Multiplication & Division; Word Problems; Measurement, Time, & Money; and Fractions
What remediation does the text recommend for students who make column subtraction errors when solving division problems?
-Giving students a worksheet with several problems that require renaming -Guiding students through the first few problems and then having students work the remaining problems on their own
Outline the steps in the short-form algorithm for solving multi-digit division problems with a single-digit divisor.
-Students read the problem -Students underline the part of the problem that they work first -Students compute the underlined part and write the answer above the last underlined digit -Students multiply, subtract, and bring down the next numeral -Students read the "new" problem and compute a quotient -Students write the answer above the digit they just brought down -Students multiply and subtract to determine the remainder -Students say the problem and answer
Outline the steps in the short-form algorithm for solving multi-digit division problems with a two-digit divisor.
-Students read the problem -Students underline the part to be worked first -Students write the rounded-off problem -Students compute the division problem using the estimate from the rounded-off problem -Students multiply and determines if she can subtract -Students adjust the quotient if the estimate is not correct -Students compute the division problem and read the problem with the answer
Outline the steps for solving comparison word problems.
-Students read the problem and determine the bigger number -Students fill in the number-family diagram with the label in the place for total -Students complete the diagram by filling in known values -Students use number-family strategy to determine whether to add or subtract
Outline the steps for solving classification word problems.
-Students read the problem and underline the classes -Students write the biggest class in the area for total number -Students write the values for the two smaller classes, if known, in the number-family diagram -Students use the number-family strategy to determine whether to add or subtract
How would a teacher provide remediation for a student who treats a 0 as a 1 when solving multi-digit multiplication problems?
-Testing and, if necessary, teaching times-zero facts -Give students a worksheet containing 10 to 20 partially completed problems; -Lead the students through several problems beginning with the multiplication in the tens column -Have the students independently complete a worksheet similar to the original
What are the authors' recommendations regarding teaching students both measurement systems simultaneously? Explain the rationale for their recommendations.
-The two systems be taught independently of one another, preferably at different times during the year -Lower-performing students are likely to confuse facts from one system with facts from the other system when both are introduced concurrently
What example-selection guidelines apply when introducing students to addition and subtraction word problems?
-The verbs in the problems should be fairly common terms such as "buy," "give away," "make," 'break," "find," and "lose" -Problems should contain words that students are able to decode, and the problems initially should be relatively short -A random mix of addition and subtraction problems should be used so that students must discriminate between the two types of problems
Why are problems with unlike denominators included in practice sets when teaching students how to add and subtraction fractions with like denominators? What are the students instructed to do with the problems involving unlike denominators?
-They encourage students to pay attention to the denominators when working problems -Identify and cross out the problems involving unlike denominators
Briefly describe the instructional procedures found in Format 10.1: Introducing Division.
-Translating Division Problems -Structured Board Presentation -Structured Worksheet -Less Structured Worksheet
Describe the three basic forms in which multiplication and division problems are stated.
1. A problem gives the number of groups and the number in each group 2. The problem gives the total and asks either how many groups or how many in each group
Briefly describe the three types of fraction division problems. List the types in their recommended order of instruction.
1. A proper fraction is divided by a proper fraction 2. A fraction is divided by a whole number 3. Dividing a mixed number -A proper fraction is divided by a proper fraction, a fraction is divided by a whole number, and dividing a mixed number
Describe the two types of comparison problems and include an example of each type.
1. A quantity is stated describing an attribute of an object or person, such as weight, length, height, or age (e. g. Brendan is 7 years old. Colleen is 3 years older. How old is Colleen?) 2. The quantities of two objects or people are stated and the student is asked to find the difference between them (e. g. Brendan is 7 years old. Colleen is 10 years old. How much older is Colleen?)
List the four preskills required for solving multiplication problems that involve two multi-digit factors.
