The Coca-Cola Company announces it is replacing its year-old recipe with a new formula. Customers react so negatively that on July 10 the same year it reintroduces the old Coke under a new name, Coca-Cola Classic. Every minute, people around the world drink , Cokes. How many Cokes are consumed in one week? Kansas becomes the 34th state. The name Kansas comes from an Indian word meaning flat or spreading water. The state flower is the sunflower. The sunflower provides pioneer settlers in the Midwest with oil for their lamps and food for themselves and their stock. Native Americans roast sunflower seeds and ground them into flour for bread or pound them to release an oil for cooking and for making body paint.
Look at a live sunflower or a detailed picture of one. A sunflower has two distinct parallel rows of seeds spiraling clockwise and counterclockwise. The seeds are Fibonacci numbers, typically 34 going one way and 55 going the other way, although sometimes they are 55 and Find other natural examples of Fibonacci patterns. Good places to look include pinecones, pineapples, artichokes, and African daisies.
Gorman and 28 other Navajo volunteers turned their native language into a secret code that allowed Marine commanders to issue reports and orders and to coordinate complex operations. Although the highly respected Japanese code crackers broke U. All of the considerations, from storing to rolling them, were an interesting challenge. In seven weeks, we collected , pennies, and we plan to continue at least until the end of the year to see how close we get to 1,, When students bring in the pennies, they toss them into a tub that is about the size of a file drawer. That must be a million pennies.
Then we figured out that we needed more than thirty tubs of pennies to make 1,, That shocked them — and me, too! I created an open-ended activity to do with my class:. If one million fifth graders each bought a Big Grab Bag of Hot Cheetohs, the Cheetohs would completely fill three of our very high ceiling classrooms that are about 10m-bym-by If one million fifth graders lined up fifteen feet apart and passed a football from one end of the line to the other, the ball could travel from Merced, California, to Antarctica!
If one million fifth graders each ate a paper plate of lasagna and threw the plates away, the garbage would weigh as much as three blue whales and would fill a hole that is seventy-three cubic feet. Geometry comes to life in this lesson, as Rusty Bresser has fourth graders use geoboards to explore making pairs of line segments that touch exactly nine pegs, record them on dot paper, and label them as parallel, intersecting, or perpendicular. The students watched attentively as I switched on the overhead projector and showed them my clear plastic geoboard and two geobands.
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I directed one student from each table to get one geoboard and two geobands for each student at the table. As soon as the students received their materials, they got to work. Think about what you know about lines and then turn to someone sitting next to you and take turns sharing what you know. I drew on the whiteboard what I thought Hernan meant.
Hernan nodded to show his agreement. Nina added the last idea. Line segments! To introduce the problem that the students would explore, I held up a geoboard for the class to see and stretched a geoband around the five pegs in the second row. See what you notice about the line segments I make.
To verify, I touched each peg as we counted together. Next I explained what they were to do. When you have an example, copy it onto geoboard dot paper. To model recording, I quickly sketched a geoboard on the whiteboard and drew on it the two line segments I had made. I wrote the words intersecting line segments underneath my sketch on the whiteboard. I then changed the position of the geobands, making a new design to be sure that the students understood that two line segments with a common endpoint are also intersecting.
I showed the class my new design and then sketched another geoboard, drew what I had made, and wrote intersecting line segments underneath.
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Do they touch nine pegs? Is my label OK? Talk with your neighbor. I held up the geoboard for students to see. When you find a way, copy it on your geoboard dot paper and then label it as I showed. As the students got to work, I heard them using correct math terminology perpendicular, parallel, intersecting to identify their pairs of line segments; they seemed excited and challenged by the task of finding as many different arrangements as possible. However, I noticed a few students recording line segments that touched fewer or more than nine pegs. As I watched the students continue working, I noticed that some had duplicate designs drawn on their papers.
For example, Eduardo had drawn two parallel line segments on his geoboard dot paper. If he rotated that drawing, the line segments would be exactly the same as another design on his paper. Rather, I kept the focus on making intersecting, parallel, and perpendicular line segments and correctly labeling the designs. Money is a useful model for helping students make sense of tenths and hundredths, but students often have difficulty extending their knowledge to make sense of thousandths and ten thousandths.
