EDE 4123 Content Chapter Test 2 - Senior Math Methods

Explain the computational estimation--ROUNDING

-It is usually a good idea to round them to them same place value. -You keep a running sum as you round each number EX: 1. 4827, 85, & 710 (actual numbers) [textbook] 5000 + 0 + 1000 = 6000--about 6000 (thousands place) 2. 183, 466, 215 (actual numbers) [class] 200 + 500 + 200 = 900--about 900 (hundreds place)

What a proportion is and how "solving proportions" should be taught (& should NOT be taught) [chp 18]

-A proportion is... -Provide ratio and proportion tasks in a wide range of settings, since the way students approach tasks is greatly influenced by the context of the problem. -Encourage reflective thought, discussion, and experimentation in predicting and comparing ratios. Have students distinguish between proportional and non-proportional situations. -Connect proportional reasoning to existing thought processes; the concept of unit fractions is very similar to unit rates. -Recognize that the use of symbolic or mechanical methods, such as the cross-product algorithm, for solving proportions does NOT develop proportional reasoning.

What a ratio is [chp 18]

-A ratio is a multiplicative comparison of two quantities of measures; symbolically, a ratio is an ordered pair of numbers that express a comparison between the numbers. -Equal ratios result from multiplication or division, not addition or subtraction.

What a unit fraction is [chp 15 & 16]

-A unit fraction is a rational number written as a fraction where the numerator is one and the denominator is a positive integer. -A unit fraction is therefore the reciprocal of a positive integer, 1/n. EX: 1/1, 1/2, 1/3, 1/4 ,1/5, etc

How to divide fractions using the "same denominator" method (You'll have story problems like this on the test) [chp 15 & 16]

-AVOID the "mystery" of "invert and multiply" at all costs! -Go back to the meanings of division with whole numbers. Review the two meanings of division: partition (sharing or rates) and measurement (repeated subtraction). -Use a Common Denominator Algorithm which uses the measurement or repeated subtraction interpretation of division. Use analogy of quarters and nickels. REFER TO ACCORDING HANDOUT [PP]

3 Types of Fraction Models and examples of each [chp 15 & 16]

-AreaModels (Continuous) -Length Models (Continuous) -Set Models (Discrete)

What a Base-Ten Fraction is (fractions with denominators that are powers of 10) [chp 17]

-Begin with Area Models: Interlocking Circular Disks (called a Rational Number Wheel) and 10 x 10 Square Grids to model Base 10 Fractions -Have students model "Base 10 Fractions" and ask the following questions as they do this: >Is this fraction more or less than 1⁄2, 2/3, 3⁄4? >What are some different ways to say this fraction using tenths and hundredths? >Show two ways to write this fraction.

Identify the properties of Addition and Subtraction given a contextual story problem illustrating that property

-Commutative Property for Addition -Associative Property for Addition -Zero Property or Identity Property

The attributes of measurement taught in K-8 mathematics and an example of an informal unit of measurement that could be used to teach each attribute [chp 19]

-Conceptual knowledge of measuring includes being able to: >Understand the attribute being measured --->Make comparisons based on the attribute. >Understand how filling, covering, matching, or otherwise making comparisons of an attribute with units produces what is called a measure -->Use physical models of measuring to fill, cover, match, or make the desired comparison of the attribute with the unit. >Understand the way measuring instruments work -->Make measuring instruments and use them along with actual unit models to compare how each works. -The fact that smaller units produce larger numeric measures, and vice versa, is very hard for young children to understand. -Two Type of Measurement Objectives: >understanding the meaning & technique of measuring a particular attribute >learning about the standard units commonly used to measure that attribute -One of the worst errors in the metric measurement curriculum is having students "move the decimal point " to convert from one metric unit to another prior to a complete development of decimal notation and prior to a complete understanding of each unit.