1. Basic Multiplication Facts 2. Place Value Skills 3. Complex Addition Facts 4. Column Addition with Renaming
Describe the two stages of multiplication instruction.
1. Beginning Stage 2. Advanced Stage
Describe the two stages of division instruction.
1. Beginning Stage 2. Multi-Digit Division Stage
List the four basic steps in any conversion problem.
1. Determine whether the "new" unit is larger or smaller than the original unit 2. Determine the operation 3. Determine the equivalency fact 4. Solve the problem
Describe the two example-selection guidelines recommended for Format 9.1: Single Digit Multiplication.
1. Example selection should be coordinated with count-by instruction 2. There should be a mix of problems
List four reasons that students may compute an incomplete quotient when solving multi-digit division problems.
1. Failure to bring down a digit 2. Failure to write a quotient above the last digit of the dividend 3. Confusion caused by zeros in the quotient 4. Misalignment
List the two preskills that must be mastered before students are introduced to adding and subtracting fractions with unlike denominators.
1. Finding the least common multiple of two numbers 2. Rewriting a fraction as an equivalent fraction with a given denominator
List the three main topics in which the text recommends organizing instruction of fractions. Include examples for each topic.
1. Fraction Analysis (e. g. Analyzing wholes and parts of wholes) 2. Rewriting Fractions (e. g. Rewriting whole numbers as fractions 3. Operations (e. g. Adding and subtracting fractions with like denominators
Describe the three basic types of fraction problems involving addition and subtraction. List the types of problems in their recommended order of introduction.
1. Fractions with like denominators 2. Problems with mixed numbers 3. Fractions with unlike denominators -Fractions with like denominators, fractions with unlike denominators, problems with mixed numbers
Describe the two recommendations made by the text for improving conceptual understanding when introducing division of fractions.
1. Highlighting the relationship between fractions 2. Multiplication/division number families
What four example-selection guidelines are important when teaching metric conversion problems?
1. In all problems, one of the units should be a base unit 2. In half of the problems, the student should convert a unit to a larger unit; in the other half, the students should convert to a smaller unit 3. In half of the problems, the quantity of original units should be a whole number; in the other half, the quantity should be a decimal or mixed number 4. The amount the student multiplies or divides should vary from problem to problem, for example, 10 in one problem, 1,000 in the next, 100 in the next
List the four major preskills for telling time.
1. Knowledge of the direction in which the hands of the clock move 2. Discrimination of the hour hand from the minute hand 3. Counting by fives 4. Switching from counting by fives to counting by ones, which is needed to determine the number of minutes
List the three preskills that must be mastered before introducing equivalency problems.
1. Knowledge of the terms numerator and denominator 2. The ability to multiply fractions 3. The ability to construct a fraction that equals one whole
List the five skills the text recommends teaching for building conceptual understanding during early fraction instruction.
1. Learning part/whole discrimination 2. Writing a numerical representation for a diagram or figure of whole unites divided into equal-sized parts, and vice versa 3. Reading fractions 4. Determining whether a fraction equals, exceeds, or is less than one whole 5. Reading mixed fractional numerals
Describe the two algorithms for solving multi-digit factor problems that are commonly taught.
1. Long-form or low-stress algorithm 2. Short form
List the three preskills required for solving multiplication problems that involve a single-digit and a multi-digit factor.
1. Multiplication facts 2. Place value skills, including expanded notation and placing a comma in the proper position when writing an answer in the thousands 3. Complex addition facts in which a single-digit number is added to a two-digit number
Briefly describe the three types of fraction problems that involve multiplication. List the types of problems in their recommended order of introduction.
1. Multiplying proper fractions 2. Multiplying fractions and whole numbers 3. Multiplying mixed numbers -Multiplying proper fractions, multiplying fractions and whole numbers, and multiplying mixed numbers
Fractions are often challenging for students. Describe the three characteristics of fractions described in the Skill Hierarchy section that the authors believe contribute to the complexity of fractions.
