Module 2 - Revised Multiplication and Division Facts

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Module 2 - Revised Multiplication and Division Facts Powered By Docstoc
					Harcourt Grade 4            Harcourt Grade 4              Harcourt Grade 4


  S D          SAN DIEGO CITY SCHOOLS
  C S




               Instructional Module to Enhance the Teaching of

                                HARCOURT



                             Math
                              California Edition




                 Grade 4
        Module 2 - Revised
        Multiplication and Division
                   Facts
                           - WORK IN PROGRESS -
Revised 9/03
Harcourt Grade 4               Harcourt Grade 4                    Harcourt Grade 4
               Module 2: Multiplication and Division Facts
                                     3 WEEKS


Key Mathematics Concepts:
      • Understand the relationship between multiplication and division-
      • Understand multiplication as “repeated addition” and division as “repeated
         subtraction-
      • Recognize division as the inverse of multiplication-
      • Understand and use “strategies” to find products and quotients mentally-
      • Understand and use the Commutative (called the Order Property in Harcourt)
        and the Associative Properties to understand and know multiplication and
        division facts-
      • Understand parentheses and use them for grouping; recognize that with more
        than 1 operation-completing them in different order will give different
        answers-
      • Understand that a problem can be solved using a variety of operations.
      • Recognize that “cube” is a model/representation of multiplying 3 factors –
        3 factors can be multiplied in any order and the products are equal-



      Unit 3: MULTIPLICATION AND DIVISION FACTS
Chapter 8: Practice Multiplication and          Chapter 9: Algebra: Use Multiplication
             Division Facts                                  and Division Facts
   Lesson 1: Relate Multiplication and Division Lesson 1: Expression with Parentheses
Lesson 2: Multiply & Divide Facts Through 5 Lesson 2: Match Words and Expressions
Lesson 3: Multiply & Divide Facts:              Lesson 3: Multiply Equals by Equals
          Through 10                            Lesson 4: Expressions with Variables
Lesson 4: Multiplication Table Through 12       Lesson 5: Equations with Variables
Lesson 5: Multiply 3 Factors                    Lesson 6: Find a Rule
Lesson 6: Problem Solving Skill                 Lesson 7: Problem Solving Strategy




Revised 9/03                                                                         2
                      Harcourt Grade 4                    Harcourt Grade 4                  Harcourt Grade 4
GRADE 4                                                     MODULE 2: Multiplication and Division Facts
Harcourt Mathematics                                                3 Weeks of Instruction
                                                                   Unit 3: Chapters 8-9

Day 1:                        Day 2:                        Day 3:                      Day 4:                      Day 5:
Chapter 8: Practice
Multiplication and Division
Facts
                              Lesson 8.2                    Lesson 8.3                  Lesson 8.4                  Lesson 8.5
Lesson 8.1                    Multiply and Divide Facts     Multiply and Divide Facts   Hands-On: Multiplication    Multiply Three Factors
Relate Multiplication and     Through 5                     Through 10                  Table Through 12
Division
Day 6:                        Day 7:                        Day 8:                      Day 9:                      Day 10:
Chapter 8                     Chapter 9: Algebra:
                              Use Multiplication and
                              Division Facts
                                                            Lesson 9.2                           Lesson 9.3         Lesson 9.4
Lesson 8.6                    Lesson 9.1                    Match Words and             Multiply Equals by Equals   Expressions with
Problem Solving Skill:        Expressions with              Expressions                                             Variables
Choose the Operation          Parentheses


Day 11:                       Day 12:                       Day 13:                     Day 14:


Lesson 9.5:                        Lesson 9.6:              Lesson 9.7:                 CA Connection
Equations with Variables                                    Problem Solving Strategy:   Unit Review
                                   Find A Rule              Work Backward               Assessment




Day 14: CA Connection and Unit Review/Assessment: This Module does not contain a lesson for Day 14.




Revised 9/03                                                                                     3
  Harcourt Grade 4           Harcourt Grade 4             Harcourt Grade 4
                  UNIT 3: MULIPLICATION AND DIVISION FACTS
                                               Module 2: Chapter 8


                             DAY 1: LESSON: 8.1 , pp. 140-141

Materials:               *Transparency 8.1; Color Tiles - 1 bag of 50 tiles per groups of 3 – 4 students;
LESSON FOCUS:            Relate Multiplication and Division Facts
CALIFORNIA               Number Sense:
                         3.0: Students solve problems involving addition, subtraction, multiplication and division of whole numbers
STANDARDS:               and understand the relationships among operations.
                          Mathematical Reasoning: 1.1, 2.0, 2.1, 3.0, 3.2
                         1.1: Analyze problems by identifying relationships, distinguishing relevant from irrelevant information,
                         sequencing and prioritizing information and observing patterns.

Purpose of Lesson:       Understand how to relate multiplication and division:
                         Fact families have 4 different equations, 2 for each operation; the 2 factors in
                         multiplication can be interchanged to form another equation-
                         Division is the inverse of multiplication; it breaks the product apart in two different
                         ways-the factors become the divisor and quotient-
LAUNCH:                  Warm up: Number of the Day – discuss students’ equations
*Transparency 8.1                 Ask if anyone used fractions? Division? Multiplication?

