comparing two fractions - education development … · ... the smaller the value of the fraction...

$: Comparing Two Fractions - Education Development … · ... the smaller the value of the fraction (e.g., they see 1/3 as smaller than 3/5). ... Less than ()$
This research was supported by the IES US DOE through Grant 305A110306 to EDC Inc The opinions expressed are those of the authors and do no represent views of the IES or the US DOE

Education Development Center Inc | EM2 copy2015 All Rights Reserved

wwwem2edcorg

Eliciting Mathematics Misconceptions

ASSESSMENT

Understanding Fractions

The Comparing Two Fractions assessment is designed to elicit information about several common misconceptions that students have when comparing two fractions

bull Misconception 1 Viewing a Fraction as Two Separate Numbers Applying Whole-Number Thinking

bull Misconception 2 An Over-Reliance on Unit Fractions A Focus on ldquoSmaller Is Biggerrdquo

bull Misconception 3 Numerator and Denominator Have an Additive Relationship A Focus on the Difference from One Whole

Although you can access the assessment here at any time we strongly recommend that you reference the information below to learn more about these misconceptions including how they appear in student work and how to score the pre- and post-assessments once you have given them to students

raquoraquo Contents

Topic Background Learn about concepts related to comparing fractions 2

Common Misconceptions Learn about student misconceptions related to the topic 4

Administering the Pre-Assessment Learn how to introduce the pre-assessment to your students 7

Scoring Process Learn about the scoring process by reviewing the Scoring Guide 9

Sample Student Responses Review examples of student responses to assessment items 44

Administer the Post-Assessment Learn how to introduce the post-assessment to your students 54

Comparing Two Fractions

Comparing Two FractionsASSESSMENT

2 Eliciting Mathematics Misconceptions | Assessment 1 Comparing Two Fractions

Topic Backgroundraquoraquo Learn about concepts related to comparing fractions

Before they can successfully compare two fractions students need to have a foundational set of conceptual understandings of fractions for example

bull An understanding of fraction language and symbols

bull An understanding of iterating (making multiple copies of) fractionsmdashin other words they know that a fraction is 15 when five copies of it make one whole

bull A sense of the relative size of fractions

bull The ability to recognize and understand fraction notation

The comparison of fractions is often initiated by using unit fractions in order to focus on the relationship between the size of the pieces when partitioningmdashfor example understanding that 13 is greater than14 because thirds are larger pieces than fourths Learning often progresses to comparing other fractions with specific characteristics for example

bull Fractions with the same denominator (eg comparing 25 to 35) to understand that the number of parts matters when comparing fractions with same-size parts

bull Fractions with the same numerator (eg comparing 24 to 23) to understand that when you have the same number of pieces (same numerator) you must consider the size of the pieces (denominators)

bull Fractions that are slightly more or less than key benchmarks such as 14 12 and 34 (eg 316 35 and 78 respectively)

bull Fractions that are close to one whole (eg 1112 67 910)



raquoraquo Topic Background

Connections to Common Core Standards in Mathematics (CCSS)

The CCSS outline specific understandings that students should be able to meet at each grade level

At grade 3 students should be able to

raquo Understand the size of a fraction in relation to the whole and to understand that fractions with different numbers can be equivalent to one another

raquo Develop understanding of fractions as numbers

raquo 3NF3 Explain the equivalence of fractions in special cases and compare fractions by reasoning about their size Compare two fractions with the same numerator or the same denominator by reasoning about their size Recognize that comparisons are valid only when the two fractions refer to the same whole Record the results of comparisons with the symbols lt = or gt and justify the conclusions (eg by using a visual fraction model)

At grade 4 students should also be able to

raquo Build on prior understandings of fraction size and comparison of fraction pairs with the same numerators or the same denominators

raquo Extend understanding of fraction equivalence and ordering

raquo 4NF2 Compare two fractions with different numerators and different denominators for example by creating common denominators or numerators or by comparing to a benchmark fraction such as 12 Recognize that comparisons are valid only when the two fractions refer to the same whole Record the results of comparisons with symbols lt = or gt and justify the conclusions (eg by using a visual fraction model)



Common Misconceptionsraquoraquo Learn about student misconceptions related to the topic

When students are developing the understandings described above (see Topic Background) they can develop flawed understanding leading to misconceptions about how fractions are compared Once students have been exposed to fraction pairs that have different numerators and different denominators and to a variety of strategies that can help them compare fractions many overgeneralize confuse or misapply strategies

Three particular misconceptions noted in the research on studentsrsquo mathematical reasoning about fractions are targeted in the Comparing Two Fractions assessment

raquo Misconception 1 (M1) Viewing a Fraction as Two Separate Numbers Applying Whole-Number ThinkingOften students do not perceive a fraction as a single quantity but rather see it as a pair of whole numbers and they apply whole-number thinking by comparing the size of the numbers in the denominators the numerators or both

Access the website to watch a brief video clip for a fuller description of this misconception httpem2edcorgportfoliocomparing-two-fractions

raquo Misconception 2 (M2) An Over-Reliance on Unit Fractions A Focus on ldquoSmaller Is BiggerrdquoStudents with this misconception consistently compare only the denominators of the two given fractions They apply what they know about unit fractions to reason that the larger the denominator the smaller the value of the fraction (eg they see 13 as smaller than 35) These students have overgeneralized the concept that ldquosmaller is biggerrdquo to all cases without consideration of the numerator




raquoraquo Common Misconceptions

raquo Misconception 3 (M3) Numerator and Denominator Have an Additive Relationship A Focus on the Difference from One WholeStudents with this misconception understand that itrsquos important to pay attention to the relationship between the numerator and denominator but they believe that this relationship is expressed through addition or subtraction As a result they will compare fractions by focusing on the difference between the numerator and the denominator

Many students apply this reasoning only when the numerator and denominator of each fraction have a difference of one For example when comparing 89 and 45 students reason that since 89 is only one piece away (19) from 99 or one whole and 45 is also one piece (15) away from 55 or one whole the two fractions must be equal as they are each ldquoone awayrdquo from a whole


To see additional examples of student work illustrating this misconception go to the ldquoSample Student Responsesrdquo tab on the website or refer to p44 of this document




References

Hannula M S (2003) Locating fractions on a number line In N A Pateman B J Dougherty amp J Zilliox (Eds) Proceedings of the 2003 Joint Meeting of PME and PMENA Vol 3 (pp 17ndash24) Honolulu HI CRDG College of Education University of Hawaii

Harel G amp Confrey J (1994) The development of multiplicative reasoning in the learning of mathematics Albany NY State University of New York Press

Hiebert J amp Behr M (Eds) (1988) Number concepts and operations in the middle grades Reston VA National Council of Teachers of Mathematics

Martinie S amp Bay-Williams J (2003) Investigating Studentsrsquo Conceptual Understanding of Decimal Fractions Using Multiple Representations Mathematics Teaching in the Middle School 8(5) 244

Roche A amp Clarke D (2004) When does successful comparison of decimals reflect conceptual understanding In I Putt R Faragher amp M McLean (Eds) Mathematics Education for the Third Millennium Towards 2010 Proceedings of the 27th Annual Conference of the Mathematics Education Research Group of Australasia Townsville (pp 486ndash493) Sydney Australia MERGA

Stafylidou S amp Vosniadou S (2004) The development of studentsrsquo understanding of the numerical value of fractions Learning amp Instruction 14(5) 503ndash518 doi101016jlearninstruc200406015

Steinle V amp Stacey K (2004) A longitudinal study of studentsrsquo understanding of decimal notation An overview and refined results In I Putt R Faragher amp M McLean (Eds) Mathematics Education for the Third Millennium Towards 2010 Proceedings of the 27th Annual Conference of the Mathematics Education Research Group of Australasia Townsville (pp 541ndash548) Sydney Australia MERGA



Administering the Pre-Assessmentraquoraquo Learn how to introduce the pre-assessment to your students

About This Assessment

These EM2 diagnostic formative pre- and post-assessments are composed of items with specific attributes associated with student conceptions that are specific to comparing fractions Each item within any EM2 assessment includes a selected response (multiple choice) and an explanation component

While many different fraction pairs can be compared this assessment targets proper non-unit fractions (Proper fractions are those where the numerator is less than the denominator non-unit fractions are those with numerators not equal to 1) This is due to the particular difficulties that these pairs elicit as identified in the mathematics research The fractions being compared in this assessment are confined to the following

bull Proper fractions with denominators less than or equal to 24

bull Fractions pairs where both the numerator and the denominator of the first fraction have the same relationship with the numerator and denominator of the second fractionmdashfor example

frac12 lt frac34The first numerator is 1 which is less than the second numerator 3 and the first denominator is 2 which is less than the second denominator 4

The learning target for the Comparing Two Fractions assessment is as follows

The learner will accurately compare two fractions with different numerators and different denominators when the two fractions refer to the same whole

Prior to Giving the Pre-Assessment

bull Arrange for 15 minutes of class time to complete the administration process including discussing instructions and student work time Since the pre-assessment is designed to elicit misconceptions before instruction you do not need to do any special review of this topic before administering the assessment (See the ldquoStudent Misconceptionsrdquo tab for information and a video that describes this misconception You can also refer to p 4ndash6 of this document)

Pre-Assessment [Student Version]

Less13 than13 (lt)

Greater13 than13 (gt)

Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Comparing13 Frac=ons13 Pre13 Assessment13

EDCLearningtransformslives

Explain13 your13 thinking

Name_________________________13

Date__________13 Class___________

Compare13 the13 two13 fracons13 provided13 Select13 the13 choice13 that13 shows13 the13 relaonship13 between13 the13 two13 fracons

13 13 13 Copyright13 copy13 201513 Educa=on13 Development13 Center13 Inc13 All13 rights13 reserved



1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

See Appendix A for the student

version of the Pre-Assessment



raquoraquo Administering the Pre-Assessment

Administering the Pre-Assessment

bull Inform students about the assessment by reading the following

Today you will complete a short individual activity which is designed to help me understand how you think about comparing fractions

bull Distribute the assessment and read the following

The activity includes seven problems For each problem choose your answer by completely filling in the circle to show which answer you think is correct Because the goal of the activity is to learn more about how you think about fractions itrsquos important for you to include some kind of explanation in the space provided This can be a picture or words or a combination of pictures and words that shows how you chose your answer

You will have about 20 minutes to complete all the problems When you are finished please place the paper on your desk and quietly [read work on ____] until everyone is finished

bull Monitor the students as they work on the assessment making sure that they understand the directions Although this is not a strictly timed assessment it is designed to be completed within a 15-minute timeframe Students may have more time if needed When a few minutes remain say

You have a few minutes left to finish the activity Please use this time to make sure that all of your answers are as complete as possible When you are done please place the paper face down on your desk Thank you for working on this activity today

bull Collect the assessments

After Administering the Pre-Assessment

Use the analysis process (found in the Scoring Guide PDF document under the ldquoScoring Processrdquo tab and found on p 9 of this document) to analyze whether your students have one or more of three possible misconceptions

raquo Misconception 1 Viewing a Fraction as Two Separate Numbers Applying Whole-Number Thinking

raquo Misconception 2 An Over-Reliance on Unit Fractions A Focus on ldquoSmaller Is Biggerrdquo

raquo Misconception 3 Numerator and Denominator Have an Additive Relationship A Focus on the Difference from One Whole



Scoring Processraquoraquo Learn about the scoring process by reviewing the Scoring Guide

The Comparing Two Fractions assessment is composed of seven items with specific attributes associated with different misconceptions that are directly related to comparing two fractions We encourage you to carefully read the Scoring Guide to understand these specific attributes and to find information about analyzing your studentsrsquo responses

How to Use This Guide

This Scoring Guide is intended for use with both the pre-assessment and the post-assessment for Comparing Two Fractions To use this guide we recommend following these steps

bull Read the Misconceptions Description below and be sure you understand what the misconceptions are You may want to view the videos found under the ldquoStudent Misconceptionsrdquo tab Numerous examples of student work illustrating the misconceptions are included in this guide but you may also want to refer to the additional examples of student work found under the ldquoSample Student Responsesrdquo tab and found on p 44 of this document

bull Familiarize yourself with the seven assessment items and what they assess

bull Consider completing the optional scoring practice items and checking your scoring against the answer key

bull Score your studentsrsquo work using the Pre-Post-Assessment Analysis Process described below

bull Refer to the various examples found here and under the ldquoSample Student Responsesrdquo tab for guidance when you are unsure about the scoring

