Categories: All - estimation - division - algorithms - multiplication

by Allison Biagi 12 years ago

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Estimation and Computation

The content focuses on the essential concepts of estimation and computation tailored for students in grades 3-6. It delves into various multiplication and division algorithms, explaining each method in detail.

Estimation and Computation

Estimation & Computation

The goal of this map is to outline everything you need to learn about estimation and computation for grades 3-6.

Algorithms for Addition and Subtraction

Subtraction

  • A subtraction algorithm is a set of rules, calculations or operations to be followed to attain a difference.

    • Base-Ten Blocks Algorithm

    Subtraction using base-ten blocks, is complicated to explain and can only be fully understood if seen visually.

    The link posted next to the note will show you how to use this algorithm

    When using the expanded algorithm method for subtraction, you start with the greatest number and from there subtract as much as you can do in your head.

    Ex: 345

    - 16

    Since there are no hundreds to subtract

    start with tens: 345-10=335

    subtract 5 ones: 335-5=330

    subtract 1 ones: 330-1=329

    So your answer is 329

    When you use the standard algorithm to subtract, you always start with the ones and continue subtracting with the help of regrouping, from right to left.

    Check the link posted for an example

    Addition

  • An addition algorithm is a set of rules, calculations or operations to be followed to attain a sum.

    • Base-Ten Blocks Algorithm

  • The base ten blocks algorithm, is a great way to physically visualize the addition of two sums. To start, grab the number of necessary blocks for each sum. For example: If you one of the sums you are adding is 23, then you would want to grab 2 ten bars and 3 ones cubes. Next, you would grab the blocks needed for the second sum. When all the blocks have been grabbed, combine hundreds, tens, and one blocks together to exchange up for bigger blocks. This is a hard concept to explain so I have a link posted that you can look at to visually see this method at work.

    • Expanded algorithms involve adding the values of each place value first, then adding all the sums together to get the final answer

    Ex: 525

    +123

    Add the hundreds column first: 500+100=600

    Add the tens column second: 20+20=40

    Add the ones column third: 5+3=8

    Finally add all the sums together to get the final answer:

    600

    40

    + 8

    648

    Standard Algorithm

  • When you use the standard algorithm you always add your ones first. If there are more than 10 ones than you regroup ten ones as 1 ten and add to the tens column. Next you add the tens column. If there are 10 or more tens, then you regroup 10 tens to make 1 hundred. You will keep using this strategy if you have more columns of numbers to add.

    1

    Ex: 525

    + 126

    651

    • Estimation

    What is it?

  • A rough calculation of a quantity, measurement, or answer.

    • Estimation Games!!!
    Estimating a measurement

  • Finding how much there is of: length, distance, time, etc.

    • Estimating a Quantity

  • Finding how many cars, people, books, etc.

    • Estimating an Answer

  • Finding a quotient, difference, sum, or product.

    • Estimation Techniques

    Computational Estimation

  • Making a reasonable guess to the actual answer.

    • Clustering

    When you might use this technique:

    You may use this technique if you have a group of numbers that cluster around the same number

    How to use:

    1) Find the number that all the addends are closest to and that is easy to compute

    2) Replace all the addends with the new number you have chosen

    3) Use multiplication to compute the total of all the addends

    Ex: Number of Dancers

    May 23

    June 26

    July 24

    Aug 22

    Since all the numbers are around 25, I will take 25 and multiply it by the number of months that are presented. Since there are only 4 months represented in this problem, take 4*25 to get your answer.

    The answer is 100 dancers

    Front-End Estimation

    How to use this technique:

    1) Change all the digits except for the leading or front-end digit to zeroes

    2) Calculate the total with the numbers you have estimated

    EX: 551+376

    500+300=800 (change the digits to zeroes except the leading or front-end digits)

    51+76=about 125

    So 551+376 is about 925

    Substitution of Compatible numbers

    You can use this technique if numbers you estimate are close to the original numbers and are easier to solve the problem with.

    To use this technique:

    1) Identify the numbers in the original problem that could be replaced with numbers that are close to the original number, but are easier to compute.

    EX: original problem- 23*49

    since 20 is close to 23 and 50 is close to 49

    estimate and solve for 20*50 instead of 23*49

    2) Solve the new problem

    20*50=1000

    Rounding

    Rounding is based on the halfway point between consecutive multiples. For example the halfway point between 10 and 20 is 15. Any numbers that are less than 15 should round down because they are closer to 10. Numbers that are bigger than 15 should round up because they are closer to 20.