                         Discuss: How are multiplication and division related?
                                  Chart student responses
Materials: Color Tiles
1 bag of 50 tiles        Pg. 140B: Use “Alternative Teaching Strategy & Vocabulary Strategy,
per groups of 3 –4              Pg. 139: Discuss arrays - Problems: 15- 17
students                        Students name multiplication and division expressions:
                                                        5

                                          3             15                        Factors: 3 & 5
                                                                                  Product: 15

                                          3 X 5 = 15                       15 ÷ 3 = 5
                                          5 X 3 = 15                       15 ÷ 5 = 3

                         Introduce the notation: 3 X 6 = N
                         Discuss the meaning of “N” in this equation – generate others with
                         students including: 2 X N = 12 – What is “N” in this case?
EXPLORE:                 Question: If there are 16 students in the band, show all the different
Work with the concept.   ways they can line-up with same number of students in each line:
Focus on student’s          Students work with partners and use color tiles to find all the
“doing mathematics.”
                            possible arrangements, and sketch and label arrangements.
                            Discuss results, Record the multiplication and division “facts”
Revised 9/03                                                                                                                4
  Harcourt Grade 4                 Harcourt Grade 4                    Harcourt Grade 4
PRACTICE:              Pg. 141: Problems 2-5: Do with students: Represent the solution and
Focus on Communication the multiplication and division fact.
and Representation     Pg. 140: Use term “inverse operation”

                      Pg. 141: #s: 18 – 20: Partner work:
                      Students show representation and facts.
SUMMARIZE:            Pg. 141: Assess: “Discuss” – When given the equation 4X3 = 12,
Connect purpose to    describe how you would find the other equations in the fact family.
activities
CLOSURE:              Pg. 141: Assess: “Write” – Describe the relationship between
Assess                multiplication and division. Use the vocabulary: inverse operations
Individually/group    Chart student responses
HOMEWORK:             Pg. 141: Problems 19;
                      Write a word problem for 6 X 7 = 42 and 42 divided by 6 = 7


ROUTINE:
Today’s Number:
  Introduce Routine: See attached description of this routine.
  Students work individually/with a partner.

    Discuss students’ strategies for developing number sentences/representations for
    “Today’s Number”.
    Encourage equations that use more than 1 operation.

    Suggestions for 4th Grade -Today’s Numbers:
            10, 20, 24,25, 36, 50, 75, 100
    This routine develops’ students ability to “decompose” numbers so they can more easily
    operate with numbers if they have strategies for working with equivalent components of
    numbers.
         Ex: 24 = 10 + 10 + 4    6X4            1/2 of 48




Revised 9/03                                                                                 5
 Harcourt Grade 4                 Harcourt Grade 4                       Harcourt Grade 4
                              Routine: Today’s Number

       Today’s Number is a routine that gives students an opportunity to practice and extend
understanding of familiar and developing number concepts. Students need a journal to use
everyday that is separate from their math journal. Give them a chance to decorate the
journal with the title and their name. As students integrate new concepts into the routine,
they will add new words and symbols to the cover of the journal. You can begin Today’s
Number with number one on the first day of school and continue counting days throughout
the year, or you can use another sequence of numbers. The important thing is that students
can apply the new ideas they learn one day, to the work they do the next day.
       The daily routine should extend no longer than 20 minutes. The first ten minutes,
students are silently brainstorming equations that equal Today’s Number on their own. As
students begin their own work each day, the teacher can begin constructing the class chart
which will be used for sharing at the end of the session. As the teacher walks around the
room, observing student thinking, she/he can look for students who are pushing themselves
and experimenting with different methods of expressing Today’s Number. Mini-conferences
are encouraged as they provide a chance to teach students individually. When students come
up with new or interesting ways of representing the number, the teacher can add the idea to
the class chart during the brainstorm. This gives the teacher a chance to honor individual
thinking and also encourages other students to consider new ideas.
       After ten minutes, students share with a partner or with their table group, adding
their favorites from their partner or group to their own work. The last 6-8 minutes are
spent as a whole class, the teacher recording student’s ideas on the class chart. As the
teacher opens up the share, he/she can talk to the class about the equations that already
appear on the chart. This gives the teacher an opportunity to clarify misconceptions about
rules and procedures (parentheses, order of operations, numerical notation, etc.) The share
also provides a useful record of the collective knowledge of the class and gives the teacher
information about what new concepts students are ready to start thinking about. Today’s
Number also serves as ongoing assessment of students’ individual progress.

Suggestions for introducing Today’s Number: Each week, increase the level of thinking
required for Today’s number. Maintain a class chart, so that students can keep track of
weekly additions. This list is only a suggestion. You can change the order if necessary to
make sure new concepts correlate with instruction. The italicized words and corresponding
symbols can be added to the journal cover as they are introduced.
Week 1:      Students practice brainstorming equations that equal Today’s Number. Clarify
             the definition of the word, equation.
Week 2:      Encourage students to include at least two subtraction equations if they aren’t
             already doing so.



Revised 9/03                                                                             6
  Harcourt Grade 4          Harcourt Grade 4             Harcourt Grade 4
                  UNIT 3: MULIPLICATION AND DIVISION FACTS

                                                 Module 2: Chapter 8


                              DAY 2:                LESSON: 8.2, pg. 142-143

MATERIALS:                 *Transparency 8.2; Color Tiles: 1 bag of 50 per 3-4 students; Transparency of
                           Challenge 8.2
LESSON FOCUS:              Multiplication and Division Facts Through 5
CALIFORNIA                  Number Sense:
                           3.0: Students solve problems involving addition, subtraction, multiplication and division of whole numbers
STANDARDS:                 and understand the relationships among operations.
                            Mathematical Reasoning: 2.0, 2.3, 3.0, 3.2, 3.3

Purpose of Lesson:         Understand how to multiply and divide using facts from 0 through 5:
                           Connect skip-counting models to fact families and inverse operations.
                           Multiplication chart to find “a X b” and “c divided by d”
                           See Pg. 142A: Math Background
LAUNCH:                    *Warm-up: Number of the Day:
*Transparency 8.2           Students explain how they found all the possible equations.
                            HW: Students share problems with partner


Color Tiles: 1 bag of 50   Pg. 142 B: Use Alternative Teaching Strategy:
per groups of 3-4          Students use color tiles to model equations: 4 X 3 = N
students                                              3X4=N
                                                      N divided by 3 = _____
                                                      N divided by 4 = _____
                                      Students should show 3 groups of 4 or 4 groups of 3 for
                                      Each equation.
                           Question: What are all the different “models” you can use to show
                           multiplication and division? Record student responses.
                           Add those shown on Pg. 142 (Learn) that students did not mention.
                           Label each of the models: “groups of”; number line or skip counting;
                           arrays-
EXPLORE:                   Pg. 143: Partner work: Problems 2-7;
Work with the concept.     Discuss students’ responses.
Focus on student’s
                           Ask for explanations about how they found the “related” fact and
“doing mathematics.”
                           justify their response with a model.