Misconceptions Description

With the introduction of rational numbers students are faced with a new representation of numbers that can lead to a variety of misconceptions Some students do not initially develop a strong conceptual understanding of what fractions are or what the representation means As a result they donrsquot understand that the fraction bar represents division or that a fraction has a discrete value This can lead to a variety of other misconceptions

bull Some of these misconceptions stem from students using whole-number thinking that they apply to the rational numbers in flawed and often interesting ways For example prior to their introduction to fractions and other rational numbers larger numbers have meant more and smaller numbers have meant less Given this prior knowledge when students are introduced to numbers such as eighths and thirds they inaccurately assume that eighths are greater than thirds because eight is greater than three

bull Some of these misconceptions result from partial or flawed conceptual understandings of fractions and fraction comparison strategies For example when students compare two unit fractions referring to the same whole such as 13 and 19 they discover that the fraction with the larger denominator has lesser value due to the size of the piece If they overgeneralize this idea and apply it to other examples such as comparing 23 to 89 they may mistakenly identify 23 as greater than 89



raquoraquo Scoring

bull Some of these misconceptions grow out of misunderstandings of procedural strategies that students learn for comparing and ordering fractions One example is the strategy of looking at how close a fraction is to a whole the closer a number is to the whole the larger it is However if students determine the difference from the whole without considering the unit size (ie the size of the piece missing from the whole) they can draw flawed conclusions For example when comparing 23 to 89 the student might infer that the fractions are equal because they each lack one part from the wholemdash23 is only 13 away from 1 and 89 is only 19 away from 1 The student is disregarding the relative size of 13 and 19 and does not realize that in fact 89 is much closer to 1 than 23 since 19 is a smaller missing piece than 13

The EM2 assessments target three common misunderstandings and misconceptions related to comparing fractions that have been identified in mathematics research


raquo Misconception 2 (M2) An Over-Reliance on Unit Fractions A Focus on ldquoSmaller Is BiggerrdquoStudents with this misconception consistently compare only the denominators of the two given fractions They apply a partial understanding of fractions to reason that the larger the denominator the smaller the value of the fraction and the smaller the denominator the greater the value of the fraction These students have overgeneralized the concept that ldquosmaller is biggerrdquo to all cases without consideration of the numerator

raquo Misconception 3 (M3) Numerator and Denominator Have an Additive Relationship A Focus on the Difference from One WholeStudents with this misconception consistently use addition or subtraction to find the relationship between the numerator and the denominator of each given fraction Many students apply this reasoning only when the numerator and denominator of each fraction have a difference of one For example students reason that since 89 is only one piece away (19) from 99 or one whole and 45 is also one piece (15) away from 55 or one whole the two fractions must be equal as they are each ldquoone awayrdquo from a whole



raquoraquo Scoring

PRE-ASSESSMENT

Pre-Assessment Items

The assessment is composed of seven items with specific attributes associated with understandings and misunderstandings related to comparing fractions Each item may elicit information about the studentsrsquo understanding of fractions as a single quantity and the meaning of the numerator and denominator

Note Students with misconceptions may show evidence of one two or all three misconceptions on different items in the probe For instance a student may show evidence of M1 thinking on several items and M3 thinking on several other items

In particular you may see students apply Misconception 3 inconsistently Students with Misconception 3 sometimes apply M3 thinking only to items 2 4 andor 6 since those problems include fractions that are close to one whole

Refer to the Pre-Post-Assessment Analysis Process for guidance on how to determine whether a student has a particular misconception

Item Understandings and Misconceptions

Correct Response Less than (lt)

Item may elicit evidence ofbull Lack of understanding of a fraction as a single quantity instead using whole-number reasoning

on the individual numbers in the fractions Students with Misconception 1 will reason that 512 is larger because the denominator 12 is larger than the denominator 5 andor because the numerator 5 is larger than the numerator 3

bull An overgeneralization about the size of the fraction based on the size of the denominator (ie smaller denominators mean larger pieces and therefore a larger fraction) Students with Misconception 2 may incorrectly believe that 35 is the larger fraction because fifths are larger than twelfths

bull Using an additive relationship to compare fractions Students with Misconception 3 may reason that since 12 ndash 5 = 7 and 5 ndash 3 = 2 the larger difference of 7 indicates that 512 is the larger fraction The student could also reason in the opposite waymdashthat a smaller difference means a larger fraction and therefore the difference of 2 indicates the larger fraction


Item may elicit evidence ofbull Lack of understanding of a fraction as a single quantity Students with Misconception 1 will reason

that since 7 is larger than 5 andor since 6 is larger than 4 45 is less than 67 While itrsquos true that 45 is less than 67 the underlying reasoning is flawed

bull An overgeneralization about the size of the fraction based on the size of the denominator (ie smaller denominators mean larger pieces and therefore a larger fraction) Students with Misconception 2 will reason that 45 is greater than 67 because fifths are larger than sevenths

bull Using an additive relationship to compare fractions Students with Misconception 3 may reason that since 45 is one part away from one whole (55) and 67 is also one part away from one whole (77) the fractions must be equivalent




Correct Response Equivalent (=)


that since 8 is larger than 4 andor since 6 is larger than 3 68 is greater than 34

bull An overgeneralization about the size of the fraction from the size of the denominator (ie smaller denominators must mean larger pieces and therefore a larger fraction) Students with Misconception 2 will reason that since fourths are larger than eighths 34 is greater than 68

bull Using an additive relationship to compare fractions Students with Misconception 3 will reason that since 68 is two parts away from one whole (88) and 34 is one part away from one whole (44) the fraction that is closer to a whole (34) is larger The student could also reason in the opposite waymdashthat a larger difference means a larger fraction and therefore the difference of 2 indicates the larger fraction

Correct Response Greater than (gt)

Item may elicit evidence ofbull Lack of understanding of a fraction as a single quantity Students with Misconception 1 will reason that

since 9 is larger than 5 andor since 7 is larger than 3 79 is the larger fraction While itrsquos true that 79 is greater than 35 the underlying reasoning is flawed

bull An overgeneralization about the size of the fraction from the size of the denominator (ie smaller denominators must mean larger pieces and therefore a larger fraction) Students with Misconception 2 will reason that since fifths are larger than ninths 35 is greater than 79

bull Using an additive relationship to compare fractions Students with Misconception 3 will reason that since 79 is two parts away from one whole (99) and 35 is also two parts away from one whole (55) the fractions must be equivalent



that since 7 is smaller than 9 andor since 3 is smaller than 4 37 is less than 49 While itrsquos true that 37 is less than 49 the underlying reasoning is flawed

bull An overgeneralization about the size of the fraction from the size of the denominator (ie smaller denominators must mean larger pieces and therefore a larger fraction) Students with Misconception 2 will reason that since sevenths are larger than ninths 37 is greater than 49

bull Using an additive relationship to compare fractions Students with Misconception 3 will reason that since 37 is four parts away from 77 or one whole and since 49 is five parts away from one whole (99) then 37 is closer to one whole (77) and therefore larger The student could also reason in the opposite waymdashthat a larger difference means a larger fraction and therefore the difference of 5 indicates the larger fraction

raquoraquo Scoring






since 8 is larger than 6 andor since 7 is larger than 5 78 is greater than 56 While itrsquos true that 78 is greater than 56 their underlying reasoning is flawed

bull An overgeneralization about the size of the fraction from the size of the denominator (ie smaller denominators must mean larger pieces and therefore a larger fraction) Students with Misconception 2 will reason that since sixths are larger than eighths 56 is greater than 78

bull Using an additive relationship to compare fractions Students with Misconception 3 will reason that since 78 is one part away from one whole (88) and 56 is also one part away from one whole (66) the fractions must be equivalent



since 7 is less than 21 andor since 4 is less than 12 47 is less than 1221

bull An overgeneralization about the size of the fraction from the size of the denominator (ie smaller denominators must mean larger pieces and therefore a larger fraction) Students with Misconception 2 will reason that since sevenths are larger than twenty-firsts 47 is greater than 1221

bull Using an additive relationship to compare fractions Students with Misconception 3 will reason that since 47 is three parts away from one whole (77) and 1221 is nine parts away from one whole (2121) then 47 is closer to one whole and is greater than 1221 The student could also reason in the opposite waymdashthat a larger difference means a larger fraction and therefore the difference of 9 indicates the larger fraction

raquoraquo Scoring



raquoraquo Scoring

Pre-Assessment Analysis Process

Some important things to know about the analysis process for this diagnostic assessment

bull This diagnostic assessment has been validated to reliably predict the likelihood that a student has Misconception 1 2 or 3

o Items 1 3 5 and 7 are most likely to surface both Misconception 1 and Misconception 2

o Items 2 4 and 6 are most likely to surface Misconception 3

bull If a student is determined to show evidence of a misconception on even just one of items the student is likely to have that misconception

bull For each item you need to look at both the selected response choice and the explanation Students will show evidence of a misconception only if they select the corresponding response choice and have an explanation that supports the misconception To learn more about how to tell whether an explanation supports a particular misconception go to the ldquoStudent Misconceptionsrdquo tab and watch the videos provided

bull You can weigh the relative likelihood that your student has any one of these misconceptions by considering whether the studentrsquos written response provides ldquoStrong Evidencerdquo or ldquoWeak Evidencerdquo of each misconception

bull An optional Scoring Guide Template is provided for your use when you score your own studentsrsquo diagnostic assessments In each row of the assessment write a studentrsquos name then circle the appropriate information for each item on the pre-assessment (shaded) and later the post-assessment (in white) If a studentrsquos response does not fit Correct or any of the misconceptions draw a strike-through line through the item

How to Determine If a Student Has One or More of the Misconceptions

1 For each item use the table provided to determine what the selected response might indicate

Say that a student responds ldquoGreater thanrdquo for item 1 Looking at Table 1 below we see that ldquoGreater thanrdquo might indicate the presence of M1 or of M3



raquoraquo Scoring

Note that some responses for a particular item may apply to more than one misconception In other cases a correct response may also apply to one or more of the misconceptions (for example a response of ldquoLess thanrdquo on item 5 could indicate correct understanding or M1 or M3)

Therefore it is particularly important to also consider the studentrsquos explanation in order to determine whether a misconception is present and if so which one

Table 1 Response Patterns for the Pre-Assessment

G = Greater than L = Less than E = Equivalent

Item Item Correct M1 M2 M3

1 lt (L) gt (G) lt (L) gt (G) or lt (L)

2 lt (L) lt (L) gt (G) = (E)

3 = (E) gt (G) lt (L) gt (G) or lt (L)

4 gt (G) gt (G) lt (L) = (E)

5 lt (L) lt (L) gt (G) gt (G) or lt (L)

6 gt (G) gt (G) lt (L) = (E)

7 = (E) lt (L) gt (G) gt (G) or lt (L)



raquoraquo Scoring

What if therersquos no multiple-choice response selected

In that case carefully consider the explanation the student gives If the explanation leaves no doubt that the student would have chosen the misconception selected response and about how the student is reasoning you can code it as ldquoStrong Evidencerdquo of the appropriate misconception However if the explanation leaves some question about what the student was thinking code it as ldquoWeak Evidencerdquo of the misconception For additional guidance on determining the strength of the evidence see the ldquoWhat counts rdquo information in step 2 belowrdquo

2 For each item carefully consider the studentrsquos explanation to determine what it indicates and note whether the evidence from the explanation is strong or weak

If the student provides a response on any item that aligns with a misconception look next at the studentrsquos explanation to determine whether it supports M1 M2 or M3

A Caution

Table 1 shows that some responses indicate only one possibility for example a response of ldquoEquivalentrdquo for item 2 indicates only the possibility of M3 However it is still necessary to check the studentrsquos explanation to confirm evidence of this misconception It is not unusual for a student to choose a response that appears to point to a particular misconception but then provide an explanation that appears to be contradictory

The upshot Always check both the explanation and the selected response

An explanation can be categorized as ldquoStrong Evidencerdquo of a misconception ldquoWeak Evidencerdquo of a misconception or ldquoNo Supporting Evidencerdquo of a misconception

What counts as ldquoStrong Evidencerdquo of a misconception in the pre-assessment

In general responses with strong evidence of a misconception include a clear indication that the student is exhibiting the reasoning typical for that misconception There is no need to make inferences about what the student is thinking the thinking is quite clear from the combination of the selected response and the explanation

Below are three examples of student responses with strong evidence of a misconception using pre-assessment items To see additional examples of student responses that illustrate these misconceptions go to the ldquoSample Student Responsesrdquo or review the information on p 44ndash53



raquoraquo Scoring

Example A Strong Evidence of M1

For students with M1 the explanation will include clear evidence that the student is reasoning about the numbers in the fraction as if they were separate whole numbers (For a more detailed description of this misconception see the video found under the ldquoStudent Misconceptionsrdquo tab)

For item 1 this student chooses ldquoGreater thanrdquo which could indicate either M1 or M3 (see Table 1) The studentrsquos explanation showing subtraction between both the two numerators and the two denominators provides strong evidence that the student is using whole-number reasoning which is indicative of M1

