Math Methods Exam 2

Describe current curricular recommendations regarding the teaching of computation. What is your personal position on computational instruction? What are the implications of your position for classroom instruction? Use all four (mental, written, estimation, and calculator) together in teaching instruction to help students best understand the concept. They build off each other and work together to best benefits the students.

- mental computation "Done all in the head without tools such as calculator or paper/pencil" Many strategies - especially ones using rounding. - written computation writing out the equation and solving using paper and pencil - computational estimation rounding numbers - computation with calculators doing mathematical equations on a calculator

Describe and apply thinking strategies that you would use with children in learning the basic facts of addition, subtraction, multiplication and division. Be able to recognize common strategies and to give examples of strategies you would use with children.

-Basic Addition Facts: involves two one-digit addends and their sum. +ask students to look for patterns as they fill out addition tables (ask them Q's!) +Commutativity: use a variety of combinations with adding numbers so students understand that order does not affect the outcome +adding one and zero: students often come with this understanding but to reinforce it they should have experiences with objects, followed by paper and pencil activities. Recognition of the patterns should also be encouraged. Zero added to any number does not change the number...this understanding comes from many concrete examples. It is hard for students to picture adding nothing. +Adding Doubles and Near Doubles:Students benefit from working with objects (ex: egg cartons show 6=6, calendars show 7+7) followed by drawings. Can help with learning near doubles, which are facts that are 1 more or 1 less than doubles +counting on: start with the larger number (ex: 9+2 ....9, 10, 11) Should only be used when at least one of the numbers is small +Combinations to 10: (pair up numbers in a large group that add up to 10) +Adding 10 and beyond: with this strategy of adding to 10 and beyond, one number is broken apart so that one part of it can be used with the other addend to make 10. Then the remaining part of the first number is added to the 10 to go beyond to the final teen sum. -Basic Subtraction Facts: rely on the inverse relationship of addition and subtraction. The subtraction facts result from the difference between one addend and the sum for all one-digit addends. +Subtracting one and zero:easy once students know how to add 1 and how to add 0, students benefit from working with materials and observing patterns +Doubles:children need to know how to do doubles for addition +Counting back:related to counting on in addition, most efficient when the number is to be subtracted by 1 or 2 +Counting on: used best when the difference is 1 or 2. (Example: 8-6 = ?.... Think: 6...7,7 --Well that's 2 numbers that I counted on) -Basic Multiplication Facts: each involve two-digit factors and their product +Commutativity: students should try many combinations to realize that order is irrelevant +Skip Counting: works best for the multiples children know best, 2s and 5s, but can be applied to other numbers +Repeated Addition: works best when one of the factors is less than 5. Student changes multiplication example to an addition problem. +Splitting the product into known parts: students can use their known facts to drive others. This strategy is based on the distributive property of multiplication. It can be approached in terms of "one more set," "twice as much as a known fact," or "known facts of 5" +multiplying by one and zero: generally learned from experience working with multiplication. Children need to be able to generalize that "multiplying by 1 does not change the other number" and that "multiplying by 0 results in a product of 0" +patterns: patterns can be helpful. For example 9! -Basic Division Facts: rely on the inverse relationship of multiplication and division +division is the inverse of multiplication +think about things that are close to multiplying together for example what is 49 divided by 6? Well 6x8 is 48...so 49 divided by 6 is 8 with 1 remainder. +Change it into a multiplication problem (example 42/6=? → 6x_=42 +children need to recognize that division is just repeated subtraction +skip counting backward (but it is more difficult strategy to use) +6 divided by 0 is impossible

​Describe how to develop each of the following concepts in teaching rational numbers.

-Partitioning: start with even divisions and then move into odd divisions -Ordering: unit fractions (½, ⅓, ¼, ⅕); non-unit fractions (¾ = ¼ + ¼ + ¼) -Equivalent Fractions (3/6 = ½)

Describe appropriate uses of alternative algorithms. Be able to illustrate your discussion with specific examples of alternative algorithms.

