Блог Анкара

In This Chapter

^ Understanding complementary medicine diagnosis

^ Asking questions about your health

^ Examining the tongue and pulse for clues to your health

^ Exploring complementary medicine testing and diagnostic devices

^ Looking at the evidence for diagnoses

Complementary medicine practitioners use a range of techniques to assess your health and diagnose the root cause of your illness or imbalance. They use this diagnostic information to determine the best course of treatment for you.

Some diagnostic approaches are more art than science, with their roots going back hundreds or even thousands of years. These approaches include tongue and pulse diagnoses, as used by practitioners of traditional Chinese, Japanese, Tibetan, and Ayurvedic (Indian) medicine. These diagnostic methods involve great sensitivity and skill on behalf of the practitioners who claim they’re really useful tools in ‘reading’ the body. However, many modern medics reject these techniques because they haven’t been scientifically proven.

Other diagnostic methods are more firmly rooted in science and some 21st century complementary medicine practitioners use many diagnostic techniques that are also used by orthodox medical doctors, such as laboratory tests, blood analyses, and x-rays.

In this chapter, I give you a guided tour of the different types of complementary medicine diagnosis and point out how reliable they may (or may not) be.

Finding Out about Diagnosis in Complementary Medicine

In this section, I discuss the following main forms of diagnosis used by complementary practitioners:

U Questioning: Includes questions about your physical and emotional health, symptoms, lifestyle, diet, and sleep patterns

U Observation: Includes examination of your tongue, your general appearance, and gait, as well as iris diagnosis and micro-diagnosis

U Palpation: Includes checking pulse and pressure points, Hara (abdominal) diagnosis, and foot reflexology

U Physical diagnosis: Includes muscle testing and mobility testing

U* Clinical signs: Includes examining changes in the appearance of your skin, hair, nails, and eyes

U Charting and calculations: Includes food diaries, sleep diaries, and Tibetan medical astrology

U Testing: Includes laboratory tests and electro-diagnostic measurement devices

U* ‘Energetic’ diagnosis: Includes methods for sensing the electro-magnetic field around your body

Questioning

Many practitioners start by asking you to describe any current symptoms that you’re concerned about. They’re interested in when the symptoms started, their nature and severity, and what alleviates or aggravates them. Your practitioner will also want to know of any treatment you may have had so far and its effect. The practitioner will also ask about your medical history, including any previous ailments, accidents, or surgery, and details of any present or past medication.

Always remember to inform your practitioner of any pharmaceutical drugs, nutritional supplements, and herbal, homeopathic, or other medicine you’re currently taking, or any other therapies you’re undergoing, in case of any possible interactions.

Jane came to see me suffering from persistent, unexplained tiredness (a common medical syndrome called TATT, or ‘tired all the time’). Her doctor had carried out various medical investigations but nothing abnormal had revealed. Jane had been advised to take some time off and get some rest. She’d followed this advice but felt no better. A dietary analysis showed that she was eating and drinking a huge amount of energy-sapping foods, that is, foods that are low in nutrients and high in stimulants. She lived on coffee, fizzy drinks, sugary sweets, pizzas, and ready meals. She was also unhappy in her job and felt stressed because she had the workload of two people after a colleague had left and not been replaced. Her sleep habits were also erratic because she often worked late into the night.

Tongue diagnosis revealed nodules at the rear and a red tip, which are signs in Traditional Chinese Medicine (TCM) that the person is under stress and has kidney/adrenal weakness. The adrenals are small glands that sit on top of your kidneys and control the release of the hormone adrenalin, which speeds up your heart rate and increases the oxygen supply to the muscles in times of need such as fight, flight, or fright.

Palpation of the pulse revealed a weak pulse overall and a deep, faint pulse in both the ‘kidney positions’ on her wrists (TCM practitioners take pulses for each of the internal organs at different points on the wrist). A laboratory saliva test confirmed that she had adrenal insufficiency – that is, her adrenals were worn out due to continued over-stimulation from dietary stimulants and stress.

Thus, from a holistic point of view her tiredness could be seen as due to a combination of (i) dietary insufficiency (a lack of essential nutrients), (ii) adrenal insufficiency (over-stimulation of the adrenal glands due to stress and dietary stimulants), (iii) inadequate rest and poor sleep, and (iv) work stress and job dissatisfaction.