1. One-to-one and one-to-many correspondences involve fractional numbers 2. The incompatibility of different units 3. The necessity to transform or rewrite fractional numerals
Describe the four errors students commonly make when solving single-digit multiplication problems.
1. Skip counting incorrectly 2. Answers are consistently off by one count-by number 3. Students confuse the multiplication and addition operations 4. Students confuse regular and missing-factor multiplication
When does this text recommend introducing the concept of a remainder?
After students have learned about 20 division facts
When does the text recommend using activity-based strategies to solve word problems?
As an introduction to the concepts in word problems rather than the primary strategy for solving them
What rule does the text recommend introducing when teaching students to count change?
Begin counting with the largest valued coins
How can teachers introduce students to the concept of money in their classrooms?
Developing classroom or school stores
What method does this text recommend for introducing the concept of division? Describe that method.
Disjoint Sets: teaches students to remove equivalent sets by circling groups of lines
How would a teacher use pictures to introduce students to solving word problems like the following: There were six children. Two children went home. How many were left?
Drawing 6 lines representing children and crossing out 2 of the lines to represent the 2 children who went home and counting the remaining lines to determine the answer
How are measurement, time, and money related?
Each one uses numerical representations for concrete and abstract constructs required for general and workplace math literacy
Describe the preskill that students must master before introducing the number-family strategy for solving word problems.
Figuring out the missing number when two of the three numbers in a fact family are given
What activity does the text recommend for introducing students to the concept of time?
Having students close their eyes and raise their hands when they predict a minute has gone by
Briefly describe the procedures for introducing students to solving word problems using multiplication and division in Format 11.8: Introduction to Multiplication and Division Word Problems.
Introduce word problems requiring multiplication and division using coins
Why is the number-family strategy for solving multiplication and division word problems initially introduced with coins?
It helps students understand the concepts involved in solving multiplication and division word problems
Why are addition and subtraction word problems introduced together? Why are multiplication and division word problems introduced together?
It provides discrimination practice
Compare and contrast the long-form and short-form algorithms for solving multi-digit division problems.
Long-Form Algorithm: Presents a clear interpretation of what is involved in division Short-Form Algorithm: has relatively easy set of preskills that must be mastered prior to introducing division problems, but students may not understand why it works
A student is having difficulty coordinating touching her fingers as she counts to solve single-digit multiplication problems. Describe the correction procedures for this type of error.
Model, then lead by actually guiding the student's hand to coordinate touching and counting, and then test by watching while the student touches and counts alone
Describe the procedure that can be used to teach students to read units to the nearest quarter-inch on a ruler.
Point out that since there are four parts between each inch, each inch is divided into fourths then model and test, reading the ruler starting at the 1/4-inch mark: 1/4, 2/4, 3/4, 1, 1 1/4, 1 2/4, 1 3/4, 2, and so on
How does the text recommend preparing students for comparing fractions with a numerator of one in the early grades?
Providing pictorial demonstrations illustrating that the more parts a unit is divided into, the smaller the size of each part
Why does this text recommend introducing division through the use of disjoint sets?
Removing equivalent disjoint sets easily illustrates the relationship between multiplication and division as well as the concept of a remainder
When working the problem 2/3 x 12, a student converts 12 to the fraction 1/12. This student frequently makes this type of error. Describe the appropriate remediation for this type of error.
Reteaching the earlier-taught component skill
How are students initially taught to read multiplication problems in part B of Format 9.1: Single Digit Multiplication? Why does the text recommend teaching students to read multiplication problems this way in the beginning stage? When are students taught to read multiplication problems in the conventional way?
Saying "count by" Students know exactly what to do to derive an answer After several weeks
Explain why the value of 4/5 does not change when multiplied by 3/3.
The identity elements for multiplication is 1
What is the distributive property?
The product of a multiplier and a multiplicand will be the same as the sum of a series of products from multiplying individual number pairs
List the five steps in the basic procedure for introducing students to new common units and their equivalencies during the early grades.