                           Pg. 143: Problems 18 – 19: Solve and discuss with students.
PRACTICE:                  Pg. 143: Problems 8 – 13: Solve and write at least 1 story problem for
Focus on Communication     one of the equations, include a “model/diagram” as part of the answer.
and Representation         Students share problems and match equation to story problem.
Revised 9/03                                                                                  7
  Harcourt Grade 4                    Harcourt Grade 4                     Harcourt Grade 4
Overhead: Challenge 8.2   Challenge 8.2: Solve problems and discuss strategies.

SUMMARIZE/                Pg. 143: Assess: Discuss: Students describe their strategy (ies) for
CLOSURE:                  solving the problem.
Connect purpose to
activities
                          Student discuss/write how the different representations (On Pg. 142)
                          from Emma, Blake, Carlos, and Latoya and the multiplication chart -
Assess Individually       connect multiplication to division.

                          Ask students to identify which is the most useful/makes the most
                          “sense” to them.
HOMEWORK:                 Pg. 143: Problems 20 – 24.
                          Students provide models/representations for Problems 22 – 24.




ROUTINES:

                                Continue with Routines from DAY 1.




Revised 9/03                                                                                     8
  Harcourt Grade 4           Harcourt Grade 4             Harcourt Grade 4
                  UNIT 3: MULIPLICATION AND DIVISION FACTS
                                              Module 2: Chapter 8


                         DAY 3:                LESSON: 8.3, pg. 144 - 145

MATERIALS:              *Transparency 8.3; CM Grid paper – TR 60 1 per students or color tiles
LESSON FOCUS:           To multiply and divide facts from 6 – 10
CALIFORNIA              Number Sense:
                        3.0: Students solve problems involving addition, subtraction, multiplication and division of whole numbers
STANDARDS:              and understand the relationships among operations.
                        4.0: Students know how to factor small whole numbers.
                        Mathematical Reasoning: 1.2 and 2.3

Purpose of Lesson:      Understand/learn multiplication and division facts from 6 – 10:
                        Recognize that multiplication and division are inverse operations, every
                        multiplication fact in the table has a corresponding division fact.
                        Recognize and use the doubling strategy in multiplication as an
                        approach to multiplication as repeated addition.
                        Harcourt Math calls the Commutative Property the “Order Property” –
                        students should learn the term: Commutative Property
LAUNCH:                 *Warm-up: Problem of the Day
*Transparency 8.3       Students to describe their approach for solving this problem.

                        HW problems: Partner discussion. Share models/diagrams.
CM Grid paper – TR 60
1 per students or
color tiles             Show a 6X8 array on board/overhead.
                        (Optional: Students use color tiles rather than grid paper.)
                                                              8



                                             6


                        Question: Can you find a way to break this grid into 2 pieces with a
                        straight line? Describe the 2 new arrays. Ex: 2 X 6 and 6 X 6 –
                        If you figure/find the product for the 2 smaller arrays and add them
                        together - will this equal to 6 X 8?
                            Model the process for recording.
EXPLORE:                Have students try this and then justify their explanation.
                           Ask: If you break this array into 2 small arrays “that you know”
                           (4 X 6) or (3 X8), how does this help you find the product of 6 X 8?
                          Ask students to “explain” their response.
                        Try this with another array such as 6 X 7 or 6 X 9.
Revised 9/03                                                                                   9
 Harcourt Grade 4                Harcourt Grade 4                     Harcourt Grade 4
                    Discuss strategies with students
                    Ask students for other strategies that they use for learning
                    multiplication and division facts.
                      See Pg. 145 for examples:
                      Record students’ response and label, give example according to the
                      “type”:
                          Inverse-
                          Commutative Property-
                          Using a pattern-
                          Break apart-
PRACTICE:           Pg. 146, Problems: 7 – 8: Students sketch and label each of the
                    arrays.

                    Pg. 146 and 147: Partner work: Problems: 31 – 33
                      Discuss problems and approaches with students.
SUMMARIZE/          Pg. 147: Problems 28 – 29. Chart students’ response
CLOSURE:            Pg. 147: Assess: “Discuss”
                    Pg. 146: Discuss: Problem 27
HOMEWORK:           Pg. 147: Link-up to Health, Problems 1- 3 and 40 - 41



ROUTINE:
    T.E. PG. 138 G: Teach: Practice Game: “Cross It Out!”




Revised 9/03                                                                           10
  Harcourt Grade 4           Harcourt Grade 4             Harcourt Grade 4
                  UNIT 3: MULIPLICATION AND DIVISION FACTS
                                              Module 2: Chapter 8


                         DAY 4:                LESSON: 8.4, pp. 148 - 149

MATERIALS:              * Transparency 8.4; Blank Multiplication Chart – 1 per student
LESSON FOCUS:           Multiplication Table through 12
CALIFORNIA              Number Sense:
                        *3.0: Students solve problems involving addition, subtraction, multiplication and division of whole
STANDARDS:              numbers and understand the relationships among operations.
                        4.0: Students know how to factor small whole numbers.
                        Mathematical Reasoning: 1.2, 2.0 and 2.3

Purpose of Lesson:      Write multiplication facts through 12. Understand and use strategies
                        to find products and quotients mentally.
LAUNCH:                 *Warm up: Problem of the day – partner check
*Transparency 8.4
                        Do skip counting patterns: By 10s to 120
                                                   By 11s to 132
                                                   By 12s to 144
                        Students record patterns and discuss “what they notice” about them.