Jane’s immediate symptoms were remedied by means of dietary change, nutritional supplementation, adrenal support, and natural sleep aids. Jane also made the decision to change her job and to join a gym. Within a month she was bursting with energy, her tongue and pulse were normal, her sleep was good, and she was happily in training for a charity trek on Mount Kilimanjaro!

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The practitioner may also be interested to know about the health history of your immediate family (in particular your parents, grandparents, and siblings), and may use this to determine your health risk factors and predisposition to disease. For example, women who have more than one female relative diagnosed with breast cancer have an increased risk of developing it themselves. In these situations, the practitioner can then advise you on steps to take to decrease whatever your risk may be. In some traditional medical systems, such as Traditional Chinese Medicine (TCM), knowledge of your parents’ and grandparents’ health history, and the order of your birth in relation to your siblings, is used to determine how much ‘source Qi (pronounced Chee) - that is, core vitality – you have.

Other questions are likely to cover your diet, lifestyle, stress levels, mental and emotional health, and your daily habits such as sleep, exercise, and recreation. Homeopaths, for example, use answers to lifestyle questions to build a picture of your ideal constitutional remedy (for more about this, see Chapter 10 on homeopathy).

Most complementary practitioners consider you from a holistic viewpoint. They use these different bits of information to build a picture of your current symptoms in the context of your overall health and well-being.

You don’t have to have a health complaint to go to a complementary medicine practitioner. Some people are symptom-free but choose complementary medicine because they believe that it can help them optimise their health and prevent disease. In such cases, the questioning may be more focused on your existing lifestyle and health and desired optimal health goals.

Observation

In observational diagnosis, the practitioner may examine a particular part of the body to identify signs that reflect something going on in another part of the body, or may observe the movement and appearance of the whole body to determine more general underlying factors. This section includes descriptions of some of the more common types of observational diagnosis.

Tongue diagnosis

Tongue diagnosis is used by most traditional Chinese, Japanese, Tibetan, and Ayurvedic (Indian) medicine practitioners. You’re asked to stick out your tongue so that the practitioner can observe its shape, colour, coating, and any movement. According to these traditional medicine systems, different tongue shapes and appearances can give valuable information about specific internal organ function as well as general health. For example, a thick, yellow coating in the middle of the tongue is considered indicative of sluggish bowels and constipation.

Chinese and Japanese tongue diagnoses are the most detailed, with more than 100 different types of tongues identified, all relating to different patterns of individual organ balance. Tibetan and Ayurvedic medicine use a more simplified form of tongue diagnosis that concentrates on establishing the general balance within the body’s constitution, rather than focusing on individual internal organs. Western medicine also uses a simple form of tongue diagnosis.

Little scientific evidence supports the more detailed form of tongue diagnosis used in the traditional medical systems and by many complementary medicine practitioners today. Chapter 3 has more details on tongue diagnosis.

Iris diagnosis

Also known as Iridology, In this form of diagnosis the practitioner examines your iris (the coloured part of your eye) by means of either a special measurement device or a hand-held magnifying glass designed for eye inspection.

The practitioner looks for specific marks and signs in particular parts of the iris that are believed to correspond with specific parts of the body. This form of diagnosis is often used by naturopaths or nature cure practitioners and sometimes by people solely trained in this technique. Some slight evidence supports the use of this technique but more research is needed.

For more about this intriguing form of diagnosis, read Chapters 8 and 13 on nature cure and naturopathy.

Micro-diagnosis

The face, ears, abdomen, back, feet, fingers, and toes can be used diagnostically to reflect the body as a whole. This is called Micro-diagnosis. For example, in TCM face diagnosis, the areas under the eyes are linked to the kidneys, so grey bags under the eyes are believed to be indicative of adrenal exhaustion. In Japanese finger diagnosis the thumbs are linked to the lungs, so deformed or weak thumb nails may be indicative of past or underlying lung conditions such as asthma. However, the evidence for this is only anecdotal so this form of diagnosis is not accepted by many doctors.

More on this type of diagnosis appears in Chapter 7 on Japanese medicine.