CCSSM domains

-Counting and Cardinality [K] -Operations and Algebraic Thinking [K-5] -Number and Operations in Base Ten [K-5] -Measurement and Data [K-5] -Geometry [K-8] -Number and Operations--Fractions [3-5] -Ratios & Proportional Relationships [6-7] -The Number System [6-8] -Expressions and Equations [6-8] -Statistics and Probability [6-8] -Functions [8]

How we should introduce and teach decimals [chp 17]

-Decimal concepts should be taught with fraction concepts. -Decimal numbers are simply another way of writing fractions. Connections between fraction and decimal symbolism can help in understanding both. -Extending the Place Value System: Concepts of Base 10 should also be fundamental to the teaching of decimal numbers - that is between any two place values, there is a ten to one ratio. -Percents are simply hundredths and as such are a third way of writing fractions and decimals. -Begin with rational numbers that are easily represented by decimals: tenths, hundredths, and thousandths -Read and say decimals in ways that support student understanding - use "and" followed by a "fraction"; i.e. 2.5 is read "two AND five-tenths" -Use visual models (Rational Number Wheels, Grid Paper, Base 10 Blocks) to help students make meaningful translations between fractions and decimals.

Reasoning strategies for Multiplication basic facts [x]

-Doubles (one of the factors is 2) EX: 7 x 2 = 14 -Fives Facts EX: 3 x 5 = 15 -Zeros and Ones EX: 4 x 0 = 0; 4 x 1 = 4 -Nifty Nines EX: 9 x 4 = 36 -Derived Facts: Using Known "Helping" Facts to Derive Remaining Facts EX: 3 x 4 = ? Student plug in numbers that they do know, 2 x 4 = 8 + 4 = 12

Identify the various problem structures for multiplication

-Equal Groups -Multiplicative Comparison -Product of Measures -Combinations

2 Requirements for Fractional Parts (same size parts & correct number of parts) and how to assess these through "incorrect" drawings of each [chp 15 & 16]

-Fractional parts must be the same size (not necessarily the same shape or arrangement) -The number of equal-sized part that can be partitioned within the unit determines the fraction amount; that is, the number of parts making up the whole determines the names of the fractional parts.

How to use fraction tiles to add and subtract fractions (Have your fraction tiles with you to take the test and matching colored pencils or crayons, you'll have to solve addition and subtraction story problems on the test using fraction tiles like we did in class) [chp 15 & 16]

-Have students add & subtract fractions using a fraction model. Results should come completely from the use of the model. -If students have a good foundation with fraction concepts, they should be able to add or subtract like fractions immediately; it's just like adding or subtracting whole numbers. -For unlike denominators, first have students use models to show the original problem and then the "converted" problem, where one fraction can be easily changed to have the same denominator as the other one. Then try some examples where both fractions need to be changed. Focus attention on "rewriting the problem" in a form where the parts of both fractions are the same. REFER TO ACCORDING HANDOUT

How comparing fractions should be taught (using what we did in class) [chp 15 & 16]

-If the "rules" for comparing fractions are taught before allowing students to think about the relative size of various fractions, there is little chance that any number sense about fraction size will develop. -Begin by comparing Unit Fractions -Conceptual Thought Patterns for Comparing Any Fractions: >Same-size Whole >Same number of parts but different-sized wholes. >More than/less than 1⁄2; More than/less than one whole >Distance from 1⁄2; Distance from one whole -Other methods of comparison: Convert to equivalent fractions, decimals or percents REFER TO ACCORDING HANDOUT

Identify the various problem structures for addition & subtraction

-Join -Separate -Part-Part-Whole -Compare

How the attributes of length should be introduced [chp 19]

-Length is usually the first attribute students learn to measure; however, it is not immediately understood and therefore not the easiest of the attributes for students to understand. -Students should begin their study of measuring length with comparison activities (direct comparison of the lengths of two or more objects). -Common misconceptions if taught procedurally (see pg. 461) -Have students make their own rulers using a unit of their choice as described in Activity 19.13 (pg. 465- 4664).

