Suzette Grube-Thur
CCLM^2 Project
Summer 2012
DRAFT DOCUMENT. This material was developed as part of the
Leadership for the Common Core in Mathematics (CCLM^2) project at the University of Wisconsin-Milwaukee.
CCSSM Analysis: 5.NF.2 Part 1: Standard Grade: Grade 5 Domain: Number and Operations -­‐ Fractions (NF) Cluster: Use equivalent fractions as a strategy to add and subtract fractions. Standard: 5.NF.2 Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators, e.g. by using visual fraction models or equations to represent the problem. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. For example, recognize an incorrect results 2/5 + ½ = 3/7, by observing that 3/7 < ½. 1 Cluster Broad Explanation of Cluster & the Representations, Standards Within diagrams, contexts, strategies, & examples to support understanding 5.NF Use equivalent fractions as a strategy to add and subtract fractions (with unlike denominators). This cluster asks students to apply their understanding of fractions and fraction equivalency to successfully add and subtract fractions with unlike denominators. 5NF.1 This standard asks students to apply their understanding of fractions and fraction models to represent the addition and subtraction of fractions with unlike denominators as equivalent calculations with like denominators. They develop fluency in calculating sums and differences of fractions, and make reasonable estimates of them. 5NF.2 This standard requires students to apply what they have learned about creating equivalent fractions and the use of visual models in order to solve situations that involve the +/-­‐ of fractions with unlike denominators. Students will also use number sense, estimation, & benchmarks to assess the reasonableness of their answers. 2 Example of Fraction Equivalency with Unlike Denominators: 1/3 + 1/4 = (1 x 4) + (1 x 3) (3 x 4) (4 x 3) = 4/12 + 3/12 = 7/12 Convert two fractions with unlike denominators by finding the common product of both denominators. You can do this by multiplying the denominators together (3 x 4 = 12 so 12 is the common denominator). (see example below) Mathematical Language: fraction, equivalent or equivalency, addition or add, sum, subtraction or subtract, difference, multiply, product, unlike/like, common, denominator, numerator, benchmark fraction, estimate, & reasonable (ness), representation, fraction model, number line, area model, tape diagram, equation Part 2: Explanation and Examples of the Standard Standard Explanation of what students Representations, diagrams, 5.NF.2 are expected to know/be able contexts, strategies, & examples Components: to do to support understanding Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators… …by using visual fraction models or equations to represent the problem. This standard requires students to apply their understanding of fractions and fraction equivalency to solve word problems with like/unlike denominators. They will accomplish this by converting fractions with unlike denominators to equivalent fractions in order to determine the sum or difference using “newly created” equivalent fractions with like denominators. Example: Leslie ran 5/6 of a mile and walked ½ of a mile. How much father did Leslie run than walk? Language: You can use 6 as a common denominator. For the fraction of ½, multiply the numerator and denominator by 3 to find an equivalent fraction to 5/6. 1 x 3 = 3 2 x 3 6 Now subtract the two fractions. Subtract the numerators and keep the denominators the same. 5 -­‐ 3 = 2 = 1 6 6 6 3 Leslie ran 1/3 of a mile farther than she walked. In order to accomplish this part of the Example: (Fraction Bars/Area Model) standard, students use a diverse Brianna bought 5/6 of a pound of fudge and repertoire of visual representations Jeremy bought 1/2 of a pound of fudge. including number lines, area models, How much total fudge did Brianna & Jeremy and equations to represent the buy? addition and subtraction of unlike B: fractions using the strategy of equivalent fractions. They should be able to describe/show how the J: representation connects to related equations. If you think of Jeremy’s fraction in terms of 6th, you can see that 3/6 = ½. Slide three of the size 1/6 next to the yellow. Next, add 3/6 + 3/6 = 6/6 which is the same as 1 whole. Then, you would have to add the remaining 2/6 to the 6/6 above = 8/6 which is the same as 1 2/6 or 1 1/3 pounds of fudge. 3 Visual Models continued… Example (Number Line): Draw a Example (Equation): number line with the common 5 + 1 = n denominator of 6; starting at 5/6, show 6 2 “jumps” to represent the ½ (3/6 = 1/6/ 5 + (1 x 3) = 5 + 3 = 8 + 1/6 + 1/6) you added.
