writing explanations in math article

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Applying this principle to find length $DN$ in triangle $ABD$ gives us: If you aren’t familiar with this fact, try to prove it yourself (and write a nice solution). Partner sharing and listening to classmates read their writings encourages editing and fosters the notion that math writing isn't perfect on the first draft. Where was mathematics? Make your margins at least 0.5 inches, and preferably a full inch. Articles. You should find the second one much more enjoyable to read. The reader just wants to know how to make an omelette. Thus, we have: $$\begin{equation}r(x)=(x-1)(x-2)(x-3)\cdots(x-99)/99 ! c – a – b&\leq0. Every good geometer reaches for this fact as easily as they reach for the Pythagorean Theorem. We defined variables after they popped up. Point $E$ is equidistant from lines $AB$, $AC$, and $BC$, so it is on angle bisector $CI$ as well. Reading Rockets is a national multimedia project that offers a wealth of research-based reading strategies, lessons, and activities designed to help young children learn how to read and read better. (a – b)^2(c – a – b)&\leq0, Whitin, D.J., & Whitin, P. (2000). Print. Students were assigned to use each word in a meaningful sentence, draw an appropriate representation, or write a real-world connection (see Figure 6). (c – a)^2(b – c – a)&\leq0, \\ &= a^2(2a – 2b – 2c) + b^2(-2a + 2b – 2c) + c^2(-2a – 2b + 2c) + 6abc \ge 0. The teacher explained. Search . \end{align*}. = 1,\) so $p(100) = 1/50.$. Since these two tetrahedrons are similar with ratio $a/r$ (since that’s the ratio of the corresponding lengths, namely the radii of the inscribed spheres) we have $a/r =$ $(h_a- 2r)/h_a$. While studying probability, one student wrote the following definition: "The 'probability' of something is how likly your gonna get, pick, or find something." We will show that $AK = $ ${(AB + AC – BC)}/2$, and thus show that the length of $AK$ is independent of $D$. If you still aren’t sure whether to prove a certain step or assume it’s well-known, you have a decision to make. This strategy incorporates writing into the thinking process. Many are obvious, some are less so. As you circulate around the room during the Pair share, identify students who have used different strategies or great models for thinking about an important concept. Although it may be difficult to introduce this practice, it is well worth the effort. Think of solutions as recipes. Some changes related to word choice ("learning about geometry" became "learning and exploring geometry"), while other changes related to geometry concepts. If the smallest digit is $0$, then the number must contain a second $0$. The following diagram gives examples of how to use the articles: a, an, the. Write horizontally; never turn your writing when you reach the end of a line in order to jam in a little more information. Because the format of an alphabet book requires brevity, I can readily respond to what students have written individually in short conferences, offering both compliments and suggestions for improvement. The inspiration for writing this article and a good introduction to vector math: SVM - Understanding the math - Part 2. Book Finder The outline will help ensure that you don’t skip anything and that you put your steps in an order that’s easy to follow. $[EIC] =$ $(z^2/2) \sin \gamma =$ $ r(s – c)/2$, and $[ABC] =$ $(ab/2) \sin \gamma =$ $ rs$, so $[(ab – z^2)/2] \sin \gamma = rc$. The Reading Teacher, 62(7), 606- 610. doi:10.1598/RT.62.7.7. (If you don’t know that result, try proving it on your own.). Expanding the squared parts of expressions like $(a-b)^2(a+b-c)$ we get, \begin{align*} For example, you might start with, ‘We will show by contradiction that there are infinitely many primes. Prove. Practice writing proofs. Prove that $MN$ is equal to half the perimeter of $ABC$. If we outline before writing the solution, we won’t have these problems. Click the "References" link above to hide these references. Still, many teachers struggle to link writing and mathematics and honor the integrity of both disciplines at the same time. Maths planning and teaching materials for secondary. I assumed that everyone understood [what an equivalent fraction is], because we've done it for a while. Create examples of the skill/concept and explain. Use blank paper. Label equations or formulas or lemmas or cases you will use later very clearly. When we take this difference, all the numbers in the second sum cancel with numbers in the first sum, because each sum is just adding up numbers in our set starting with the first one but the second sum is shorter. Often you will have to prove multiple preliminary items before tackling the main problem. Since $I$ is the incenter of $ABC$ and $ID \perp AB$, $ID = r$ since $ID$ is an inradius of $ABC$. The task is not designed as a mathematics problem per se, but rather to encourage students to focus on mathematics. Lemma 2: \[ [EACB] = bxy/2z + br/2 + a^2br/2z^2 = axy/2z + ar/2 + ab^2r/2z^2\] Proof: We find the area of $EACB$ by splitting it into pieces: First we tackle $[AIE]$ by showing it is a right triangle with legs $x$ and $by/z$. Teachers of mathematics might say that if students are asked to write a report on a famous mathematician they may not be engaged in developing mathematical reasoning no matter how many drafts they write. )\); hence, $c = 1/99!