                        Investigation: Multiplication Table: What do you know about it?
Blank Multiplication
                             How is it used? How is it useful to you?
Chart – 1 per student
                        Students complete the blank table with the “facts” that they already
                        know.
                          Ex: 1s, 2s, 5s, 10s, doubles – Student draw a line through facts
                          they know.

                        Complete the remainder of the chart.
                        Discuss the “facts” that are difficult.
                        Highlight: Commutative Property. Students circle the products that
                        are difficult for them or that they do not “know”.
                        Discuss strategies for learning facts and make a list (chart) of them:
                           - Use an easier fact and “add on”
                           - Skip counting
                           - For 9s, multiply by 10 and then subtract
                           - Break apart

                        Pg. 149: Discuss the connection to division – “Connect”
EXPLORE:                Partner work: Try a strategy from the class chart to “practice” 2
                        different facts that they do not know.
                        Students explain how the strategy was helpful or not.
Revised 9/03                                                                                                                  11
  Harcourt Grade 4             Harcourt Grade 4                        Harcourt Grade 4
PRACTICE:          Pg. 149: Problems: 16 –23.
                   Confer with small groups of students, ask them to model/represent
                   their thinking and explain their strategy for solving the problems.
SUMMARIZE/         Discuss students’ approaches and solutions to Problems 22 and 23.
CLOSURE:           Ask students to try other’s approach to solving these problems.

                      Pg. 149: Assess: Discuss how the break-apart strategy would work
                      for Mrs. Richards’ problem.
HOMEWORK:             Students record the multiplication facts that are difficult for them.
                      Make a plan for “learning” all difficult multiplication facts.
                      Describe how you will practice.
                      Select a “target” date for learning all the facts.
                      List the ways that the teacher/classmates can help them learn the
                      facts.

ROUTINES:

Continue with Routines from previous lessons.




Revised 9/03                                                                              12
  Harcourt Grade 4           Harcourt Grade 4             Harcourt Grade 4
                  UNIT 3: MULIPLICATION AND DIVISION FACTS
                                              Module 2: Chapter 8


                        DAY 5 :               LESSON: 8.5, pp. 150 - 151

MATERIALS:              Locking cubes: 25 per each group of 2 students or tiles; *Challenge
                        8.5 – copies for Early Finishers;
LESSON FOCUS:           Multiply 3 Factors
CALIFORNIA              Number Sense:
                        *3.0: Students solve problems involving addition, subtraction, multiplication and division of whole
STANDARDS:              numbers and understand the relationships among operations.
                        4.0: Students know how to factor small whole numbers.
                        Mathematical Reasoning: 2.0, 3.0, 3.2, 3.3
                                                          Algebra and Functions: 1.0 and 1.2

Purpose of Lesson:      Understand how to multiply 3 factors using the Associative Property
                        of Multiplication: for any numbers: a, b, and c:
                         (a X b) X c = a X (b X c). Parentheses are used for grouping and
                        determine the order of operations. A “cube” models/represents the
                        product of three factors. Multiplying 3 factors using the Grouping
                        Property of Multiplication - Grouping is Harcourt’s term, students
                        should also use the term: Associative Property.
LAUNCH:                 Warm up: Pg. 154: Check Skills, Problems: 4 – 10
                                   Discuss homework plans

                        Question: How long does it take Kim to play all of her favorite songs
                        if she plays 4 songs that are each 5 minutes long and each song is
                        sung by 2 different artists?
                        Think individually about how to approach this problem for 1-2 minutes;
                        then discuss/share with a partner.

                        Students model/represent their solution.
                              Record students thinking about this problem.
                              Ex: songs X artists X 5 minutes each or
                                   4 songs X 5 minutes each X 2 artists
                              Are these the “same”? Can you provide a convincing argument?
EXPLORE:                Pg. 150 B: Alternative Teaching Strategy: Give students cubes to
                        model problems with 3 factors.
                        Students can use tiles and draw solutions.
Locking cubes: 25 per
each group of 2
students or tiles       Students create their own situation using 3 factors and model with the
                        cubes or tiles.
                        Describe and record the factors and the total number of cubes, or
Revised 9/03                                                                                                                  13
  Harcourt Grade 4                Harcourt Grade 4                       Harcourt Grade 4
                      tiles.
PRACTICE:             Pg. 151: Problems 6 & 9 – ask students to try to use parentheses
                      around different pairs of numbers.
                      Remind students about parentheses and order of operations.
                      Does this change the answer?
                      Is this always the case?

*Challenge 8.5        Problems: 32 – 35: Discuss with students.
                      *Use Challenge 8.5 for “early finishers”
SUMMARIZE/            Pg. 151: Discuss: Problems 39 and 40.
CLOSURE:              Pg. 151: Assess: Write – chart/share students’ responses.
HOMEWORK:             Pg. 151: Problems 27 – 31:
                      Choose 1 equation and write a story problem for it. Solve it.

                      Be ready to explain how you know that the equation fits the story
                      problem.



ROUTINES:

Continue with Routines from previous lessons.




Revised 9/03                                                                              14
  Harcourt Grade 4           Harcourt Grade 4             Harcourt Grade 4
                  UNIT 3: MULIPLICATION AND DIVISION FACTS
                                            Module 2: Chapter 8


                      DAY 6:                LESSON: 8.6, pp. 152 - 153

MATERIALS:            Counters; *Challenge 8.6 – copies for students: This is optional class
                      work or Homework;
LESSON FOCUS:         Choose the correct operation
CALIFORNIA            Number Sense:
                      *3.0: Students solve problems involving addition, subtraction, multiplication and division of whole
STANDARDS:            numbers and understand the relationships among operations.
                      Mathematical Reasoning: 1.1, 1.2, 2.0 and 3.2
                      1.0: Students make decisions about how to approach problems.