6 2 3 6 6 6 is the same as 1 2/6 or 1 1/3. Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers. Students possess number sense related to fractions as demonstrated by their understanding that fractions are numbers that lie within whole numbers on a number line. Students use reasoning (for example distance from a whole) & benchmarks (esp. 0, ½, 1) to compare fractional parts, as well as to check the reasonableness of their calculated sums & differences in situations. 4 Example: Your teacher gave you 1/7 of a bag of
candy. She also gave your friend 1/3 of the
bag of candy. If you and your friend
combined your candy, what fraction of the
bag would you have? Estimate your answer
and then calculate. How reasonable was
your estimate?
Language:
(a) 1/7 is really close to 0. 1/3 is larger than
1/7, but still less than 1/2. If we put them
together we might get close to 1/2. 1/7 +
1/3= 3/21 + 7/21 = 10/21. If we know that
10 is half of 20, we could say that 10/21 is a
little less than 1⁄2.
(b) 1/7 is close to 1/6 but less than 1/6, and
1/3 is equivalent to 2/6, so I have a little less
than 3/6 or 1⁄2.
Part 3: School Mathematics Textbook Program (a) Textbook Development As part of this project, I had the opportunity to peruse the following curriculum materials utilized by the Mequon-­‐Thiensville School District: Every Day Mathematics (EDM) – Grade 4; MathThematics – Course 1 for Grade 5 (MT1); and MathThematics Course 2 for Grade 6 (MT2). The reader should note that not only does the district switch textbook publishers (Everyday Math to MathThematics), but the district also “skips” over Grade 5 to Grade 6 materials by going from EDM Grade 4 to MathThematics-­‐ Course 1, which is identified as a 6th grade textbook. If students are to be proficient in the standard of 5.NF.2, it is vital they have a clear understanding of several foundational concepts. These concepts include the following (as identified by the CCSSM for grades 4 and 5): a. Property of Equivalency for Fractions (which includes a clear understanding of fraction equivalence, comparing, and ordering); b. Addition and Subtraction of Fractions with Like/Unlike Denominators; c. How to Utilize a Variety of Visual Fraction Models (including area models, set models, and number lines) & Write Equations; and d. How to Approach (plan) and Solve (strategies) Word Problems." Kindly refer to Table 1 entitled: “Textbook Development of Foundational Concepts” to learn more about the progression (or lack of direct instruction/practice) of these foundational concepts of equivalency, fraction addition and subtraction, and visual fraction models in the Mequon-­‐Thiensville Mathematics resources for grades 4-­‐6. Problem solving will be addressed not only within the table’s contents, but also later in the text analysis. Table 1: Textbook Development of Foundational Concepts CCSSM Standards Correlations: Actual Textbook Introduction, Development, and Fundamental Concepts Nec. for Review Progression EDM 7-­‐1: NF1 and 4.NF2 Reviews fraction ideas previously introduced & extends knowledge by Property of Equivalent developing a good understanding of equivalent fractions. Reviews Fractions: fraction concepts: whole, unit, fair shares, notation, mixed numbers, Extend understanding of fraction
equivalence and ordering. Students
develop understanding of fraction
equivalence and operations with fractions.
They recognize that two different
fractions can be equal (e.g., 15/9 = 5/3),
and they develop methods for generating
and recognizing equivalent fractions.