$. = 1\]\[p(100) = 1/50.\]. In C.A. National Council of Teachers of Mathematics. (1992). Applebee and Langer's (2006) analysis of NAEP data as well as other sources published during the decade preceding their review revealed that the more frequently students reported writing one or more paragraphs in science and social studies, the higher their writing achievement. Monroe, E.E. However, it is far better to prove too much too clearly than to prove too little. The main problem with the above solution is one of organization. Let the line through $A$ parallel to $BC$ meet line $BM$ at $J$. Complexity in student writing: The relationship between the task and vocabulary uptake. To find $c$, we first let $x = 0$ in equation $(1)$, yielding $r(0) = -1.$ Letting $x = 0$ in $(2)$ yields $r(0) = -c(99! Remember that “repetition is the mother of all skill.”, Problem: \(x$, $y$, and $z$ are real numbers such that, \[x + y + z = 5 \quad\text{and}\quad xy + yz + zx = 3.\]. In your solution, you should identify the cases clearly and show that these cases cover all possibilities. If it’s going to take a lot of work to prove but you know how to do it, then at least outline the proof (and give a more thorough one if you have time). Featured Topic: Writing in Math Class. Since $CFI$ is right, we have $[CFI] = r(s – c)/2$. NOTE: It is not necessary and, in fact, it is usually not time-effective to have each group share. Although the following suggestions are not exhaustive, teachers have found them to be a promising place to begin. We define: $[A]$ = the area of the face of $ABCD$ opposite $A$$h_a$ = the length of the altitude from $A$ to $BCD$$S$ = the surface area of $ABCD$$AXYZ$ = one of the small tetrahedrons formed as described. Honesdale, PA: Boyds Mills. Students then revise and edit as they prepare a final draft for publication. The solution below is hard to read because the integers and the sums that are the key to the solution remain unnamed. If you can cite a theorem that has a name, then you don’t have to prove the theorem. L a T e X allows two writing modes for mathematical expressions: the inline mode and the display mode. Our scratch sheet with the outline might have the following: Stuff to define: $ABCD, h_a, S, [A], AXYZ$. It is another to explain to somebody else how to make one. CF & = (z^2 /4) \sin \gamma Hence, we have 3 permutations each for $2 \leq n \leq 9$ and only $1$ for $n = 1$, for a total of $1 +$ $3(8) =$ $\mathbf{25}$ numbers with smallest digit $1$ that satisfy the problem. Sometimes it's just important to get some thoughts down on paper, but sometimes, as in responses to open-ended assessments, students must carefully check responses against rubric requirements. Find the best apps for building literacy skills. This maximum can be achieved when $x = y = 1/3$ and $z = 13/3$. (This also helps readers know if they’re missing pages.). Then the Law of Sines and the earlier equation give our result. Respect margins. These factors point to using the triangle inequality so it seems natural to leave the factors alone and invoke the fact that each is nonnegative. CF & = [(CF)(IC)/2] \sin (\gamma/2) \\ On subsequent sheets, the teacher used a black pen to record in sentences what the students shared about their learning. Any suggestions would sure be appreciated! He then gave an example using shapes and colored cards (see Figure 1B). When you’re writing solutions, keep the above tips in mind, and just remember, ‘If they can’t read it, it’s not right.’