Basic Facts, Addition & Subtraction, and Multiplication & Division
Describe the procedures for determining individualized performance criteria for students on written performance tasks.
- Determine a student's writing ability by giving a 1-minute timed test -The student is instructed to write the numerals 1 through 9 as many times as he can -The student's writing rate is determined by counting the number of digits written during this 1-minute period. -By multiplying that number by 2/3, the teacher can estimate how many digits a student should be able to write as answers during a 1-minute fact timing.
Describe basic addition, subtraction, multiplication, and division facts?
-Basic Addition Facts: all possible combinations in which each of the addends is a whole number under 10 -Basic Subtraction Facts: all possible combinations in which the subtrahend and the difference are one-digit numbers. -Basic Multiplication Facts: all possible combinations in which each of the factors is a single-digit number. -Basic Division Facts: all possible combinations in which the divisor and quotient are single-digit numbers.
Outline the steps students follow in the crossing-out strategy using.
-Draw the number of lines for the minuend and then subtract by crossing out the number of lines indicated by the subtrahend -Count the remaining lines and draw an equal number of lines on the other side of the equal sign -Write the numeral representing that set of lines
Describe memorization activities that may be used when teaching math facts.
-Paired Practice: students work with each other -Teacher-Directed Practice: the teacher presents facts to a group -Worksheet Exercises -Flash Card Exercises -Fact Games -Computer-Based Activities
Outline the steps in addition the fast way.
-Start counting at the number represented by the numeral in the first added position -Count the lines -Write the numeral representing the sum in the box on the other side of the equal sign
Outline the steps in the beginning addition strategy (addition the slow way).
-Students read equation: 5+2= how many? -Students recite equality rule: "We must end up with the same number on this side AND the other side of the equal sign" -Students draw lines under the first addd and then under the second addend -Students count all lines on that side of the equal sign -Students apply the equality rule and make lines under the box on the other side of the equal sign -Students write a numeral to represent the number of lines
Outline the steps addition with renaming.
-Students read the problem -Students begin adding in the ones column -If the sum of the ones column is 10 or more, students determine that they must rename -Students use expanded notation to determine the number of tens and ones in the sum of the ones column -Students rename by putting the tens in the tens column and the ones in the ones column -Students add the first two tens in the tens column and then add the next ten to that sum -Students write the sum of tens in the tens column
Outline the steps in the cross-out strategy for solving missing subtrahend subtraction problems using 7 - how many = 3.
-Students read the problem: 7- how many =3 -Students draw lines under the minuend: IIIIIII -Students determine the number they must end with on both sides -Students circle three of the seven lines, since they must end with three to make the sides equal: 7- how many =3 (III)IIII -Students cross out uncircled lines: 7- how many =3 (III)-I-I-I-I -Students count crossed-out lines and write numeral in the box: 7-4=3 (III)-I-I-I-I
Describe the types of problems that should be included on the practice worksheets for Format 8.3: Subtraction with Renaming.
-Subtraction with and without renaming -Addition
Explain the approach that this text recommends for introducing students to the concept of beginning subtraction using 6 - 4 = .
-The student first draws the number of lines for the minuend and then subtracts by crossing out the number of lines indicated by the subtrahend: 6-4= how many II-I-I-I-I -The student counts the remaining lines and draws an equal number of lines on the other side of the equal sign: 6-4= how many II-I-I-I-I II -The student writes the numeral representing that set of lines: 6-4=2 II-I-I-I-I II
Describe the procedures for teaching students to solve column addition problems that do not require renaming.
-The teacher has the students read the problem and then points out the place value columns, telling students to first add the ones and then add the tens -Students add the ones and then the tens, writing the sum for each column
What preskills need to be mastered before introducing column addition problems requiring renaming?
-Working addition problems without renaming -Reading and writing numerals -Knowing basic addition facts -Using expanded notation with teen and tens numbers
Describe the three types of errors students may make during the advanced subtraction stage.
1) Basic fact errors 2) Errors caused by failure to rename 3) Errors involving the procedures of renaming
Describe the two main categories of errors that students often make during the beginning addition stage.
1) Component-skill errors: a difficulty on one or more of the component skills or preskills used in the strategy 2) Strategy errors: a fundamental lack of understanding of the strategy
List the five components required for building mastery of math facts.
1) Designing a coherent sequence for introducing facts as previously discussed 2) Coordinating relationship activities with memorization activities 3) Establishing specific performance criteria that indicate when new facts can be introduced 4) Providing intensive and systematic review 5) Implementing record-keeping procedures for monitoring student performance and increasing motivation
Describe the four steps in diagnosing and remedying errors.
