A teacher's experience with Math Mammoth Geometry 1

I intend to publish several of these "stories" or "reports" from teachers who have been using Math Mammoth. Here's the first one, from Megan in Belgium. She used some lessons from my Math Mammoth Geometry 1 book.

Presentationof the Class and the School

I am a1^st,5^thand 6^thgrade teacher in south Belgium. My school is what we call an“immersive school”. The students are taught in French, theirmother tongue, 14 periods a week. During the other 10 periods eachweek, class is given in English – the students are “immersed”in the language from the age of 5 and learn to speak this language ina very natural manner.

Becauseof these awkward hours, we teachers “share” the classes. In orderto have a full schedule, I give class in English 10 hours a week in5^thgrade, 10 hours a week in 6^thgrade and 4 hours a week in 1^stgrade. We also split the subjects between French and English – andGeometry is one of the subjects seen in English.

Now,the immersive program is not without its downfalls, one of whichbeing that we must consecrate two times as many periods per week onLanguage Arts, as we have both the French and the English language toteach. As a result, we have less time to teach everything else.

I amalso a first-year teacher, thrown into this system with no life-linesto cling to! I was searching desperately for a guide, something tohelp me, but also for something that would help my studentsunderstand this subject that is such a part of our everyday lives,yet seemingly so abstract. When I saw Maria Miller’s offer – one“free” e-book in exchange for a report of my experiences – Icouldn’t turn it up! So, without further ado, here is what I haveto say about Math Mammoth.

Session1

Istarted the year teaching lines and angles to both the 5th and 6thgraders. Most of the 5^thgraders did poorly on the test and some of the 6^thgraders needed a refresher course.

Forthe 5^thgraders, I dove right into the Geometry 1 Elementary Math book inorder to review the different lines and angles, as well as how todraw them (chapters Lines, Rays andAngles and DrawingRight Angles). I did not print out andphotocopy the pages exactly as they were, I chose instead to pick andchoose the exercises I found most appropriate for my students at themoment. I must say though, I felt no need to modify any of theproblems, as I usually do with workbooks.

Thereaction from my students was a good one, especially concerning theterm ray.Most of them had such a hard time remembering what a ray is and theimage of the sun and its rays was just the trick they needed! Theirresults were very satisfactory – though they all had a hard timewith the construction of the squares and the rectangles! I don’tthink they’ve passed a lot of time on that as of yet, and I hope itis a point that MM covers more thoroughly later on in the program.All in all, the weaknesses I noticed from the test have now beenovercome, except for one or two exceptions of course.

Agreat point about these exercises is that I could let most of theclass work independently while I spent a bit more time helping theslower/weaker students. Everything is very clearly explained and easyto follow and understand, even for 5^thgraders whose native language is not English.

Forone of my 6^thgraders, I started him off with the chapters MeasuringAngles and DrawingAngles as I noticed he had difficultiesusing his protractor. Once again, I did not give him all the pages inthese chapters, nor did I alter anything. (As a side note, it isquite nice to have such a great bank of problems at the ready – Iusually spend more time perfecting my worksheets than the childrenspend working on them!) He whizzed through the exercises and trulysurprised me with his improvement. It’s as if a light bulb went offin his head. I don’t know if it’s because of the book’sexplanations or not, but I’m glad it happened! He’s now at thesame level as the others.

Myother 6^thgraders did not receive sheets from this book during this period.

Sessions2 and 3

Forthis work period, I combined the chapters Measuring Angles and DrawingAngles for the 5^thgraders. I let several children do some examples on the board andthen students worked quietly on the worksheets (which, again, I didnot modify, save the page-layout). They finished these quickly,though that was because of laziness from most of them. They wereconvinced that the angle was still alright; even it was a few degreesoff! The papers were returned to them the next day and the redoneangles were more than acceptable.

I amnot sure about this, but I am getting the impression that Belgianstudents see certain geometry concepts earlier than their Americancompatriots. I am not saying that these concepts are known andmastered by Belgian children (far from it! :p), but they do have somebases to work from. I have yet to explain a point starting from zeroknowledge, though the book sometimes seems to suggest that theconcept is completely new.