                                    Statistics, Data and Probability: 1.0, 1.3
Purpose of Lesson:    Understand that choosing the correct operations includes using
                      multiplication as repeated addition and division as repeated
                      subtraction. Recognize that when more than one operation is involved,
                      the order of operations is important,
LAUNCH:               HW: Partner share/discuss problems.
                      Students share problems with entire class - highlight strategies.

                      Pg. 152 B: Alternative Teaching Strategy: Students complete
Counters
                      problems together.
                      Work in groups of 4: Make a list of situations that use each of the
                      operations. Share with class.
                      Example: Totaling many different quantities.

EXPLORE:              Pg. 152: Sketch the graph: “Orange County Rainfall” on
Students have books   board/overhead.
closed                Pg. 152: Discuss Problems: A – D and the following question:
                      Can different operations be used to solve the same problem?
                      Discuss with partner and have a class discussion.
PRACTICE:             Pg. 153: Problems 1-6. Discuss strategies and solutions.
                      Students provide a “convincing” argument about their approach.
                      Ask students to draw pictures to help solve, where appropriate.

*Challenge 8.6        Optional: Challenge 8.6
SUMMARIZE/            Review the operations and which types of situations require which
CLOSURE:              operation.
                      Student reflection: What new understanding do you have about
                      knowing which operation to choose when solving a problem?

Revised 9/03                                                                                                                15
  Harcourt Grade 4                  Harcourt Grade 4                                Harcourt Grade 4
HOMEWORK:               Pg. 153: Problems 7 – 11 OR Challenge 8.6
*Challenge 8.6 – 1 copy
per student


ROUTINES:
• “Guess My Number”:
  Students give number clues to determine the “selected number”.
  Introduce 1-100 Chart as tool for recording information.
  Example: 76 is the chosen number.
     Question 1: Is the # less than 50? “No”
      Mark 50 on the chart to indicate the number is greater
      than 50.
    Question 2: Is the # less than 95? “Yes”
       Mark 95 on the number line to indicate that the number is
       between 50 and 95.
    Question 3: Is the # greater than 70? “Yes”
       Students discuss what they know about the number and
       strategize the next “guess”.
       Continue until the number is “discovered”.
  Discuss the strategies students used for developing questions.
  Discuss the number with students:
     Example: What’s 10 more or 10 less than this number?
                20 more: 20 less?
                100 more? 100 less?
                How much more to 200? 500?
  Encourage students to use the number grid as tool for answering these questions.
  Use strategies to keep all students participating in this discussion rather than just the
  student giving the question.
  Have students explain their strategies for comparing these numbers.

    As students develop strategies for numbers between 1 – 100, use larger numbers.

    Add the number line as another tool for students to use with this Routine.

•   Strategies: Addition and Subtraction Facts
Use “10 Frame” (next page) for recording/compiling students’ addition and subtraction fact strategies.
   (Facts through 20.)
    Students should have facility with these facts—but they may need to ‘review’ them and
    use strategies to recall them easily.
    The “10 Frame” provides visual models for adding by 1, 2, 3 or 4 that can be recorded as
    “generalizations” for known facts.

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  Harcourt Grade 4                     Harcourt Grade 4                             Harcourt Grade 4

   Discuss strategies for:
      “Doubles” – 5 +5 or 50 + 50, etc.
      “Doubles plus 1 that use the double facts” – 5 + 6 or, 50 + 60
       Adding by 5, 6, 7, 8 and 9
           Example: When is it easier to add by 2 and 3 when adding?
                      By 5? Or to add 3 and 3 when adding by 6?

   Record students’ strategies on a chart—students need to identify which strategies to use
   for the facts that are difficult to learn.

Connect addition facts to subtraction facts as another strategy for developing computational fluency.

   This is a Routine designed for 2-3 week “review”.


Continue re-visiting Routines from Days 1-5 and Module 1 as needed.




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 Harcourt Grade 4   Harcourt Grade 4   Harcourt Grade 4




Revised 9/03                                          18
 Harcourt Grade 4           Harcourt Grade 4             Harcourt Grade 4
                 UNIT 3: MULIPLICATION AND DIVISION FACTS
                                          Module 2: Chapter 9


                       DAY 7:             LESSON: 9.1, pp. 158 - 159

MATERIALS:           *Transparency 9.1; Counters
LESSON FOCUS:        Expressions with Parentheses
CALIFORNIA           Algebra and Functions: 1.0
                     *1.2 Interpret and evaluate mathematical expressions that now use parentheses
STANDARDS:           *1.3: Use parentheses to indicate which operation to perform first when writing expressions containing
                     more than two terms and different operations.
                     Mathematical Reasoning: 2.4, 3.0, 3.2, 3.3

Purpose of Lesson:   Understand how to evaluate expressions that contain parentheses.
                     Learn the order of operations and that multiplication and division are
                     performed before addition and subtraction. Understand that when
                     parentheses appear in an expression, the operation in parentheses is
                     completed first. Learn that varying the placement of the parentheses
                     can change the value of an expression.
LAUNCH               *Number of the Day. Share responses.
*Transparency 9.1    HW: Partner check

                     Pg. 158 B: Alternative Teaching Strategy
                           Write an expression on the board/overhead:
                                    (8 X 3) – 9 = _____ Read with students.
                           Discuss expression. Students find a situation represented by
                           expression. Check with students to see if they are convinced that
                           the situation fits the expression.
                     Pg. 158: Bunches of Bears problems – complete together AND
                     Pg. 159: “Choose an expression”
EXPLORE:             Pg. 159: “More Examples”: Discuss the difference in the answer because of
                     the parentheses.