The teacher: 1. Tells the function of the specific unit 2. Illustrates the unit 3. Demonstrates how to use measuring tools, measuring to the nearest whole unit 4. Presents application exercises in which the students determine the appropriate tool to use when measuring an object 5. Presents an equivalency fact, such as 12 inches equals one foot
Why is performing operations with customary units more difficult than performing operations with metric units?
There is not a consistent base to use when renaming
Why does this text recommend introducing proper and improper fractions at the same time?
This feature prevents students from learning the misrule that all fractions are proper fractions
What times are particularly difficult for students to write when learning to write time? Include an example.
Times representing less than 10 minutes after the hour, since a zero must be added when the time is expressed both verbally and in written form (e. g. 8 minutes after 6 is written as 6:08 and states as six oh eight)
What strategies are students taught to determine if they solved a division problem correctly?
To compare the remainder and divisor to determine the accuracy of their answer
Explain why an understanding of equivalency is critical to mastering fractions concepts and skills.
Without understanding equivalency, students will be able to apply very few of the skills they learn
How are students initially taught to read division problems? Why does the text recommend teaching students to read problems in this way?
"5 goes into 20 how many times?": it draws attention to the divisor, since it specifies the size of the equivalent groups, the critical feature in using lines to solve division problems
What numerals would be underlined in the problem 4291/7? Why does the text recommend underlining these numerals?
-4 and 2 -Division problems are worked a part at a time
Why is analyzing language critical to successfully solving word problems? How does the number-family strategy support students to analyze language correctly?
-A name must be determined for the total -Highlighting the relationships between the concepts in the word problem and the values that are given
Describe a temporal sequence word problem. Provide two examples - one that requires addition and one that requires subtraction.
-A problem in which a person starts out with a specified quantity, and then an action occurs that results in the person ending up with more or less -Carlos had 7 apples. His sister gave him 3 more. How many does he have in all? or Carlos had 7 apples. He gave 3 to his sister. How many does he have left?
Outline the procedures for teaching students the meaning of "quarter after" and "half past." Be sure to include the final task after both "quarter after" and "half past" have been taught.
-Another way of saying 15 minutes after 2 is quarter after two -What's another way of saying 15 minutes after 8? -What's another way of saying 15 minutes after 6?, What's another way of saying 30 minutes after 6?, It it's half past 4, how many minutes after 4 is it?, It it's a quarter after 4, how many minutes after 4 is it?
Describe the number-family strategy and include an example using the numbers 5, 2, and 7.
-Based on the concept that three numbers can be used to form four math statements -The numbers 2, 5, and 7 yield 2+5=7, 5+2=7, 7-5=2, and 7-2=5
What amounts are students likely to find difficult when writing about money using decimal notation? Why are students likely to find these amounts difficult?
-Between 1 and 9 cents -The need to place a zero after the decimal
Outline the sequence for introducing conversion problems with customary units. Include an example of each type of problem.
-Converting a quantity of a specified unit into a quantity of the next larger or smaller unit (e. g. 28 days equal _____ weeks) -Converting a unit into a mixed number containing the next larger or smaller quantity (e. g. 27 inches equal _____ feet ____ inches) -Converting a unit into a unit twice removed (e. g. 2 yards equal _____ inches)
Outline the steps for solving temporal sequence word problems.
-Determine whether the person in the problem starts or ends with the total -Determine whether the total is give -Add or subtract to find the answer
Outline the discriminations related to time that, if not properly taught, tend to cause errors.
-Discriminating the direction the clock hands move -Discriminating the minute hand from the hour hand -Discriminating minutes from hours -Discriminating time-related vocabulary; for example, when to use "after" and when to use "before"
Describe the strategy for rounding off multi-digit divisors when solving multi-digit division problems.