Addition Type of Algorithm Sample Problem Description Standard 26 +17 ------ 43 Add 6 and 7 ones. Record 3 ones and regroup 1 ten. Add 1+2+1 tens. Record 4 tens. Front End 42 +26 ------ 68 Work left to right Partial Sums 26 +17 ----- 6+7=13 + 20+10=30 ------ 43 Record each partial sum and add. This simplifies regrouping. (Good for students struggling with regrouping) Rounding 26-->23 + 17-->20 ___ 43 Round 17 and 20. Since 3 is added to 17 to equal 20, subtract 3 from 26. Rewrite the problem and add. (good for students struggling w/regrouping) Low Stress Check the stupid chart! ;) Ain't no way I can put this down! haha Record each step of the addition sequence. Subtraction Type of Algorithm Sample Problem Description Standard 42 -17 ------ 25 Regroup 4 tens and 3 tens and 12 ones, then perform subtraction. Equal Additions 42 ← add a group of 10 -17 ← add a group of 10 ------ 25 Since 1 ten is added to make 12, 1 ten is also added to make 27. Austrian Additive What number is added to 7 to equal 2? We know the 2 must stand for 12. So, what number added to 7 equals 12? Record 5. Add 1 ten from 12 to 17, making 27. Rounding 42 → 45 -17 → -20 ----- ----- 25 Round 17 to 20. Since 3 added to 17, 3 is added to 42. Nines Complement 42 → 42 -17 +82 ---- ---- 25 124 Find the nine complement of Find the nine complement of 17 (17+82+99). Rewrite the problem as addition and move the "misplaced one." Low Stress Do all regrouping and then perform subtraction.

Describe how to develop a conceptual and algorithmic understanding of the following operations with fractions. (You should be able to discuss how you would use materials to develop these ideas.)

Addition and subtraction with like denominators Addition and subtraction with unlike denominators Fractions greater than one Multiplication of a whole number times a fraction Multiplication of a fraction times a whole number Multiplication of a fraction times a fraction Division of a fraction by a whole number

Define each of the four basic operations

Addition: Finding the total, or sum, by combining (adding) two or more numbers.. It means, "finding how many in all". Subtraction: Taking one number away from another. Finding the difference between at least two numbers/quantities Multiplication: The basic idea of multiplication is repeated addition -- the process of adding a number to itself a certain number of times Division: Division is splitting into equal parts or groups. It is the result of "fair sharing".

Define and give a number sentence to illustrate each of the following properties

Commutative property of addition Definition: Changing the order of the addends does not affect the sum Number Sentence: 5+8=13, 8+5=13 (p. 197) ​commutative property of multiplication Definition: Changing the order of the factors does not affect the product Number Sentence: 4x6=24, 6x4=24 (p. 204) associative property of addition​ Definition: When I am adding three or more numbers, it does not matter which number I start with. Number Sentence: (2+3)+ 4 = 2 +(3+4) (p. 193, P.S. The chart on this page is very helpful!) associative property of multiplication Definition: When I am multiplying three or more numbers, it does not matter which number I start with. Number Sentence: (2x3)x 4 = 2 x(3x4) (p. 193) identity elements of addition/multiplication/subtraction/division Addition: "Finding how many in all" Inverse operation of subtraction Subtraction: Having one quantity, removing a specified quantity from it, and noting what is left Inverse operation of addition Not associative or commutative Multiplication: Repeated addition Inverse operation of division Division: Repeated subtraction Inversion operation of multiplication Not associative or commutative (p. 189-190) distributive property of multiplication over addition Because multiplication is simply repeated addition, number sentences such as this can occur (p.193): 8(2 + 3) = (8 x 2) + (8 x 3)

Describe four different models for representing multiplication concepts. Be able to give an example to illustrate each one

Equal-Groups Problems: most common type of multiplicative structure, "where you are dealing with a certain number of groups, all the same size. When both the number and size of the groups are known (but the total is unknown), the problem can be solved by multiplication". Example: "Andrew has two boxes of trading cards. Each box holds 24 cards. How many cards does he have altogether?" The problem would be written as 2x24=48, the first factor (2) tells us how many groups or sets of equal size are being considered, and the second factor (24) tells us the size of each set, and the third factor (48) is the product. Comparison Problems: "involve two different sets, but the relationship is not one-to-one" (compare two different sets). "In multiplicative comparison situations, one set involves multiple copies of the other". Example: "Hilary spent $35 on Christmas gifts for her family. Geoff spent 3 times as much. How much did Geoff spend?" Hilary's spending is being compared to Geoff's and involves multiplication to solve it. (How many times as much-involves comparative multiplication) Combination Problems: "Here the two factors represent the sizes of two different sets and the product indicates how many different pairs of things can be formed, with one member of each pair taken from each of the two sets". Example: "Consider the number of different sundaes possible with four different ice cream flavors and two toppings if each sundae can have exactly one ice cream flavor and one topping" (create a chart). Area and Array Problems: (common multiplicative structure) "The area of any rectangle (in square units) can be found either by covering the rectangle with unit squares and counting them all individually or by multiplying the width of the rectangle (number of rows of unit squares) by the length (number of unit squares in each row)" (helps students visualize multiplication). Example: In a classroom of rectangular array, there are 6 rows with 8 chairs in each row. How many chairs are there total in the classroom? ("the total number of objects can be found by multiplying the number of rows by the number of objects in each row) 6 rows of chairs x 8 chairs in each row= 48 chairs total.