                     Pg. 159: Do problems 1 – 5 with students – discuss solutions.
                     Discuss how #2 would be different without parentheses.
PRACTICE:            Pg. 160: Problems: 26 – 30 – solve each problem and “prove” that the
                     answer is correct.
                     Choose 1 equation and write a story problem that fits.
SUMMARIZE/           Pg. 160: Problem 34 and Pg. 161: Problem 40
CLOSURE:             Discuss with students.
                     Pg. 161: Assess: Write – students write individually
HOMEWORK:            Pg. 160: Problems: 36, 38 and 39.
                     Be sure students show the expressions used to solve the problem.
                     Pg. 161: Problems: 41, 48 and 49, choose 1 from “Thinker’s Corner”
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  Harcourt Grade 4           Harcourt Grade 4             Harcourt Grade 4
                  UNIT 3: MULIPLICATION AND DIVISION FACTS
                                              Module 2: Chapter 9


                          DAY 8:              LESSON: 9.2, pp. 162 - 163

MATERIALS:               Play money from manipulative kit; Challenge 9.2 – 1 per student;
LESSON FOCUS:            Match Words and Expressions
CALIFORNIA               Algebra and Functions: 1.0
                         *1.2 Interpret and evaluate mathematical expressions that now use parentheses
STANDARDS:               *1.3: Use parentheses to indicate which operation to perform first when writing expressions containing
                         more than two terms and different operations.
                         Mathematical Reasoning: 11, 2.0, 3.1, 3.2

Purpose of Lesson:       Understand and learn to write expressions for given word problems.
                         Know that the mathematical operations chosen for the expression
                         must match the meaning of the words. Understand that parentheses
                         are used to insure that the order of performing the operations
                         matches the events in the problem.
LAUNCH:                  Warm up: Discuss homework and strategies for solving the “Valuable
                         Plates” problem from Thinkers Corner.
                         Complete using parentheses and operation signs differently and
                         compute results. Discuss with students.

“Play money” from        Pg. 162: Question – “What’s left?” on board/overhead
manipulative kit         Pg. 162B: Use “Alternative Teaching Strategy”: Students model
                         the problem.
                         Pg. 162: Record expression and connect to problem, use Guided
                         Instruction Questions
                         PG. 162: Do “Example ” Problem: “Sue bought 6 tickets…”
EXPLORE:                 Pg. 163: Problems 1-6: Work with students.
                         Discuss solutions and strategies for finding the correct expression.
                         Chart: “Strategies for Finding the Expression”
PRACTICE:                Pg. 163: Problems 7 – 9
Focus on Communication   Choose 2-3 expressions from Challenge 9.2 and put on
and Representation       board/overhead-
                         Students write and share problems. The class matches the problem to
                         the expression.
                         Students provide convincing arguments or justification for their
                         choice
SUMMARIZE/               Pg. 163, Problem 10: Discuss – students explain/justify response.
CLOSURE:                 Pg. 163: Assess: Write: Students response individually in writing
HOMEWORK:                Pg. 163: Mixed Review, Problems 11 – 15;
Challenge 9.2             Challenge 9.2: Write 1 word problem for 1 of the expressions
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  Harcourt Grade 4           Harcourt Grade 4             Harcourt Grade 4
                  UNIT 3: MULIPLICATION AND DIVISION FACTS
                                                Module 2: Chapter 9


                              DAY 9 :              LESSON: 9.3, pp. 164 - 167

MATERIALS:                 Transparency 9.3; Color Tiles: 50 for each group of 2 – 4 students;
                           “Play Money” from manipulative kit;
LESSON FOCUS:              Multiply Equals by Equals
CALIFORNIA                 Algebra and Functions: 1.0
                           *2.0: Students know how to manipulate equations.
STANDARDS:                 *2.2: Know and understand that equals multiplied by equals are equal.
                           Mathematical Reasoning: 2.0, 2.3, 2.4, 3.0, 3.2, 3.3

Purpose of Lesson:         Understand that both sides of an equation are equal when both sides
                           are multiplied by the same number. (Property of Equality)
                           An equation is a number sentence that shows that two amounts are
                           equal. If both sides of an equation are multiplied by the same number,
                           the values on both sides change but remain equal.
LAUNCH:                    *Warm up: Number of the Day
*Transparency 9.3
                           HW: Discussion problems with partners– discuss and share results
                           with class.

                           Pg. 164B: Use English Language Learners Strategy: To introduce
                           concept.
                           Pg. 164B: Use “Alternative Teaching Strategy”
EXPLORE:                   Pg. 166: Use: “Alternative Teaching Strategy” to have students
                           “model” multiplying both sides of an equation by the same number.
                           Students discuss what they notice and have them provide their own
Color Tiles: 50 for each
                           “generalizations”/explanation. Students explain what they notice.
group of 2 –4 students

“Play Money” from          • Question: What are all the different combinations of pennies,
manipulative kit               nickels and dimes that equal $ .25?
                               How do you know if you have all possible combinations?
                               How would you organize your answers to be convinced that you
                               found all the possible combinations?
                           Pg. 164: Use “Learn”: Students select 2 combinations for $ .25.
                               Multiply each combination by 2:
                               Is the value of the 2 amounts equal?
                               Have students try with other amounts.
                           Reminder: Focus on the “value” rather then the number of coins.


Revised 9/03                                                                                       21
  Harcourt Grade 4                 Harcourt Grade 4                      Harcourt Grade 4
                      Each group of 4 gives an equation: Ex: (1 X 6) = (2 X3)
                          Students multiply each side by a chosen number and check results.
                          Discuss and develop “generalizations” using students’ terms and
                          vocabulary.
PRACTICE:             Pg. 166: Model: Record and “show”: Ex: 9 = (3 X 3)
                             Multiply both sides by 9: 9 X 9 = 9 X (3 X 3)
                             Solve
                      Students do Problems 8 – 15
                             Monitor and confer with groups of students as they work.
SUMMARIZE/            Pg. 166: Problems 20 and 21
CLOSURE:                  Discuss, explain/justify responses (Why/How do they know?)
                      Pg. 167: Assess: Have students write individually or chart with class
HOMEWORK:             Pg. 167: Lesson Quiz: Transparency 9.3 – Problems 1 – 5
Transparency 9.3      Pg. 167: Problems 23 and 24




ROUTINES:

Continue with Routines from previous lessons.