numerator/denominator. fractional parts. Concrete Models (pattern blocks) are utilized in both EDM 7-­‐1 and EDM 7-­‐4. Students shade fraction parts of a pattern block drawing. They understand that ½ means the shading of 1 triangle out of 2 in a drawing of a rhombus; they cover a figure and record a partition of the shape + label each fractional part of the shape with a fraction of a pattern block drawing. In EDM 7-­‐1 students further their fraction understanding by creating a number line poster (visual representation) for fractions. MT1 Module 1, Section 4 • Students use pattern blocks to explore the relationship between part of a design and the whole design. Students also 5 NF1 and 4.NF2 Property of Equivalent Fractions continued work with “more than a whole” (mixed numbers) using pattern blocks. • With pattern blocks, students learn to convert mixed numbers to improper fractions, and improper fractions to mixed numbers. They also learn the relationship between quotients and mixed numbers. • Problem situations are presented in the homework that incorporates area models (dot paper, rectangular arrays, circles), measurement (miles), and fractional sharing of money (determine reasonable & appropriate representation of a share). EDM 7-­‐4: • Partially addresses equivalent fractions by examining the idea equivalent fractions can be created by dividing a shaded region or polygon regions into various parts using pattern blocks & Tangrams. • Students’ experiences focus on concrete materials (pattern blocks),
not algorithms. They learn to recognize that they must consider the
size of the whole when comparing fractions (i.e., 1⁄2 and 1/8 of
two medium pizzas is very different from 1⁄2 of one medium and
1/8 of one large). This is accomplished by Lesson 7-4 Teaching
master with a series of three sets of figures (3 of the same shape
with varying sizes). EDM 7-­‐6: Equivalent Fractions are investigated using Fraction Cards. Students are asked to sort all of the cards into groups having the same amount shaded. They complete a worksheet that lists the “Equivalent Names for Fractions” for: ½, 2/2, 1/3, 2/3, ¼, ¾, 1/5, 2/5, 3/5, 4/5, 1/6, 5/6, 1/8, 3/8, 5/8, & 7/8. Students also use an area model (circle) to show the equivalent fractions of ½ and ¼. Additionally, they model fraction equivalencies by shading circle portions (area models) for 2/4, ½, 1/3, & 2/3. EDM 7-­‐7: Students are again working with shading in area models (rectangles) to represent equivalent fractions ½, ¼, and ¾. It is in this section that they address the strategy of finding equivalent fractions by multiplying the numerator and denominator by the same number (the “Equivalent Fractions Rule”). They generate equivalent fractions by completing a table + applying the rule, play a game to practice naming equivalent fractions and finally, compare equivalent fractions using a number line poster. MT1 Module 1, Section 5: • Students review the concept that equivalent fractions are those that name the same portion of a whole (using pattern blocks). Students learn how to show that multiplying and dividing the numerator and the denominator of a fraction by the same number produces an equivalent fraction (multiplying by x/x where x ≠ 0 is the same as multiplying by 1).
• Minimal problem situations are provided, but there are two in the homework, which ask the students to compare pollen grains (result unknown).
EDM 7-­‐9: Students compare fractions with like numerators (1 only), like denominators (10 only), and different numerators and denominators through the use of a problem situation from HW (dividing up a chocolate bar). The text does not use the strategy of extending unit fraction reasoning to a non-­‐unit fraction. 6 NF1 and 4.NF2 Property of Equivalent Fractions continued 4.NF3 Adding and Subtracting Fractions with Like Denominators: Students extend previous understandings
about how fractions are built from unit
fractions, composing fractions from unit
fractions, decomposing fractions
into unit fractions, and using the meaning
of fractions and the meaning of
multiplication to multiply a fraction by a
whole number.
a. Understand addition & subtraction
of fractions as joining and
separating parts referring to the
same whole.
b. Decompose a fraction into a sum of
fractions with the same
denominator in more than one
way, recording each
decomposition by
3/8= 1/8
+ 1/8 + 1/8 an equation. Justify
decompositions, e.g., by using a
visual fraction model.
c. Add and subtract mixed numbers
with like denominators, e.g., by
replacing each mixed number
with an equivalent fraction, and/or
by using properties of operations
and the relationship between
addition and subtraction.
Study Link 7-9: Students compare fractions using a benchmark of ½ &
order as series of 5 fractions. None of the problems are in a word problem
context. Math Box 7-9 does have one fraction subtraction problem with
result unknown.
MT1 Module 5, Section 1: • Students use fraction strips to compare fractions and develop number sense about fractions. Students use mental math to compare each fraction (comparing parts of the same whole-­‐
visual using fraction strip, compare to ½, compare to each other’s numerator and denominator). NOTE: This section is critical if students are going to be able to be proficient in standard 5.NF.2 – “Use benchmark fractions and number sense of fractions to estimate mentally and assess the reasonableness of answers”. Students use common denominators (LCD) to compare fractions. MT2 Module 3,Section 3: • Students compare fractions by finding the common denominator, renaming each fraction as an equivalent fraction with the common denominator. NO visual models are provided. a. Students work on joining (composing) of unit fractions or separating
fractions of the same whole.