Example B Strong Evidence of M2

For students with M2 the explanation will include clear evidence that the student is paying attention almost exclusively to the size of the denominators and is disregarding the numerators (For a more detailed description of this misconception see the video found under the ldquoStudent Misconceptionsrdquo tab)

ldquoThe smaller the bottom number the bigger the peices [pieces]rdquo

For item 2 this student chooses ldquoGreater thanrdquo which may indicate M2 (see the table above) The studentrsquos explanation specifically refers to ldquothe bottom numberrdquo which provides strong evidence of M2 thinking the student is paying attention only to the size of the pieces indicated by the denominators



raquoraquo Scoring

Example C Strong Evidence of M3

For students with M3 the explanation will include clear evidence that the student is focusing on the difference between the fraction and one whole (For a more detailed description of this misconception see the video found under the ldquoStudent Misconceptionsrdquo tab)

ldquoBoth need 1 filled in to be a wholerdquo

For item 2 this student chooses ldquoEquivalentrdquo which may indicate M3 (see Table 1) The studentrsquos explanation clearly shows that the student is paying attention to the difference between the fraction and one whole This makes it ldquoStrong Evidencerdquo of M3

Can a correct response be considered to have ldquoStrong Evidencerdquo

Yes a correct response can also have ldquoStrong Evidencerdquo ldquoWeak Evidencerdquo or ldquoNo Supporting Evidencerdquo as well While it is not necessary to categorize correct responses as strong weak or non-existent for the purposes of this diagnostic assessment you may want to note this on your scoring template for your own purposes

What counts as ldquoWeak Evidencerdquo of a misconception in the pre-assessment

Responses with weak evidence of a misconception include some indication that the student is exhibiting the reasoning typical for that misconception However these responses also generally require making more inferences about what the student was thinking or they leave some question or doubt about whether the misconception is present or to what degree it is present

Below are three examples of student responses with weak evidence of a misconception using pre-assessment items To see additional examples of student responses that illustrate these misconceptions go to the ldquoSample Student Responsesrdquo or review the information on p 44ndash53



raquoraquo Scoring

Example A Weak Evidence of M1

For item 1 this student chooses ldquoGreater thanrdquo which indicates the possibility of either M1 or M3 (see the table above) In the explanation the student draws arrows pointing to the 5 and 12 and says ldquois greaterrdquo and then draws arrows pointing to the 3 and 5 and says ldquois lessrdquo However it is unclear what the student is comparing and what the arrows mean without having to make inferences about what the student is thinking This makes it ldquoWeak Evidencerdquo of M1


ldquoBecause the denometers [denominators] are big and the smaller the danometer [denominator] means thatrsquos going to be itrdquo

For item 7 this student chooses ldquoGreater thanrdquo which indicates the possibility of either M2 or M3 (see Table 1) The studentrsquos explanation focuses on the denominators and suggests that the student is thinking that the smaller denominator is the larger fraction which is indicative of M2 However it is unclear precisely what the student is thinking without having to make an inference so it is considered ldquoWeak Evidencerdquo of M2


For item 2 this student chooses ldquoEquivalentrdquo which indicates the possibility of M3 The studentrsquos explanation shows diagrams of each fraction with 89 and 67 shaded Itrsquos not clear what the arrows indicate without having to make an inference about what the student is thinking so it is considered ldquoWeak Evidencerdquo of M3



raquoraquo Scoring

What counts as ldquoNo Supporting Evidencerdquo in the pre-assessment

If a student selects an M1 M2 or M3 response choice but provides no explanation at all this counts as ldquoNo Supporting Evidencerdquo of the misconception If a studentrsquos response choice suggests a possible misconception but the explanation does not support it the item is not considered to be indicative of the misconception and can also be scored as ldquoNo Supporting Evidencerdquo

3 After you have analyzed each item for a student use the guidelines below to determine whether the student has any of the misconceptions

This diagnostic assessment has been validated to predict the possible presence of M1 M2 or M3 for a student If a student is determined to show evidence of the misconception on even just one of the items the student is likely to have that misconception regardless of whether the evidence is coded as ldquoStrongrdquo or ldquoWeakrdquo The relative number of items with weak or strong evidence gives you information about how strongly the misconception may be present for the student

What if my student has only one item coded as M1 M2 or M3 with ldquoWeak Evidencerdquo and the rest are correct

Even if your student has only one item with ldquoWeak Evidencerdquo of a misconception this diagnostic assessment is validated to predict that it is likely your student has that misconception However the presence of only one item with ldquoWeak Evidencerdquo of the misconception suggests that the misconception may not be very deeply rooted in this studentrsquos thinking

You may want to keep an eye on this student during regular classwork to watch for other evidence of this misconception

What if the studentrsquos explanation is contradictory to the multiple-choice response chosen

If you come across a response in which the explanation seems to contradict the response choice it is considered a possible indication of the misconception Look for additional evidence either on these assessments or from the studentrsquos comments in class



raquoraquo Scoring

(Optional) Scoring Practice ItemsmdashPre-Assessment

The following sample student responses are provided as an optional practice set If you would like to practice scoring several items to further clarify your understanding of the scoring process you may try scoring the following 10 items

We recommend scoring one or two at a time and checking your scoring as you go against our key found on p 24

Practice Example 1

[Written above 78] ldquoGreater numberrdquo

Written above 56] ldquoSmaller numberrdquo

Practice Example 2

ldquo68 amp 34 are equivalentrdquo

Practice Example 3



raquoraquo Scoring

Practice Example 4

ldquoThe fraction 34 has the smallest denominator out of both fractionsrdquo

Practice Example 5

ldquo68 is greater than because 68 is higher than 34 amp also itrsquos a better numberrdquo

Practice Example 6

ldquoEach fraction needs to add 15 or 17 and that means they are the samerdquo

Practice Example 7

ldquoItrsquos more out of the bottom num rdquo



raquoraquo Scoring

Practice Example 8

ldquo35 is greater than 512 because 35 has a lower denominatorrdquo

Practice Example 9

Practice Example 10

ldquoBoth need 2 to be wholerdquo



raquoraquo Scoring

Scoring Practice Items Answer KeymdashPre-Assessment

Practice Example 1


[Written above 56] ldquoSmaller numberrdquo

This is an example of Correct with ldquoWeak Evidencerdquo The explanation shows that the student may be focusing on the individual numerators in the fraction comparing them as greater or smaller

Practice Example 2


This is a ldquoCorrectrdquo example with ldquoStrong Evidencerdquo (though making any distinction between strong and weak correct responses is not necessary for this diagnostic assessment it simply gives you more information about your student) The student draws an accurate diagram for each fraction showing that they are equivalent

Practice Example 3

This is an example of M3 with ldquoWeak Evidencerdquo The student subtracts two from each denominator to equal each numerator We can infer that the student is showing that since the difference between each numerator and denominator is the same the fractions are equivalent However because we have to infer this from the work it is considered ldquoWeak Evidencerdquo of M3



raquoraquo Scoring

Practice Example 4


This is an example of M2 with ldquoStrong Evidencerdquo The studentrsquos response choice combined with the explanation stating that 34 ldquohas the smallest denominatorrdquo provides strong evidence that the student is focusing on the smaller denominator being the larger fraction

Practice Example 5


This is an example of M1 with ldquoWeak Evidencerdquo The explanation suggests that the student is paying attention to individual larger numbers (ldquo68 is higher than 34rdquo) but leaves ambiguity about what the student means by ldquoa better numberrdquo

Practice Example 6


This is an example of M3 with ldquoStrong Evidencerdquo The studentrsquos explanation clearly focuses on the remaining one piece (15 or 17) of each fraction and says this means that the fractions are the same



raquoraquo Scoring

Practice Example 7


This is a ldquoCorrectrdquo example with ldquoWeak Evidencerdquo (though making any distinction between strong and weak correct responses is not necessary for this diagnostic assessment it simply gives you more information about your student) The student selects the correct response (ldquoLess thanrdquo) and then draws two very different (but accurate) representations of the fractions However itrsquos unclear what the student means by ldquoItrsquos more out of the bottom [number]rdquo making it ldquoWeak Evidencerdquo that the student is thinking correctly

Practice Example 8


This is an example of M2 with ldquoStrong Evidencerdquo The explanation clearly states that the student is focusing on the smaller denominator as an indication of the larger fraction

Practice Example 9

This is an example of M1 with ldquoWeak Evidencerdquo Although the student selects the M1 response (ldquoGreater thanrdquo) the explanation is unclear The representations of the fractions are not drawn with equal-size wholes suggesting that the student is using whole-number thinking However the lack of clarity about the studentrsquos thinking makes it ldquoWeak Evidencerdquo of M1



raquoraquo Scoring

Practice Example 10


This is an example of M3 with ldquoStrong Evidencerdquo The studentrsquos explanation (ldquoboth need 2 to be wholerdquo) clearly indicates that the student is focusing on each fraction needing two more pieces to make one whole since theyrsquore each missing the same number of pieces they are equivalent



raquoraquo Scoring

POST-ASSESSMENT

Post-Assessment Items

The post-assessment is structured exactly the same as the pre-assessment comprising seven items with specific attributes associated with understandings and misunderstandings related to comparing fractions Each item may elicit information about the studentsrsquo understanding of fractions as a single quantity and their understanding of the meaning of the numerator and denominator



Item may elicit evidence ofbull Lack of understanding of a fraction as a single quantity and instead using whole-number reasoning


bull An overgeneralization about the size of the fraction from the size of the denominator (ie smaller denominators mean larger pieces and therefore a larger fraction) Students with Misconception 2 may incorrectly believe that 58 is the larger fraction because eighths are larger than twelfths




that since 9 is larger than 7 andor since 8 is larger than 6 89 is greater than 67 While itrsquos true that 67 is less than 89 the underlying reasoning is flawed

bull An overgeneralization about the size of the fraction from the size of the denominator (ie smaller denominators mean larger pieces and therefore a larger fraction) Students with Misconception 2 will reason that 67 is greater than 89 because sevenths are larger than ninths




raquoraquo Scoring





bull An overgeneralization about the size of the fraction from the size of the denominator (ie smaller denominators mean larger pieces and therefore a larger fraction) Students with Misconception 2 will reason that since thirds are larger than sixths 23 must be greater than 46

bull Using an additive relationship to compare fractions Students with Misconception 3 will reason that since 46 is two parts away from one whole (66) and 23 is one part away from one whole (33) the fraction that is closer to one whole is larger




bull An overgeneralization about the size of the fraction from the size of the denominator (ie smaller denominators mean larger pieces and therefore a larger fraction) Students with Misconception 2 will reason that since sevenths are larger than elevenths 57 is greater than 911




that since 5 is smaller than 9 andor since 2 is smaller than 4 then 25 is less than 49 While itrsquos true that 25 is less than 49 the underlying reasoning is flawed

bull An overgeneralization about the size of the fraction from the size of the denominator (ie smaller denominators mean larger pieces and therefore a larger fraction) Students with Misconception 2 will reason that since fifths are larger than ninths 25 is greater than 49

bull Using an additive relationship to compare fractions Students with Misconception 3 will reason that 25 is three parts away from one whole (55) and 49 is five parts away from one whole (99) therefore 25 is larger because itrsquos closer to one whole






since 10 is larger than 7 andor since 9 is larger than 6 910 is greater than 67 While itrsquos true that 910 is greater than 67 the underlying reasoning is flawed

bull An overgeneralization about the size of the fraction from the size of the denominator (ie smaller denominators mean larger pieces and therefore a larger fraction) Students with Misconception 2 will reason that since sevenths are larger than tenths 67 is greater than 910





bull An overgeneralization about the size of the fraction from the size of the denominator (ie smaller denominators mean larger pieces and therefore a larger fraction) Students with Misconception 2 will reason that since sevenths are larger than twenty-firsts 37 is greater than 921

bull Using an additive relationship to compare fractions Students with Misconception 3 will reason that 37 is four parts away from one whole (77) and 1221 is nine parts away from one whole (2121) therefore 37 is greater because itrsquos closer to one whole

raquoraquo Scoring



raquoraquo Scoring

Post-Assessment Analysis Process

The post-assessment uses the same scoring process as the pre-assessment If you are not already familiar with the steps for scoring the assessment please review that section starting on p 14

How to Determine If a Student Has the Misconception

1 For each item look at the table provided to determine what the selected response might indicate

Table 2 Response Patterns for the Post-Assessment




2 lt (L) lt (L) gt (G) = (E)


4 gt (G) gt (G) lt (L) = (E)


6 gt (G) gt (G) lt (L) = (E)




raquoraquo Scoring


In that case carefully consider the explanation the student gives If the explanation leaves no doubt that the student would have chosen the misconception response and about how the student is reasoning you can code it as ldquoStrong Evidencerdquo of the appropriate misconception However if the explanation leaves some question about what the student was thinking code it as ldquoWeak Evidencerdquo See the guidelines for evidence below

2 For each item carefully consider the studentrsquos explanation to determine what the response indicates and note whether the evidence from the explanation is strong or weak

If the student provides a response on any item that aligns with a misconception look at the studentrsquos explanation to determine whether it supports M1 M2 or M3

A Caution

Table 2 shows that some responses indicate only one possibility for example a response of ldquoEquivalentrdquo for item 2 indicates only the possibility of M3 However it is still necessary to check the studentrsquos explanation to confirm evidence of the misconception It is not unusual for a student to choose a response that appears to point to a particular misconception but then provide an explanation that appears to be contradictory