Revised 9/03                                                                             22
  Harcourt Grade 4          Harcourt Grade 4             Harcourt Grade 4
                  UNIT 3: MULIPLICATION AND DIVISION FACTS
                                          Module 2: Chapter 9


                     DAY 10:                LESSON: 9.4, pp. 168 - 169

MATERIALS:           *Transparency 9.4;
LESSON FOCUS:        Expressions with Variables
CALIFORNIA           Algebra and Functions: 1.0, 1.1
                     *1.o: Students use and interpret variables, mathematical symbols and properties to write and simplify
STANDARDS:           expressions and sentences.
                     Mathematical Reasoning: 1.1, 2.0, 2.3, 3.0, 3.2, 3.3

Purpose of Lesson:   To understand how to write and evaluate an expression with a variable
                     (and know the meaning of “variable”). Know when using a variable how
                     to determine what the variable will represent. Understand that
                     letters used in expressions are called variables; in equations (number
                     sentences) those variables are called “unknowns”.
LAUNCH:              HW: Collect and correct: Lesson Quiz

*Transparency 9.4    *Pg. 168A: Number of the Day
                     *Pg. 168A: Problem of the Day

                     What’s the Expression?
                     Pg. 168: Question: Learn: “B is for Basket”: Tara made some 3 point
                     baskets playing basketball. We need to write an expression to
                     represent this situation and figure the points she earned.
                     We do not know “how many baskets” she made.
                     Ask students for a “number expression” or description of this
                     situation.

                     Write: b X 3 - Discuss with partners – what does “b” represent?
                     What does 3 represent?
                           (Note: “b “ represents the number of baskets)
                           Try different numbers for b and discuss results.
                           Make a table to show the different possible results.
                     Ask students to discuss and form some “generalizations” about the
                     meaning of and use of “variables”-
EXPLORE:             Pg. 169: Do “Check” Problems (1 –5) with students
                         Discuss results, have them create situations for the equations
PRACTICE:            Pg. 169: Problems 10 –12 and 14 – 17
                     Discuss results
                     Do Problems 18 – 19 together
Revised 9/03                                                                                                             23
  Harcourt Grade 4             Harcourt Grade 4                       Harcourt Grade 4
SUMMARIZE/         Pg. 169: Problems 20 and 21
CLOSURE:           Create chart similar to “Special Needs”, pg. 168 B with students.
                   Do not include equations such as “5 x n = 10”
                   Pg. 169: Problem 25: Students write response to question
HOMEWORK:          Pg. 169: Problems 22 – 24, 26 - 30

ROUTINES:

Continue with Routines from previous lessons.




Revised 9/03                                                                         24
  Harcourt Grade 4           Harcourt Grade 4             Harcourt Grade 4
                  UNIT 3: MULIPLICATION AND DIVISION FACTS
                                     Module 2: Chapter 9


                       DAY 11:         LESSON: 9.5, pp. 170 - 171

MATERIALS:           *Transparency 9.5; Counters
LESSON FOCUS:        Equations with Variables
CALIFORNIA           Algebra and Functions: 2.0
STANDARDS:           1.0: Students use and interpret variables, mathematical symbols and properties
                     to write and simplify expressions and sentences.
                     1.1: Use letters, boxes, or other symbols to stand for any number in simple
                     expressions or equations.
                     Mathematical Reasoning: 1.1, 2.3
Purpose of Lesson:   Understand how to write equations with variables and know that an
                     equation is a complete sentence (with a verb that shows equality).
                     Recognize that the equal sign in an equation serves as a balancing point
                     and the expressions on either side are equal to each other.
                     Learn that a variable can be used to represent an unknown number,
                     amount/quantity.
LAUNCH:              *Pg. 170: Problem of the Day: Share students’ approaches.
*Transparency 9.5
                     Discussion Question: “What is a variable?”
                     Students think individually for 1 - 2 minutes and then discuss with partner
                     Whole class discussion – collect/note students’ responses

                     Pg. 170 B: “Alternative Teaching Strategy”: Model/do with students
Counters
                     Pg. 170: Learn: Put equations: A, B, C on board/overhead
                         Ex: 6 X 8 = t “t” = total number of marbles in the bag
                                         6 = marbles in each bag
                                         8 = number of bags
                         Analyze and discuss equations B and C
EXPLORE:             Pg. 170: Check – Do with students
                       Students identify “what” the variable represents in each equation.
                       Students represent/make a diagram for situation along with the
                       equation.
                       Discuss students’ strategies/solutions.
PRACTICE:            Pg. 171: Problems: 5 – 12
                       Students work with partners. Confer with groups as students are working.
                       Share and discuss solutions with class
SUMMARIZE/           Pg. 171: Assess: “Write” – Chart responses
CLOSURE:              Discuss Transparency 9.5 with students or use a written assessment
HOMEWORK:            Pg. 171: Problems 13 – 15 & Mixed Review: Problems 16 -20
Revised 9/03                                                                                       25
 Harcourt Grade 4                     Harcourt Grade 4                Harcourt Grade 4

ROUTINES:
• In and Out Boxes
  See example: T.E. Pg. 143 – Challenge 8,2
  Model using horizontal and vertical tables for organizing “In and
  Out” box numbers.

   Example:
   Weeks         1     2       3       4      5   6
 Days in a Wk    7     14      21      28

   OR:
    Weeks       Days in a Wk
    1           7
    2           14
    3           21
    4           28
    5
    6
    7

   Discussion and Generalizations:
   Ask students how they used the information from the table to
   determine the number of “Days in a Week”.
   Generate other “in and out box” tables with students.
   Develop generalizations for the patterns:
      1. “The number of weeks multiplied by 7 equals the number of
          days.”
      2. N weeks X 7 = the number of days



  Continue re-visiting Routines from Days 1-10 and Module 1 as
                                    needed.