Appearance and gait

Before diagnosis, the practitioner observes you from the moment you enter the room, noticing your gait and general appearance in case either has a bearing on your symptoms. For example, you may visit an osteopath because of back pain and as you walk in, the practitioner may observe that you have a very lopsided gait. Further physical investigation may reveal that one leg is longer than the other and that this leg discrepancy causes pelvic and lower-back imbalance, which causes the back pain. By re-adjusting pelvic alignment and recommending the use of special inner soles for your shoes, called Orthotics, Your back pain may be dramatically relieved.

Alternatively, a practitioner may observe that a person’s gait is very stooped and their appearance dishevelled, prompting the practitioner to ask questions about mood and self-care. This combination of observational diagnoses and questioning may reveal that the person is actually suffering from depression even though this had not been mentioned as a symptom. Depression can then be addressed by the practitioner.

This type of observational diagnosis can have preliminary importance but then needs to be backed up with other diagnosis to confirm the significance of the observation.

Palpation

Palpation Involves applying light pressure to different parts of the body to determine the body’s response, or to feel for areas of tenderness, pain, heat, and so on. The most commonly used forms of palpation are pulse diagnosis, pressure point diagnosis, and abdominal diagnosis.

Pulse diagnosis

Traditional Chinese, Japanese, Ayurvedic, and Tibetan medicine practitioners rely strongly on pulse diagnosis. In Western medicine, only a single pulse is taken at the wrist to determine the rate of blood flow with each heartbeat. However, in the traditional medical systems, pulses are taken at three points on each wrist and at different depths. Each position is said to correspond to a different internal organ and to give information about its functioning.

The well-known Tibetan physician, Dr Yeshe Donden, was invited to carry out a ward round at an American hospital. He was presented with patients without being given any information about their medical condition or medical history and invited to diagnose them simply with his pulse-taking skills. The American doctors were amazed when, without any questions to the patients, Dr Donden was able to accurately diagnose health problems, including a faulty heart valve.

Western medicine finds it hard to comprehend this concept of tuning into the body using just the artery on the wrist and because this technique is difficult to measure and verify scientifically, many medics dismiss it.

More information on different types of pulse-taking appears in the chapters of Part II.

Pressure points

^W-f^ Many complementary medicine practitioners palpate the skin in order to determine points of sensitivity or pain known as Pressure points. Pressure ‘ A ‘ points can give information about the condition of underlying tissues and organs but may also be used to give information about related internal organs according to Meridian Or other body maps. For example:

Osteopaths, chiropractors, and massage therapists palpate to locate areas of inflammation and to get information about structural alignment and the health of the joints and surrounding tissues.

IU Reflexologists use painful points on the feet to diagnose internal organ health according to a zone system that links specific points on the feet with specific internal organs and body parts.

I Acupuncturists palpate to locate points where the flow of vital energy, or Qi, May be blocked in the acupuncture meridians (energy channels).

Painful pressure at certain points is believed to indicate malfunction in related internal organs. For example, a point on the upper side of the foot, above the groove between the big and the fourth toes (known as the Liver 3 point) is said to become very sensitive if a person has a hangover and a weakened liver due to excess alcohol!

Little evidence supports these forms of diagnosis, other than the fact that they’ve been used for thousands of years. Many medics reject these forms of diagnosis, and more research is needed.

For more on these approaches see the relevant chapters that cover osteopathy, acupuncture, and reflexology.

Abdominal palpation

Another form of palpation is Abdominal diagnosis, Also known as Hara Diagnosis or Ampuku. This involves firm pressure to different parts of the abdomen, which is used both diagnostically and to give treatment. No real scientific evidence supports the use of this technique, although it’s been used clinically for hundreds of years, which many feel is a testimony to its effectiveness.

For more on abdominal diagnosis check out Chapter 7 on Japanese medicine.

Physical diagnosis

Physical examination is an important part of diagnosis for several types of practitioners, especially osteopaths and chiropractors. These practitioners usually ask you to bend your spine and move your joints in different ways to determine your level of mobility and your range of movement.

