of 3. Is he correct? Explain. Compute (-15) (-9) Week 4 SP18

21 4.3 Integers: Multiplication and Division 2 SKILL BUILDER 3B Compute (-5) 13. (-4) (-9) + (-2) Use the distributive property to compute Danesha divided 255 by 15 and got 17 as her answer. Use an area model and a multiplication computation procedure to verify that this problem is solved correctly. 17. Using an area model. 18. Using a computation procedure. Week 4 SP19

22 4.3 Integers: Multiplication and Division 2 TEST PREPARATION 4 Show your work on a separate sheet of paper and choose the best answer. 1. Compute 18 x 64. A. 1,200 B. 6,912 C. 1,152 D The product of a negative number and a positive number is: E. Always positive F. Always negative G. Zero H. Sometimes negative 3. The product of a positive number and a positive number is: A. Always positive ,506 B. Always negative C. Zero D. Sometimes negative E. 5,358 F. 53,580 G. 5,558 H. 5, = A. 4 B. 3 C. -4 D What are the coordinates of point G?. E. (2, 3) F. (3, 2) G. (6, 5) 5 G H. (5, 6) 5 Week 4 SP20

23 4.3 Integers: Multiplication and Division 2 KNOWLEDGE CHECK 4 Show your work on a separate sheet of paper and write your answers on this page. 4.1 Integers: Multiplication and Division 1 Compute. 1. (-8)(-8) (-5) 4.2 Fourfold Way: Numbers, Pictures, Words, and Symbols 3. The perimeter of a figure is the distance around it. Find the perimeter of the figure in each step. Write a rule in words for the perimeter of the figure at any step. Step 1 Step 2 Step 3 Step # Perimeter of figure 3 4 Rule in words 4.3 Integers: Multiplication and Division 2 Compute (-7) Highlighted Review: Whole Numbers: Multiplication, Properties of Multiplication 6. Multiply (3 7) = (5 3) 7 is an example of the property. Week 4 SP21

24 4.3 Integers: Multiplication and Division 2 Home-School Connection 4 Here are some questions from this week s lessons to review with your young mathematician. Compute. 1. What is the product of 3 and -5? (-6) 3. Complete the table and find a rule that fits the pattern. Input Output x Parent (or Guardian) signature Selected California Mathematics Content Standards NS AF MG MR Solve addition, subtraction, multiplication, and division problems, including those arising in concrete situations, that use positive and negative integers and combinations of these operations. Students investigate geometric patterns and describe them algebraically. Draw the points corresponding to linear relationships on graph paper (e.g., draw 10 points on the graph of the equation y = 3x and connect them by using a straight line). Use a variety of methods, such as words, numbers, symbols, charts, graphs, tables, diagrams, and models, to explain mathematical reasoning. Week 4 SP22

UNIT 8: LINEAR FUNCTIONS WEEK 31: Student Packet

Name Period Date UNIT 8: LINEAR FUNCTIONS WEEK 31: Student Packet 31.1 Introduction to Systems of Equations Use variables to write equations and systems of equations. Solve problems involving rate, distance,