    Games for Rounding

    Algorithms for Multiplication and Division

    Division

    A division algorithm is a set of rules, calculations or operations to be followed to attain a quotient.

  • The standard algorithm for divison involves sharing and is useful for simple/complex problems. It is also sometimes referred to as "long divison." This algorithm does involve the subtraction of the dividend from the divsor to get the quotient just like the expanded algorithm, but this algorithim should work on any dividend that does not go into the divisor perfectly.

  • A link is posted to show you just exactly how this algorithm differs from the expanded algorithm.

    • The expanded algorithm for division involves the repeated subtraction of the dividend from the divisor, until you can't subtract anymore.

    Ex: 25/5

    ____

    5l 25

    - 5

    20

    - 5

    15

    - 5

    10

    - 5

    5

    - 5

    0

    So we know that 5 goes into 25, 5 times

    Base-Ten Blocks

    The Base-Ten block algorithm for dividing, is a visual way of attaining the desired answer. Using this algorithm you will need to trade when necessary to acquire needed blocks.

    A link has been posted for this algorithm

    Multiplication

  • A multiplication algotrithm is a set of rules, calculations or operations to be followed to attain a product.

    • Lattice Multiplication

  • The Lattice multiplication algorithm uses a box to break down basic multiplication facts, that are then added together at the end to attain the desired answer.

  • A video is attached to this topic to show you how to use this algorithm.

    • Standard Algorithm

    This algorithm includes the formation of only two partial products.

    Ex: 32

    *14

    Multiply: 4*32=128

    Multiply: 10*32=320

    Add the two partial products: 128

    +320

    =448

    Expanded Algorithm

    This algorithm involves breaking apart all the numbers in the problem according to the numbers place value and multiplying every digits place value to get the partial product.

    Ex: 23

    *12

    Multiply 2*3=6

    Multiply 2*20=40

    Multiply 10*3=30

    Multiply 10*20=200

    When you are finished finding the partial products, add all the sums together to get the final answer.

    Answer: 6

    40

    30

    +200

    =276

    Area Algorithm

  • Using the area model for multiplication requires the use of base-ten blocks. A video is posted to show how you would use this algorithm to solve a multiplication problem.

    • Techniques
    Use Compensation

    With this technique you first want to substitute a compatiable number for the original number so that it is easier to solve.

    EX: you want to buy three packs of toothpaste that cost $2.95 a piece.

    A compatiable number for $2.95 would be $3.00

    So take 3.00 * 3 (the number of toothpastes you want to buy) to get $9.00

    Adding $0.05 to $2.95 three times requires compensating by taking $0.15 away from $9.00.

    This leaves you with the answer: $8.85

    Break Apart Numbers

  • This method involves breaking numbers down into more workable parts to be able to use communative, associative, and distributive properties.
  • Example: 232+440

    • 200+32

    400+40

    600+72=672

    Choose Compatible Numbers

  • Numbers that are easy to compute are called compatible numbers. They can be numbers easy to add such as 10 and 20 or they can be numbers that are easy to multiply like 10 and 32. For most people compatible numbers that involve multiples of 10's or 100's are easy to add, subtract, and multiply. Ex: 40+60.

    • Count On and Count Back

  • The count on technique is accurate when you are only adding 1, 2 or 3; 10,20 or 30; or 100, 200, or 300. To solve using this technique, first look at the problem. If we are adding 25+30, start with 25 and count upwards by tens. Ex: 25, 35, 45, 55. Since we are adding 30 to 25, we only counted by tens three times to get 55 as our answer.
  • The count back technique is accurate when you are subtracting 1, 2, or 3; 10, 20, or 30; or 100, 200, or 300. For the problem 54-2, start with the larger number, 54, and count back by ones. Since you are subtracting two ones total, it would look like this: 54, 53, 52. Therefore, the final answer is 52.

    • Mental Computation

    What is it

  • It is the process of attaining an answer mentally without pencil, paper, calculator, etc.

    • Games

  • Next to this note is a site posted that involves mental computation games. The games consist of quick addition and subtraction problems that pop up on the screen and you have to solve them in your head right away.

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