Kinesiologists and Touch for Health practitioners use muscle testing as part of their examination. This testing involves ‘challenging’ a muscle by applying gentle pressure that you have to resist. Kinesiologists believe that the muscles are linked to specific internal organs and body systems and that muscle weakness reflects internal organ or body system weakness.

Chapters 14 and 15 cover osteopathy and chiropractic, and Chapter 16 explains kinesiology.

Clinical signs

Clinical signs are like signposts on the road to a destination (the diagnosis). These signs may be changes in the appearance of your skin, hair, nails, eyes, and so on, or perhaps a bloated abdomen or swollen glands. All these signs may be linked to specific diseases or imbalances according to the various complementary medicine approaches. So, for a practitioner of Traditional Chinese Medicine, spots around the mouth would be linked to problems with the large intestine and digestion, according to the meridian pathways. Ridged nails may indicate a zinc deficiency to a nutritionist.

Complementary medical practitioners who’ve completed a full, professional training course of several years (usually a degree course that also includes practical training in clinical skills) are also carefully trained to recognise specific physical, emotional, or mental signs that may indicate serious diseases requiring medical attention – for example, excessive thirst and weight loss are potentially linked to diabetes, while chest pain and breathlessness may be linked to heart disease.

Sometimes, clinical signs may all be linked together to confirm a diagnosis. To an acupuncturist, for example, headaches, irritability, abdominal fullness and bloating, a love of sour foods, and an aggravation of symptoms in the spring are all linked to a liver imbalance. This diagnosis is according to the system of correspondences of Five Element Theory (to find out more about this, see Chapter 4) and makes perfect sense to a TCM practitioner, but wouldn’t make a lot of sense to an orthodox medic with no background in TCM.

Charting and calculations

Several complementary medicine disciplines use charting or paper calculations as part of their diagnosis. A nutritionist is likely to ask you to complete a food diary and uses it to analyse your nutritional status and the suitability of your diet. Tibetan and Ayurvedic physicians sometimes calculate your birth chart and use astrology to determine what sort of diseases you may be predisposed to. Naturopaths may ask you to keep sleep, food, and lifestyle records that may be analysed for patterns as part of the diagnostic process.

Testing

Laboratory or home-kit testing used by complementary practitioners can include blood, saliva, hair, urine, and stool tests. These tests may be used to determine many things including vitamin and mineral status, allergies, food intolerance, hormone balance, the presence of parasites, and more.

Manipulative therapists, such as osteopaths and chiropractors, may use x-rays and scans to check on structural balance; they may also use equipment to check blood pressure or small hammers to test muscle reflexes.

Many of these forms of testing are also used by medical doctors and are well-accepted in orthodox medicine although some of the newer tests, such as tests for food intolerances remain controversial despite growing scientific evidence to support their use.

Other methods of testing may involve specially designed devices such as the equipment used to examine the eyes in iridology or the electro-acupuncture devices used to measure the flow of vital energy (qi) In the acupuncture meridians (see Chapter 22 for details on these energy medicine devices). These are less well-accepted in orthodox medicine and have little scientific evidence to support their use.

Energetic diagnosis

Energetic diagnosis involves assessing the Aura Or electro-magnetic field believed to surround each person and may be determined using the hands or a measurement device.

In manual diagnosis the practitioner runs their hands above the surface of your body to detect subtle changes in the electro-magnetic field according to changes in sensations felt in the hands that may indicate health problems.

In other cases a diagnostic measurement device may be used. An example of this is Kirlian photography, whereby images of the energetic field around the body are alleged to be captured photographically, (see Chapter 22 for more about such energy medicine techniques).

These methods of ‘energetic’ diagnosis are controversial and aren’t accepted within conventional medicine because no real scientific evidence supports them.

Complementary versus Orthodox Medical Diagnosis

Many different forms of diagnosis are used in complementary medicine. Different therapies employ different techniques and the diagnoses may not always agree. The diagnoses can also be quite different from orthodox medical ones.

Practitioners must take great care to ensure that they don’t miss spotting any serious medical conditions.

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Taking the necessary precautions

Certain symptoms are regarded as red flags that require proper medical investigation. A well-trained complementary medicine professional should spot these and refer you on to your GP or other medical specialist. However some complementary therapists who have only undergone a very short training, such as a short course in a particular type of massage, may not know how to spot these warning signs.

