The solutions below capture what I'm thinking when I coach these questions. For many younger children, including my own, getting past square one in the first few quantitative problems starting at #21 was really a challenge. It took about 2 to 3 weeks.

**Solution Commentary**

**Questions 1-20 Whole Language Math**

- The foundation of math is observing, comparing, grouping, splitting, etc. This question is the first step.

The foundation of a competent mathematician (or writer, scientist, etc) is the brain capacity to take in a lot of new information at once and work through it, so these problems are going to slowly ramp up 'new' and 'a lot'. - I'm not sure where your child is in terms of 'before' and 'after' or the names of shapes. This question could be a quick, or it could take a long time. What is that thing on the left? Wait for answer. Is it a ladder? Is it a slide? I ask questions like these to warm up the child for how we will do math from now on, which is to explain to the dumb parent (me) what a problem is asking because I just don't get it. It takes practice for both of us, but by 2nd grade that will be my official response to when my child asks 'Can you explain this to me?'
- Notice that these are still one step problems, just a lot of them. I would like a child to be able to handle much longer problems but she's only 4 so we have to start somewhere. What is that dark shape on the left of this diagram? It looks like a finger. What's a diagram? It's a math word for picture, and I love new words, so I think I'll start using it. My stubborn kids eventually get it but they refuse to use 'diagram' in a sentence.
- When you see a diagram that is not organized, you have to stop and ask the child to group it however they want. (Long pause, maybe 10 minutes before you speak again.) Can they group it by color? Can they group it by shape, what about size? Am I missing anything? Yes, count, and in the case of shapes, the number of sides. Please take a moment to explain that a circle has zero sides. They may not get this or any of it, but you have the rest of the book to work with. This diagram will be repeated on question 35 (the question will include variation of course) with a warm up repeat on a different diagram on question 32. I don't expect any children to get anything in the first few weeks, or even after that. They all do, but 'not expecting' it is part of the magic that will produce outstanding results.
- This diagram is obviously grouped by color, but what would happen if you grouped the diagram by shape? First, it might make this problem take 20 minutes, which is good if the child has that much stamina (unlikely) or it might make this the problem to start again the next day. To make it doable, start with number of sides, because you'll get 4,4,4 and 5, which is easier on working memory.
- Can you group the 4 triangles, or are there really 2 different types of triangles (right and equilateral)? How are these triangles the same and different? What is the difference between the two types of triangles? One is wider, but they are both the same height. One has sides that are equal in length (thus equilateral) and one has a square corner (which I henceforth refer to as a square triangle instead of a right triangle which is not intuitive). Here's what I expect to happen from experience:
- You ask the question and then sit there for minutes of silence. How many minutes you wait depends on how good of an academic coach you are. Let's say 5.
- Your child just sits there saying nothing.
- Then you start asking more detailed questions (which is wider).
- Your child doesn't know what wide means so you hold up your arms to demonstrate. Or you just go with long or whatever because this is covered thoroughly later in the book.
- You're still getting silence so you just start talking the child through all of this. You can do that with 4 year old children. I forbid it after the age of 8.