Nevertheless,the MM book is so clearly laid out and the explanations are soconcise and understandable that I feel more confident myself inexplaining the different concepts to my students. Also, it is a loteasier for students to look at a piece of paper right in front ofthem and follow along with the pictures than just doing the exampleson the board alone. The drawn protractors act as a sort of teacher’saide – I don’t have to walk around to verify that each child hasplaced their protractor how they should because they can clearly seehow the protractor is supposed to be on their paper.

Also,it was very practical as I make sure to have the papers for the nextlesson printed out for my “ace” students that always finish wellbefore the others. Because of their clarity, these students can oftenget right to work without any additional explanations.

The6^thgraders are busy doing precision drills and going over drawing basicgeometric shapes (I’m surprised at how many of them have a hardtime drawing a square!), though I am not using MM for their work.

Session4

Thissession was dedicated to the EstimateAngles chapter in MM. This was a breezefor my 5^thgraders – much to my surprise! We started the lesson out by doing afew examples “in real life” using chalk on the playground outsideand our giant chalkboard protractor. Walking the lines and making theturns themselves really helped them understand what the directionsmeant when they said “turn 45° to the left”.

I mustadmit though, the last exercise had everyone baffled! I still don’tknow what the secret message is (I could look at the answer key, butwhere is the fun in that?)! Nevertheless, the children enjoyed trying to figure it out, andthen trying to find where they went wrong in their drawings. Itbecame a very rich activity in which the children analyzed and foundtheir own errors, as well as the errors of others.

Sessions5 and 6

Weworked on the chapter Parallel andPerpendicular Lines for these last twosessions. These concepts had already been touched upon in theearlier chapters, so it was more of an in-depth review than a newconcept for them. This really helped set things straight in theirheads and served as a great closing lesson for the first part of theschool year, before leaving for a week of vacation. Most of the workduring these two sessions was done individually while I worked with afew children in need of an extra boost.

Evaluations

Iincluded three evaluations in this first part of the MM curriculum:one on Lines, Rays and Angles;another on Measuring, Drawing and Estimating Angles; and the last one onParallel and Perpendicular Lines.These were tests that I created myself, based on the concepts coveredin the MM chapters.

Inever had more than three or four students out of twenty fail thetests, but these are also the students who present very importantweaknesses in nearly every subject matter. The other results weremore than satisfactory and provided for a very impressive firstreport card. (As a first-year teacher, I was very proud to have beenable to show the parents that their children had indeed learnedsomething with me.)

Continuation

I planon using MM for the rest of the year with my 5^thgraders, while adding my own touches. We’re going to start with thetriangles, but before using MM, I would like to use a more activeapproach for the discovery of the different types (let them createthe different triangles themselves, for example).

FinalWord

I findMM to be very complete, though a bit traditional. Perfect forhomeschooling parents who have little or no pedagogic experience, butI think it could benefit from the addition of a few hands-ondiscovery sessions. (Though I do believe that Maria speaks of acurriculum that shows how math is useful in everyday life… thiscould also be very interesting for a hands-on approach…) I have noproblem with the hands-on approach, but I was rather blocked by thetheory part of the geometry… MM is my perfect complement! I wouldhave been very lost this year if not for this amazing find.

Iwould like to thank Maria for this wonderful opportunity, apologizefor the absence of photos (I always get so caught up in my lessonsand with my students that the camera stays inactive on my desk!) andI do hope that my words and my experiences can help convince othersof MM’s wonderful structure, clarity and presentation.

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Click the link to read more about the book Math Mammoth Geometry 1 and to see its FREE sample pages!

Free book - Geometry in Art

I was told about this free download of the book Geometry in Art, by Hilton Andrade de Mello.

On the page, you need to scroll down to the words "free downloading".

This book is a basic introduction to geometry in art, with topics such as polygons, spirals, polyhedrons, tessellations, perspective, the golden ratio, symmetry, geometry and symbolism, and geometry and informatics. It has lots of illustrations and artwork by various artists, and can serve as a nice introduction for anyone who hasn't studied these topics before.