-Expressing Numerals As Tens Units -Structured Board Presentation -Structured Worksheet
Describe the errors related to renaming that students may make when solving multi-digit multiplication problems.
-Fact Errors -Component-Skill Errors -Strategy Errors
Describe five possible causes of errors when solving word problems and how each type of error would be addressed.
-Fact Errors: Extra practice on memorization of basic facts -Calculation Errors: Remediation on the calculation procedure -Decoding Errors: Previewing difficult words before students encounter them in word problems or reading the problems to the student -Vocabulary Errors: Presenting the meaning of critical vocabulary words prior to presenting the word problems -Translation Errors: A worksheet exercise using the less structured format for that particular problem type
Describe the example-selection guidelines that apply to teaching students to round off multi-digit divisors when solving multi-digit division problems.
-Half of the numbers to be rounded off should have a numeral less than 5 in the ones column, while the other half should have 5 or a numeral greater than 5 -About two-thirds of the examples should be three-digit numbers and one-third, two-digit numbers, so that practice is provided on the two types of numbers students will have to round off -Numbers that may cause particular difficulty for students should not be included in initial exercises
How are the students initially taught to interpret what the denominator and numerator in a fraction represent? Contrast this strategy with the traditional way of reading fractions.
-In 3/4 the 4 tells "four parts in each whole unit," and 3 tells "three parts are used" -3/4 as "three-fourths"
Why is teaching students to convert quantities from one unit to another more difficult in the customary system than in the metric system?
-In metric conversions the students always multiply or divide by a multiple or 10, but in conversion problems with customary units, the number to multiply or divide by varies from problem to problem -The procedures used when the converted unit is not a multiple of the original unit -Conversions are sometimes made to a unit two or more steps removed
What is a distractor? Why do distractors prove problematic for students?
-Information given in a word problem that is irrelevant to finding the solution to the problem -The additional information may distract them from determining the correct equation
How does the text recommend introducing the concept of a remainder to students?
-Introducing Remainders -Structured Worksheet -Less Structured Worksheet
Explain why problem b is more difficult than problem a: a. Antonio wrote 6 sentences. The teacher crossed out 2 of them. How many are left? b. When the teacher read Ariana's paper, she deleted 15 sentences and 3 commas. She had initially written 52 sentences. How many sentences did she have at the end?
-It contains more difficult vocabulary -The amount that Antonio because with is stated first in problem a, but in problem b the amount Ariana began with is stated after the amount of the decrease -The presence of the phrase "and 3 commas," a distractor that must be ignored -The numerals in problem b are larger, and the operation requires renaming
What preskills must students have mastered before division is introduced?
-Knowledge of basic multiplication facts -Column subtraction with renaming
Outline the steps for converting an improper fraction to a mixed number and the steps for converting a mixed number to an improper fraction.
-Look at the fraction and determine whether it is less than one whole, equal to one whole, or more than one whole -If the fraction equals more than 1, they are instructed to divide and write the answer as a mixed numeral -Determine the fraction equivalent to the whole number -Then they add the fraction portion of the mixed number
What preskills are required before introducing multiplication?
-Mastery of three count-by series (twos, fives, nines) -Reading and writing all numerals between 1 and 99
Describe activities recommended by the text for introducing length, weight, and capacity.
-Measuring the length of various strips of paper using paper clips as the unit of measurement -Introducing students to standard weights against which other objects can be measured -Setting out a number of empty cups, presenting a water-filled container, and pouring the contents into the cups, one at a time, to determine the container's capacity
Why isn't the introductory strategy of translating word problems phrase by phrase into an equation a reliable long-term strategy? What strategy is recommended after the initial introduction of word problems?
-Most word problems cannot be translated phrase by phrase into an equation -One that encourages students to integrate their knowledge of fact-number-family concepts with basic language skills to solve temporal sequencing, comparison, and classification problems
Briefly describe the procedures found in Format 9.4: Two-Digit Factor Times Two-Digit Factor for teaching students to solve multiplication problems that involve two multi-digit factors.