- This book is going to methodically cover math vocabulary like the terms top and bottom, and just keep going. 'Seeing' becomes possible when the child learns conceptual vocabulary to articulate relationships.
- How many of each shape are there? In addition to the squares, the triangles, and the rectangles, there are 17 circles, and I expect children to ignore the black circles and once I point these out, struggle to count to 17. Even counters might need a few tries or maybe this is too big of a number to count. Which row (top or bottom) has more or fewer black circles? There's so many important skills going on with this question but I just want to highlight one: getting a question wrong is totally awesome. A wrong answer is a signal that we're on the verge of learning. I love wrong answers. Do not lose this attitude because once you expect a correct answer, you will kill learning.
- My favorite math book is Every Day Math Grade 2. This book teaches number sense, a small but mighty math skill. What I set out to do from page 1 to the end is build visual number sense, in case you were wondering, and seeing 6 + 1 in this diagram (not on question #9 of course, but by the end of the book) is what I'm after. In question 10, the child might see 6-1-1. It takes a while, typically most of the book.
- Did you notice in question #9 that the white square is lower than the other shapes but not lower than the door? Did you ask about this? If you didn't notice and didn't ask, then you are going through these questions too fast to maximize visual cognitive growth not to mention another important academic skill which is taking a long time to read (aka view) the question. I can't tell if the blue oval is lower. Maybe. Are all of the rectangles the same width? Of course not. But if you set the blue rectangle upright will it have the same width? The answer is irrelevant but the question and the thinking are golden. What about the 'small red circle'? What about it? Nothing, I just want you to get into the habit of saying 'small red circle' instead of 'circle'.
- I love the question about blue ovals. Ask it with no commentary please. You may see a child who just sits there confused, but I see a brain that just went into overdrive. Again, how else can we group these shapes?
- How is your child doing on vocabulary? Which diagram on the two facing pages has more square triangles? At this age, my kids managed to forget what an oval was about 7,239 times so I just started using the terms ellipse and oval interchangeably.
- Warm up problem to set the plot. Ideally, you do this one first and 14 next on the same day.
- 'Which shape is on top of the stack that has 6 shapes?' Now we're getting into working memory building territory. The gold standard of working memory by 3rd or 4th grade is 3 problems that all have to be solved to get the final answer. 3 memory buckets is about the limit. But, back to age four, we want 'small red circle' and a question with 3 parts like this one. Take plenty of time and repeat the question as many times as you need to.
- Some repetition. Repetition is important when you want you child to actually know something from memory, which is not what this book is about. This book is about thinking. But it's hard to throw in a multi-part question if your child forgets what 'between' or 'wider' means. Plus, they're only 4 years old.
- More repetition.
- How do we know what the 2 sets are. Ask your child if he can tell you why. Maybe you get silence, but this is the brain at work and you have to start somewhere. How else can these shapes be grouped? Is it possible to come up with a grouping strategy that has 2 groups? 4 groups?
- I'm not crazy about the phrasing of 'if you group these shapes into sets by shapes' but it gets the job done, and this holds open the ambiguity (which is a key strategy of the COGAT by the way and I love it) that we're talking about number of sides (triangle) or dimension (equilateral and right). Any decision that the child makes on this is correct. The subskill to promote is decisiveness and if the child is wrong because you are thinking something else you are not helping. But we also love wrong answers. What to do as a parent? "Awesome answer!" then do the quantitative part at the end and ask if there are any other ways to group the set by 'shape'. Ah! We are learning to classify. We've got a set of 6 triangles and this is made up of 2 sets of types of triangles. But of course, this is very advanced and we're just trying to introduce the child that there are other ways of thinking. You can help by telling the answer after a suitable pause. No telling after age 8.

**Questions 21-10 Quantitative Visual Number Sense**

These question types may need ample start up time (in our case a few weeks) before the child actually gets it. The most common challenge is just getting the child to understand that the picture on the left is before, the picture on the right is after, and something changed.

I tried dozens of approaches, but the one that worked the best is to tell the story of a party. Cover the 2nd square, announce that some fish are having a party, then cover the 1st square and say here's the party again? What changed? Or who just arrived? If you still need help, draw some pictures, act this out with stuffed animals. If you still need help, keep reading.

In all cases go slowly. Ask your child to do the next one and say nothing. I'll wait 5 minutes here if I can get the child to stare at the picture that long. Then read the question. The longer the child is staring at 5 items in some organized layout, the higher probability that the brain is turning the 5 into a visual representation of the quantity 5 instead of a certain number of items that must be counted one by one.

On wrong answers, play the Blind Coach. Get out some paper, announce that you can't see, and tell your child that you are going to draw this diagram one step at a time and the child will tell you what to draw. Amazingly, I've never been asked how a blind coach can draw. For question 21, announce that you are going to draw the first box or the first picture and point to it, then go back to your paper. "How many do I draw?" Then announce "I'm ready to draw the next picture. How many more (or fewer) do I draw?" Yes, the blind coach will say "more" or "fewer" but you are using the force so it's OK.