What counts as ldquoStrong Evidencerdquo of a misconception in the post-assessment

In general responses with strong evidence of a misconception include a clear indication that the student is exhibiting the reasoning typical for that misconception There is no need to make inferences about what the student is thinking it is quite clear from the combination of the selected response and the explanation

Below are three examples of student responses with strong evidence of a misconception using post-assessment items To see additional examples of student responses that illustrate these misconceptions go to the ldquoSample Student Responsesrdquo or review the information on p 44ndash53



raquoraquo Scoring



ldquoIt is greater because the denominater is bigger than the otherrsquosrdquo

For item 1 this student chooses ldquoGreater thanrdquo which could indicate either M1 or M3 (see Table 2) The explanation however provides clear evidence that the student is thinking of the denominators as whole numbers reasoning that the larger denominator indicates the larger fraction This is ldquoStrong Evidencerdquo of M1



ldquo37 is bigger because it has biger peices [bigger pieces]rdquo

For item 7 this student chooses ldquoGreater thanrdquo which could indicate either M2 or M3 However the studentrsquos explanation focuses on sevenths being bigger than twenty-firsts indicating the ldquosmaller number means larger fractionrdquo reasoning that is characteristic of M2



raquoraquo Scoring


For students with M3 the explanation will include clear evidence that the student is paying attention to the difference between the fraction and one whole (For a more detailed description of this misconception see the video found under the ldquoStudent Misconceptionsrdquo tab)

ldquo25 has less pieces until itrsquos a wholerdquo

For item 5 the student chose ldquoGreater thanrdquo which could indicate either M2 or M3 The explanation however makes it clear that the student is focusing on the number of pieces needed to make a whole (35 vs 59) indicating M3 thinking

What counts as ldquoWeak Evidencerdquo of a misconception in the post-assessment


Below are three examples of student responses with weak evidence of a misconception using post-assessment items To see additional examples of student responses that illustrate these misconceptions go to the ldquoSample Student Responsesrdquo or review the information on p 44-53


For item 3 this student chooses ldquoGreater thanrdquo which indicates the possibility of either M1 or M3 (see the table above) In the explanation the student draws arrows pointing to the 5 and 12 and uses the ldquoGreater thanrdquo symbol However it is unclear what the student is comparing and what the arrows mean without having to make inferences about what the student is thinking This makes it ldquoWeak Evidencerdquo of M1



raquoraquo Scoring

Example B Weak Evidence of M2

For item 2 this student chooses ldquoGreater thanrdquo which indicates the possibility of M2 The studentrsquos explanation lists only ldquo7thrdquo which possibly refers to the size of the pieces in the greater fraction It is unclear precisely what the student is thinking without having to make an inference so it is considered ldquoWeak Evidencerdquo of M2

Example C Weak Evidence of M3

This For item 1 this student uses the ldquoGreater thanrdquo symbol in hisher explanation but does not select a response Using ldquoGreater thanrdquo indicates the possibility of either M1 or M3 (see the table above) The studentrsquos explanation shows diagrams of each fraction 712 and 58 shaded with numbers in the unshaded pieces Since the student is likely focusing on the missing pieces rather than the shaded pieces it is considered ldquoWeak Evidencerdquo of M3

What counts as ldquoNo Supporting Evidencerdquo in the post-assessment




raquoraquo Scoring


This diagnostic assessment has been validated to predict the possible presence of M1 M2 or M3 for a student If a student is determined to show evidence of any these misconceptions on even just one of the items the student is likely to have that misconception regardless of whether the evidence is coded as ldquoStrongrdquo or ldquoWeakrdquo The relative number of items with weak or strong evidence gives you information about how strongly the misconception may be present for the student

What if my student has only one item indicating one of the misconceptions with ldquoWeak Evidencerdquo and the rest are correct

Even if your student has only one item with ldquoWeak Evidencerdquo of a misconception this diagnostic assessment is validated to predict that it is likely your student has that misconception However the presence of only one item with weak evidence suggests that the misconception may not be very deeply rooted in this studentrsquos thinking You may want to keep an eye on this student during regular classwork to watch for other evidence of this misconception





raquoraquo Scoring

(Optional) Scoring Practice ItemsmdashPost-Assessment



Practice Example 1

Practice Example 2

ldquo7 is greater than 21 in this matter so 37 is greater than 921rdquo

Practice Example 3

ldquoI cannot double 5 to 7 so 7 is still more than 5 so 45 is smaller than 57rdquo



raquoraquo Scoring

Practice Example 4

Practice Example 5

ldquoBigger peice [piece] missingrdquoldquoSmaller peice [piece] missingrdquo

Practice Example 6

ldquoTwo thirds has one more piece left and four sixths has two pieces leftrdquo

Practice Example 7

ldquoBoth of the top numbers plus two equal the bottom numbersrdquo



raquoraquo Scoring

Practice Example 8

ldquoYou have more pieces with 9 11thsrdquo

Practice Example 9

ldquoI choose less than because 12 is bigger than 8 so that means smaller piecesrdquo

Practice Example 10



raquoraquo Scoring

Scoring Practice Items Answer KeymdashPost-Assessment

Practice Example 1

This is an example of M3 with ldquoStrong Evidencerdquo The student selects ldquoEquivalentrdquo and then circles 6 out of 7 circles and 9 out of 10 circles showing 1 left over in each case

Practice Example 2


This is an example of M2 with ldquoWeak Evidencerdquo Though the combination of the selected response and the explanation suggests M2 thinking it is not clear from the studentrsquos explanation what is meant by ldquo7 is greater than 21 in this matterrdquo without making inferences about the studentrsquos work This lack of clarity makes it ldquoWeak Evidencerdquo of M2

Practice Example 3


This is an example of M1 with ldquoStrong Evidencerdquo The studentrsquos explanation clearly mentions that ldquo7 is still more than 5rdquo suggesting whole-number thinking



raquoraquo Scoring

Practice Example 4

This is an example of M2 with ldquoWeak Evidencerdquo The student selects ldquoGreater thanrdquo which could indicate either M2 or M3 for this item However the explanation shows fifths as bigger pieces and ninths as smaller pieces suggesting that the student is employing the ldquosmaller number means bigger fractionrdquo thinking that is typical of M2 Since it is difficult to know exactly what the student is thinking without making inferences this evidence is considered weak

Practice Example 5


This is a ldquoCorrectrdquo example with ldquoStrong Evidencerdquo (though making any distinction between strong and weak correct responses is not necessary for this diagnostic assessment it simply gives you more information about your student) The student selects the correct response and then draws reasonably accurate representations of the fractions noting that 67 has a bigger piece missing (17) and 89 has a smaller piece missing (19)

Practice Example 6


This is an example of M3 with ldquoStrong Evidencerdquo The student is paying attention to the number of pieces needed in each fraction to make a whole and concludes that the fraction that needs fewer pieces is larger



raquoraquo Scoring

Practice Example 7

ldquoBoth of the top numbers plus two equal the bottom numbers

This is an example of M3 with ldquoStrong Evidencerdquo The student selects ldquoEquivalentrdquo which indicates the possibility of M3 for this item and focuses on the equal difference between the numerators and denominators in each fraction (comparing 911 to 1111 and 57 to 77) This attention to the difference between the fractions and one whole is ldquoStrong Evidencerdquo of M3

Practice Example 8

ldquoYou have more pieces with 911thsrdquo

This is an example of M1 with ldquoStrong Evidencerdquo The student selects ldquoGreater thanrdquo which is correct though it could also indicate M1 for this item The explanation which shows 9 out of 11 circles circled and 5 out of 7 circles circled and states ldquoYou have more pieces with 9 11thsrdquo clearly indicates the whole-number thinking that is strong evidence of M1

Practice Example 9


This is an example of M2 with ldquoStrong Evidencerdquo The student selects ldquoLess thanrdquo which is the correct answer but could also indicate M1 or M3 for this item The explanation clearly mentions that the denominator 12 is bigger so it has ldquosmaller piecesrdquo which is strong evidence of M2 thinking



raquoraquo Scoring

Practice Example 10

This is an example of M1 with ldquoWeak Evidencerdquo The student selects ldquoLess thanrdquo which can indicate either M1 or M3 for this item and draws diagrams with different-size wholes shading in the number of pieces in the numerator While this suggests whole-number thinking it is not fully clear exactly how the student is thinking making it ldquoWeak Evidencerdquo of M1



Sample Student Responsesraquoraquo Review examples of student responses to assessment items

To determine the degree of understanding and misunderstanding in the student work itrsquos important to consider both the answer to the selected response and the explanation text and representations The example above is one of many student work samples that provide insight into student thinking about one of three different misconceptions targeted in these diagnostic assessments (see ldquoStudent Misconceptionsrdquo for more information and videos about these misconceptions)

We encourage you to look at the collection of student work examples provided here

The Comparing Two Fractions diagnostic assessment focuses on three particular misconceptions that students have regarding how to compare fractions Sample student responses indicative of each misconception are provided separately below along with samples of correct student responses To determine the degree of understanding and misunderstanding itrsquos important to consider both the studentrsquos answer to the selected response and the studentrsquos explanation text and representations

raquo Misconception 1 (M1) Viewing a Fraction as Two Separate Numbers Applying Whole-Number ThinkingOften students do not perceive a fraction as a single quantity but rather see it as a pair of whole numbers and they apply whole-number thinking by comparing the size of the numbers in the denominators or numerators or both (For more information go to the ldquoStudent Misconceptionsrdquo tab)



raquoraquo Sample Student Responses

Item Sample Student Responses withEvidence of Misconception 1 Notes

Pre-Assessment 1

bull The misconception selected response is chosen

ANDbull The subtraction shows the student using

whole-number thinking to compare the fractions

Post-Assessment 1



ANDbull The explanation clearly states that the

student is focusing on the size of the whole numbers in the denominators

Post-Assessment 2

ldquo67 is less than 89 because the 9 is bigger than the 7rdquo

bull The misconception selected response is chosen (while it is the correct response it can also indicate this misconception)

ANDbull The explanation (ldquothe 9 is bigger than the

7rdquo) shows that the student is comparing the denominators as whole numbers

Pre-Assessment 4


ANDbull The circled amounts correspond to the

numerators which are being compared as whole numbers





Pre-Assessment 5

ldquoI chosed [chose] this because I saw two couldnrsquot be bigger than four So I realized 49 was biggerrdquo


ANDbull The explanation compares the numerators

as whole numbers noting that 4 is bigger than 2

Pre-Assessment 6

ldquoGreater numberrdquo ldquoSmaller numberrdquo



as whole numbers the student draws arrows pointing to 7 (ldquogreater numberrdquo) and 5 (ldquosmaller numberrdquo)




raquo Misconception 2 (M2) An Over-Reliance on Unit Fractions A Focus on ldquoSmaller Is BiggerrdquoStudents with this misconception consistently compare only the denominators of the two given fractions They apply what they know about unit fractions to reason that the larger the denominator the smaller the value of the fraction (eg they see 13 as greater than 35) These students have overgeneralized the concept that ldquosmaller is biggerrdquo to all cases without consideration of the numerator (For more information go to the ldquoStudent Misconceptionsrdquo tab)


Pre-Assessment 1



ANDbull The explanation focuses on the lower

denominator as the reason for the fraction being larger

Pre-Assessment 2


ANDbull The diagram indicates that fifths are bigger

slices and sevenths are smaller slices suggesting that a larger denominator always results in a smaller fraction

Pre-Assessment 2



ANDbull The explanation focuses solely on the size

of the denominator stating ldquoThe smaller the bottom number the bigger the piecesrdquo





Pre-Assessment 3



ANDbull The explanation identifies the larger

fraction by focusing solely on the size of the denominators

Pre-Assessment 4

ldquoI rather char [share] a candy bar with 5 pepol [people] than 9 pepolrdquo



of the denominators to determine which fraction is larger stating that a candy bar divided among 5 people will result in larger pieces than one divided among 9 people

Pre-Assessment 1

ldquoThink of a chocolate bar and think which one will have bigger bricks (Hersheys)rdquo


ANDbull The explanation uses the example of

a candy bar and focuses on which will have ldquobigger bricksrdquo indicating that the student is considering only the size of the denominator this is further reinforced by the diagram which shows that twelfths are smaller than fifths



raquo Misconception 3 (M3) Numerator and Denominator Have an Additive Relationship A Focus on the Difference from One WholeStudents with this misconception understand that itrsquos important to pay attention to the relationship between the numerator and denominator but they believe that this relationship is expressed through addition or subtraction As a result they will pay attention to the difference between the numerator and denominator in order to compare fractions

Many students apply this reasoning only when the numerator and denominator of each fraction have a difference of one For example when comparing 89 and 45 students reason that since 89 is only one piece away (19) from 99 or one whole and 45 is also one piece (15) away from 55 or one whole the two fractions must be equal as they are each ldquoone awayrdquo from a whole (For more information go to the ldquoStudent Misconceptionsrdquo tab)