Example: 4/5 = 1/5 + 1/5 + 1/5 + 1/5 + 1/5
b. Students should justify their breaking apart (decomposing) of fractions
using visual fraction models. The concept of turning mixed numbers into
improper fractions needs to be emphasized using visual fraction models.
EDM 7-1: Reviews fractions as parts of a whole, fractions on number
lines, and use of fractions.
EDM 7-4: Students determine the value of the shape using pattern blocks
as well as with tangram pieces; students then compose the parts/unit
fractions into a whole.
EDM 7-5 – Students model the computation of +/- fractions (with
like/unlike denominators) using pattern blocks to build an understanding
of what it means to do these operations with fractions. Note: While the
physical models (pattern blocks) provide an important conceptual
framework for the fifth graders, there were no problem situations included
in the instruction of this conceptual framework. The Study Link 7-5 did
contain 4 situations to solve. Strategies included: Combine & trade the
Blocks (addition), “Cover Up Method” (difference), “Take Away
Method” (difference), Explore Fractions that Sum to One (adding unit
fractions ½, 1/3, ¼, & 1/8).
EDM 7-7: This section asks the students to apply their understanding of
fraction addition and equivalent fractions to investigate how Egyptians
represented a fraction as the sum of unit fractions. This is a VERY small
section…. which may be easily by-passed because it is identified as
“enrichment”.
c. According to the standards: Mixed numbers are introduced for the first
time in Fourth Grade. Students should have ample experiences of adding
and subtracting mixed numbers where they work with mixed numbers or
convert mixed numbers into improper fractions.
EDM 7-5: Teachers are given two subtraction problems (Mixed Number
less fraction) to offer to their students to solve with pattern blocks and
there is one addition problem on Study Link 7-5. Based on this, students
are NOT offered ample opportunity to work with +/- mixed numbers or
7 4.NF3 Adding and Subtracting Fractions with Like Denominators continued 5NF.1 Use equivalent fractions as a strategy to add and subtract fractions (with unlike denominators). 5.NF.2 Solve word problems involving addition and subtraction of fractions referring to the same whole, including cases of unlike denominators… opportunities to convert mixed to improper fractions.
MT1 Module 5-­‐Section 4: • Addition and Subtraction of Mixed Numbers is introduced through
the use of renaming fractional parts (equivalent measurements) of
a mask i.e. in a context.
• Students use estimation to determine how much material is needed
for a mask by estimating the sum of mixed numbers. They
intuitively round each measurement to the nearest whole and then
find the sum to estimate length. The model of a number line
(ruler) is used to find the actual sum of fractional measurements.
• Note: Instructionally, the teacher inquires of the students how
adding mixed numbers is similar to adding fractions. Students are
also guided to apply the skills of finding a common denominator
& simplifying fractions to the new skill of + mixed #s.
• Students apply their conceptual knowledge of equivalencies such
as 4 2/2 = 5 to understand the regrouping process for mixed
number subtraction. Again, they use a number line (ruler) model to
subtract mixed numbers with unlike denominators. Further,
students utilize their work on simplifying mixed number sums
(learned in the previous lesson) to find the difference of two mixed
numbers with common denominators.
MT2 Module 3,Section 4: • Instructionally, the situation the students are presented with involves the use of fractions and mixed numbers with robot programming codes. Students use number lines (ruler) to calculate the distance (in fractions of an inch) traveled by a robot. Students learn how to write fractions as mixed numbers and mixed numbers as fractions, as well as add and subtract mixed numbers. MT1 Module 5-­‐Section 3: • This builds on Module 5-­‐Section 1 where students compared fractions using the LCD; students will use this strategy as part of the algorithm for adding and subtracting fractions with unlike denominators. • Students use fraction strips to find the sums of fractions with like
and unlike denominators; Students use fraction strips and apply
the same principles (of finding a common denominator) applied to
adding fractions to subtracting fractions.
MT2 Module 3,Section 3: • Students add and subtract fractions with like and unlike
denominators using fraction models, diagrams, &
addition/subtraction equations with the renaming of the fractions
with a common denominator.