The Lobster Problem presents students with a problem-solving experience that helps them learn about extending decimals beyond hundredths and provides them practice with identifying decimals that come in between other numbers. How much do you think the lobster weighs? Fifty-seven—hundredths comes right after fiftysix—hundredths. How much could the lobster weigh? How much could the lobster weigh then? I changed the question again, this time using a context that was more familiar. How much money could Matthew have? He could have two sixty-one, two sixty-two, all the way up to two seventy.
And he could have even more combinations between two eighty and two ninety. The students seemed clear about this, so I presented another version of the lobster question. Decide if there could indeed be measurements between two and five-tenths pounds and three pounds and, if so, discuss the possible weights of the lobster. Many students felt that the lobster could weigh anywhere between 2. Make sure you both agree on the numbers you come up with, know how to say them, and are able to explain your thinking.
Many students were able to make the assumption that they could add another decimal place behind the last number as they had done when the lobster weighed more than 2. However, they were having difficulty reading the possible solutions. The students did so easily. I repeated this for the second number, and again the students were able to do this.
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We read the number aloud, and then I asked the students for solutions to the question about the in-between weights. As students called out answers, I wrote them on the board. Most agreed that the least the lobster could weigh was 2. Blaire agreed. She came up and wrote 2. Then, under the number 2. At the end of the first 2. At the end of the second 2. Blaire continued down the list, creating a sequence of numbers each with four decimal places:. When you have four digits after the decimal point, the number refers to ten thousandths. To end the class period, I gave a homework assignment.
Answer this question. Are there an infinite number of possible weights between 2. Explain your thinking and give examples. Figure 1. Justin agreed with Blaire and was clear about the infinite number of weights between 2. Figure 2. Figure 3. Matthew wrote that there are weights between 2. Their new book helps students calculate with multidigit divisors and dividends using a method that makes sense to them! The lesson presented below teaches students a game that reinforces all of these goals.
Be sure to write both your names on your recording sheet. What divisor would be a good choice? Five times twenty is one hundred. He has to cross off nineteen from the list of divisors. We only get to use each number listed once in a game. I wrote 95 under the Start Number column. Skylar looked overwhelmed, so I called on Kenzie for advice.
He reminded me to circle the remainder of 15, to put his initial beside the problem, and to cross off I did. Joanna suggested I use eighteen. Eight times four is thirty-two. So forty and thirty-two make seventy-two. Eighty minus eight is seventy-two. Skylar decided to use eleven as his divisor. So seventy-two divided by eleven is six with a remainder of six. I get a score of six.
After a few minutes, I noticed Sean and Lucas were involved in an intense discussion. What can be a remainder depends on the divisor. I noticed as I continued to circulate that Skylar and Sasha showed how they did the dividing and, after two rounds, changed their way of recording their work.
Skylar and Sasha changed how they recorded the game but showed their thinking clearly. Lupe made an error when she divided 40 by At the end, you can get a lot of leftovers once the starting number gets below twenty. We got to thirteen as the starting number, and fourteen was left as a divisor, so I took fourteen and got all thirteen because thirteen divided by fourteen is zero with a remainder of thirteen. Everett and Derek added up all their leftovers together and it came out to one hundred. When Kenzie and I added ours up, it was only eighty-eight.
I suggested to Beth and Kenzie that it was possible they had made an error somewhere and perhaps they needed to go back and check their work. They reported on the subtraction and division errors they had made and commented on what they had discovered. There would be a remainder of seven for that problem.
Look where it says eighty divided by eighteen equals four remainder eight. Four times eighteen is seventy-two, which is the next start number. There were no other comments. Over the next several days, children continued to play when they had free time. It was wonderful to see them so happily engaged while getting practice with division. In previous lessons, students built rectangular prisms using cubic units and determined the volume of the prisms by counting cubes. Students started to devise methods for finding the volume of any rectangular prism without counting.
In this lesson, students continue their work on developing a method for determining the volume of any rectangular prism. They share their methods with each other and discuss similarities and differences between the methods. One goal of the lesson is to help students articulate how volume can be determined by finding the number of cubes in each layer of a prism.