1) Diagnosis: The teacher analyzes worksheet errors and hypothesizes about the cause of each error 2) Confirmation: The teacher interviews the student to determine the cause of the error if it is not obvious 3) Reteaching: The teacher provides reteaching through board and/or worksheet presentations of a component skill or a strategy 4) Assess: The teacher tests the student on a set of problems similar to the ones on which the original errors were made
Describe the three guidelines that were followed when the sequences in Figures 6.1-6.4 were constructed.
1) Easier facts are introduced first. 2) Related facts are introduced together. 3) The reverse of specific series of facts is taught relatively soon after the initial series was presented.
Describe three common types of errors when teaching multi-digit addition.
1) Fact Errors 2) Component-Skill Errors 3) Strategy Errors
List the six preskills must master before addition the slow way is introduced.
1) Identifying and writing the numerals 0-10 and the symbols +, -, =, and box 2) Equality rule 3) Reading an equation 4) Drawing the appropriate number of lines to represent a numeral 5) Counting the lines in two groups 6) Writing the numeral that represents a set of objects
Describe the two stages of subtraction instruction.
1) Introducing the concept 2) Multi-digit subtraction
Describe the three types of relationship activities.
1) Plus-One Facts 2) Series Saying 3) Three-Number Fact Families
What two strategies are included in the teaching procedures for relationship activities?
1) Presenting facts in a related series 2) Introducing number fact family exercises that utilize inverse relationships between addition and subtraction and between multiplication and division
Outline the three groups into which column addition may be divided. Provide an example of each type of problem.
1) Problems Not Requiring Renaming (e. g. 24+15=) 2) Problems Requiring Renaming (e. g. 424+317=) 3) Three or More Addends (e. g. 671+424+317=)
Lisa has made a pattern of errors like the following. 37 48 34 +26 +28 +26 513 616 510 Outline the four steps for providing diagnosis and remediation.
1) Reteaching the format for that particular type of problem 2) Presenting several problems using a structured board presentation 3) Leading the students through several worksheet problems using the structured parts of the format 4) Leading the students through several worksheet problems using the less structured parts of the format
Describe the two stages of addition instruction.
1) Strategies designed to establish a conceptual understanding of the process of addition 2) Mental computation rather than relying on representations of concrete objects
Describe the two types of errors that are common during the beginning subtraction stage.
1) Strategy Errors: A fundamental lack of understanding of the strategy 2) Component-Skill Errors: A deficit in one or more of the component skills or preskills used in the strategy
List the three preskills that are required for solving subtraction problems with renaming.
1) The place-value related skills inherent in reading and writing numerals over 10 2) Knowledge of at least six facts that can be used for renaming 3) A conceptual understanding of renaming
Describe the three basic types of column subtraction problems.
1) The subtrahend is smaller than the minuend in each column 2) One or more columns have a subtrahend that is larger than the minuend 3) Column subtraction problems that require renaming
Describe two motivational practice activities that may be used to promote mastery.
1) The teacher or tutor making flash cards for particularly difficult facts 2) Using the flash cards prior to the oral worksheet practice and the math fact race
Describe the exercise suggested to provide a conceptual understanding of renaming in subtraction. How long would this format be presented before the mechanics of renaming are taught?
A diagram showing several packages, each of which contains 10 objects and several single objects is presented and the teacher tells a story that involves giving away some of those objects for several days before the mechanics of renaming are taught
How long after introducing addition facts does the text recommend introducing subtraction facts?
A month or more
Describe how addition the fast way differs from addition the slow way.
Addition the fast way is taught as a transitional step between the strategy in which students draw lines for each member of the sets represented by each addend and later exercises in which students memorize addition facts, while addition the slow way is the first problem-solving strategy taught.
When does the text recommend beginning to teach students to memorize subtraction facts?
As soon as students reach the 80% to 90% accuracy criterion during supervised practice of one-digit minus one-digit subtraction problems
What new preskill is integrated into Format 8.1: Subtraction with Lines for teaching the cross out strategy for subtraction?
Crossing out lines and counting the remaining lines
Compare and contrast the homogenous and heterogenous group programs for teaching basic math facts.
Homogenous Group Program: Teacher-directed instruction with students completing daily exercises on a fact worksheet Heterogenous Group Program: Teachers working with a group of students who demonstrate significant differences in their knowledge of facts
Why does the text recommend waiting at least a month after introducing the facts for one operation before introducing the facts for another operation?
In their observations of lower-performing students, they have found that students have more difficulty when a set of addition facts and the inverse subtraction facts are introduced concurrently.
Why does the worksheet in part D of Format 8.1: Subtraction with Lines include a mix of addition and subtraction problems?
Instructionally-naive students often have difficulty discriminating which of the two similar problem-solving strategies to use
Why are students initially taught to use "minus" as a verb?