Geometry games online

I have updated the list of online geometry resources and games at HomeschoolMath.net. It became quite a long list of links! The links are categorized by topics, such as

Shapes & polygons 
Area & perimeter 
Angles 
Solids, volume, & surface area
Coordinate plane 
Congruent transformations 
Similar figures 
Circle / Pi 
Constructions 
Pythagorean Theorem 
General geometry websites 
Worksheets 
Books 
Advanced topics.

I hope this is helpful!

Math Mammoth Geometry 2

I have just finished writing the material for Math Mammoth Geometry 2 book. The material in it is suitable for grades 6-7. Download price is $5.80.

The main topics in the book include:

* angle relationships
* classifying triangles and quadrilaterals
* angle sum of triangles and quadrilaterals
* congruent transformations, including some in the coordinate grid
* similar figures, including using ratios and proportions
* review of the area of all common polygons
* circumference of a circle (Pi)
* area of a circle
* conversions between units of area (both metric and customary)
* volume and surface area of common solids
* conversions between units of volume (both metric and customary)
* some common compass-and-ruler constructions.

I've included several complete lessons from the book as samples (PDF). Feel free to download these and use with your students!

Angles in Polygons
Review: Area of Polygons, 1
Surface Area

Besides those, there are two other sample pages:

Area and Perimeter Problems
Basic Compass and Ruler Constructions, 1

What is next?

• See more information
• Purchase a download at Kagi
• Purchase a download at Currclick

Review of Harold Jacobs Geometry

I have just posted a comprehensive review of this book that is popular among homeschoolers for a high school geometry course. Click to read:

Review of Harold Jacobs Geometry book

Angles in a parallelogram and a triangle

This is a set of three geometry videos dealing with angles.

First, showing that vertical angles are equal:



Next, finding out about the angles in a parallelogram. I start out with two parallel lines and a transversal (line that intersects them both). We explore the angles formed, which some of them are corresponding angles, some are vertical angles. I draw a new line, and get a parallelogram.



Lastly, here is a short and easy proof about the angles in a triangle.

Using a protractor

I've finally ventured into geometry topics with my latest video:



It shows how to use a protractor to measure angles. I show where the "base line" of the protractor is for three different protractors, and use it to find out the angle measure of different kinds of angles (including a reflex angle).

Paper models for polyhedra

Have you ever wanted to fold a pyramid or some of the other interesting polyhedron out of paper?

At least, take a look at the pictures! www.korthalsaltes.com provides hundreds of free printable paper models for various solids, including platonic solids, pyramids, and lots of polyhedra. Some are available as colored versions also.

Folding and gluing one or two is a great summer math project also, and will keep kids busy for a while.

Carnival picks

I have a collection of interesting links for you today; mostly some picks from Math Teachers At Play #2 blog carnival, which you SHOULD visit. Denise always has good stuff and really nice photos to spice up the reading.

Here's a neat way to practice multiplication tables by Math Mojo.

I also liked some simple dissection puzzles; you'll even get a free worksheet download.

Nick has posted two geometry "Wrapper" problems

This one is hilarious - lots of pictures of mathematical clocks!

Those were all from the carnival; please check out all the other posts as well; it is a good carnival!

Lastly, I visited this yesterday, and I think ALL of us should check out what a trillion dollars looks like.

Geometry problem: a tiling

UPDATED!

Today I have a geometry problem for you.

Just yesterday I showed you a proportion problem and its solution.

(I want to insert a note here, especially for parents reading this.

Don't feel that you always need to use the problems I present here with your students. Instead, as a homeschooling parent, consider them part of your "math teacher training" that you never got. Reading through problems and their solutions will help you become a better math teacher. Consider yourself an apprentice observing a master completing a task, in this case mathematics problem solving.

It just takes you a minute or two to read this, think it through, try it - and understand (I hope). Your students' or child's book will have similar problems later on when it's time for geometry studies, but if you read these that I present, you will have learned more and can better tackle those problems at that time.)

OK, here we go. Look at the tiling below. Here you can see the tiling in real life, on a floor in Germany.



The original problem asks you about the area of the small square and of the parallelogram, given that the area of the large square is 1 square meter and that the ANGLE between the large and the small square is 45°.