Revised 9/03                                                                         26
  Harcourt Grade 4           Harcourt Grade 4             Harcourt Grade 4
                  UNIT 3: MULIPLICATION AND DIVISION FACTS
                                           Module 2: Chapter 9


                       DAY 12:               LESSON: 9.6, pp. 172 - 173

MATERIALS:            Transparency of Challenge 9.5; *Challenge 9.6 – copies for early
                      finishers;
LESSON FOCUS:         Find a Rule
CALIFORNIA            Algebra and Functions:
                      1.0: Students use and interpret variables, mathematical symbols and properties to write and simplify
STANDARDS:            expressions and sentences.
                      1.5: Understand that an equation such as y + 3x + 5 is a prescription for determining a second number
                      when a first number is given.
                      Mathematical Reasoning: 1.0, 1.1,2.4, 3.1

Purpose of Lesson:    Understand how to find a rule when input and output numbers are
                      given. Recognize that finding a pattern can help identify the rule to
                      write an equation. Learn how to apply a rule to each input number to
                      get the corresponding output number. Understand that when
                      equations have two variables, an input/output table is a tool that can
                      organize the information to see how the numbers are related.
LAUNCH:               HW: Discuss: Problems 13 –15
                          Partner check: Mixed Review: Problems 16 – 20

Transparency of       Challenge 9.5 – Do and discuss Problems 3-6 with students.
Challenge 9.5


                      Pg. 172: Learn: Put data in “Input/Output” box on board/overhead
Students have books
                         Partner discuss – students determine “rule” for “Input/Output” box
closed.
                         Students explain how they found the rule.
                         Apply rule to students’ suggestions; check by working “backward” and
                         model:
                                  X divided by 5 = Y
                                  X = Input and Y = Output
EXPLORE:              Pg. 173: Discuss: “Check”:
                            Problems 1 – 3: Model how to write rule using equation and words.
                            Ex: X divided by 7 = Y   To find output, divide input by 7.
                            Ask students to explain how “working” backwards could be used
                            to “check” this rule.
PRACTICE:             Pg. 173: Problems 4 – 7
                       Choose 2-3 problems from #s: 8 – 13
*Challenge 9.6         Challenge 9.6 for “early finishers”
SUMMARIZE/            Pg. 173: Assess: Discuss
CLOSURE:              Pg. 173: Assess: Write: Students write individually or create a class chart
                      with responses.
Revised 9/03                                                                                                              27
  Harcourt Grade 4                  Harcourt Grade 4                          Harcourt Grade 4
HOMEWORK:               Pg. 173: Problems 14 – 17; write a situation for problem 15

ROUTINES:

Continue with Routines from previous lessons.




Revised 9/03                                                                                 28
  Harcourt Grade 4           Harcourt Grade 4             Harcourt Grade 4
                  UNIT 3: MULIPLICATION AND DIVISION FACTS
                                          Module 2: Chapter 9


                     DAY 13:              LESSON: 9.7, pp. 174 - 175

MATERIALS:           *Transparency 9.7; Practice 9.7: 1 copy per 4 students;
LESSON FOCUS:        Problem Solving Strategy: Working Backward
CALIFORNIA           Algebra and Functions: 1.0
                     *2.0: Students know how to manipulate equations.
STANDARDS:           *.2.2: Know and understand that equals multiplied by equals are equal.
                     Mathematical Reasoning: 1.0, 1.1,2.6, 3.1, 3.2

Purpose of Lesson:   To understand how to use the “working backward” strategy to solve
                     problems. Recognize that when working backward – inverse operations
                     are used in the reverse order. Addition and subtraction are inverse
                     operations. Multiplication and division are inverse operations. Learn
                     how a diagram or drawing can provide “visual” help to understand the
                     problem.
LAUNCH:              * Pg. 174A: Number of the Day
*Transparency 9.7
                     HW check: Discuss problems students wrote for Problem 15.
                        Divide by 12 – a divided by 12 = b
                        Discuss strategies for changing the problem so that it matches
                       the equation if they do not.

                     Question: When “solving a problem” – what are the “steps” you follow?
                       Include in discussion: A “problem” is a situation where you do
                       not immediately know/have a solution.
                     List students’ steps and discuss situations where you might choose 1
                     step over another and “why”.

                     Pg. 174 B: Use: Alternative Teaching Strategy and Reading Strategy
                     Pg. 174: “Tony’s Problem” on board/overhead
                      Use Alternative Teaching Strategy to “model” process and record
                        process on board.
                     Pg. 174: Discuss: “Guided Instruction” Questions
                        Discussion: Explain why this process makes sense? Does not make
                        sense?
                      Explain/show other strategies that might have been used to solve
                        this problem.
                        Highlight: Pg. 174: Check: How can the expression be used to
                           check the solution? Explain why this process works.

Revised 9/03                                                                                  29
  Harcourt Grade 4             Harcourt Grade 4                       Harcourt Grade 4
EXPLORE:           Pg. 175: Problems 1 and 2 – Do with students and discuss approaches.
                   Students explain how they know if the answer makes sense for the
                   problem.
PRACTICE:          Practice 9.7 – Students work in groups of 2 – Solve #s: 1 - 6 of
                   Practice page problems.
                   Students: solve, model/represent and explain or justify solutions.
Prepare Practice 9.7
1 sheet per 4 students
                         Students share solutions at tables or with entire class.
SUMMARIZE/               Pg. 175: Assess: Discuss: Explain how to find the value of the
CLOSURE:                 variable in the equation q + (8 X 5) = 65.
                         Ask students to discuss/share a variety of approaches to solving this
                         problem: include “working backward” is one approach.

                         Pg. 175: Assess: Write
                         Students individually respond to question.
HOMEWORK:                Pg. 175: Mixed Strategy Practice: Problems 7 –10

                         Or: Page 182 or 183: California Connection,




ROUTINES:

Continue with Routines from previous lessons.




Revised 9/03                                                                                 30