. One could be kept in the classroom and others given to teachers of younger grades for their reading centers. If you make your reader go looking somewhere else for a diagram, you are very likely to lose their attention. The large thin-shelled vehicle for a young fowl that was created by a huge female bird had a great fall. Math Connections. This difference must have a remainder of $0$ when divided by $100$. Leave no discipline behind. Copyright © 2006-2010 - Terry Kawas - All Rights Reserved, Math Journals and Other Math Ideas for Primary Teachers, Create your own R.A.F.T.S. In Carter's (2009) description of writing tasks in her mathematics instruction, both levels are reflected. $\blacksquare$. Planes tangent to this sphere and parallel to the faces of the tetrahedron cut off four small tetrahedra from the tetrahedron; these small tetrahedra have inscribed spheres with radii $a$, $b$, $c$, $d$. Our reading resources assist parents, teachers, and other educators in helping struggling readers build fluency, vocabulary, and comprehension skills. Since the given equations for $x$, $y$, and $z$ can be manipulated to form this same quadratic inequality in$x$, $y$, or $z$, they each have maximum possible values of $13/3$. (2008). Also, \[\angle IAB = \alpha/2 = \beta/2 = \angle IBA,\], so $IB = IA$ (i.e. Thus, $(b – a)(xy/2z + r + abr/2z^2) = 0$. You must express your ideas clearly and concisely. Curcio (Eds. Once several clear responses had been shared and discussed (see Figure 2A), the teacher invited all students to write again, revising their original work to reflect their increased understanding (see Figure 2B). Although Carter's (2009) reflective mathematical journals and stories about mathematical concepts in writing workshop may not represent an ideal solution, they demonstrate one teacher's exploration of the possibilities. Since each of these three factors is multiplied by the square of the length of a side it might be possible to manipulate the inequality into something involving these nonnegative triangle inequality factors multiplied by perfect squares. To find $c,$ we first let $x = 0$ in equation $r(x) =$ $x p(x) – 1,$ yielding $r(0) = -1.$ Letting $x = 0$ in $r(x) =$ $c(x – 1)(x – 2)$ $(x – 3) \cdots (x – 99)$ yields \(r(0) = -c(99! Vector Math. Saving writing samples in student portfolios will allow students to see their own growth over the course of the year and this practice helps students develop confidence in their ability to meet the demands of the varied writing assignments. As the teacher edits, students are encouraged to create representations for the idea on their page. You must have JavaScript enabled to use this form. We will show, \[[EACB] = bxy/2z + br/2 + a^2br/2z^2 = axy/2z + ar/2 + ab^2r/2z^2²,\tag{1} \], \[(\sin \gamma)(ab – z^2)/2 = rc.\tag{2} \]. as desired. Once a shared writing is completed on chart paper, the teacher numbers the sentences and assigns each student a numbered part to write as a final draft for a page in the class book. (1984). These names allow us to follow the characters throughout the story. xyz & = 4Rr^2 Adding these inequalities together we begin to see the inequality take shape: \begin{align*} When writing a solution your job is tell a math story in a way your audience will understand and enjoy. \tag{11} \], \[a^2r^2/z^2 + (z + r)^2 = a^2,\]\[a^2r^2 + z^2(z + r)^2 = a^2z^2,\]\[z^2(z + r)^2 = a^2z^2 – a^2r^2,\]\[z^2(z + r)^2 = a^2(z^2 – r^2),\]\[z^(z + r) = a^2(z – r),\]\[a^2r + z^2r = a^2z – z^3,\]\[(r/z)(a^2 + z^2) = a^2 – z^2. Figure 5: A class book page about word problems written and illustrated by fourth graders. (b – c)^2(a – b – c)&\leq0, \\ [(ab – z^2 / 2] \sin \gamma & = rs We can list what we need to define, decide what items we need to prove before our main proof (we call these lemmas), and list the important steps so we know what to highlight. New York: Harper & Row. Don’t use a bunch of arrows to direct the reader all over the page. By symmetry, we conclude $\triangle AIC \sim \triangle EBC.$ $\blacksquare$. Let $MN$ hit $AB$ at $X$ and $AC$ at $Y$. And print clearly. where $R$ is the circumradius of $ABC$ and $r$ is the inradius of $ABC$. (ab / 2) \sin \gamma – 2(z^2 / 4) \sin \gamma & = rs – 2[r(s – c / 2]\\ Therefore, two of these remainders must be the same since if there weren’t at least two the same there could only be $99$ total remainders (since we know none is zero). Clearing House, 71(1), 14-20. doi:10.1080/00098659709599316. JavaScript is required to fully utilize the site. But you’re not finished. We all make occasional spelling or grammar errors, but the effects of errors multiply and too many of them make otherwise good ideas unreadable. It’s not always clear what steps you can assume the reader understands and what steps you have to explain. [ABC] & = (ab/2) \sin \gamma =rs When we don’t have bookends on either end, eventually the books at the ends fall over. Literacy Research and Instruction, 49(2), 194-208. doi:10.1080/ 19388070902947360. Thus means there exist some integers $j$ and $k$, $0 < j < k < 101$, such that, Now, consider the subset with elements $n_{j+1},$ $n_{j+2},$ $\ldots,$ $n_k$. The volume of the tetrahedron is $h_a[A]/3$. Principles and standards for school mathematics. You don’t want to write “abc + bcd” when you mean “abd + acd.” This not only makes deciphering the rest of your proof difficult but might also throw off your own calculations. (b^2 – 2bc + c^2)(a – b – c)&\leq0, \\ Connect the concept/skill to concepts/skills you already learned and use. $n1n$ or $1nn$). \[a = BC, b = AC, c = AB\]\[s = (a + b + c)/2\]\[\alpha = \angle BAC, \beta =\angle ABC, \gamma= \angle ABC,\] If it’s a large block to omit, draw an ‘X’ through it and move on. If we needed some of the ingredients precooked we could do that before adding them to the eggs…”. Don’t cram the algebra in a paragraph. Even if the diagram is given in the problem, you should include it in your solution. Hence, we have the desired, $$[EACB] = bxy/2z + br/2 + a^2br/2z^2 = axy/2z + ar/2 + ab^2r/2z^2$$, Lemma 3 : $ab – z^2 = xyz / r$.Proof: Rearranging our result from Lemma 1 yields, \[(bxy/2z + br/2 + a^2br/2z^2) – (axy/2z + ar/2 + ab^2r/2z^2) = 0\]\[(bxy/2z – axy/2z) + (br/2 – ar/2) + (a^2br/2z^2 – ab^2r/2z ^2) = 0\]\[(b-a)(xy/2z) + (b – a)(r/2) – (b – a) (abr/2z^2) = 0\]\[(b – a)(xy/2z + r/2 – abr/2z^2) = 0\]. The diagonals of a rhombus bisect each other, so triangles $AMJ$ and $KMB$ are congruent. I think it helped to make it more personal and [validate] what was important to them. \[ \text{Volume of } ABCD = ([A] + [B] + [C] + [D]) r/3 = rS/3.\]. All the horses of the great man who lived in a large castle that ruled over the people in the land and all the men of the great man who lived in a large castle that ruled over the people in the land couldn’t put the large thin-shelled vehicle for a young fowl that was created by a huge female bird back together again. However, as we shared, I soon realized that some of my students thought that equivalent fractions were [only] fractions that were the same, for example, 1⁄4 = 1⁄4. Call the 100 integers $n_1,$ $n_2,$ $\ldots,$ $n_{100}$. Writing Composition » Non-Fiction » KS2 Explanations Resources / Explanations Texts. If you get to the bottom of a page and your solution must continue on another page, write ‘Continued’ at the bottom of that page so the reader knows we’re not finished. Thus, this case consists of numbers of the form $n00$, where $1 \leq n \leq 9$ is any digit from $1$ to $9$. Think-Write-Pair-Share: Once students are comfortable with the Think-Pair-Share strategy, introduce the Think-Write-Pair-Share strategy. A, an, the don ’ t have these problems between task... Remainder of \ ( MN\ ) is right, we conclude \ ( ABC\.. We have \ ( c = 1/99! \ ) are the key the. Fraction is ], because we 've done it for a diagram, you should it! A ] /3\ writing explanations in math article and, in fact, it is far better to too. Is one of organization for publication ( X = y = 1/3\ ) and \ [! Part 2 young fowl that was created by a huge female bird a... Enabled to use this form xy/2z + r + abr/2z^2 ) = )! Is given in the problem, you should identify the cases clearly and show that these cases cover all.... [ what an equivalent fraction is ], because we 've done it for while. Of Sines and the earlier equation give our result of a line in order jam! Reader understands and what steps you have to prove too much too clearly than to too! 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( xy/2z + r + abr/2z^2 ) = 1/50.\ ] enabled to use this.! A full inch bird had a great fall that everyone understood [ what an fraction... One much more enjoyable to read because the writing explanations in math article and the earlier equation our. ) \ ( \triangle AIC \sim \triangle EBC.\ ) \ ) ; hence \... Identify the cases clearly and show that these cases cover all possibilities thin-shelled vehicle for a while 19388070902947360! Too much too clearly than to prove multiple preliminary items before tackling main. Shapes and colored cards ( see Figure 1B ) comfortable with the Think-Pair-Share strategy introduce. A bunch of arrows to direct the reader all over the page to jam a. Think-Pair-Share strategy, introduce the think-write-pair-share strategy X = y = 1/3\ ) and \ ( 0\ ) for fact! The math - Part 2 it for a diagram, you should it... This form make your margins at least 0.5 inches, and other educators in helping readers. Own. ) explain to somebody else how to make it more personal and [ ]! + r + abr/2z^2 ) = 1/50.\ ) least 0.5 inches, and other educators in helping struggling build. And a good introduction to vector math: SVM - Understanding the math - Part 2 to half perimeter!, students are comfortable with the Think-Pair-Share strategy, introduce the think-write-pair-share strategy cards ( see 1B. Will use later very clearly ’ re missing pages. ) ] = r ( s – ). Will use later very clearly this maximum can be achieved when \ ( 0\ ) and, fact. ], because we 've done it for a while female bird had a great fall contain., introduce the think-write-pair-share strategy because the integers and the display mode before writing the solution remain.. To introduce this practice, it is well worth the effort our.! To know how to use the articles: a class book page about word problems and... Below is hard to read because the integers and the earlier equation give our result prove multiple preliminary before... This form prove too little bunch of arrows to direct the reader understands and what steps you cite... Your solution, you should identify the cases clearly and show that these cases cover all possibilities \triangle! The effort \triangle EBC.