But, I simply ask you (or your student) to DRAW this picture. Just take out blank paper, a ruler, and a protractor, and DRAW it. Can you draw it exactly how it is in my picture?

Solution:

This tiling will work (it will tile the plane), no matter what the size of the "little square". So, it is NOT possible for you to draw it exactly like I did, unless you knew something more about the side lengths.

Here's an example where use the same information, but made the small square smaller than in the previous picture:



To solve the original problem of finding the area of the little square and of the parallelogram, one would need some more information, such as a side length or ratio of side lengths.

SO, all in all, it's a problem with incomplete information to solve it...

However, someone suggested in the comments a solution to the original problem where we assume that the parallelogram is cut into two right triangles by its diagonal. With that additional information, the area calculation would be solvable.

(BTW, For my first picture, I used golden ratio as the ratio of the side of the large square to the side of the small square.)

Two new books in the Blue Series

64 pages
(of which 37 are lesson pages)
Price: $2.50 download

Buy at Kagi (PDF download)

Sample pages (PDF)

Contents and Introduction
Shapes
Right Angles
Line Symmetry
Perimeter
Solids

1. Math Mammoth Early Geometry covers geometry topics for the early elementary grades (approximately grades 1-3).

The first lessons in this book have to do with shapes - that is where geometry starts. Children learn the names of the common shapes, and also put several shapes together to form new ones, or divide an existing shape into new ones. They practice using a ruler to draw various shapes and are introduced to tilings.

Next children learn the concepts of parallel lines and lines that are at a right angle (perpendicular lines). The book also has beginner lessons about symmetry, area, perimeter, and solids.

After studying these early geometry lessons, you can continue the study of geometry with Math Mammoth Geometry 1 book. In it, children will learn to classify figures (during grades 4-6) according to their sides and angles, and learn much more about area, perimeter, and volume.

52 pages
(of which 40 are lesson pages)
Price: $3.00 PDF download



Sample pages (PDF)
Contents
Counting One, Two,and Five-Cent Coins
Counting Coins Review
Euros, part 2
Practising Shopping
Mental Math and Money Problems

2. Math Mammoth European Money is a worktext that covers money-related topics usually encountered during grades 1-3. The book contains both textbook explanations and exercises, and is designed to be very easy to teach from, requiring very little teacher preparation (you do need to find practice coins before the
lessons).

The book starts with first-grade topics such as counting coins with cent-amounts and easy problems about
change.

From there, the lessons advance toward second-grade, and finally to third grade topics, such as practicing with euro amounts, and figuring out total bills and change. Therefore, you can also let your child work the pages of this book in different time periods, and not go through it all at once, depending on your child's
current level.

Are these really parallelograms - answers

These are answers to my earlier post where I asked if certain figures necessarily are parallelograms.

The question was: Does the given information in each diagram guarantee that each is a parallelogram?

Figure 1:
This one you can't get around; it ends up being a parallelogram, actually a dandy rhombus. Let's prove it. You can notice it has lots of sides of the same length. If we draw a diagonal, we get two triangles with all kinds of same sides:

The two triangles ABD and BCD end up having all three sides the same. So by the SSS triangle congruence theorem, they are congruent triangles. Hence, their corresponding angles are the same.

I've marked the corresponding angles with the same colors. Actually the triangles are even isosceles so the blue and purple angles are even congruent... but we don't need that fact.

To prove ABCD is a parallelogram, we need to prove its two sides are parallel. And for that, it's often handy to use the corresponding angle theorem: if corresponding angles are equal, the lines are parallel. So image that we continue the line segment CD. Notice the additional green angle that I've marked:


How do I know it actually is a "green angle" (congruent with the other green angle)? It's because the three angles, being angles of a triangle, add up to 180:
+ + = 180. And the three angles being there along the same line (the continuation of CD), it must be. This sounds a little too complicated as I'm typing it. Perhaps I shouldn't have marked it green. Anyhow, since it IS congruent with the other green angle at C, then the line segments BC and AD must be parallel.

A similar argument would prove the other two sides parallel.