If students know the number of cubes in one layer, they can multiply that amount by the number of layers, or height of the rectangular prism. The method of multiplying the dimensions is then connected to the idea of layers. Measurement and Data: Standard 5. MD Understand concepts of volume and relate volume to multiplication and addition. Begin this lesson by reviewing the terms rectangular prism, volume, and cubic unit. Give the groups interlocking cubes to help them develop their methods. Withhold comments or corrections. If groups have identical methods, post the methods both times.
With your partner, see if you can apply each of the methods displayed to a three-by-four-by-five-unit prism. If you get stuck, think about how you might edit the method so that it works. If the method does work, think about why it works and whether it will work for all rectangular prisms. Briefly discuss which methods may need a little revision or editing.
If a student makes a suggestion on how to revise a method that he or she did not write, check back with the students who wrote it. Do you think it makes sense to add it to what you wrote? Pick the methods you would like everyone to discuss based on the mathematics. Any time a student uses language associated with layering, ask at least two other students to repeat it.
Ask students if they agree or disagree that the method would work for all rectangular prisms and why. Ask other students to explain why they agree or disagree that this method would work for all rectangular prisms. How is it different? Summarize the key mathematical points. They all involve finding the volume by determining the number of cubes in one layer and then multiplying that by the number of layers.
In this lesson, fourth and fifth graders gain experience multiplying by ten and multiples of ten as they make choices about the numbers to use to reach the target amount of three hundred. In this game you will be multiplying by ten, twenty, thirty, forty, or fifty. The goal of the game is to be the player closest to three hundred. So the player with three hundred ten wins. Remember, you want to get closest to three hundred, and you must take all six turns. I called on Ben because I knew he had a good grasp of multiplying.
Ben did the same on his. I went first so I could model out loud my thinking process as well as how to record. I rolled a 1. If I multiply one by ten that will only give me ten. That seems like a lot. Maybe I should multiply by thirty; one times thirty equals thirty. Thirty is closer, but I still have two hundred seventy to go. Do you agree that one times fifty equals fifty?
Ben nodded. I recorded my turn on my side of the chart. Once Ben had recorded my turn on his chart, I handed him the die, indicating it was his turn. Ben rolled a 2. This time I rolled a 4. That gives me eighty for this turn. Add the two hundred to the fifty from your first turn and that would be two hundred fifty. You could almost win on your second turn. Several students put their hands up to respond. I called on Cindy. This is Mrs. If she got two hundred fifty by the end of her second turn, then she could only get fifty more to get three hundred!
I decided to move on rather than continue to discuss this point. I handed the die to Ben. Ben rolled a 1. Now I have fifty. He gave me the die. What would work better? Hands immediately went up. I called on Allie. Subtract that from three hundred and you still have three turns to get one hundred ten more points. That equals two hundred fifty. Two hundred fifty and fifty is three hundred! That only equals fifty, so my total is one hundred.
After our next two turns, I had and Ben had There are six sides on a die. One is on only one side of the die so it has one out of six chances of being rolled. He could get forty by rolling a one and multiplying by forty, or getting a two and multiplying by twenty, or getting a four and multiplying by ten. Ben looked delighted.
Giggling with delight and anticipation of getting exactly , Ben rolled. He got a 3. The class cheered and Ben did a little victory dance. I waited for a few moments for the students to settle down and then showed them what else to record when they played. I wrote on the board under my chart:. The students played the game with great enthusiasm and involvement as partners participated in every turn.
In this two-person game, students take turns identifying factors of successive numbers, continuing until one of them can no longer contribute a new number. To play the game you need a partner. One of the partners begins by picking a number greater than one and less than Can anyone tell me a number that goes evenly into 36? Another way to think about it is by skip counting. Which numbers can you skip count by and get to 36? By introducing several ways to think about factors, I hoped to explain the game more quickly. Several students nodded or vocalized their assent. I pushed for more of a commitment.
Those are the two main rules of this game. Can you think of any other factors of two? Chrissy had confused factors and multiples. I was glad she had made the multiplication connection, but I needed to prompt her a bit to get her back on track. Like 36 is a multiple of six because six times six is The class consensus was no. I raised my eyebrows in feigned surprise as I looked at the numbers on the overhead.
I wonder if that always happens in this game. I hoped that in subsequent games students would pay more attention to patterns in general as they played. Looking for patterns is a powerful way to build number sense, particularly when students have opportunities to think about the patterns and their relationships to numbers and operations. I referred to the string of numbers on the overhead, which now looked like this:. I also wanted the students to see that math involves taking time to think.