Reasoning strategies for Addition basic facts [+]

-One-More-Than and Two-More-Than EX: 3 + 1 = 4; 7 + 2 = 9 -Facts with Zero / Adding Zero EX: 5 + 0 = 5 -Doubles EX: 6 + 6 = 12 -Doubles Plus One / Near Doubles EX: 7 + 8 = 15 -10 Frame Facts / Combinations of 10 EX: 3 + 7 = 10; 4 + 6 = 10 -Make-a-Ten / Making 10 EX: 9 + 4 = 13 -Doubles Plus Two EX: 3 + 5 = 8 -Remaining Facts: Using 5 as an Anchor EX: 3 + 6 = 9; 3 + 5 = 8

Various types of ratios [chp 18]

-Part to Whole (fractions, percentages, probabilities) -Part to Part (of the same whole; odds) -Quotients -Rates (a comparison of the measures of two different things or quantities; the measuring unit is different for each value)

Identify the various problem structures for division

-Partition -Measurement

Base-10, 4 partial products

-Partition is the BEST to use when modeling to students. 4 PARTIAL PRODUCTS: -Make three separate columns -Label the first column: DIMENSIONS OF 4 SMALL RECTANGLES USING BASE-10 LANGUAGE, the second column: ACTUAL PIECES, & the third column: NUMERIC SYMBOLS -Look at the area model drawn to fill in your charts -Under the first column write down the dimensions in hundreds, tens, & ones (1 ten x 2 tens; 7 tens x 1 one) -Under the second column write down the number amount and its value place ( 2 hundreds; 7 tens) -Under the third column write down the amount using numbers (200; 70) -Finally add up all the numbers and put it into the unit amount squared (405 ft^2) EX: 15ft x 27 ft = ? D.O.4.S.R.U.B.10.L Actual Pieces Numeric Symbols 1 ten x 2 tens = 2 hundreds = 200 2 tens x 5 ones = 10 tens = 100 1 ten x 7 ones = 7 tens = 70 5 ones x 7 ones = 35 ones = + 35 ----------- 405 ft^2

Base-10 area model

-Partition is the BEST to use when modeling to students. AREA MODEL: -First you look at the 2 multi-digit number being multiplied -With the first number, you look at the tens place & draw that amount as lines. (horizontally) Make sure to put small spaces in between them. -Then look at the ones place of the first number and draw that amount as smaller lines. (horizontally) Make sure to put small space in between them. -Do the exact same process for the second multi-digit number but do this vertically. -Then once you do this, create a rectangle -Next, draw lines in between each of the spaces. -Finally, label which sections are hundreds, tens, or ones.

What proportional reasoning is [chp 18]

-Proportional reasoning is a way of thinking about and recognizing multiplicative situations in realistic situation. It goes well beyond "setting up and solving proportions". -Involves the ability to compare ratios and to predict or produce equivalent ratios -Requires the ability to compare mentally different pieces of information and to make comparisons, not just of the quantities involved, but of the relationships between the quantities , as well. -Involves both quantitative and qualitative thinking. -Is not dependent on a skill with a mechanical or algorithmic procedure.

Using Benchmarks for fractions (0, 1⁄2, 1) to compare fractions and develop fraction number sense [chp 15 & 16]

-Requires that students have some intuitive feel for fractions; they should know "about" how big a particular fraction is & be able to tell easily which of two fractions is larger -Children have a tremendously strong mind-set about numbers that causes them difficulties with the relative size of fractions. -The inverse relationship between the number of parts & the size of parts cannot be told but must be a creation of each student's own thought process -Benchmarks of Zero, One-Half, and One - Understanding why a fraction is close to 0, 1⁄2, or 1 is a good beginning for fraction number sense.

The 6 conceptual thought patterns for comparing fractions & when each is used (be able to identify which conceptual thought pattern should be used, given a pair of fractions) [chp 15 & 16]

-Same-size Whole -Same number of parts but different sized wholes -More than/less than 1⁄2 -More than/less than one whole -Distance from 1⁄2 -Distance from one whole

Explain the computational estimation--COMPATIBLE NUMBERS

-Sometimes is useful to look for 2 or 3 compatible numbers that can be grouped to equal benchmark values. -If numbers in the list can be adjusted slightly to equal these amounts, that will make finding an estimate easier. -In subtraction, it is often possible to adjust only one number to produce an easily observed difference. EX: 1. 41, 29, 63, 17, 65, 48, & 85 (actual numbers) [textbook] 41 + 63 = 100; 29 + 65 = 100; 17 + 85 = 100 100 + 100 + 100 = 300 I didn't use the 48, so I'll change that to 50 300 + 50 = 350--about 350 2. 183, 466, 215 (actual numbers) [class] 150 + 450 + 220 = 820--about 820 (hundreds place)