It allows students to understand the concept of subtraction without the additional demand of new vocabulary
Why is record-keeping critical to successfully teaching basic math facts?
It helps the teacher know when a student needs additional practice and when a students is ready to progress to the next set of facts
Why does the teacher ask about the number of tens rather than the quantity represented by the tens numeral when teaching students to solve column addition problems?
It reminds students they are working in the tens column
What preskill is required for solving subtraction problems that involve zeros? When is this preskill taught?
Learning the tens-numbers-minues-one facts about a week prior to introducing problems that involve renaming numbers with zeros
What preskills are required before introducing column addition problems that do not require renaming?
Reading and writing numerals through 99 and being able to mentally determine answers to about 25 basic addition facts
Describe the remediation procedure for students who demonstrate a pattern of confusing the addition and subtraction signs.
Reintroducing the less structured worksheet format 8.1, Part D, and instructing students to circle the sign before solving the problem
Contrast relationship activities and mastery activities.
Relationship Activities: make fact memorization easier and more efficient Mastery Activities: are designed to facilitate fact memorization
What activities promote conceptual understanding when teaching basic facts?
Relationship and mastery activities
Describe the strategy for teaching students to rename when zeros are involved.
Rename several digits at once
Describe the error that is common when students solve subtraction problems that require renaming in consecutive columns.
Students become confused of the crossed-out digits
Describe the homogenous group program for teaching basic math facts.
Teacher-directed instruction with students completing daily exercises on a fact worksheet
Write a script showing how Format 7.3: Solving Missing Addends can be modified when the missing addend is zero.
Teacher: Touch problem a. Read the problem. Students: 8+ how many =8 Teacher: Touch the side you start counting on. (Students touch side with 8.) Teacher: Make the lines under the 8. (Students make 8 lines.) Teacher: Touch the side that says 8+ how many." (Students touch that side.) Teacher: How many do you have on that side now? Students: 5 Teacher: Make five lines under the 5. (Students make five lines.) Teacher: This is a special kind of problem. The sides are already equal. Eight on both sides. So you shouldn't make any lines. You plus zero lines. Write a zero in the box. (Check.) Teacher: Eight plus how many equals 8? Students: Zero Teacher: Say the whole statement. Students: 8+0=8
Describe the heterogenous group program for teaching basic math facts.
Teachers working with a group of students who demonstrate significant differences in their knowledge of facts
How can Format 8.3: Subtraction with Renaming be modified for renaming hundreds or thousands?
The teacher first asking the students to identify what they are starting with and taking away in the tens column and then ask if it is necessary to rename to work the problem then leading students through solving the problem
Why are relationship activities use to teach students basic math facts?
They make fact memorization easier and more efficient.
How does the equality rule support students to solve missing addend problems?
This strategy is based on the equality rule and students are presented this form of problem to enhance their understanding of the equality principle and to demonstrate that the equality principle may be used to solve a variety of problem types
How does this text recommend building conceptual understanding of addition? Why is this approach recommend?
Using lines as semi-concrete objects to represent members of sets because drawing lines graphically demonstrates equality, and teachers can more readily monitor student performance on written work because the lines provide a written record of student performance, which makes diagnosis of errors easier
What is the equality rule? How, why, and when is it taught?
We must end with the same number on this side and the other side of the equal sign. The teacher introduces the equal sign and equality rule, demonstrates examples in which the sides are equal and examples in which the sides are not equal, and leads the students in determining whether an equal sign would be drawn between the circles because a grasp of equality is necessary for success in more complicated exercises during the beginning state before addition is introduced.
When does the text recommend introducing column subtraction problems?
When students have memorized approximately 12 subtraction facts
Specify a diagnosis and remediation for the following student errors. 5 + 4 = 8 2 + 5 = 6 6 + 3 = 8 |||| ||||| |||
Component-skill errors in which the student may have touched the first line under the second addend while saying the get-it-going number and then counted on. The teacher presents practice exercises on the component skill in isolation for several lessons before returning to the more advanced problems such as presenting an exercise on coordinating counting and touching lines.
How much time does the text recommend devoting to math facts instruction? Why is it important for teachers to allocate time for math facts instruction?
15-20 minutes per day because students who know facts will be able to compute efficiently and are more likely to achieve success in later problem-solving activities
When does the textbook recommend introducing multiplication facts? What rationale is provided for this recommendation?
Beginning in third grade because knowledge of basic multiplication facts is a critical prerequisite for more advanced operations and should be mastered no later than the end of third grade
How can parents be involved in supporting basic math facts instruction?
By committing to work with their children on facts at home for about 10 minutes three or four days a week