Pre-Assessment 2



ANDbull The explanation focuses on both fractions

being one part away from a whole

Pre-Assessment 2



ANDbull Though the student correctly identifies

the remaining single fractional part the explanation shows that the student sees this as meaning ldquothey are the samerdquo

Pre-Assessment 2


ANDbull The diagrams highlight one piece remaining

to complete a whole illustrating this as the reason for the fractions being equivalent

Sample Student Responses





Pre-Assessment 4

ldquoThey are both 2 fractions away from 0rdquo


ANDbull The explanation focuses on how many parts

each fraction is from a benchmark (in this case 0 which is incorrect)

Post-Assessment 4



ANDbull The explanation focuses on the difference

between the numerator and denominator since each fractionrsquos numerator and denominator differ by 2 the student states that they are equivalent

Post-Assessment 6


ANDbull The explanation uses subtraction to show

that the difference between the numerator and denominator in each case is the same the student disregards the size of the fractional pieces

Post-Assessment 6


ANDbull The explanation shows both fractions drawn

with equal-size wholes and vertical lines drawn to highlight the one part remaining on the right illustrating this as the reason the fractions are equivalent




Incorrect Reasoning That Is Not One of These Misconceptions

There may be some cases in which the student selects the response that indicates the misconception but does not provide convincing evidence that he or she actually has the misconception In some cases the student may have a different set of difficulties than the specific misconceptions targeted by this probe Here are two examples

Item Sample Student Responses with Evidence of Correct Responses Notes

Pre-Assessment 2

ldquoI chose equivalent because 4 + 2 equals 6 and 5 + 2 equals 7rdquo

bull The M3 selected response is chosenHOWEVER

bull The explanation does not fit M3mdashthe student sees the fractions as equivalent because both numerators and denominators differ from the other by 2 (a student with M3 would instead focus on both fractions being ldquo1 awayrdquo from a whole)

Pre-Assessment 4


bull The explanation does not fit M2mdashthe student tries to compare the fractions by drawing each one but the drawings are not accurate (a student with M2 would instead focus on ninths being smaller than fifths)




Correct Reasoning

Students with correct reasoning about comparing fractions are often able to do one or more of the following

bull Multiply the numerator and denominator by a common factor to get common denominators

bull Use benchmarks (eg understand that 512 is less than 12 but 35 is greater than 12)

bull Use accurate diagrams to compare the fractions

Item Sample Student Responses with Correct Reasoning

Notes

Pre-Assessment 2

bull The correct selected response is chosenAND

bull The fractions are correctly converted into equivalent fractions with common denominators

Pre-Assessment 1


bull The explanation shows how each fraction compares to the benchmark of 12

Pre-Assessment 6


bull The diagram shows equal- size wholes ltltcorrectly partitioned andgtgtbeing compared visually

Pre-Assessment 7

ldquo3 times bigger same exact thingrdquo


bull The explanation shows that the student sees the second fraction as ldquo3 times biggerrdquo and this is the reason they are equivalent





Notes

Post-Assessment 2

ldquo17 gt 19 so 67 (77 ndash 17) is less than 89 (99 ndash 19)rdquo


bull The student notices that 67 is 17 away from 1 and that 89 is 19 away from 1 and reasons that since 17 gt 19 89 is greater because the difference between the fraction and 1 is less



Administering the Post-Assessmentraquoraquo Learn how to introduce the post-assessment to your students

If the Comparing Two Fractions pre-assessment shows that any of your students have one or more of the misconceptions outlined in the Scoring Guide plan and implement instructional activities designed to increase studentsrsquo understanding The post-assessment provided here can then be used to determine if the misconception has been addressed

Prior to Giving the Post-Assessment

bull Arrange for 20 minutes of class time to complete the administration process including discussing instructions and student work time Since the post-assessment is designed to elicit a particular misconception after instruction you should avoid using or reviewing items from the post-assessment before administering it

Administering the Post-Assessment

bull Inform the students about the assessment by reading the following

Today you will complete a short individual activity which is designed to help me understand how you think about comparing fractions a topic we have been working on in class


This activity includes seven problems For each problem choose your answer by completely filling in the circle to show which answer you think is correct Because the goal of the activity is to learn more about how you think about fractions itrsquos important for you to include some kind of explanation in the space provided This can be a picture or words or a combination of pictures and words that shows how you chose your answer



You have a few minutes to finish the activity Please use this time to make sure that all of your answers are as complete as possible When you are done please place the paper face down on your desk Thank you for working on this activity today


Post-Assessment [Student Version]

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Comparing13 Frac=ons13 Post13 Assessment13



Name_________________________13 Date__________13 Class___________

Compare13 the13 two13 fracons13 provided13 Select13 the13 choice13 that13 shows13 the13 relaonship13

between13 the13 two13 fracons





67

712

58

89

46

23

911

57

1)

2)

3)

4)


version of the Post-Assessment



raquoraquo Administering the Post Assessment

After Administering the Post-Assessment





Some students who previously had the misconception will no longer have itmdashthe ideal case Consider your instructional next steps for those students who still show evidence of the misconception

Student Pre 1 Pre 2 Pre 3 Pre 4 Pre 5 Pre 6 Pre 7 Likelihood

Cor M1 M2 M3 Cor M1 M2 M3 Cor M1 M2 M3 Cor M1 M2 M3 Cor M1 M2 M3 Cor M1 M2 M3 Cor M1 M2 M3 M1 M2 M3

Str Wk Str Wk Str Wk Str Wk Str Wk Str Wk Str Wk None

Post 1 Post 2 Post 3 Post 4 Post 5 Post 6 Post 7 Likelihood



























Comparing Two Fractions Scoring Guide

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________


13 13 Copyright13 copy13 201513 Educa=on13 Development13 Center13 Inc13 All13 rights13 reserved

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)



Topic Backgroundraquoraquo Learn about concepts related to comparing fractions

Before they can successfully compare two fractions students need to have a foundational set of conceptual understandings of fractions for example

bull An understanding of fraction language and symbols

bull An understanding of iterating (making multiple copies of) fractionsmdashin other words they know that a fraction is 15 when five copies of it make one whole

bull A sense of the relative size of fractions

bull The ability to recognize and understand fraction notation

The comparison of fractions is often initiated by using unit fractions in order to focus on the relationship between the size of the pieces when partitioningmdashfor example understanding that 13 is greater than14 because thirds are larger pieces than fourths Learning often progresses to comparing other fractions with specific characteristics for example

bull Fractions with the same denominator (eg comparing 25 to 35) to understand that the number of parts matters when comparing fractions with same-size parts

bull Fractions with the same numerator (eg comparing 24 to 23) to understand that when you have the same number of pieces (same numerator) you must consider the size of the pieces (denominators)

bull Fractions that are slightly more or less than key benchmarks such as 14 12 and 34 (eg 316 35 and 78 respectively)

bull Fractions that are close to one whole (eg 1112 67 910)

































References






















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35





































raquoraquo Scoring








raquoraquo Scoring

PRE-ASSESSMENT



































raquoraquo Scoring














raquoraquo Scoring



raquoraquo Scoring















raquoraquo Scoring







2 lt (L) lt (L) gt (G) = (E)


4 gt (G) gt (G) lt (L) = (E)


6 gt (G) gt (G) lt (L) = (E)




raquoraquo Scoring





A Caution









raquoraquo Scoring










raquoraquo Scoring












raquoraquo Scoring










raquoraquo Scoring












raquoraquo Scoring




Practice Example 1



Practice Example 2


Practice Example 3



raquoraquo Scoring

Practice Example 4


Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9

Practice Example 10




raquoraquo Scoring


Practice Example 1




Practice Example 2



Practice Example 3




raquoraquo Scoring

Practice Example 4



Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9




raquoraquo Scoring

Practice Example 10





raquoraquo Scoring

POST-ASSESSMENT
















raquoraquo Scoring






























raquoraquo Scoring



raquoraquo Scoring









2 lt (L) lt (L) gt (G) = (E)


4 gt (G) gt (G) lt (L) = (E)


6 gt (G) gt (G) lt (L) = (E)




raquoraquo Scoring





A Caution









raquoraquo Scoring











raquoraquo Scoring












raquoraquo Scoring









raquoraquo Scoring









raquoraquo Scoring




Practice Example 1

Practice Example 2


Practice Example 3




raquoraquo Scoring

Practice Example 4

Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9


Practice Example 10



raquoraquo Scoring


Practice Example 1


Practice Example 2



Practice Example 3





raquoraquo Scoring

Practice Example 4


Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9





raquoraquo Scoring

Practice Example 10













Pre-Assessment 1




Post-Assessment 1





Post-Assessment 2





Pre-Assessment 4








Pre-Assessment 5





Pre-Assessment 6










Pre-Assessment 1





Pre-Assessment 2




Pre-Assessment 2









Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)

































References






















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35





































raquoraquo Scoring








raquoraquo Scoring

PRE-ASSESSMENT



































raquoraquo Scoring














raquoraquo Scoring



raquoraquo Scoring















raquoraquo Scoring







2 lt (L) lt (L) gt (G) = (E)


4 gt (G) gt (G) lt (L) = (E)


6 gt (G) gt (G) lt (L) = (E)




raquoraquo Scoring





A Caution









raquoraquo Scoring










raquoraquo Scoring












raquoraquo Scoring










raquoraquo Scoring












raquoraquo Scoring




Practice Example 1



Practice Example 2


Practice Example 3



raquoraquo Scoring

Practice Example 4


Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9

Practice Example 10




raquoraquo Scoring


Practice Example 1




Practice Example 2



Practice Example 3




raquoraquo Scoring

Practice Example 4



Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9




raquoraquo Scoring

Practice Example 10





raquoraquo Scoring

POST-ASSESSMENT
















raquoraquo Scoring






























raquoraquo Scoring



raquoraquo Scoring









2 lt (L) lt (L) gt (G) = (E)


4 gt (G) gt (G) lt (L) = (E)


6 gt (G) gt (G) lt (L) = (E)




raquoraquo Scoring





A Caution









raquoraquo Scoring











raquoraquo Scoring












raquoraquo Scoring









raquoraquo Scoring









raquoraquo Scoring




Practice Example 1

Practice Example 2


Practice Example 3




raquoraquo Scoring

Practice Example 4

Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9


Practice Example 10



raquoraquo Scoring


Practice Example 1


Practice Example 2



Practice Example 3





raquoraquo Scoring

Practice Example 4


Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9





raquoraquo Scoring

Practice Example 10













Pre-Assessment 1




Post-Assessment 1





Post-Assessment 2





Pre-Assessment 4








Pre-Assessment 5





Pre-Assessment 6










Pre-Assessment 1





Pre-Assessment 2




Pre-Assessment 2









Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35





































raquoraquo Scoring








raquoraquo Scoring

PRE-ASSESSMENT



































raquoraquo Scoring














raquoraquo Scoring



raquoraquo Scoring















raquoraquo Scoring







2 lt (L) lt (L) gt (G) = (E)


4 gt (G) gt (G) lt (L) = (E)


6 gt (G) gt (G) lt (L) = (E)




raquoraquo Scoring





A Caution









raquoraquo Scoring










raquoraquo Scoring












raquoraquo Scoring










raquoraquo Scoring












raquoraquo Scoring




Practice Example 1



Practice Example 2


Practice Example 3



raquoraquo Scoring

Practice Example 4


Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9

Practice Example 10




raquoraquo Scoring


Practice Example 1




Practice Example 2



Practice Example 3




raquoraquo Scoring

Practice Example 4



Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9




raquoraquo Scoring

Practice Example 10





raquoraquo Scoring

POST-ASSESSMENT
















raquoraquo Scoring






























raquoraquo Scoring



raquoraquo Scoring









2 lt (L) lt (L) gt (G) = (E)


4 gt (G) gt (G) lt (L) = (E)


6 gt (G) gt (G) lt (L) = (E)




raquoraquo Scoring





A Caution









raquoraquo Scoring











raquoraquo Scoring












raquoraquo Scoring









raquoraquo Scoring









raquoraquo Scoring




Practice Example 1

Practice Example 2


Practice Example 3




raquoraquo Scoring

Practice Example 4

Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9


Practice Example 10



raquoraquo Scoring


Practice Example 1


Practice Example 2



Practice Example 3





raquoraquo Scoring

Practice Example 4


Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9





raquoraquo Scoring

Practice Example 10













Pre-Assessment 1




Post-Assessment 1





Post-Assessment 2





Pre-Assessment 4








Pre-Assessment 5





Pre-Assessment 6










Pre-Assessment 1





Pre-Assessment 2




Pre-Assessment 2









Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)



































raquoraquo Scoring








raquoraquo Scoring

PRE-ASSESSMENT



































raquoraquo Scoring














raquoraquo Scoring



raquoraquo Scoring















raquoraquo Scoring







2 lt (L) lt (L) gt (G) = (E)


4 gt (G) gt (G) lt (L) = (E)