How measurement estimation should be experienced in the elementary classroom [chp 19]

-Strategies for Estimating Measurements: >Develop and use benchmarks or referents for important units. >Use "chunking" or subdivision when appropriate. >Iterate a unit mentally or physically. -Tips for Teaching Measurement Estimation: >Help students learn strategies by having them use a specified approach. >Discuss how different students made their estimates. >Accept a range of estimates. >Do not promote a "winning" estimate. >Encourage students to give a range of measures that they believe includes the actual measure. >Make measurement estimation an ongoing activity. >Be precise with your language - use "measure" and "estimate" appropriately.

How the formulas of rectangles, parallelograms, triangles & trapezoids should be taught (as completed in class) [chp 19]

-Students should never use formulas without first participating in the development of those formulas. -All of the standard area formulas should be developed together as an integrated whole - using base times height for all of them in some capacity. -Misconceptions to keep in mind: >Confusing linear and square units. >Difficulty in conceptualizing the meaning of height and base. -Guided Activity: Use Color Tiles, scissors, scotch tape and the rectangles, parallelograms, triangles and trapezoids given to you to discover the area formula for each.

How equivalent fractions should be taught (using what we did in class) [chp 15 & 16]

-The CONCEPT: Two fractions are equivalent if they are representations for the same amount or quantity; if they are the same number -The PROCEDURE: To get an equivalent fraction, multiply (or divide) the top and bottom numbers by the same nonzero number -GENERAL APPROACH: For a conceptual understanding of equivalent fractions, have students use models to generate different names for models of fractions -When teaching equivalent fractions: >Refer to the "renaming" process as "simplifying" >Avoid penalizing students for not simplifying fraction as this sends the message that the two fractions are not equivalent or the same; yet, they are REFER TO ACCORDING HANDOUT

The meanings of the top and bottom numbers in a fraction (part under consideration or being counted and parts in the whole) [chp 15 & 16]

-The number on the top (the numerator) tells how many "things" are being considered; this number counts the parts or shares. -The number on the bottom (the denominator) tells what kind of "thing" is being considered; this number tells what is being counted.

How to multiply fractions using a rectangular area grid (You'll have story problems like this on the test) [chp 15 & 16]

-There is a greater need to review the meaning of the operation with whole numbers - as in so many sets of a certain size, not multiplication makes bigger. >For example, 3 x 5 means "3 sets of size 5"; so 1/3 x 1⁄2 is "1/3 of a set of size 1⁄2"; -Remember the first factor tells how much of the second factor you have or want. -To teach multiplication of fractions conceptually, begin with an area model (similar to using an area model for multi-digit whole number multiplication), but where the "area" is the whole. REFER TO ACCORDING HANDOUT [PP]

Reasoning strategies for Subtraction basic facts [-]

-Think Addition EX: 15 - 8 = ? You need to use 7 + 8 = 15 -Doubles & Near Doubles EX: 8 - 4 = 4; 13 - 7 = 6 -Facts with Zero EX: 6 - 0 = 6 -One-less-than & Two-less-than EX: 6 - 1 = 5; 8 - 2 = 6 -Ten-Frame Facts (Minuend equals 10) EX: 10 - 3 = 7 -Sums Greater than 10 (Minuend is greater than 10): Take from 10 & Add

Explain the computational estimation--FRONT END

-This approach is reasonable for addition & subtraction when all or most of the numbers have the same number of digits -Only the front (leftmost) number is used & the computation is then done as if there were zeros in the other positions. -After adding or subtracting the front digits, an adjustment is made to correct for the digits or numbers that were ignored. EX: 1. 398, 4250, & 2725 (actual numbers) [textbook] 0 + 4 + 2 = 6 --about 6000 (thousands place) 3 + 2 + 7 = 12 --about 7200 (hundreds place) 2. 183, 466, 215 (actual numbers) [class] 1 + 4 + 2 = 7--about 700 (hundreds place) 8 + 6 + 1 = 15--about 850 (tens place)