- The first goal is to get the child to see that the 2 fish were joined by a turtle. The two became three. There were two, and now there are three. That's one more. We're going to call that "plus 1" and I'll circle the answer. If you just extrapolate this one person show for a few hundred more words, you have me coaching this question. Then you have to announce in the second diagram that "one left" means that "there is one fewer ocean going creature" and we call that in math "minus one" which you point to. That's a lot of vocabulary that the child will get the way children get vocabulary, using it repeatedly. By the way, my little brainiac did ZERO problems in the first few weeks, and then maybe 1 diagram a day after that before we started to zip along.
- We use see double/half/triple/cut in thirds, start pointing this out. Lesson 42 through 44 will explore these concepts in depth the way I coach them, and then after that it becomes part of the question. For now, just start using the words without worrying about comprehension or recognition.
- Go slow on this picture. Don't read or ask for the question until you look at it - in silence. Ask your child to explain it to you. Note that the red frogs are pointed right, then they move. Wouldn't it be great if the child said "The answer is zero changed in the count, but the 2 red frogs are in the bottom row in the first box, and then one red frog moves to the middle of the top row." This level of articulation is unlikely. However, if you don't give your child the change, this level of articulation will be impossible. Unlikely is better than impossible. Be prepared to help.
- These crabs are organized into rows and columns. How many rows? How many columns? At this point, we've not only got more vocabulary to introduce, but we're spending a lot of time just looking at the picture and describing it before we decide to ask and answer the question. This approach is so important to tests and academic work that it is one of the 4 Core Cognitive skills. If you add "Seeing", which is more of a bucket of subskills instead of a skill on it's own, you end up with the Big Five Skills. These questions are designed to build "Seeing".
- These are jelly fish and blob fish. Are either of these really fish? Why are they named fish? Time for daddy to get on wikipedia and find out. Then in the bottom picture, the turtle has spots on it's back. Rows, columns, spots, 2 shades of brown, or is orange a shade of brown? More unanswerable questions, more wiki. Is that a dot or a spot? I think a dot is what a tiny circle is on a diagram, but a spot is a tiny circle on an animal.
- I am methodically presenting diagram elements in special groupings to slowly build visual number sense. 5 is probably the most important concept to get because it anchors arithmetic so 5 appears often in different configurations. 6 appears in the 2nd diagram. Ask your child to count to six on one hand using his fingers. This child will not be able to this because there are only 5 fingers. But this is not correct. Every child has an invisible 6th finger that is there for this purpose. This invisible finger will come in handy later, but this exercise is helpful for visual number sense on its own.
- Appreciate the beauty of the rows and columns of the 4 penguins on the top right. See how the 5 penguins stack up. If a test date isn't looming, get some blocks and stack them in a pyramid fashion starting with 1, 2, 3... Notice how even numbers are always missing the top item, but odd numbers always complete the pyramid. Odd and even are beyond the scope of the book. How is the second diagram organized? What do these shapes have in common? It is not organized and there are only 2 yellow shapes, none have the same number of sides, none are the same shape, but some might have the same height or width. Shape Size Color Count.
- What is that little black thing in the top right of the car? Go outside or to the garage to find out. I call this a field trip. Stretching out a question or problem is called "learning". Trying to get your child through as many problems as possible as fast as possible is called "training". Cognitive skills tests measure learning.
- How are these pentagons arranged? Which way do they point? It looks like they point left. Or up and to the left, or down and to the right. They point 5 ways. OK, I normally get no recognition of any of this so we go back to rows and columns. Which pentagon points in a different direction?
- Can you name these animals? The gray fish is a bass, the yellow fish is an abstraction of an angle fish. The frog is a poison dart frog. The turtle could be a box turtle or a sea turtle. Look at pictures of each. I'm leaning toward box turtle but your child could argue sea turtle. The crab is an abstraction. How many animals are in each row and column of each diagram? It's hard to say because the rows and columns don't line up perfectly.

Take a moment before question 31 to become an academic coach. A parent wants the child to get the question correct quickly, but an academic coach wants the child to experience a session of learning. These are 2 completely opposite goals. A coach doesn't help. A coach answers a question with a question. A coach answers a question with "You tell me."