If you have any of the symptoms listed below, or are in any doubt about a diagnosis you’ve been given, seek a second opinion from a medical practitioner or another well-qualified complementary medicine practitioner:

I Chest pain or discomfort, especially if accompanied by sudden shortness of breath.

I Persistent cough or coughing up of blood.

I Prolonged difficulty breathing or swallowing.

Unexplained bleeding, such as from the vagina or anus. I Persistent headaches or head pain.

Blurred vision or other persistent visual disturbance.

Unexplained lumps, swellings, or persistent pain.

I Changes in size or appearance, or presence of lumps, in breasts or testicles, plus any discharge.

I Unexplained dizziness, mental changes, or memory loss.

I Persistent stomach or abdominal pain.

I Changes in size, shape, or colour of any mole on the skin or any sudden itching or bleeding from a mole.

I Sudden skin changes, such as a rash, especially if accompanied by high fever.

I Unexplained swelling of the legs or abdomen.

I Severe allergic reactions.

Prolonged, unexplained weight loss and/or loss of appetite and/or excessive thirst.

I Difficulty urinating, such as experiencing blockage and being unable to maintain flow (men) or increased frequency and painful, burning urination (women).

I High fever or loss of consciousness.

I Prolonged, unexplained fatigue.

I Compulsive repetition of simple daily tasks, excessive mood swings, paranoia, or delusions.

All these symptoms may be indicative of serious health conditions and need to be investigated fully.

Weighing up the evidence

Although many of the complementary medicine diagnostic techniques have little, or no, scientific evidence to support their use, don’t forget that many of them have been used for hundreds or even thousands of years.

Many practitioners who’ve practised for years to refine these diagnostic skills are convinced of their effectiveness and feel it may be only a matter of time before scientific research is able to confirm their accuracy.

Mmmmmmmmmmmmmmmmmmmmmm•••••«•••••••••#••••••••mm ♦

In This Chapter

► Discovering your three-dimensional body

► Getting a feel for muscles and bones

► Exploring the organ systems

Ost people would prefer to leave the interior of the human body a mys -

# W P tery, like the ingredients in a Hostess Twinkie. You’re better off just enjoying the thing, they figure, and not asking too many questions. This attitude works fine for most applications in life, such as walking around, going to the movies, eating pizza, and so forth, but once you decide to massage somebody, you’ll benefit by knowing a little about human anatomy.

Here’s why:

You become aware of certain areas that are delicate or sensitive and should therefore beavoided (see Chapter 10 for more information on this issue).

You develop an idea of what’s going on internally when someone complains about specific aches and pains.

You discover how certain strokes on the surface are acting on deeper structures, such as the circulatory system, the lymph system, and more.

^ You come to understand how your touch is affecting the body as a whole.

The purpose of this chapter is to give you a very basic understanding of how your touch is felt, not just on the surface of the body, but into its depths as well.

Michelangelo’s inner vision

Have you ever been to Florence and visited the Galleria della Accademia where Michelangelo’s famous statue of David stands? Well, let me tell you, it’s worth it. You enter and walk down a long corridor filled with half-finished sculptures before you come to the high glass-domed chamber where David awaits. The power of the piece is not just in its mass and detail, but also in its fullness. David seems to be filled to the brim with life, as if he’s about to burst out of his skin at any moment

How did Michelangelo know exactly how each of the muscles and bones were arranged inside the human body in order to recreate such reality? Did he study anatomy at the medical school of Florence? Did he look it up in books?

The fact is that studying anatomy the time-honored way (using cadavers) was illegal during Michelangelo’s lifetime, outlawed by the Catholic Church as sacrilegious. Undaunted, the artist found a sympathetic priest who gave him a key to one of the city’s morgues where Michelangelo would break in at night to perform illicit autopsies. Even though he had to risk his own life to do it, exploring the interior of the human body proved invaluable to the creation of his art.

To give a massage that feels like a work of art, you’ll want know what’s inside the human body, too. And these days, nobody’s going to arrest you for wanting to find out

Wovtl9 That’s Deep

Perhaps the most fundamental misconception people have as they first set out to massage somebody is that the human body is a two-dimensional object rather than a three-dimensional object. How is that possible, you say? Everyone knows we’re not flat. Right?