Be able to identify if a proportional story problem should be solved with a unit rate or a factor of change method and in either case, be able to identify what the unit rate IS or what the factor of change IS [chp 18]

-Unit Rates - used with a ratio of two measures in the same setting (different referents) -Scale Factors - used with a ratio of two corresponding measures in different situations (same as "Factor of Change"; same referents) -Guide students to analyzing the quantities in a proportional situation and then based on these quantities, use either a unite rate or a scale factor to solve the proportion. REFER TO ACCORDING HANDOUT

How the attributes of area should be introduced [chp 19]

-When comparing two areas, there is added consideration of shape that causes difficulties not present in length measures. Shapes can differ; length is always the measure of a "line segment". -One of the purposes of early comparison activities with areas is to help students distinguish between size (or area) and shape, length, and other dimensions. -Cutting a shape into two parts and reassembling it in a different shape can show that the before and after shapes have the same area, even though they are different shapes. -Your objective in the beginning is to develop the idea that area is a measure of covering (not to discuss formulas).

How simplifying fractions should be taught (using what we did in class) [chp 15 & 16]

....... REFER TO ACCORDING HANDOUT

Contrast invented strategies vs traditional algorithms--contrast the 2 by discussing their 3 differences and be able to *explain each of the 3 differences

1. Invented strategies are NUMBER-ORIENTED rather than digit-oriented (245 is seen as 2 hundreds, 4 tens, 5 ones) 2. Invented strategies are LEFT-HANDED rather than right-handed (computation starts with the left most digit [largest value], not the right most digit [smallest value]) 3. Invented strategies are FLEXIBLE rather than rigid (the strategy is based on the numbers)

What a measurement is [chp 19]

A measurement is a number that indicates a comparison between the attribute of the object being measured and the same attribute of a given unit of measure. This attribute is a continuous.

Zero Property or Identity Property [+ & -]

Adding or subtracting zero to a number doesn't change its identity

Associative Property for Addition [+]

Adding three addends in several different orders

Basic Facts--"Helping Facts"

EX: 6 x 7 = 42 --helping fact: 5 x 7 = 35 4 x 3 = 12 --helping fact: 4 x 2 = 8 or 3 x 5 = 15

Commutative Property for Addition [+]

Pair contextual problems that have the same addends

How to use each of the three computational estimation strategies to determine decimal placement in decimal computations [chp 17]

REFER TO ACCORDING HANDOUT

Set Model, Cuisenaire Rod and Circular Region Problems like we completed in class [chp 15 & 16]

REFER TO ACCORDING HANDOUT

How to use a Rational Number Wheel to convert a fraction to a decimal (bring your Rational Number Wheel with you) - will have to explain how to model any given fraction using a Rational Number Wheel and how to "convert" the modeled fraction into Base 10 fractions and then decimals (like on handout completed in class) [chp 17]

REFER TO ACCORDING HANDOUT Explanation:

How to use Base-10 Blocks to model decimals [chp 17]

REFER TO ACCORDING HANDOUT [PP]

Be able to explain what the answer to a proportional comparison problem would be if solved additively and if solved multiplicatively [chp 18]

REFER TO TEXTBOOK...

Multiplicative Comparison problems & examples [x]

The answer is the same type of thing as one of the numbers in problem, but two quantities are being compared EX: 1. Caleb and Joe both have marble collections. Caleb has 20 marbles. Joe has two times as many marbles than Caleb. How many marbles does Joe have? [class] 2. Jeff read 12 books during the month of August. He read four times as many books as Paul. How many books did Paul read? [class]

Equal Groups problems & examples [x]

The answer is the same type of thing as one of the numbers in the problem EX: 1. Anna has 2 packs of gum. If each pack contains 6 pieces of gum, how many pieces of gum does Anna have? [class] 2. Every time Jacob does his chores he gets 10 dollars. If Jacob has done his chores 7 times, how many dollars will he have? [class]