Figure 2: This isn't necessarily a parallelogram, but it IS always a trapezoid:



Figure 3:
This one is trickier, but it isn't necessarily a parallelogram. I used a compass to find a way to make this into a trapezoid:ABCD is a trapezoid with the non-parallel sides 5 units long.

Figure 4: This one is actually a repetition of the Figure 1, because it has the opposites sides of same length. We can use the identical argument to prove it is a parallelogram.

Are these really parallelograms?

Continuing with the idea in my post about Squares that aren't squares?, let's look at the following "parallelograms".

The question is the same:

Does the given information in each diagram guarantee that each is a parallelogram?

If you don't think so, your mission is to draw a quadrilateral with the given information but that clearly does NOT look like a parallelogram.

Figure 1:


Figure 2:


Figure 3:


Figure 4:


Again, these problems let students practice logical reasoning, and also learn about parallelograms, of course. See answers here.

Squares that aren't squares?

Updated with solutions!

Today I want to highlight a square problem I saw at MathNotations. I hope Dave Marain doesn't mind me showing this picture and problem on my blog... I have no problem acknowledging it's from his blog. I COULD just tell you all to "go read it at Dave's blog....

BUT I don't feel that's the best way, IF I want you to think about this. I can just guess that most of the folks would feel too lazy to click on the link and go read it there (would you?). So I want to show it here.


Figures not drawn to scale! And this is important!

Now here's the question:

Does the given information in each diagram guarantee that each is a square?

If you don't think so, your mission is to draw a quadrilateral with the given information but that clearly does NOT look like a square.

The IDEA is to make our students THINK LOGICALLY, or practice their deductive reasoning skills. A great little problem.

The answers:


Figure 1 is not necessarily a square. The upper left corner angle can be of any size. The upper side can be of any length. And so on. See here two examples.

Figure two is not necessarily a square either since the "top" side can be of any length. But it is a rectangle.

Figure 3 in the original problem IS always a square!

Now, I'll write another post where we'll extend this idea to some parallelograms.

Icosahedron from picnic supplies

This was such a fun video to watch! This guy makes an icosahedron from plastic plates, and then from plastic cups - and one more from just plastic and duct tape.

It can serve as a fun summer math project for kids who love explorations, cutting, gluing, building, and that sort of stuff!

He is hilarious! See the video:


Video from Makezine.com

Read the instructions for the icosahedrons

Find out what is an icosahedron

High school geometry - a review

It's done! Finally! Took me some time to finish this review, perhaps because it involved three products:

• The book Geometry: A Guided Inquiry. As the name suggests, this book is based on letting students learn about theorems and their proofs in the setting of "guided inquiries" or interesting problems. It is quite unique in its approach.


• A Home Study Companion which includes solutions and about 300 interactive demonstrations


• Geometer's Sketchpad - dynamic geometry software.

This review isn't just what you typically find on the web; someone called it an "exquisite in-depth review". It's fairly long... with sample pages and other pictures, examples, and more.

I encourage you to read it even if you don't need a high school geometry book right now... because you'll get valuable insight just HOW GOOD geometry instruction can be, how the book handles proof, or what to think about an axiomatic vs. discovery based geometry text.

Review of Geometry: A Guided Inquiry with Geometer's Sketchpad and Home Study Companion.

Measure the circumference of the earth - contest

I got word of an interesting contest where school children will form teams and attempt to measure the circumference of the Earth using the same method as Eratosthenes used back in ancient times.

Any students from USA, Mexico, and Peru can form these teams, whether homeschooled, after-schooled, public schooled or whatever.

Whether you will participate or not, go see the animation that explains the method Eratosthenes used (in the left sidebar).

This sounds like an exciting opportunity to connect geometry, measuring, and math history in a project!

And here's some more information:

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Please help us get the word out on this new, exciting student centered event!

Measure Your World!

Join us this fall as we pilot a new student-centered project where teams from the United States, Chile, and Mexico partner to replicate the technique introduced by Eratosthenes to determine the circumference of the Earth. Around 240 BC, Eratosthenes used trigonometry and knowledge of the angle of elevation of the Sun at noon in Alexandria and in Syene to calculate the size of the Earth. Windows to the Universe, Educared, and CREA are working together to offer school children in the U.S., Chile, and Mexico the opportunity to form partnerships, take local measurements, and collaborate using the Eratosthenes method to Measure Your World.