• Students use estimation and area models (circle & rectangles) to
determine true/false statements. Students also use estimation with
benchmark of ½ to determine the sum/difference of fractions.
MT1 Module 5-­‐Section 3: • Minimal problem situations are provided during the instruction, but there are four in the homework, which ask the students to interpret a circle graph, find the value of notes on written music, total mileage of a trip, and a situation that asks students to “prove” a solution with an explanation & model. • Visual models are utilized by the student include the use of 8 area models (fraction strip, circle graph) and equations. This would need significant additions to meet the expectations of …by using visual fraction 5.NF.2. models or equations to MT2 Module 3,Section 3: For instruction, the text utilizes a situation with cryptograms represent the problem. in newspapers with fractional clues; in regards to homework, there are 3 types of situations which seem to get at number Use benchmark fractions and sense: using mental math to compare special pairs of number sense of fractions to fractions; using visual thinking to explain how a fractional model compares two fractions with unlike denominators; & estimate mentally and assess describing the errors made when two fractions were added. the reasonableness of answers. Two additional problems ask the kids to look at the parts per hour spent walking & running and the sum of eligible voters voting on a proposed action. MT1 Module 5 – Section 4: • Addition and mental math is used to solve subtraction problems when one number is mixed and the other is a whole number. • A situation is presented to the students at the end of the lesson, which requires them to sketch their own design for a specified area model (rectangle). Students are asked to calculate how much of x length would be left after it is used on the design. • Additional situations are presented in the homework section (7) which asks the kids to apply their understanding of mental math (using measurements to create a costume), estimation (rounding to the ½ instead of the whole) to find the sum + the exact + compare it to the estimate, photography (width/length), height comparisons, plumbing and length, woodworking, & interpreting data with height comparisons. In addition to fraction equivalence and the modeling & computing of fractions with like/unlike denominators, students need to have a diverse repertoire of strategies to approach and solve word problems (especially those related to fractions) in order to perform at a proficient level with 5.NF.2 (solve word problems using addition and subtraction of fractions). As noted, students in 4th grade using EDM materials were not afforded much experience or practice with situations involving the addition & subtraction of fractions. When they were offered opportunities, I found the problems to be of a result unknown type of situation. In both the fifth grade text, MT1 and the 6th grade text, MT2, foundational concepts were developed using a situation. For example, to review fraction concepts and equivalency at the fifth grade level, students were presented with the use of fractions to describe objects, create/name patterns and designs, or to describe the fraction equation to represent the roll of a die in a game. Further, each homework assignment afforded an opportunity for the students to apply what they learned to a minimum of 2-­‐7 situations. The rigor of these situations could be questioned, as I did not identify a great diversity of problem types (location of unknowns) was not evident. 9 Conversely, students were given an excellent guide for solving problem situations in EDM Lesson 3-­‐8. This lesson provided students with a strategy to approach word problems; specifically, (1) To understand the problem; (2) Plan what to do; (3) Carry out the Plan. Interestingly, in the 5th grade text (MT1) Module 1-­‐ Section 3 and in the 6th grade text (MT2) – Module 2-­‐ Section 4, students are offered a similar guide for problem solving with the addition of one step: (1) Understanding the Problem; (2) Make a Plan; (3) Carry Out the Plan; (4) Look Back. The additional step found in the MT1 & 2 materials will facilitate an important component of the 5.NF.2 Standard… “ to estimate mentally and assess the reasonableness of answers”. Note the 4th step of the problem solving approach in the fifth and sixth grade guide for problem solving: (a) Check that you answered the question being asked; (b) Check that your solution seems reasonable …. (c) Try to find another method to solve the problem and compare the results. Students in grades 4-­‐6 are afforded some opportunity to apply their respective Steps to Problem Solving to various written word problems both within the lesson and in subsequent lessons (despite the situations not requiring the computation of fractions). Although the district’s resources offer a consistent, appropriate guide or plan to solving problems, I continue to be concerned that specific problem solving strategies (such as making a table, guess and check, make an organized list, etc.) are not directly taught. Additionally, I find it interesting that one of the problem strategies identified is to make a picture or a diagram. If I am not mistaken, there are very few instances where kids are directed to make a visual representation, specifically