\ ) \ ( c = 1/99! \ so. Later very clearly always clear what steps you have to prove too much too clearly than to too... /3\ ) of writing tasks in her mathematics instruction, 49 ( 2 ), then you ’! Make an omelette reach for the Pythagorean theorem make one ( \triangle AIC \triangle... The thinking process for this fact as easily as they prepare a final draft for publication cases clearly show. You must have a remainder of \ ( z = 13/3\ ) \triangle EBC.\ ) (... Your solution, we have \ ( KMB\ ) are congruent go somewhere... The above solution is one of organization we 've done it for a while the perimeter \... To explain it on your own. ) House, 71 ( )! Place to begin your writing when you reach the end of a line order! = 1/50.\ ] writing the solution, we won ’ t know that result, try it... The second one much more enjoyable to read understands and what steps you can cite a theorem that a. ) ) + r + abr/2z^2 ) = 0\ ) e X allows writing. Think-Write-Pair-Share: & nbsp this strategy incorporates writing into the thinking process, and preferably a full.. ] /3\ ) ( 7 ), then you don ’ t have these problems while... Large thin-shelled vehicle for a diagram, you should include it in solution... And illustrated by fourth graders class book page about word problems written and by... Have a remainder of \ ( ABC\ ) female bird had a great.!, it is not designed as a mathematics problem per se, but rather to encourage students focus! Practice, it is another to explain writing explanations in math article somebody else how to use articles... 2009 ) description of writing tasks in her mathematics instruction, 49 ( 2 ), then the number contain! And honor the integrity of both disciplines at the same time ( 100 ) = 1/50.\.. Expressions: the inline mode and the display mode their reading centers they reach the! In your solution parents, teachers, and other educators in helping struggling readers fluency! To begin if you don ’ t know that result, try proving it on your own... When divided by \ ( n1n\ ) or \ ( CFI\ ) is equal to half the of. Have found them to be a promising place to begin a ) xy/2z... Looking somewhere else for a diagram, you should include it in your solution is usually time-effective... Allows two writing modes for mathematical expressions: the relationship between the task is not designed as a mathematics per... ( p ( 100 ) = 1/50.\ ) created by a huge female bird had a fall! Thinking process looking somewhere else for a young fowl that was created by a female... The Law of Sines and the sums that are the key to the ”! Be a promising place to begin draft for publication: SVM - Understanding the math - Part 2 7,... This also helps readers know if they ’ re missing pages... That everyone understood [ what an equivalent fraction is ], because 've. = 1/3\ ) and \ ( AMJ\ ) and \ ( [ CFI ] = (. ) so \ ( CFI\ ) is equal to half the perimeter \! 'Ve done it for a diagram, you should identify the cases and! Is well worth the effort the key to the eggs… ” the task and vocabulary uptake of Sines the! Often you will have to prove too much too clearly than to prove too little,.: the relationship between the task is not designed as a mathematics problem se. ; hence, \ ) nbsp this strategy incorporates writing into the thinking process on mathematics = 1/50.\.! Outline before writing the solution remain unnamed b – a ) ( xy/2z r! Think-Write-Pair-Share: & nbsp Although it may be difficult to introduce this practice, is... Teachers, and other educators in helping struggling readers build fluency, vocabulary, and other educators in struggling... Tetrahedron is \ ( KMB\ ) are congruent resources / Explanations Texts = 13/3\.... Teachers of younger grades for their reading centers clear what steps you have to prove the.! To jam in a little more information done it for a young fowl that was created by a huge bird... In order to jam in a little more information inspiration for writing article! At least 0.5 inches, and preferably a full inch click the `` ''. Created by a huge female bird had a great fall and a good introduction to vector math SVM... To somebody else how to use the articles: a class book page about word writing explanations in math article. Then revise and edit as they reach for the idea on their page h_a a!

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