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    Bridges in Mathematics (

    Parents and teachers may reproduce for classroom and home use.) ©The Math Learning Center

    16

    x 12

    32

    + 160

    192

    1

    Standard

    Algorithm

    Grade 5, Unit Two: Seeing & Understanding

    Multi-Digit Multiplication & Division

    In this unit your child will:

    • use multiplication and division facts through 12’s fluently

    • multiply and divide by multiples of 10 (e.g., 40 x 70)

    • review double-digit multiplication using a variety of strategies

    including the standard algorithm

    • divide 2- and 3-digit numbers by 1- and 2-digit numbers

    using a variety of strategies

    • solve story problems involving multiplication and division with remainders

    • measure using metric units of length (centimeters, meters, etc.), mass

    (grams, kilograms, etc.), and capacity (milliliters, liters, etc.) and perform

    conversions between units

    Your child will learn and practice these skills by solving problems like those shown

    below. Keep this sheet for reference when you’re helping with homework.

    Problem Comments

    Write the answer to each problem below.

    37 30 40

    x 10 x 12 x 20

    370 360 800

    Students are able to multiply fluently by

    multiples of 10 when they know their basic

    facts and when they have a solid

    understanding of place value. Being able to

    calculate mentally with multiples of 10 is

    useful in and of itself, and it also helps

    students estimate reasonable answers before

    multiplying multi-digit numbers.

    Write a story problem for 12 x 16.

    There are 12 bottles of juice in each case.

    Luis bought 16 cases for his party. How

    many bottles of juice is that altogether?

    Make a labeled sketch to solve your problem.

    12

    10

    2

    10 6

    16

    10 x 16 = 160

    2 x 16 = 32

    160

    + 32

    192

    Luis has

    192 bottles

    of juice in

    all.

    It is important that students are able to

    multiply multi-digit numbers. They must also

    understand the meaning of multiplication

    well enough to write problems that can be

    solved by multiplying.

    In this unit, students review the strategies for

    multiplying multi-digit numbers they learned

    in fourth grade: making sketches, finding and

    adding partial products, and

    using the standard algorithm. If

    you look closely at the labeled

    sketch at left, you can see how it

    helps students understand why

    the standard algorithm works.

    Bridges in Mathematics (

    Parents and teachers may reproduce for classroom and home use.) ©The Math Learning Center

    Write a story problem for this division problem.

    260 ÷ 20

    Malia’s soccer team has 20 players. They have

    $260 to spend on team shirts. How much can they

    spend per shirt?

    Make a labeled sketch on the grid below to

    solve the problem.

    20

    200

    13

    200 + 20 = 220

    220 + 20 = 240

    240 + 20 = 260

    20 x 13 = 260

    so 260 ÷ 20 = 13

    They could spend $13 on each shirt.

    Students use rectangles to think about

    division as the opposite of multiplication.

    They also use the pictures to solve division

    problems by adding equal groups until they

    get to the dividend (the number being

    divided, in this example, 260). This serves as

    the foundation for the numerical methods

    and algorithm they will learn in Unit Four.

    There are 97 people on the swim team. They are

    riding in vans to the swim meet in another city.

    Each van carries 12 swimmers. How many vans

    will they have to take?

    8 vans can carry 96 swimmers because 12 x

    8 = 96. There’s still one more swimmer, so

    they need another van. They need 9 vans

    altogether.

    Students continue to solve division problems

    with remainders, as they did in fourth grade

    and in Unit One. In this unit, however, the

    problems involve larger numbers.

    A student who is fluent with facts through

    12’s can apply that knowledge to solve this

    problem. Other students might add up by

    12’s or use a collection of related

    multiplication facts to solve a problem like

    this one.

    Frequently Asked Questions about Unit Two

    Q: Why do students use sketches to solve multiplication and division problems?

    A:

    Pictures help students see why different strategies, including the algorithms, work. An

    algorithm is a set of steps for performing a particular calculation with specific kinds of numbers.

    Algorithms are important because when they are used accurately and with understanding, they

    are reliable, efficient, and universally applicable. Difficulties arise when students attempt to use

    an algorithm for multiplying or dividing without having mastered the basic facts, when they

    don’t understand why the algorithm works, when they forget the steps, or when they can carry

    out the steps yet are unable to use their estimation skills to judge whether their final answer is

    reasonable. The understanding of number relationships that students develop by using sketches

    ensures that they will be able to use the algorithms correctly.

    Q: When will students learn an algorithm for long division?

    A:

    In this unit, students review and consolidate methods for multiplying multi-digit numbers, and

    they also gain solid foundations for understanding long division. They will learn an algorithm for

    long division in Unit Four.