6 gt (G) gt (G) lt (L) = (E)




raquoraquo Scoring





A Caution









raquoraquo Scoring










raquoraquo Scoring












raquoraquo Scoring










raquoraquo Scoring












raquoraquo Scoring




Practice Example 1



Practice Example 2


Practice Example 3



raquoraquo Scoring

Practice Example 4


Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9

Practice Example 10




raquoraquo Scoring


Practice Example 1




Practice Example 2



Practice Example 3




raquoraquo Scoring

Practice Example 4



Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9




raquoraquo Scoring

Practice Example 10





raquoraquo Scoring

POST-ASSESSMENT
















raquoraquo Scoring






























raquoraquo Scoring



raquoraquo Scoring









2 lt (L) lt (L) gt (G) = (E)


4 gt (G) gt (G) lt (L) = (E)


6 gt (G) gt (G) lt (L) = (E)




raquoraquo Scoring





A Caution









raquoraquo Scoring











raquoraquo Scoring












raquoraquo Scoring









raquoraquo Scoring









raquoraquo Scoring




Practice Example 1

Practice Example 2


Practice Example 3




raquoraquo Scoring

Practice Example 4

Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9


Practice Example 10



raquoraquo Scoring


Practice Example 1


Practice Example 2



Practice Example 3





raquoraquo Scoring

Practice Example 4


Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9





raquoraquo Scoring

Practice Example 10













Pre-Assessment 1




Post-Assessment 1





Post-Assessment 2





Pre-Assessment 4








Pre-Assessment 5





Pre-Assessment 6










Pre-Assessment 1





Pre-Assessment 2




Pre-Assessment 2









Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)



raquoraquo Scoring

PRE-ASSESSMENT



































raquoraquo Scoring














raquoraquo Scoring



raquoraquo Scoring















raquoraquo Scoring







2 lt (L) lt (L) gt (G) = (E)


4 gt (G) gt (G) lt (L) = (E)


6 gt (G) gt (G) lt (L) = (E)




raquoraquo Scoring





A Caution









raquoraquo Scoring










raquoraquo Scoring












raquoraquo Scoring










raquoraquo Scoring












raquoraquo Scoring




Practice Example 1



Practice Example 2


Practice Example 3



raquoraquo Scoring

Practice Example 4


Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9

Practice Example 10




raquoraquo Scoring


Practice Example 1




Practice Example 2



Practice Example 3




raquoraquo Scoring

Practice Example 4



Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9




raquoraquo Scoring

Practice Example 10





raquoraquo Scoring

POST-ASSESSMENT
















raquoraquo Scoring






























raquoraquo Scoring



raquoraquo Scoring









2 lt (L) lt (L) gt (G) = (E)


4 gt (G) gt (G) lt (L) = (E)


6 gt (G) gt (G) lt (L) = (E)




raquoraquo Scoring





A Caution









raquoraquo Scoring











raquoraquo Scoring












raquoraquo Scoring









raquoraquo Scoring









raquoraquo Scoring




Practice Example 1

Practice Example 2


Practice Example 3




raquoraquo Scoring

Practice Example 4

Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9


Practice Example 10



raquoraquo Scoring


Practice Example 1


Practice Example 2



Practice Example 3





raquoraquo Scoring

Practice Example 4


Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9





raquoraquo Scoring

Practice Example 10













Pre-Assessment 1




Post-Assessment 1





Post-Assessment 2





Pre-Assessment 4








Pre-Assessment 5





Pre-Assessment 6










Pre-Assessment 1





Pre-Assessment 2




Pre-Assessment 2









Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)



















raquoraquo Scoring














raquoraquo Scoring



raquoraquo Scoring















raquoraquo Scoring







2 lt (L) lt (L) gt (G) = (E)


4 gt (G) gt (G) lt (L) = (E)


6 gt (G) gt (G) lt (L) = (E)




raquoraquo Scoring





A Caution









raquoraquo Scoring










raquoraquo Scoring












raquoraquo Scoring










raquoraquo Scoring












raquoraquo Scoring




Practice Example 1



Practice Example 2


Practice Example 3



raquoraquo Scoring

Practice Example 4


Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9

Practice Example 10




raquoraquo Scoring


Practice Example 1




Practice Example 2



Practice Example 3




raquoraquo Scoring

Practice Example 4



Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9




raquoraquo Scoring

Practice Example 10





raquoraquo Scoring

POST-ASSESSMENT
















raquoraquo Scoring






























raquoraquo Scoring



raquoraquo Scoring









2 lt (L) lt (L) gt (G) = (E)


4 gt (G) gt (G) lt (L) = (E)


6 gt (G) gt (G) lt (L) = (E)




raquoraquo Scoring





A Caution









raquoraquo Scoring











raquoraquo Scoring












raquoraquo Scoring









raquoraquo Scoring









raquoraquo Scoring




Practice Example 1

Practice Example 2


Practice Example 3




raquoraquo Scoring

Practice Example 4

Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9


Practice Example 10



raquoraquo Scoring


Practice Example 1


Practice Example 2



Practice Example 3





raquoraquo Scoring

Practice Example 4


Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9





raquoraquo Scoring

Practice Example 10













Pre-Assessment 1




Post-Assessment 1





Post-Assessment 2





Pre-Assessment 4








Pre-Assessment 5





Pre-Assessment 6










Pre-Assessment 1





Pre-Assessment 2




Pre-Assessment 2









Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)














raquoraquo Scoring



raquoraquo Scoring















raquoraquo Scoring







2 lt (L) lt (L) gt (G) = (E)


4 gt (G) gt (G) lt (L) = (E)


6 gt (G) gt (G) lt (L) = (E)




raquoraquo Scoring





A Caution









raquoraquo Scoring










raquoraquo Scoring












raquoraquo Scoring










raquoraquo Scoring












raquoraquo Scoring




Practice Example 1



Practice Example 2


Practice Example 3



raquoraquo Scoring

Practice Example 4


Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9

Practice Example 10




raquoraquo Scoring


Practice Example 1




Practice Example 2



Practice Example 3




raquoraquo Scoring

Practice Example 4



Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9




raquoraquo Scoring

Practice Example 10





raquoraquo Scoring

POST-ASSESSMENT
















raquoraquo Scoring






























raquoraquo Scoring



raquoraquo Scoring









2 lt (L) lt (L) gt (G) = (E)


4 gt (G) gt (G) lt (L) = (E)


6 gt (G) gt (G) lt (L) = (E)




raquoraquo Scoring





A Caution









raquoraquo Scoring











raquoraquo Scoring












raquoraquo Scoring









raquoraquo Scoring









raquoraquo Scoring




Practice Example 1

Practice Example 2


Practice Example 3




raquoraquo Scoring

Practice Example 4

Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9


Practice Example 10



raquoraquo Scoring


Practice Example 1


Practice Example 2



Practice Example 3





raquoraquo Scoring

Practice Example 4


Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9





raquoraquo Scoring

Practice Example 10













Pre-Assessment 1




Post-Assessment 1





Post-Assessment 2





Pre-Assessment 4








Pre-Assessment 5





Pre-Assessment 6










Pre-Assessment 1





Pre-Assessment 2




Pre-Assessment 2









Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)



raquoraquo Scoring















raquoraquo Scoring







2 lt (L) lt (L) gt (G) = (E)


4 gt (G) gt (G) lt (L) = (E)


6 gt (G) gt (G) lt (L) = (E)




raquoraquo Scoring





A Caution









raquoraquo Scoring










raquoraquo Scoring












raquoraquo Scoring










raquoraquo Scoring












raquoraquo Scoring




Practice Example 1



Practice Example 2


Practice Example 3



raquoraquo Scoring

Practice Example 4


Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9

Practice Example 10




raquoraquo Scoring


Practice Example 1




Practice Example 2



Practice Example 3




raquoraquo Scoring

Practice Example 4



Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9




raquoraquo Scoring

Practice Example 10





raquoraquo Scoring

POST-ASSESSMENT
















raquoraquo Scoring






























raquoraquo Scoring



raquoraquo Scoring









2 lt (L) lt (L) gt (G) = (E)


4 gt (G) gt (G) lt (L) = (E)


6 gt (G) gt (G) lt (L) = (E)




raquoraquo Scoring





A Caution









raquoraquo Scoring











raquoraquo Scoring












raquoraquo Scoring









raquoraquo Scoring









raquoraquo Scoring




Practice Example 1

Practice Example 2


Practice Example 3




raquoraquo Scoring

Practice Example 4

Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9


Practice Example 10



raquoraquo Scoring


Practice Example 1


Practice Example 2



Practice Example 3





raquoraquo Scoring

Practice Example 4


Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9





raquoraquo Scoring

Practice Example 10













Pre-Assessment 1




Post-Assessment 1





Post-Assessment 2





Pre-Assessment 4








Pre-Assessment 5





Pre-Assessment 6










Pre-Assessment 1





Pre-Assessment 2




Pre-Assessment 2









Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)



raquoraquo Scoring







2 lt (L) lt (L) gt (G) = (E)


4 gt (G) gt (G) lt (L) = (E)


6 gt (G) gt (G) lt (L) = (E)




raquoraquo Scoring





A Caution









raquoraquo Scoring










raquoraquo Scoring












raquoraquo Scoring










raquoraquo Scoring












raquoraquo Scoring




Practice Example 1



Practice Example 2


Practice Example 3



raquoraquo Scoring

Practice Example 4


Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9

Practice Example 10




raquoraquo Scoring


Practice Example 1




Practice Example 2



Practice Example 3




raquoraquo Scoring

Practice Example 4



Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9




raquoraquo Scoring

Practice Example 10





raquoraquo Scoring

POST-ASSESSMENT
















raquoraquo Scoring






























raquoraquo Scoring



raquoraquo Scoring









2 lt (L) lt (L) gt (G) = (E)


4 gt (G) gt (G) lt (L) = (E)


6 gt (G) gt (G) lt (L) = (E)




raquoraquo Scoring





A Caution









raquoraquo Scoring











raquoraquo Scoring












raquoraquo Scoring









raquoraquo Scoring









raquoraquo Scoring




Practice Example 1

Practice Example 2


Practice Example 3




raquoraquo Scoring

Practice Example 4

Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9


Practice Example 10



raquoraquo Scoring


Practice Example 1


Practice Example 2



Practice Example 3





raquoraquo Scoring

Practice Example 4


Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9





raquoraquo Scoring

Practice Example 10













Pre-Assessment 1




Post-Assessment 1





Post-Assessment 2





Pre-Assessment 4








Pre-Assessment 5





Pre-Assessment 6










Pre-Assessment 1





Pre-Assessment 2




Pre-Assessment 2









Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)



raquoraquo Scoring





A Caution









raquoraquo Scoring










raquoraquo Scoring












raquoraquo Scoring










raquoraquo Scoring












raquoraquo Scoring




Practice Example 1



Practice Example 2


Practice Example 3



raquoraquo Scoring

Practice Example 4


Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9

Practice Example 10




raquoraquo Scoring


Practice Example 1




Practice Example 2



Practice Example 3




raquoraquo Scoring

Practice Example 4



Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9




raquoraquo Scoring

Practice Example 10





raquoraquo Scoring

POST-ASSESSMENT
















raquoraquo Scoring






























raquoraquo Scoring



raquoraquo Scoring









2 lt (L) lt (L) gt (G) = (E)


4 gt (G) gt (G) lt (L) = (E)


6 gt (G) gt (G) lt (L) = (E)




raquoraquo Scoring





A Caution









raquoraquo Scoring











raquoraquo Scoring












raquoraquo Scoring









raquoraquo Scoring









raquoraquo Scoring




Practice Example 1

Practice Example 2


Practice Example 3




raquoraquo Scoring

Practice Example 4

Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9


Practice Example 10



raquoraquo Scoring


Practice Example 1


Practice Example 2



Practice Example 3





raquoraquo Scoring

Practice Example 4


Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9





raquoraquo Scoring

Practice Example 10













Pre-Assessment 1




Post-Assessment 1





Post-Assessment 2





Pre-Assessment 4








Pre-Assessment 5





Pre-Assessment 6










Pre-Assessment 1





Pre-Assessment 2




Pre-Assessment 2









Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)



raquoraquo Scoring










raquoraquo Scoring












raquoraquo Scoring










raquoraquo Scoring












raquoraquo Scoring




Practice Example 1



Practice Example 2


Practice Example 3



raquoraquo Scoring

Practice Example 4


Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9

Practice Example 10




raquoraquo Scoring


Practice Example 1




Practice Example 2



Practice Example 3




raquoraquo Scoring

Practice Example 4



Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9




raquoraquo Scoring

Practice Example 10





raquoraquo Scoring

POST-ASSESSMENT
















raquoraquo Scoring






























raquoraquo Scoring



raquoraquo Scoring









2 lt (L) lt (L) gt (G) = (E)


4 gt (G) gt (G) lt (L) = (E)