Talk at your tables for a minute or two and see what you can come up with. Four and two are used already. You want to get your partner stuck so she or he is unable to add a number to the string. The important part of the game is the mathematical thinking that you do. I played one more game with the whole class.
The factor concept had been reinforced, the term multiple had been introduced in context, and the students knew how to identify prime numbers. The students were ready to play with their partners. In addition to having practice with multiplication facts, students who play One Time Only search for winning strategies by thinking about relationships among numbers and factors. In so doing, they build their number sense. How are these numbers alike? Students may offer that the numbers are all less than Accept this, but push students to think about the factors of the numbers.
Following are several possible responses: All have a factor of one. All have exactly two factors one and itself. Each can be represented by two rectangular arrays. All of them are prime. This question has one right answer the least common multiple for the numbers 4 and 6 is 12 ,but students may arrive at the answer in different ways. But because 4 and 6 both share 2 as a factor, the least common multiple is less than the product of this pair of numbers.
If numbers do not have a common factor, however, then the least common multiple is their product. To help students think about these ideas, consider presenting additional questions for them to ponder:. Can you find pairs of numbers for which the least common multiple is equal to the product of the pair? Can you find pairs of numbers for which the least common multiple is less than the product of the pair? This question provides students with a real-world context—telling time—for thinking about a situation that involves numerical reasoning.
The problem also provides a problem context for thinking about multiples. You might also ask students if they think it makes sense to have an amount other than 60 minutes in each hour, perhaps minutes, for example. What effect would such a decision have on how time is displayed on watches and clocks?
Discuss the meanings of the math terms they use and the relationships among them. This question can be asked for any set of four numbers. As an extension, ask students to choose four numbers for others to consider. Then have them list all the ways the numbers differ from one another. Use their sets of numbers for subsequent class discussions. Adding consecutive odd numbers produces the sums of 4, 9, 16, 25, 36, 49, 64, 81, , and so on.
Of these, only some are reasonable predictions for Mr. All of these sums, however, are square numbers. Using different-colored square tiles or by coloring on squared paper, represent square numbers as squares to help students see that they can be represented as the sum of odd numbers. Start with one tile or square colored in. Then, in a different color, add three squares around it to create a 2-by-2 square, then five squares to create a 3-by-3 square, and so on.
Discuss the terms prime and relatively prime and the distinction between them. Then have students work on answering the question. Finally, ask students to write their own definitions of prime and relatively prime. Have them share their ideas, first in pairs and then with the whole class. This question aims to help students generalize about the relationship between the sign of the sum and the numbers in an integer addition problem. Students may need to make a list of integer addition problems whose sums are negative and look for commonalities among them in order to answer this question.
Throughout the year, I continued helping the class learn ways to compare fractions. As always, I learned a great deal from the students, especially from their written work. Most revealing to me was the variety of strategies that students developed for comparing fractions. Below, I describe some of what I learned from their writing and offer suggestions for how you can use writing with your students. In my early years teaching mathematics, I taught students to compare fractions the way I had learned as an elementary student—convert the fractions so they all have common denominators.
However, in my more recent teaching of fractions, I do not teach one method. Instead, I prod students to think, reason, and make sense of comparing fractions, helping them learn a variety of strategies that they can apply appropriately in different situations.
To help students learn to compare fractions, I used several types of lessons. I gave students real-world problems to solve, such as sharing cookies or comparing how much pizza different people ate, and had class discussions about different ways to solve the problems. I gave them experiences with manipulative materials—pattern blocks, color tiles, Cuisenaire rods, and others—and we explored and discussed how to represent fractional parts. I taught fraction games that required them to compare fractions, and we shared strategies.
At times I just gave them fractions, and we discussed different ways to compare them. We talked a good deal about fractions. By expressing their own ideas and hearing ideas from others, children expand their views of how to think mathematically. Also, talking and listening helps prepare them for writing, which I have them do individually several times a week in class and often for homework as well. My students had many opportunities to explain in writing how they compared fractions.