Describe how you would teach the standard algorithms for addition, subtraction, multiplication, and division. How would you build a bridge between conceptual understanding and algorithmic proficiency? Be able to describe appropriate methods and materials to use in this process.

For any standard algorithm, always start with manipulatives and allowing the children to explore and learn the concept through manipulation. The students need to know the concept of addition, subtraction, multiplication, and division before they can perform any algorithm. That must be the basis of our understanding of how we teach standard algorithms. Addition: Subtraction: Multiplication: Division:

Distinguish between division as measurement and division as partitioning. Be able to give an example to illustrate each type of division.

Measurement Division: (repeated subtraction) You know how many objects are in each group and you must determine the number of groups. Example: Jenny had 12 candies. She gave 3 to each person. How many people got candies. How to solve: Make candy piles of 3 repeatedly until all the candies are gone. You are measuring how many groups of 3 she can make from the original pile of 12. Partitioning Division: (Sharing) A collection of objects is separated into a given number of equivalent groups and you seek the number in each group. In contrast with measurement situations, here you already know how many groups you want to make, but you don't know how many objects must be put in each group. Example: Gil had 15 shells. If he wanted to share them equally among 5 friends, how many should he give to each friend? How to solve: Imagine Gil passing out the shells to his five friends (one for you, one for you, one for you, etc., and then a second shell to each person, and so on)until they are all distributed, and then checking to see how many each person got. -This is hard to show in a diagram, but relatively easy for children to act out-Dealing cards for a game is another instance of a partition situation.

Identify three different interpretations of rational numbers (numbers written as a/b with b not equal to zero).

Part-whole ¾ of a pizza (three ¼ pieces) Quotient division: three cookies divided between 4 people Ratio relationship: 3 girls for every 4 boys

Recognize different forms of joining and separating problems. Be able to give examples of how you would support students in developing meaningful conceptual understanding of these forms of problems.

Problem Type Join (Result Unknown) Ann had 5 shells. Maria gave her 8 more shells. How many shells does Ann have altogether? (Change Unknown) Ann has 5 shells. How many more shells does she need to have 13 shells? (Start Unknown) Ann had some shells. Maria gave her 5 more shells. Now she has 13 shells. How many shells did Ann have to start with? Separate (Result Unknown) Nathan had 13 shells. He gave 5 to Juan. How many shells does Nathan have left? (Change Unknown) Nathan had 13 shells. He gave some to Juan. Now he has 5 shells left. How many shells did Nathan give to Juan? (Start Unknown) Nathan had some shells. He gave 5 to Juan. Now he has 8 shells left. How many shells did Nathan have to start with? Part-Part-Whole (Whole Unknown) Thomas has 5 red buttons and 8 blue buttons. How many buttons does he have? (Part Unkown) Thomas has 13 buttons. 5 are red and the rest are blue. How many blue buttons does Thomas have? Compare (Difference Unknown) Abby has 13 buttons. Ehneida has 5 buttons. How many more buttons does Abby have than Ehneida? (Compare Quantity Unknown) Ehneida has 5 buttons. Abby has 8 more than Ehneida. How many buttons does Abby have? (Referent Unknown) Abby has 13 buttons. She has 5 more buttons than Ehneida. How many buttons does Ehneida have? -Separation or take away involves having one quantity, removing a specified quantity from it, and noting what is left. This is the easiest for children to learn. But constantly using the words "take away" makes children think this is the only type of subtraction problem. -Comparison, or finding the difference, involves having two quantities, matching them one to one, and noting the quantities, matching them one to one, and noting the quantity that is the difference between them. -part-whole problems: a set of objects can logically be separated into two parts. You know how many are in the entire set and you know how many are in one of the parts. You need to find out how many must be in the remaining part.