6 gt (G) gt (G) lt (L) = (E)




raquoraquo Scoring





A Caution









raquoraquo Scoring











raquoraquo Scoring












raquoraquo Scoring









raquoraquo Scoring









raquoraquo Scoring




Practice Example 1

Practice Example 2


Practice Example 3




raquoraquo Scoring

Practice Example 4

Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9


Practice Example 10



raquoraquo Scoring


Practice Example 1


Practice Example 2



Practice Example 3





raquoraquo Scoring

Practice Example 4


Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9





raquoraquo Scoring

Practice Example 10













Pre-Assessment 1




Post-Assessment 1





Post-Assessment 2





Pre-Assessment 4








Pre-Assessment 5





Pre-Assessment 6










Pre-Assessment 1





Pre-Assessment 2




Pre-Assessment 2









Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)



raquoraquo Scoring












raquoraquo Scoring










raquoraquo Scoring












raquoraquo Scoring




Practice Example 1



Practice Example 2


Practice Example 3



raquoraquo Scoring

Practice Example 4


Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9

Practice Example 10




raquoraquo Scoring


Practice Example 1




Practice Example 2



Practice Example 3




raquoraquo Scoring

Practice Example 4



Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9




raquoraquo Scoring

Practice Example 10





raquoraquo Scoring

POST-ASSESSMENT
















raquoraquo Scoring






























raquoraquo Scoring



raquoraquo Scoring









2 lt (L) lt (L) gt (G) = (E)


4 gt (G) gt (G) lt (L) = (E)


6 gt (G) gt (G) lt (L) = (E)




raquoraquo Scoring





A Caution









raquoraquo Scoring











raquoraquo Scoring












raquoraquo Scoring









raquoraquo Scoring









raquoraquo Scoring




Practice Example 1

Practice Example 2


Practice Example 3




raquoraquo Scoring

Practice Example 4

Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9


Practice Example 10



raquoraquo Scoring


Practice Example 1


Practice Example 2



Practice Example 3





raquoraquo Scoring

Practice Example 4


Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9





raquoraquo Scoring

Practice Example 10













Pre-Assessment 1




Post-Assessment 1





Post-Assessment 2





Pre-Assessment 4








Pre-Assessment 5





Pre-Assessment 6










Pre-Assessment 1





Pre-Assessment 2




Pre-Assessment 2









Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)



raquoraquo Scoring










raquoraquo Scoring












raquoraquo Scoring




Practice Example 1



Practice Example 2


Practice Example 3



raquoraquo Scoring

Practice Example 4


Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9

Practice Example 10




raquoraquo Scoring


Practice Example 1




Practice Example 2



Practice Example 3




raquoraquo Scoring

Practice Example 4



Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9




raquoraquo Scoring

Practice Example 10





raquoraquo Scoring

POST-ASSESSMENT
















raquoraquo Scoring






























raquoraquo Scoring



raquoraquo Scoring









2 lt (L) lt (L) gt (G) = (E)


4 gt (G) gt (G) lt (L) = (E)


6 gt (G) gt (G) lt (L) = (E)




raquoraquo Scoring





A Caution









raquoraquo Scoring











raquoraquo Scoring












raquoraquo Scoring









raquoraquo Scoring









raquoraquo Scoring




Practice Example 1

Practice Example 2


Practice Example 3




raquoraquo Scoring

Practice Example 4

Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9


Practice Example 10



raquoraquo Scoring


Practice Example 1


Practice Example 2



Practice Example 3





raquoraquo Scoring

Practice Example 4


Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9





raquoraquo Scoring

Practice Example 10













Pre-Assessment 1




Post-Assessment 1





Post-Assessment 2





Pre-Assessment 4








Pre-Assessment 5





Pre-Assessment 6










Pre-Assessment 1





Pre-Assessment 2




Pre-Assessment 2









Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)



raquoraquo Scoring












raquoraquo Scoring




Practice Example 1



Practice Example 2


Practice Example 3



raquoraquo Scoring

Practice Example 4


Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9

Practice Example 10




raquoraquo Scoring


Practice Example 1




Practice Example 2



Practice Example 3




raquoraquo Scoring

Practice Example 4



Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9




raquoraquo Scoring

Practice Example 10





raquoraquo Scoring

POST-ASSESSMENT
















raquoraquo Scoring






























raquoraquo Scoring



raquoraquo Scoring









2 lt (L) lt (L) gt (G) = (E)


4 gt (G) gt (G) lt (L) = (E)


6 gt (G) gt (G) lt (L) = (E)




raquoraquo Scoring





A Caution









raquoraquo Scoring











raquoraquo Scoring












raquoraquo Scoring









raquoraquo Scoring









raquoraquo Scoring




Practice Example 1

Practice Example 2


Practice Example 3




raquoraquo Scoring

Practice Example 4

Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9


Practice Example 10



raquoraquo Scoring


Practice Example 1


Practice Example 2



Practice Example 3





raquoraquo Scoring

Practice Example 4


Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9





raquoraquo Scoring

Practice Example 10













Pre-Assessment 1




Post-Assessment 1





Post-Assessment 2





Pre-Assessment 4








Pre-Assessment 5





Pre-Assessment 6










Pre-Assessment 1





Pre-Assessment 2




Pre-Assessment 2









Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)



raquoraquo Scoring




Practice Example 1



Practice Example 2


Practice Example 3



raquoraquo Scoring

Practice Example 4


Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9

Practice Example 10




raquoraquo Scoring


Practice Example 1




Practice Example 2



Practice Example 3




raquoraquo Scoring

Practice Example 4



Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9




raquoraquo Scoring

Practice Example 10





raquoraquo Scoring

POST-ASSESSMENT
















raquoraquo Scoring






























raquoraquo Scoring



raquoraquo Scoring









2 lt (L) lt (L) gt (G) = (E)


4 gt (G) gt (G) lt (L) = (E)


6 gt (G) gt (G) lt (L) = (E)




raquoraquo Scoring





A Caution









raquoraquo Scoring











raquoraquo Scoring












raquoraquo Scoring









raquoraquo Scoring









raquoraquo Scoring




Practice Example 1

Practice Example 2


Practice Example 3




raquoraquo Scoring

Practice Example 4

Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9


Practice Example 10



raquoraquo Scoring


Practice Example 1


Practice Example 2



Practice Example 3





raquoraquo Scoring

Practice Example 4


Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9





raquoraquo Scoring

Practice Example 10













Pre-Assessment 1




Post-Assessment 1





Post-Assessment 2





Pre-Assessment 4








Pre-Assessment 5





Pre-Assessment 6










Pre-Assessment 1





Pre-Assessment 2




Pre-Assessment 2









Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)



raquoraquo Scoring

Practice Example 4


Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9

Practice Example 10




raquoraquo Scoring


Practice Example 1




Practice Example 2



Practice Example 3




raquoraquo Scoring

Practice Example 4



Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9




raquoraquo Scoring

Practice Example 10





raquoraquo Scoring

POST-ASSESSMENT
















raquoraquo Scoring






























raquoraquo Scoring



raquoraquo Scoring









2 lt (L) lt (L) gt (G) = (E)


4 gt (G) gt (G) lt (L) = (E)


6 gt (G) gt (G) lt (L) = (E)




raquoraquo Scoring





A Caution









raquoraquo Scoring











raquoraquo Scoring












raquoraquo Scoring









raquoraquo Scoring









raquoraquo Scoring




Practice Example 1

Practice Example 2


Practice Example 3




raquoraquo Scoring

Practice Example 4

Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9


Practice Example 10



raquoraquo Scoring


Practice Example 1


Practice Example 2



Practice Example 3





raquoraquo Scoring

Practice Example 4


Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9





raquoraquo Scoring

Practice Example 10













Pre-Assessment 1




Post-Assessment 1





Post-Assessment 2





Pre-Assessment 4








Pre-Assessment 5





Pre-Assessment 6










Pre-Assessment 1





Pre-Assessment 2




Pre-Assessment 2









Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)



raquoraquo Scoring

Practice Example 8


Practice Example 9

Practice Example 10




raquoraquo Scoring


Practice Example 1




Practice Example 2



Practice Example 3




raquoraquo Scoring

Practice Example 4



Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9




raquoraquo Scoring

Practice Example 10





raquoraquo Scoring

POST-ASSESSMENT
















raquoraquo Scoring






























raquoraquo Scoring



raquoraquo Scoring









2 lt (L) lt (L) gt (G) = (E)


4 gt (G) gt (G) lt (L) = (E)


6 gt (G) gt (G) lt (L) = (E)




raquoraquo Scoring





A Caution









raquoraquo Scoring











raquoraquo Scoring












raquoraquo Scoring









raquoraquo Scoring









raquoraquo Scoring




Practice Example 1

Practice Example 2


Practice Example 3




raquoraquo Scoring

Practice Example 4

Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9


Practice Example 10



raquoraquo Scoring


Practice Example 1


Practice Example 2



Practice Example 3





raquoraquo Scoring

Practice Example 4


Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9





raquoraquo Scoring

Practice Example 10













Pre-Assessment 1




Post-Assessment 1





Post-Assessment 2





Pre-Assessment 4








Pre-Assessment 5





Pre-Assessment 6










Pre-Assessment 1





Pre-Assessment 2




Pre-Assessment 2









Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)



raquoraquo Scoring


Practice Example 1




Practice Example 2



Practice Example 3




raquoraquo Scoring

Practice Example 4



Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9




raquoraquo Scoring

Practice Example 10





raquoraquo Scoring

POST-ASSESSMENT
















raquoraquo Scoring






























raquoraquo Scoring



raquoraquo Scoring









2 lt (L) lt (L) gt (G) = (E)


4 gt (G) gt (G) lt (L) = (E)


6 gt (G) gt (G) lt (L) = (E)




raquoraquo Scoring





A Caution









raquoraquo Scoring











raquoraquo Scoring












raquoraquo Scoring









raquoraquo Scoring









raquoraquo Scoring




Practice Example 1

Practice Example 2


Practice Example 3




raquoraquo Scoring

Practice Example 4

Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9


Practice Example 10



raquoraquo Scoring


Practice Example 1


Practice Example 2



Practice Example 3





raquoraquo Scoring

Practice Example 4


Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9





raquoraquo Scoring

Practice Example 10













Pre-Assessment 1




Post-Assessment 1





Post-Assessment 2





Pre-Assessment 4








Pre-Assessment 5





Pre-Assessment 6










Pre-Assessment 1





Pre-Assessment 2




Pre-Assessment 2









Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)



raquoraquo Scoring

Practice Example 4



Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9




raquoraquo Scoring

Practice Example 10





raquoraquo Scoring

POST-ASSESSMENT
















raquoraquo Scoring






























raquoraquo Scoring



raquoraquo Scoring









2 lt (L) lt (L) gt (G) = (E)


4 gt (G) gt (G) lt (L) = (E)


6 gt (G) gt (G) lt (L) = (E)




raquoraquo Scoring





A Caution









raquoraquo Scoring











raquoraquo Scoring












raquoraquo Scoring









raquoraquo Scoring









raquoraquo Scoring




Practice Example 1

Practice Example 2


Practice Example 3




raquoraquo Scoring

Practice Example 4

Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9


Practice Example 10



raquoraquo Scoring


Practice Example 1


Practice Example 2



Practice Example 3





raquoraquo Scoring

Practice Example 4


Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9





raquoraquo Scoring

Practice Example 10













Pre-Assessment 1




Post-Assessment 1





Post-Assessment 2





Pre-Assessment 4








Pre-Assessment 5





Pre-Assessment 6










Pre-Assessment 1





Pre-Assessment 2




Pre-Assessment 2









Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)



raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9




raquoraquo Scoring

Practice Example 10





raquoraquo Scoring

POST-ASSESSMENT
















raquoraquo Scoring






























raquoraquo Scoring



raquoraquo Scoring









2 lt (L) lt (L) gt (G) = (E)


4 gt (G) gt (G) lt (L) = (E)


6 gt (G) gt (G) lt (L) = (E)




raquoraquo Scoring





A Caution









raquoraquo Scoring











raquoraquo Scoring












raquoraquo Scoring









raquoraquo Scoring









raquoraquo Scoring




Practice Example 1

Practice Example 2


Practice Example 3




raquoraquo Scoring

Practice Example 4

Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9


Practice Example 10



raquoraquo Scoring


Practice Example 1


Practice Example 2



Practice Example 3





raquoraquo Scoring

Practice Example 4


Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9





raquoraquo Scoring

Practice Example 10













Pre-Assessment 1




Post-Assessment 1





Post-Assessment 2





Pre-Assessment 4








Pre-Assessment 5





Pre-Assessment 6










Pre-Assessment 1





Pre-Assessment 2




Pre-Assessment 2









Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)



raquoraquo Scoring

Practice Example 10





raquoraquo Scoring

POST-ASSESSMENT
















raquoraquo Scoring






























raquoraquo Scoring



raquoraquo Scoring









2 lt (L) lt (L) gt (G) = (E)


4 gt (G) gt (G) lt (L) = (E)