- This question is very important. The child may say zero because there is no change, or point out that it's a different shape. The answer is +1 from 3 sides to 4 sides. When a young student announces the answer and I respond 'try again' , some students go into 'guess mode', and I go into 'explain this to me and justify your answer mode'. Ask the child to explain the diagram. Become the blind coach? "I'll draw this, you tell me what to draw. I don't know what a triangle is, so be specific, as in how many sides?"
- This is a review question. Any time you see rows or columns that almost line up, ask about the count in each row and column. When your child is mentally tweaking the diagram to get rows and columns, their visual number sense is exercising itself.
- The text "(Count, size and color) is a note to me that I left in the question. Don't read this text. The last question on this page requires 2 or 3 working memory buckets to be active. My goal through Test Prep Math 3 is to burn in 3 working memory buckets plus a few dozen "working" skills. With children approaching 5, I read this question fast and wait in silence. With kids closer to their 4th birthday, I read it slowly and multiple times. Strong readers only need to hear it once. At this age, a child is a strong reader if the parent reads to the child all of the time.
- Ask your child to help the blind coach draw a pentagon. I have a lot of fun drawing this because the descriptions tend to be so bad the first time. "The shape has 5 sides." My 5 sided shape turns out to look nothing close to the equilateral pentagon in the picture. This is usually a very long exercise where he have to go one line at a time, usually with me recommending we start with the line on the bottom of the shape. A child learns to see the shape with new eyes.
- Which of these shapes has the same width? There are 3 different groups of shapes when grouped by width. The rectangle standing vertically is its own group.
- The last sentence in the paragraph is a working memory builder. I broke down the solution strategy to this in steps and find it fascinating and counter intuitive.
- How are these shapes grouped? They are not grouped. The parallelogram needs the blind coach. It is very challenging for a child of any age up to 10 to articulate that the right side slants diagonally from the right side of the base to the top right. Mainly, this exercise is just silly fun that never gets close to completion. Just a bunch of poorly drawn quadrilaterals. But putting the child on the spot like this is the very beginning of a comprehensive and powerful skill set that will emerge much later.
- No helping. I don't care how long this question takes. You can repeat Shape Size Color Count as you look to the sky if you want. Later, when you are doing a practice test, officially tell your child that any time she is stuck to say Shape Size Color Count because it is the magical incantation for becoming unstuck. I could write a whole book on this. It's not just that a nonverbal question is mainly these 4 attributes, but drawing a picture, checking wiki, and looking at the diagram before answering the question are all techniques that free the intellect from the chains of pressure and limited modes of thinking that bind it. Using this book properly will probably prevent these chains in the first place, but when school puts them back, you can break the glass and get Test Prep Math some day.
- What really annoys me about nonverbal tests is that the same diagram with no anchor (the slide) implies a left to right orientation because that is how writing is organized. I can't imagine why test makers would assume this. On any diagram, you can ask about rows and columns, and you can ask which is first, second, etc. If there is no anchor, the first one is always on the left.
- These are parallelograms. Here is the invisible finger again in its primary purpose, mainly to count to 12 on your fingers. Can you count to 12 with one hand? Yes. You have to go over the fingers twice. But its easier to count to 10 with one hand first as practice to gear up to 12. Any time between now and post doctoral studies, if your child is stuck on a hard problem, try an easier version of the problem first then work your way up to the harder problem. This is the most powerful of the Big 5 Problem Solving Strategies, not to be confused with the Big Five Core Cognitive Skills. What is the simplest version of the problem of counting to 12 on one hand? I'm not telling, but I can guarantee that you got it wrong because there is yet another simpler version you didn't think about.
- See if the blind coach makes any progress on drawing the pentagon or parallelogram. I'm guessing not. So help. Start with the base of each shape. The pentagon has a point at the top and points between the base and the point that are farther out than the ends of the base. Then you can connect the dots. For the parallelogram, on each side of the base, a side extends upward and to the right. Or for the parallelogram, draw a rectangle that is wider than it is tall, and then push the top to the right. Yes, this is more explaining than coaching, but the impact on "seeing" is extraordinary.
- Now we will formalize the concept of doubling.
- And halving.
- And not just tripling, but now you have to ask your child to demonstrate halving, doubling, tripling, cutting in thirds with 1 through 9 and all doable permutations in between. Not that I expect the child to get this right away, but you now have a crutch to refer to for the next month.
- The concept of "zero" and "no change" is important and confusing on tests and important in really hard and confusing math problems in graduate school so I start with that before launching into the first halving problem. Go slow, take a lot of time, don't help.
- 7 to 9 is "+2" but there is no halving or doubling going on, and in the second diagram nothing is happening. I hate to take away the challenge of this concept but when I set up this part of the book I was in a stressed out mood over the future GAT test and was thinking "the heck with learning, I'm just going to train a test crushing machine." I actually created a logo with "Test Crusher" and put it on a few books, but over time we made some actual progress in learning I decided to remove it. I did in fact end up with a Test Crushing Machine, but it resulted in the extraordinary cognitive growth from these exercises and not because SSCC includes every permutation of everything possible. Which I will reluctantly admit that it does.

I'm taking a break from these solutions. Questions 47 and 48 review base concepts and begin stepping through permutations of quantities and concepts. 49 is worth elaborating because the penguins are organized diagonally and the robots in the part of the diagram on the right are either a bit shorter or not quite lined up properly. I always take time to ask if there's anything not quite right about a diagram just to get the child to start looking and seeing as opposed to just counting. No help on question 50.

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