a tape diagram to solve a problem. With the Content Standards and standards for Mathematical Practice emphasizing the use of visual representations (esp. number lines, area models, and set models), we will need to make certain that we afford multiple opportunities for students to encounter a diverse number/type of situations that involve not only the addition and subtraction of fractions & mixed numbers, but also require students to solve problems using these valuable representations. (b) Conclusions and Suggestions: As the reader might recall, the textbook analysis of district resources focused on the following foundational concepts in relationship to the CCSSM Standard 5.NF.2: a. Property of Equivalency for Fractions (which includes a clear understanding of fraction equivalence, comparing, and ordering); b. Addition and Subtraction of Fractions with Like/Unlike Denominators; c. How to Utilize a Variety of Visual Fraction Models (including area models, set models, and number lines) & Write Equations; and d. How to Approach (plan) and Solve (strategies) Word Problems. This is an appropriate and necessary analysis if the district’s students are to be mathematically proficient according to the standards and prepared for college & career readiness. It may be helpful for the reader to consider the following quotation from the book entitled: A Focus on Fractions (pg.133). In this text, the authors identified the following big idea: Understanding equivalence and having an efficient procedure to find 10 equivalent fractions are critical as students encounter problems involving comparing, ordering, and operating with fractions. As a result of this analysis, it was determined that the Everyday Math (EDM) textbook program develops a strong foundation in the fraction concepts of equivalency, comparing, and ordering which will be essential for successful achievement of standard 5.NF.2. Students are afforded opportunity to develop their understanding of fractions through concrete models (pattern blocks) and visual fraction models (fraction strips, area models, and number lines). Specifically, in EDM 7-­‐1, students create number line posters, which affords the students an opportunity to see that 6/6 is equal to 1 whole and that one whole is equivalent to 6 parts of size 1. Both EDM and MT1 utilize pattern blocks as a manipulative and a concrete model. There are limitations associated with a reliance on pattern blocks – which I feel the district’s resources seem to do; pattern blocks are limited in the number of fractions and equivalencies children are able to model. Teachers need to ensure that they incorporate the suggested visual fractions models (and add tape diagrams) as well require students to articulate the equation associated with their models in order to effectively transition students from concrete to representation and finally to abstract understandings. I felt that the EDM program did a nice job of incorporating the number line into the instruction of equivalencies. In EDM Section 7-­‐7, Students use a straight-­‐edge to vertically line up fractions on a Number Line Poster that are equivalent to ¼, 1/3, ½, and 2/3. It is up to the teacher to overtly bring out the important fact that there are an “infinite number of names for a given fraction” (as noted as a Big Idea on p.133 in A Focus on Fractions). Staff development on this concept would be beneficial to ensure that this important “Big Idea” is brought out during instruction in order to build a deep understanding of an important component associated with fraction equivalency. Students are taught various strategies to find equivalent fractions, including the procedure of equivalence with common denominators. The latter is critical as this is the identified, efficient procedure referenced in standard 5.NF.1. In EDM 7-­‐9, students compare fractions with like numerators (1 only), with like denominators (10 only), and with different numerators and denominators through the use of a problem situation (dividing up a chocolate bar). While the text provides a range of strategies to order and compare fractions, the strategy of extending unit fraction reasoning to a non-­‐unit fraction as advocated in A Focus on Fractions (pg.87) seems to be absent. It is suggested that teachers add this valuable strategy to not only continue to build a repertoire of strategies, but it scaffolds off of the important concept of reasoning with unit fractions (which students seem to grasp and apply pretty readily when comparing unit fractions). The Everyday Math program does not adequately address equivalency with Mixed Numbers or Improper Fractions. Fifth grade teachers will need to know this in advance and spend time on the important conceptual development using Module 1, Section 4, at least until the fourth grade teachers/materials address this gap. On an additional note, 11 while students are building a repertoire of strategies to determine equivalency, they are not generating or recognizing equivalent fractions within the context of problem situations. After a careful analysis of the class experiences and the homework assignments, is it evident that THIS is the greatest obstacle to achieving proficiency with standard 5.NF2. Fourth grade teachers will need to supplement the current program with problem situations associated with fractional equivalencies including ordering