How would you decide which of the two fractions is larger? This is common when I read student work. But her method worked and was efficient. Laura converted the fractions so that they had common numerators. He had reasoned the same way that Laura had but expressed his thinking differently and with more detail. Jenny, however, reasoned differently. She compared both fractions to one whole. Also, it was easy for me to understand because her approach mirrored the way I had thought about the problem.
Jenny converted the fractions so that they had common numerators. As Jenny did, Donald compared both fractions to one whole and showed his understanding of equivalent fractions. But he also thought about common numerators. Donald needed to be reminded regularly to reread his papers before handing them in, and although this paper represented an improvement in his writing, more improvement was still needed.
Mariah relied on common denominators. I got this because I used common denominators. Mariah also showed how she had arrived at the converted fractions.
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She wrote:. They were doing what made sense to them as they tried to reason about the fractions. Sometimes I focus on their writing errors; other times I keep the focus just on the mathematics. Making this decision depends on the paper, the student, and the mathematics involved. For example, I asked Mariah to show the class how she arrived at fractions with common denominators.
I knew when I hit fortieths that it would work for six-eighths. Several students were eager to explain the methods they used. Raul was particularly animated. Then multiply 6 times 5 to get 30 and have that numerator for six-eighths and then multiply 4 times 8 and use 32 for the numerator for four-fifths.
I had them compare their work with partners and then had volunteers demonstrate each method so that Mariah and Raul could judge if their methods had been correctly applied. I had others read their papers as well and began to compile a class list of strategies for comparing fractions. I wrote the list on chart paper and kept it posted in the classroom for students to refer to. In my years of teaching, not all classes have come up with all strategies.
This was the first class in which common numerators was such a prevalent strategy. Reproduce one of them, give copies to students in pairs, and have them see if they can figure out why it makes sense. Have them explain the method in their own words. Then give them practice applying it to other fractions.
But, if possible, share the ideas from your own class. In this beginning lesson, students first explore arithmetic sentences to decide whether they are true or false. For each, I had a student read it aloud, tell if it was true or false, and explain why. Few students knew how to read the third sentence. I then asked the students to write examples of arithmetic equations that were true and some that were false.
A few minutes later I interrupted them. I drew two columns on the board, one for true mathematical sentences, and a second for false mathematical sentences. Then comes the equals sign. On the other side you do four plus five. I paused to give students time to think and then asked Rayna to come up to the board and write her equation. After a few moments, most students were clear that it was correct. I wrote her equation in the True column. After several other students shared their equations, even though more of the students wanted to do so, I moved on with the lesson.
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As the students watched, I wrote the following on the board:. The class was quiet. Finally a few hands went up. I called on Jazmin. I did as Lizzie instructed. The box is called a variable, because you can vary what number you put into it or use to replace it. Novels have options beyond direct speech: indirect or reported speech, summary, dialogue tags, and ways of rendering action and gestures too that can only be achieved in prose.
And again: a book is not a film, or a play for that matter. A film is something intended for consumption within a block of two hours while you sit in a dark room with strangers, or in front of your telly with your loved ones, or on your iPad by yourself. A play is something that is watched in a hushed room or further back in time an outdoor amphitheatre, again over the course of a couple of hours or a bit more maybe.
A book however can be picked up and put down over the course of many days, and in any context in bed, on the train, on the beach ; the duration of that reading experience will be several hours even for a fast reader. Theories developed for one medium need serious adaptation for a differently consumed form. Also, showing-not-telling is something of a Western bias, a colonial relic even. Which is fine, if you want to be a colonial relic, I guess. Also, however many time people try to explain beats to me I.
I think the problem I have with beats and some of these other structures is probably that they can feel to me like externally conceived structures imposed upon narrative matter. They are not arising in that intuitive way that I recently heard Anna Burns describe how she wrote Milkman. And, once again: I do observe that a book is not a film, or a play for that matter. Beats are used in writing for screen and stage, and films and plays are creations that manifest primarily as external expressions of action: performances that rely on spectacle and a choreographed management of time and space.
So: narrative theories based on screen- and scriptwriting have other limits beyond their emphasis on conflict. There are many theories on structure in film, in fact, and I find myself drawn more to the idea of the sequence. But I do note that three- or five-act structures are the ones I come across most in discussions in fiction writing. Kellie Jackson of Words Away and I have lined up two further day-long masterclasses on craft for the summer term.