6 gt (G) gt (G) lt (L) = (E)




raquoraquo Scoring





A Caution









raquoraquo Scoring











raquoraquo Scoring












raquoraquo Scoring









raquoraquo Scoring









raquoraquo Scoring




Practice Example 1

Practice Example 2


Practice Example 3




raquoraquo Scoring

Practice Example 4

Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9


Practice Example 10



raquoraquo Scoring


Practice Example 1


Practice Example 2



Practice Example 3





raquoraquo Scoring

Practice Example 4


Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9





raquoraquo Scoring

Practice Example 10













Pre-Assessment 1




Post-Assessment 1





Post-Assessment 2





Pre-Assessment 4








Pre-Assessment 5





Pre-Assessment 6










Pre-Assessment 1





Pre-Assessment 2




Pre-Assessment 2









Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)



raquoraquo Scoring

POST-ASSESSMENT
















raquoraquo Scoring






























raquoraquo Scoring



raquoraquo Scoring









2 lt (L) lt (L) gt (G) = (E)


4 gt (G) gt (G) lt (L) = (E)


6 gt (G) gt (G) lt (L) = (E)




raquoraquo Scoring





A Caution









raquoraquo Scoring











raquoraquo Scoring












raquoraquo Scoring









raquoraquo Scoring









raquoraquo Scoring




Practice Example 1

Practice Example 2


Practice Example 3




raquoraquo Scoring

Practice Example 4

Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9


Practice Example 10



raquoraquo Scoring


Practice Example 1


Practice Example 2



Practice Example 3





raquoraquo Scoring

Practice Example 4


Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9





raquoraquo Scoring

Practice Example 10













Pre-Assessment 1




Post-Assessment 1





Post-Assessment 2





Pre-Assessment 4








Pre-Assessment 5





Pre-Assessment 6










Pre-Assessment 1





Pre-Assessment 2




Pre-Assessment 2









Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)



raquoraquo Scoring






























raquoraquo Scoring



raquoraquo Scoring









2 lt (L) lt (L) gt (G) = (E)


4 gt (G) gt (G) lt (L) = (E)


6 gt (G) gt (G) lt (L) = (E)




raquoraquo Scoring





A Caution









raquoraquo Scoring











raquoraquo Scoring












raquoraquo Scoring









raquoraquo Scoring









raquoraquo Scoring




Practice Example 1

Practice Example 2


Practice Example 3




raquoraquo Scoring

Practice Example 4

Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9


Practice Example 10



raquoraquo Scoring


Practice Example 1


Practice Example 2



Practice Example 3





raquoraquo Scoring

Practice Example 4


Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9





raquoraquo Scoring

Practice Example 10













Pre-Assessment 1




Post-Assessment 1





Post-Assessment 2





Pre-Assessment 4








Pre-Assessment 5





Pre-Assessment 6










Pre-Assessment 1





Pre-Assessment 2




Pre-Assessment 2









Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)














raquoraquo Scoring



raquoraquo Scoring









2 lt (L) lt (L) gt (G) = (E)


4 gt (G) gt (G) lt (L) = (E)


6 gt (G) gt (G) lt (L) = (E)




raquoraquo Scoring





A Caution









raquoraquo Scoring











raquoraquo Scoring












raquoraquo Scoring









raquoraquo Scoring









raquoraquo Scoring




Practice Example 1

Practice Example 2


Practice Example 3




raquoraquo Scoring

Practice Example 4

Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9


Practice Example 10



raquoraquo Scoring


Practice Example 1


Practice Example 2



Practice Example 3





raquoraquo Scoring

Practice Example 4


Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9





raquoraquo Scoring

Practice Example 10













Pre-Assessment 1




Post-Assessment 1





Post-Assessment 2





Pre-Assessment 4








Pre-Assessment 5





Pre-Assessment 6










Pre-Assessment 1





Pre-Assessment 2




Pre-Assessment 2









Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)



raquoraquo Scoring









2 lt (L) lt (L) gt (G) = (E)


4 gt (G) gt (G) lt (L) = (E)


6 gt (G) gt (G) lt (L) = (E)




raquoraquo Scoring





A Caution









raquoraquo Scoring











raquoraquo Scoring












raquoraquo Scoring









raquoraquo Scoring









raquoraquo Scoring




Practice Example 1

Practice Example 2


Practice Example 3




raquoraquo Scoring

Practice Example 4

Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9


Practice Example 10



raquoraquo Scoring


Practice Example 1


Practice Example 2



Practice Example 3





raquoraquo Scoring

Practice Example 4


Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9





raquoraquo Scoring

Practice Example 10













Pre-Assessment 1




Post-Assessment 1





Post-Assessment 2





Pre-Assessment 4








Pre-Assessment 5





Pre-Assessment 6










Pre-Assessment 1





Pre-Assessment 2




Pre-Assessment 2









Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)



raquoraquo Scoring





A Caution









raquoraquo Scoring











raquoraquo Scoring












raquoraquo Scoring









raquoraquo Scoring









raquoraquo Scoring




Practice Example 1

Practice Example 2


Practice Example 3




raquoraquo Scoring

Practice Example 4

Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9


Practice Example 10



raquoraquo Scoring


Practice Example 1


Practice Example 2



Practice Example 3





raquoraquo Scoring

Practice Example 4


Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9





raquoraquo Scoring

Practice Example 10













Pre-Assessment 1




Post-Assessment 1





Post-Assessment 2





Pre-Assessment 4








Pre-Assessment 5





Pre-Assessment 6










Pre-Assessment 1





Pre-Assessment 2




Pre-Assessment 2









Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)



raquoraquo Scoring











raquoraquo Scoring












raquoraquo Scoring









raquoraquo Scoring









raquoraquo Scoring




Practice Example 1

Practice Example 2


Practice Example 3




raquoraquo Scoring

Practice Example 4

Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9


Practice Example 10



raquoraquo Scoring


Practice Example 1


Practice Example 2



Practice Example 3





raquoraquo Scoring

Practice Example 4


Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9





raquoraquo Scoring

Practice Example 10













Pre-Assessment 1




Post-Assessment 1





Post-Assessment 2





Pre-Assessment 4








Pre-Assessment 5





Pre-Assessment 6










Pre-Assessment 1





Pre-Assessment 2




Pre-Assessment 2









Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)



raquoraquo Scoring












raquoraquo Scoring









raquoraquo Scoring









raquoraquo Scoring




Practice Example 1

Practice Example 2


Practice Example 3




raquoraquo Scoring

Practice Example 4

Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9


Practice Example 10



raquoraquo Scoring


Practice Example 1


Practice Example 2



Practice Example 3





raquoraquo Scoring

Practice Example 4


Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9





raquoraquo Scoring

Practice Example 10













Pre-Assessment 1




Post-Assessment 1





Post-Assessment 2





Pre-Assessment 4








Pre-Assessment 5





Pre-Assessment 6










Pre-Assessment 1





Pre-Assessment 2




Pre-Assessment 2









Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)



raquoraquo Scoring









raquoraquo Scoring









raquoraquo Scoring




Practice Example 1

Practice Example 2


Practice Example 3




raquoraquo Scoring

Practice Example 4

Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9


Practice Example 10



raquoraquo Scoring


Practice Example 1


Practice Example 2



Practice Example 3





raquoraquo Scoring

Practice Example 4


Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9





raquoraquo Scoring

Practice Example 10













Pre-Assessment 1




Post-Assessment 1





Post-Assessment 2





Pre-Assessment 4








Pre-Assessment 5





Pre-Assessment 6










Pre-Assessment 1





Pre-Assessment 2




Pre-Assessment 2









Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)



raquoraquo Scoring









raquoraquo Scoring




Practice Example 1

Practice Example 2


Practice Example 3




raquoraquo Scoring

Practice Example 4

Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9


Practice Example 10



raquoraquo Scoring


Practice Example 1


Practice Example 2



Practice Example 3





raquoraquo Scoring

Practice Example 4


Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9





raquoraquo Scoring

Practice Example 10













Pre-Assessment 1




Post-Assessment 1





Post-Assessment 2





Pre-Assessment 4








Pre-Assessment 5





Pre-Assessment 6










Pre-Assessment 1





Pre-Assessment 2




Pre-Assessment 2









Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)



raquoraquo Scoring




Practice Example 1

Practice Example 2


Practice Example 3




raquoraquo Scoring

Practice Example 4

Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9


Practice Example 10



raquoraquo Scoring


Practice Example 1


Practice Example 2



Practice Example 3





raquoraquo Scoring

Practice Example 4


Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9





raquoraquo Scoring

Practice Example 10













Pre-Assessment 1




Post-Assessment 1





Post-Assessment 2





Pre-Assessment 4








Pre-Assessment 5





Pre-Assessment 6










Pre-Assessment 1





Pre-Assessment 2




Pre-Assessment 2









Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)



raquoraquo Scoring

Practice Example 4

Practice Example 5


Practice Example 6


Practice Example 7




raquoraquo Scoring

Practice Example 8


Practice Example 9


Practice Example 10



raquoraquo Scoring


Practice Example 1


Practice Example 2



Practice Example 3





raquoraquo Scoring

Practice Example 4


Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9





raquoraquo Scoring

Practice Example 10













Pre-Assessment 1




Post-Assessment 1





Post-Assessment 2





Pre-Assessment 4








Pre-Assessment 5





Pre-Assessment 6










Pre-Assessment 1





Pre-Assessment 2




Pre-Assessment 2









Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)



raquoraquo Scoring

Practice Example 8


Practice Example 9


Practice Example 10



raquoraquo Scoring


Practice Example 1


Practice Example 2



Practice Example 3





raquoraquo Scoring

Practice Example 4


Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9





raquoraquo Scoring

Practice Example 10













Pre-Assessment 1




Post-Assessment 1





Post-Assessment 2





Pre-Assessment 4








Pre-Assessment 5





Pre-Assessment 6










Pre-Assessment 1





Pre-Assessment 2




Pre-Assessment 2









Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)



raquoraquo Scoring


Practice Example 1


Practice Example 2



Practice Example 3





raquoraquo Scoring

Practice Example 4


Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9





raquoraquo Scoring

Practice Example 10













Pre-Assessment 1




Post-Assessment 1





Post-Assessment 2





Pre-Assessment 4








Pre-Assessment 5





Pre-Assessment 6










Pre-Assessment 1





Pre-Assessment 2




Pre-Assessment 2









Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)



raquoraquo Scoring

Practice Example 4


Practice Example 5



Practice Example 6





raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9





raquoraquo Scoring

Practice Example 10













Pre-Assessment 1




Post-Assessment 1





Post-Assessment 2





Pre-Assessment 4








Pre-Assessment 5





Pre-Assessment 6










Pre-Assessment 1





Pre-Assessment 2




Pre-Assessment 2









Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)



raquoraquo Scoring

Practice Example 7



Practice Example 8



Practice Example 9





raquoraquo Scoring

Practice Example 10













Pre-Assessment 1




Post-Assessment 1





Post-Assessment 2





Pre-Assessment 4








Pre-Assessment 5





Pre-Assessment 6










Pre-Assessment 1





Pre-Assessment 2




Pre-Assessment 2









Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)



raquoraquo Scoring

Practice Example 10













Pre-Assessment 1




Post-Assessment 1





Post-Assessment 2





Pre-Assessment 4








Pre-Assessment 5





Pre-Assessment 6










Pre-Assessment 1





Pre-Assessment 2




Pre-Assessment 2









Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)












Pre-Assessment 1




Post-Assessment 1





Post-Assessment 2





Pre-Assessment 4








Pre-Assessment 5





Pre-Assessment 6










Pre-Assessment 1





Pre-Assessment 2




Pre-Assessment 2









Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)





Pre-Assessment 5





Pre-Assessment 6










Pre-Assessment 1





Pre-Assessment 2




Pre-Assessment 2









Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)






Pre-Assessment 1





Pre-Assessment 2




Pre-Assessment 2









Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)





Pre-Assessment 3





Pre-Assessment 4





Pre-Assessment 1










Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)






Pre-Assessment 2





Pre-Assessment 2





Pre-Assessment 2









Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)





Pre-Assessment 4





Post-Assessment 4





Post-Assessment 6




Post-Assessment 6










Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)







Pre-Assessment 2




Pre-Assessment 4






Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)




Correct Reasoning






Notes

Pre-Assessment 2



Pre-Assessment 1



Pre-Assessment 6



Pre-Assessment 7








Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)





Notes

Post-Assessment 2




















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)

















Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13 Date__________13 Class___________







67

712

58

89

46

23

911

57

1)

2)

3)

4)











































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)









































Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________





1)

2)

3)

4)

45

67

68

34

79

35

Less13 than13 (lt)


Equivalent13 (=)


512

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

Less13 than13 (lt)


Equivalent13 (=)


37

49

7)

5)

6)

7)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)




Name_________________________13

Date__________13 Class___________






1)

2)

3)

4)

45

67

68

34

37

49

79

35

1)

2)

3)

4)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)



Name_________________________13

Date__________13 Class___________



Less13 than13 (lt)


Equivalent13 (=)

Less13 than13 (lt)


Equivalent13 (=)



5)

6)

78

56

47

1221

5)

6)

comparing two fractions - education development … · ... the smaller the value of the fraction...

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