All of the information necessary to participate in this pilot student project can be found on the Measure Your World Web sites (www.measureyourworld.org and www.MideTuMundo.org). Student teams must have a parent or adult sponsor to participate. At least one of the team members or adult sponsors must be fluent in both English and Spanish. This event is open to all students in the three participating countries and does not have to be affiliated with a formal K-12 school. Home-schooled children and children participating in after-school programs (e.g. the Scouts, 4-H, etc.) are welcome to participate.

In addition to taking the measurements and calculating the circumference of the Earth, student teams will be encouraged to learn more about their partners in the other participating countries. Suggested activities to promote cultural exchange can be found on the Web site.

Registration for the Measure Your World event will be open from August 13 — September 14, 2007. Student teams will be notified of their partners by September 21, 2007. The time period for taking the measurements will be September 29 till October 7, 2007.

Geometry fun with GeoMag

While on vacation, a friend of mine gave my older daughter a set of Geomag. You might already know about it, but it was new for us.

This has proved to be a fantastic learning toy! She's thoroughly enjoying building various shapes.

For example, she made a cube with sides 2 bars long and was proudly explaining to me how to do it: "First do a square, then put legs up from each corner, and then another square."

I made a tetrahedron that also had 2 bars on each side, according to the model. She thought it was neat and built that one several times herself last night.

I can see how the toy can help build geometric insight and beautifully demonstrate the common three-dimensional figures.

We've already ordered another set to accompany the small 42-piece set she got. You can find Geomag kits of various sizes and colors at Amazon.

Geometric patterns in islamic art

This was in the news recently so you might have seen it...

Physics student Peter J. Lu has discovered, after his trip to Uzbekistan, that the geometric patterns shown on the walls of old islamic shrines depicted a very complex mathematical pattern, one that was only found in the west during last century.

I feel the article at ScienceNews.org explains it all very well, and shows plenty of pictures of the patterns AND of the underlying tilings, plus has references and links for further study, so I'll just refer you there:

Ancient Islamic Penrose Tiles

Elementary geometry: how much time should you devote to it?

A geometry question from a visitor:

1. How much time should be invested teaching geometry at an elementary level?
2. How much time is actually dedicated towards geometry in a tradicional textbook

Your guidance will be extremely appreciated!

During elementary mathematics, geometry plays more of a sideline role at first. It is intimately tied with measuring topics - and really, the word "geometry" means "measuring the earth", the science to measure the land.

The goal of elementary geometry seems to be that the student be able to find perimeters, areas, and volumes of common two and three dimensional shapes.

I would add to that the goal that the student can understand and form abstract definitions, distinguish between necessary and sufficient conditions for a concept, and understand relationships between different shapes before entering 10th grade. (I've written about that before in the article Why is high school geometry difficult?.

According to the Curriculum Focal Points report recently released by National Council of Teachers of Mathematics, the following geometry topics play a major role in elementary grades:

Grade	Explanations (from Curriculum Focal Points by NCTM)
Grade 1 Geometry: Composing and decomposing geometric shapes.	Children compose and decompose plane and solid figures (e.g., by putting two congruent isosceles triangles together to make a rhombus), thus building an understanding of part-whole relationships as well as the properties of the original and composite shapes. As they combine figures, they recognize them from different perspectives and orientations, describe their geometric attributes and properties, and determine how they are alike and different, in the process developing a background for measurement and initial understandings of such properties as congruence and symmetry.
Grade 3 Geometry: Describing and analyzing properties of two-dimensional shapes.	Students describe, analyze, compare, and classify two-dimensional shapes by their sides and angles and connect these attributes to definitions of shapes. Students investigate, describe, and reason about decomposing, combining, and transforming polygons to make other polygons. Through building, drawing, and analyzing two-dimensional shapes, students understand attributes and properties of two-dimensional space and the use of those attributes and properties in solving problems, including applications involving congruence and symmetry.
Grade 4: Measurement: Developing an understanding of area and determining the areas of two-dimensional shapes	Students recognize area as an attribute of two-dimensional regions. They learn that they can quantify area by finding the total number of same-sized units of area that cover the shape without gaps or overlaps. They understand that a square that is 1 unit on a side is the standard unit for measuring area. They select appropriate units, strategies (e.g., decomposing shapes), and tools for solving problems that involve estimating or measuring area. Students connect area measure to the area model that they have used to represent multiplication, and they use this connection to justify the formula for the area of a rectangle.
Grade 5: Geometry and Measurement and Algebra: Describing three-dimensional shapes and analyzing their properties, including volume and surface area	Students relate two-dimensional shapes to three-dimensional shapes and analyze properties of polyhedral solids, describing them by the number of edges, faces, or vertices as well as the types of faces. Students recognize volume as an attribute of three-dimensional space. They understand that they can quantify volume by finding the total number of same-sized units of volume that they need to fill the space without gaps or overlaps. They understand that a cube that is 1 unit on an edge is the standard unit for measuring volume. They select appropriate units, strategies, and tools for solving problems that involve estimating or measuring volume. They decompose three-dimensional shapes and find surface areas and volumes of prisms. As they work with surface area, they find and justify relationships among the formulas for the areas of different polygons. They measure necessary attributes of shapes to use area formulas to solve problems.
Grade 7: Number and Operations and Algebra and Geometry: Developing an understanding of and applying proportionality, including similarity	Students also solve problems about similar objects (including figures) by using scale factors that relate corresponding lengths of the objects or by using the fact that relationships of lengths within an object are preserved in similar objects.
Grade 7: and Measurement and Geometry and Algebra: Developing an understanding of and using formulas to determine surface areas and volumes of three-dimensional shapes	By decomposing two- and three-dimensional shapes into smaller, component shapes, students find surface areas and develop and justify formulas for the surface areas and volumes of prisms and cylinders. ... Students see that the formula for the area of a circle is plausible by decomposing a circle into a number of wedges and rearranging them into a shape that approximates a parallelogram. They select appropriate two- and three dimensional shapes to model real-world situations and solve a variety of problems (including multistep problems) involving surface areas, areas and circumferences of circles, and volumes of prisms and cylinders.
Grade 8: Geometry and Measurement: Analyzing two- and three-dimensional space and figures by using distance and angle	Students use fundamental facts about distance and angles to describe and analyze figures and situations in two- and three-dimensional space and to solve problems, including those with multiple steps. They prove that particular configurations of lines give rise to similar triangles because of the congruent angles created when a transversal cuts parallel lines. Students apply this reasoning about similar triangles to solve a variety of problems, including those that ask them to find heights and distances. They use facts about the angles that are created when a transversal cuts parallel lines to explain why the sum of the measures of the angles in a triangle is 180 degrees, and they apply this fact about triangles to find unknown measures of angles. Students explain why the Pythagorean theorem is valid by using a variety of methods—for example, by decomposing a square in two different ways. They apply the Pythagorean theorem to find distances between points in the Cartesian coordinate plane to measure lengths and analyze polygons and polyhedra.

Notice how the focal point below for grade 8 is different: no longer is the focus on area and volume of shapes, but on reasoning with lines and angles.

(Note: The absence of a geometry focal point for grades 2 and 6 does not mean that geometry is not studied on those grades. NCTM's focal points are only three per grade so on those grades there were other three topics that were in the focus.)

In a traditional textbook, how much time is spent on geometry? I checked a few books page counts to get an idea:

3rd 24/336 = 7%
4th 23/340 = 6.7%
4th 18/196 = 9.2%
6th 42/340 = 12.3%
6th 31/224 = 13.8%
7th 56/372 = 15.1%

These did not include measuring topics, but just geometry having to do with shapes, lines, angles, area, perimeter, volume.

Of course on lower grades, measuring topics are another 'slice', usually at least about as large as geometry.

So basically you would spend from 1/12 to 1/7 of the total time on geometry topics, increasing as you proceed to higher grades (while decreasing the amount of time devoted to measuring topics). Obviously various arithmetic topics take the bulk of time in elementary mathematics instruction.