Describe four different models which may be used in teaching fraction concepts. Be able to distinguish between discrete and continuous models

Region Model [continuous]: the region of the whole (the unit) and the parts are congruent (same size and shape) example: using a rectangle or a circle to show fractions Length Model [continuous]: any unit of length can be partitioned into fractional parts of equal length example: folding a strip of paper into halves, fourths, etc. Set Model [discrete]: uses a set of objects as a whole example: asking students if 12 coins can be split equally among 4 children Area Model [continuous]: a more general version of the region model, in which the parts must be equal in area but not necessarily congruent example: having 15 marbles as the whole and then partitioning them into 5 equal groups -Continuous: all a part of the same piece -Discrete: separate pieces (counting chips)

Identify the common misconceptions and difficulties students face in understanding fractions

Students perceive the fraction as two quantities instead of one Students find ordering fractions more complex than ordering whole numbers Fraction equivalence is misunderstood in pictorial models Difficulties due to early symbolization

Identify the prerequisite skills that students need in order to begin working successfully with decimals.

Students should already have a previous understanding of common fractions and place value. Children should also be able to understand how to order and round numbers to be able to transfer that knowledge into ordering and rounding decimals.

Describe the important relationships children should recognize between the various operations (their concepts and algorithms). How can these relationships be developed?

The important relationships that children should recognize in various operations is that students need to know they conceptual understanding first so they know what they are doing. EX. When using base ten blocks, students can know how to place certain blocks in the correct spot, but they do not fully understand what they are doing conceptually or why. By having the students first understand why it is important to have the different numerical places (concept) and then are shown how to perform that with the base ten blocks( algorithms) students will fully understand the concepts better. To do so, one can use materials, place value, show students how to do a problem and ways they can check it to make sure it is correct. Overall, students need to be actively involved in constructing their own mathematical learning.

Identify the appropriate way to read decimal numbers so that students can easily recognize the place value concepts and the relationship between fractions and decimals.

Using academic language is essential so the student sees the relationship between the place values. Reading the number 32.43 as "thirty two and forty three hundredths" helps the student visualize the number and understand the relationship between tenths and hundredths. The relationship between fractions and decimals is hard for students. 2/10 is the same as 0.2 and 23/100 is the same as 0.23, they are read the same but written differently. 2/10 is the same as 2 divided by 10, which equals 0.2. this is the math computation the students need to understand to know the relationship between fractions and decimals.

Describe each of the following forms of estimation and give an example of how each one might be used in mathematics instruction

adjusting and compensating for... -front-end estimation ​- the front end strategy is basic, yet powerful. Involves 1) checking the leading or front-end digit of the number, and 2) the place value of that digit. Just use the most important digits (the first digits). This number will end up being a low estimate of the real number. -clustering - also called averaging. 2 step process: 1) estimate the average value of all #s, 2) Multiply by the number of numbers in the group -flexible rounding​ - rounding to get numbers that are easier to work with. Involves rounding all numbers to the easiest number possible and them add them -compatible numbers - mental computation; most powerful for division/multiplication; you round one of the numbers to make the computation compatible (if it is 60/8, you would round the 60 to 64 and the problem would be 64/8).

escribe effective guidelines for encouraging computational estimation in your classroom.

hen encouraging computational estimation remind students that the answer is not going to be exactly correct, but it will be close to the correct answer. This is all done mentally to get a quick idea. Give your students real life examples of when you would use this. (shopping to find %). Remind them that they need to use different strategies. Inform students what the thought process will be like, a) Before computing one can think (hmmm if a sandwich is $.99 and a pop is $.79, can I buy them for $2? b) During computing, Wait that can't be right because ⅔ and ¾ are each more than one-half, so the total must be greater than one. c) After Computing, I can use my calculator to see if I messed up

Define and recognize examples of each of the following concepts involved in teaching rational numbers

partitioning​- Process of sharing equally (½ → ¼→ ⅛, etc.) ordering- Relative size of fractions (½, ⅓, ¼, ⅕, etc) ​equivalence- Different ways to represent the same amount (¼= 2/8, ½= 3/6, etc.)

Describe how to develop an understanding of tenths, hundredths, and thousandths with students.

they can group 10 units of 1 to reach the next place value. The same is true with grouping 10 units of 10 to reach the next place value (hundreds). Students should be able to identify the tenths, hundredths, and thousandths place. Develop this by giving numbers and letting the student tell which number is in the tenths and which is in the hundredths. As the teacher, read the numbers using academic language (ex: 24.09 is read "twenty four and nine hundredths" not twenty four point oh nine). This reinforces the ties between place value meanings. For students that struggle, use place value grids to help students visually see where the decimal point goes.