-Order of Multiplying -Structured Board Presentation -Structured Worksheet
Describe the procedures recommended by the text for introducing students to the concept of multiplication in Format 9.1: Single Digit Multiplication. How long does the text recommend presenting this exercise to students?
-Pictorial Demonstration -Analyzing Problems -Structured Board Presentation -Structured Worksheet -Less Structured Worksheet At least 3 to 4 weeks
Describe the sequence of examples used when teaching students to verify change received.
-Relatively easy, including small numbers of coins -Paying a dollar for an object costing 36 cents, for which students would need to count pennies, dimes, and quarters -Involving more than one dollar
When adding fractions with unlike denominators, a student consistently adds the unlike denominators. Is this a fact error, component-skill error, or strategy error? Describe the appropriate remediation for this type of error.
-Strategy Error -Present entire format for fractions with unlike denominators over, beginning with Part A
Briefly describe the instructional procedures for teaching students to mentally compute division facts with remainders found in Format 10.3 Introducing Remainder Facts.
-Structured Board Presentation -Less Structured Board Presentation -Structured Worksheet
Briefly describe the procedures for teaching students to solving multiplication problems that involve a single-digit and a multi-digit factor.
-Structured Board Presentation -Structured Worksheet -Less Structured Worksheet
Briefly describe the teaching procedures found in Format 9.2: Missing-Factor Multiplication.
-Structured Board Presentation- Model and Test Translation -Structured Worksheet -Less Structured Worksheet
When are division facts introduced to students? What can teachers do to help their students master division facts?
A week or so after student have been introduced to division through the line-circling exercises described in Format 10.1. Presenting exercises that demonstrate the relationship between multiplication and division facts using fact number families prior to exercises that promote memorization
List two of the four errors students commonly make during the beginning multiplication stage and describe the related remediation procedures.
1. Skip counting incorrectly: the teacher provides practice on counting by nines for several lessons and the student shouldn't be required to solve any multiplication problems involving with nines until he has mastered counting by nines 2. Answers are consistently off by one count-by number: the teacher presents Part C of Format 9.1: Single- Digit Multiplication, the structured board part of the multiplication format; the teacher continues to present this part until the students can respond correctly to approximately four consecutive problems; and several days of practice on this exercise should be provided before students are given problems to work independently
Outline the four steps for multiplying fractions and whole numbers.
1. Student reads the problem 2. Student changes the whole numeral into a fraction 3. Student multiplies numerators and denominators 4. Student converts product into a whole numeral or mixed numeral
Describe the three preskills that must be mastered before students can be introduced to addition and subtraction word problems.
1. Students can work a page of simple addition and subtraction problems, using a line-drawing strategy, with 80% to 90% accuracy 2. Students can translate to symbols four key phrases: "get more," "get rid of," "end with," and "how many" 3. Students can translate a common verb to a plus or minus sign
What two preskills are required for determining the value of a group of coins? Which two types of coins does the text recommend introducing first?
1. The ability to identify and tell the value of individual coins 2. Knowledge of the 5, 10, and 25 count-by series -The penny and nickel
Describe the two example-selection guidelines for teaching students to solve multiplication problems that involve a single-digit and a multi-digit factor.
1. The basic facts included in problems should be those that the student has already mastered 2. Less structured, supervised practice, and independent worksheets should include a mixture of problems
Other than not knowing basic facts, what three errors do students commonly make during the beginning division stage?
1. Writing quotients that are either too small or too large 2. Subtracting incorrectly 3. Confusing the placement of the quotient and remainder
Describe the strategy the text recommends for reducing fractions.
A greatest-common-factor strategy
What is a multiplication map? How can they be used to support students to memorize multiplication facts?
A map designed to facilitate learning multiplication facts for a particular series. Students who can visualize the maps in their minds learn and remember facts more easily