I am right now putting together materials for new class Density and Speed. What did she mean by that?! It is related to perspective, tone, and description. And speed for me describes our movement through a piece of writing: the techniques with which a story creates pace and tension and urgency, and that keep the pages turning from the start — a commercial imperative, too. At the sentence level, as well, parts of speech play roles: at a basic level, nouns anchor us and verbs move us through.
Word counts, genre, sex, death — immortality! In June our class on revising and self-editing is a repeat of one we ran successfully last summer, and it should be of use to writers who have finished drafts, as well as people with works-in-progress. I think they have been successful so far as we have a great community of regulars coming along for intelligent and good-humoured discussions; many of us go along to Words Away salons with Kellie Jackson and Emma Darwin too.
The spirit is collaborative, and our focus is practical, and everyone brings along valuable contributions from their own writing, reading, and professional backgrounds. They are designed to stand alone, and dropping in to just one class might simply offer fresh insight or a jolt of energy to any writer wanting a bit of a boost in their creative process. So far we have appearances from not only an agent and editors, but also people from other areas of publishing talking about other aspects of the book trade: audiobooks, production, PR, rights, literary estates.
Understanding that this is not only a business but a working life can have a subtle effect on how writers think about their own books and careers. Classes usually come with preparatory reading suggestions and sometimes an advance writing exercise too. I also send follow-up notes after each class, including recommended resources, further reading, and writing prompts and exercises.
And places for these classes are going; Density and Speed is already booking up quickly. Hope to see some of you there. The English language is a richer resource than we might realise. Thanks to the influence of many languages arriving on our islands over the centuries, we have enough synonyms to warrant a thesaurus not all languages have one of these. Especially useful to the creative writer is learning to distinguish between words with Anglo-Saxon, or Germanic, roots, and those that derive from Latin, or related romance languages.
Germanic words are our heart language: they are direct, simple, concrete, and go straight to the heart with their emotional impact. Think of words like home, hearth, love, hate. They are sometimes onomatopoeic, blunt, informal or downright rude think of the best swear words. Latinate words are our head language: they are formal, academic, abstract, and appeal to the rational mind. Think of words like intellectual, superior, consideration, providence.
They are sometimes emotionally distancing, elitist, technical or jargonistic. Learning to spot which kinds of words you are using gives you the power to choose for effect. If you want to prioritise the rational over the emotional, or create distance, use Latinate. Exercise: take a scenario and write it twice, once using as many formal, Latinate language words as you can, and once using Germanic language words. Use a thesaurus or etymological dictionary if you get really stuck. You might find that the different words require different sentence structures, too. Could she share her home with this fiend?
In battle, which of them would win? She backed away, into the kitchen, to think through her next move. Would it be possible to inhabit the same space as this demon? In combat, which of them would be victorious? She retreated to the kitchen to consider her strategy. She is currently completing a PhD on folk tales in short fiction, and teaches creative writing at London Lit Lab.
You can find out about her courses on using folk tales in fiction here. Back to Andrew: Thanks, Zoe! This really is one of the best exercises — really getting down into the mire of language. And before we go: on this day of days, let this exercise honour the many languages and cultures that have always made up these islands — and always will. Last month I felt very privileged to see Anna Burns talk about writing and read from her wonderful, prize-winning novel Milkman.
It was a profound experience, and I brought away many things. She has a lovely, intuitive approach to writing. I came away most of all with an impression of someone who is not grasping: not grasping for success, but not really grasping for things in the writing either. And that lets her and her creations shine as originals. This made me think about creating character questionnaires, which are exercises we often do in creative writing, and in fact we had been cooking up some questions for one at my most recent masterclass on character and setting; I used this as the basis of a recent writing experiment.
Such activities are often necessary tasks in bringing characters to life or in researching who they might become. Many of the lovely little details that we put into character questionnaires are juicy, and we grow attached to them, and they end up in our manuscripts.
And though often they are important as telling details we find or cook up for our characters, sometimes too they can end up cluttering our stories, or simply making them feel a bit stilted, like writing by numbers. And on that dread day, the Ineffable One will summon the artificers and makers of graven images, and He will command them to give life to their creations, and failing, they and their creations will be dedicated to the flames… Auto Suggestions are available once you type at least 3 letters.
October 3rd. Next month will see the release of Peter S.