Partition problems & examples [/]

The number of groups is known, but not the size of each group (dealing cards to a group of people) EX: 1. Gabrielle has 12 eggs in her refrigerator. She wants to cook eggs for 3 people. How many eggs can each person have? [class] 2. Maya went to the pet store to buy treats for her 3 dogs. Maya buys a box of 15 treats. How many treats will each dog get? [class]

Measurement problems & examples [/]

The size of each group is known, but not the number of groups EX: 1. I have 15 hamsters and there are 3 hamsters in each cage. How many hamster cages do I have? [class] 2. Sam has collected markers over the years. One day, Sam counted his markers and saw that he had 90 markers. Sam has marker boxes that can only hold 9 markers each. How many marker boxes does Sam have? [class]

Combinations problems & examples [x]

The two numbers in a multiplicative problem represent different types of things; the answer will be a "new thing" (shirts and shorts combined form outfits) EX: 1. Vicky is picking out a new skateboard. The can be black, gray, tan, or dark brown, and the wheels can be yellow, purple, green, pink, or orange. How many different combinations does Vicky have to choose from? [internet] 2. Bridgitt is picking out her outfit for the day. She has a green, pink, and white t-shirt, and she has blue, black, and brown leggings. How many different outfits can Bridgitt create? [made up]

Product of Measure problems & examples [x]

The two numbers in a multiplicative problem represent different types of things; the answer will be in terms of something different (a 5 INCH by 7 INCH frame frames a picture that is 35 SQUARE INCHES) EX: 1. Jack uses 19 quarts more paint for the outside of a shed than for the inside of the shed. If Jack uses 89 quarts of paint for the outside of the shed, how many gallons will he use to paint the inside of the shed? [internet] 2. Mrs. Smith is building a bench and needs 6 lengths of wood that each measure 1.4 meters. How many meters of wood does he need to purchase? [internet]

Part-Part-Whole problems & examples [+ & -]

These are 2 types of 1 thing (quarters & dimes are COINS; pecans & almonds are NUTS) [Both addition & subtraction] EX: 1. Mickey Mouse has 4 cubes of swiss cheese & 7 cubes of cheddar cheese. How many cubes of cheese does Mickey have? [Addition] [PP] 2. Trevor & Kaitlyn are cooking supper for their parents tonight. They have 15 potatoes and onions in a bin. Trevor counts 6 potatoes in the bin. How many onions are there? [Subtraction] [class]

Compare problems & examples [+ & -]

These have NO ACTION [Both addition & subtraction] -Make sure that the 2 parts being added remain distinct so that once the addition is performed (the answer set is made) the student can still "see" the original 2 sets. -When modeling subtraction problems, keep the part subtracted distinct, but present, fro the part that remains. EX: 1. Chip has 9 walnuts. Dale has 4 fewer walnuts than Chip. How many walnuts does Dale have? [Subtraction] [PP] 2. Brad has 5 more bananas than Nick. Nick has 3 bananas. How many bananas does Brad have? [Addition] [class]

Join problems & examples [+ & -]

These must have a physical ACTION [mainly addition] EX: 1. Daisy Duck has several pots of flowers. After Donald gave Daisy 6 more pots of flowers, she had 15 pots of flowers. How many pots of flowers did Daisy have before Donald gave her any? [PP] 2. Jodi has 4 suckers. Max gave her 10 more suckers. How many suckers does Jodi have? [class]

Separate problems & examples [+ & -]

These must have a physical ACTION [mainly subtraction] EX: 1. Pluto gave Goofy 7 of his dog bones leaving him with 5 bones. How many bones did Pluto have before he gave any to Goofy? [PP] 2. Abby makes three cupcakes. She gives two cupcakes to her friends. How many cupcakes does she have left? [class]

Length Models (continuous)

each part has the same length and the fraction is both a distance and the point that is that distance away from 0 EX: Cuisenaire rods

Set Models (discrete)

objects are put into groups; confusion arises between the name of the share and the number of objects in the share EX: Counters

Area Models (continuous)

same size parts doesn't mean the parts have the same shape EX: Pattern Blocks