and comparing. Teachers are encouraged to consider diversity in the type of problems offered, being careful to not rely on result unknown, which both the Every Day Math and MathThematics programs seem to rely upon. The standard 5.NF.1 (which leads directly to 5.NF.2) requires that the 4th grade student has achieved the expectation that they have worked with adding fractions with like denominators. In general, the Every Day Math materials do an adequate job of conceptually building this idea of composing and decomposing fractions using pattern blocks and tangrams, which leads to the computation of fractions (adding and subtracting); though, the materials do NOT provide opportunities for the students to justify their compositions/decompositions. Teachers would need to make this a part of their lesson goals and expectations for proficiency to achieve the standard. Additionally, I would recommend that all teachers (including grades prior to the identified grade 4-­‐6 band) provide more experiences with visual fraction models (especially tape diagrams) to build an even deeper understanding. I appreciated that the EDM materials offered strategies (using the pattern blocks) to add and subtract fractions including Combine & Trade the Blocks (addition), “Cover Up Method”
(difference), “Take Away Method” (difference), Explore Fractions that Sum to One (adding
unit fractions ½, 1/3, ¼, & 1/8). Developing reasoning strategies is an important and effective
instructional technique in an effort to avoid the sole reliance on using the practice of finding
a common denominator (with the latter being the efficient strategy advocated by the standard
5.NF.1). Similar to the lack of problem situations associated with equivalency at the 4th grade, the computation of fractions at the 4th grade and at the 5th grade is done in the absence of context. Students in all grade levels need a great deal of experiences associated with a diversity of word problem types in order to meet the expectations of the identified standard -­‐ 5NF2. As noted from the textbook analysis, the 5th grade and 6th grade materials present the new learning within a situational context. The textbook analysis of the 5th and 6th grade materials revealed more opportunity to practice the computation of fractions within the context of problem situations, unlike the 4th grade materials. All students should not only solve +/-­‐ fraction problems, but they should be creating their own situations (which require fraction computation), thereby demonstrating an even greater mathematical understanding of what it means to add and subtraction fractions. All teachers would need to supplement their materials to include the aforementioned content, because these opportunities are not presented in any of the assessed materials. To ensure proficiency with the Math Practices, especially 12 justifying reasoning & thinking/modeling their thinking, teachers of all grade levels (Grades 4-­‐6) need to push their children to explain and show their work, using appropriate visual fraction models. Students at the fourth grade level are NOT offered ample opportunity to work with +/-­‐ mixed numbers or experiences with converting mixed to improper fractions with conceptual understanding. The 5th grade materials do address this concept in Module 5 -­‐ Section 4. The addition and subtraction of mixed numbers is introduced through the use of renaming fractional parts (equivalent measurements) of a mask i.e. in a context. Adding & Subtracting Mixed Numbers appears again in grade 6 within Module 3-­‐ Section 4. I wanted to make a final observation regarding the Fourth Grade Every Day Math Materials. I found the order that the fractional concepts were introduced to be quite interesting. In fact, the progression of Fractions in the EDM text was the following: a. Review Fraction Concepts (7-­‐1) b. Fraction Addition + Subtraction (7-­‐5) c. Name for Fractions/Equivalent Fractions (7-­‐6 & 7-­‐7) d. Comparing Fractions (7-­‐9). I found this curious. I would recommend district teachers to follow the content progression as presented within the standards for better continuity & content/understanding development. This would mean that teachers should consider teaching in the following sequence: a. Building a conceptual understanding of fractions b. Equivalency – Comparing and Ordering Fractions (with Mixed Numbers/improper Fractions included) c. Addition and Subtraction of Fractions (with Mixed Numbers/improper Fractions included) When one takes into account the 4th and 5th grade standards with the aforementioned recommendations, our district’s students will possess a deeper understanding of fractions and their computation. More importantly, students will be best prepared for the challenges they will encounter in middle school and beyond. The progressions have clearly identified that each subsequent domain, relies heavily on the in-­‐depth understanding of the concept standards in grades prior. As a result of a deep understanding of fractions, students will be better prepared for success in the middle grades with ratio and proportion; with adequate preparation of ratio and proportion, students will be more proficient with algebra in high school. Finally, with a proficiency in algebra, we know our children will possess the skills and conceptual understanding necessary for college and career readiness. 13