Well, that’s true, but everyone knows that a lake is three-dimensional, too, having depth as well as width and breadth, right? What do you picture in your mind when you think of a lake, though? If you’re like most people, you think of the surface of the lake, the visible area of water surrounded on all sides by the shore.

And in a similar way, even though you know there’s depth inside you, too, containing all the unfathomable mysteries of tissue and bone, you may still habitually concern yourself with the surface, because that is what you see.

The problem with this two-dimensional way of thinking is obvious if you consider what would happen were you to attempt to walk out to the middle of the lake. Quite quickly, you’d understand about the lake’s three-dimensionality. The same applies when you wade out onto the seemingly two-dimensional surface of a person’s body as you give her a massage. The mysterious liquid depths beneath the skin suddenly surge up around your fingers, and if you don’t know how to swim, you’ll drown. Or at the very least, you’ll look silly doing the dog paddle as you head desperately back to shore.

You can give a nice, pleasant rubdown without knowing a thing about what you’re doing: The mere tactile stimulation of skin-to-skin has positive therapeutic effects; but to give a good massage, one that makes people say "wow, that was incredible," you have to learn how to swim.

Proof That \/ou’re Three Dimensional

Here’s a way to prove scientifically that you are indeed a three-dimensional being and that all kinds of secrets exist below the surface of your skin. You only need two things to do this experiment: your hand and a flat surface such as a table or desk.

First, turn your hand palm-downward and hold it over the table a few inches high. Then reach down with just your fingertips to touch the surface. And finally bend your middle finger and fold it under your hand until the first two knuckles are flat on the table. Good.

What I’d like you to do now is lift your other fingers up and away from the table top one at a time while leaving your middle finger firmly planted. Go ahead and do this right along with me as you’re reading if you’d like. First try the thumb; it lifts downright easily, doesn’t it? Way up high. Next try the index finger; not quite as impressive as the thumb but still definitely off the table. Try the pinkie finger; you see how it lifts up about the same or higher than the index finger? And lastly, try the ring finger. Go ahead. I’ll wait. What’s wrong? Come on! Lift it up already. Can’t do it?

Why can’t you lift your ring finger? You may have tried this experiment before, but did you ever figure out what’s going on? Somebody showed it to me when I was in high school, but it wasn’t until I was studying anatomy as part of massage traininglhat I understood what’s happening.

The secret is this: Buried within the depths of your forearm are three tiny little muscles, one that lifts your index finger, one that lifts your pinkie finger, and one that lifts your thumb. But you have just one muscle that lifts both the ring finger and middle finger, and so when one of them is held down, the other one can’t lift up. Go ahead, try it with the ring finger on the table instead of the middle finger. Same result, right?

This example is just to show the effects of your three-dimensional depths on your two-dimensional surface. It’s important to remember this when you’re getting ready to massage someone, and I’ll remind you to "think 3-D" when you read the chapters in Part III.

Figure 4-1:

Tiny muscles deep in your forearm determine the possible movements of your hand.

Bones

-Ligament

Learning to Feet

For a moment, imagine you have a bas-relief map of the world before you in which all the landmarks are raised from the surface. Now imagine an opaque layer of rubber a quarter-inch thick covering the whole thing. Reaching down and touching this smooth surface, can you tell where your fingers are just by feeling? Where’s California? Where’s the tip of South America? Where’s the protruding peninsula of Iberia? Can you determine what it is you’re feeling, even without seeing it?

Now, making a leap in your imagination, think of the human body as that covered-up map that you are trying to identify by feeling its contours. This type of feeling-with-a-purpose is called Palpation. Many professional massage people use palpation to determine what type of massage they are going to give to an individual, based on the way the person’s body feels compared to the norm. You can get very sensitive fingers by practicing this, and in the next section, I lead you through an exercise to help you start that sensitization process.

Getting a feeling far palpation

Try this exercise to begin sensitizing your fingers to the various textures, shapes, and landmarks you will find beneath the skin.

1. Sitting in a chair, with your back straight, turn your head to the right, as if you were trying to look back over your right shoulder.

2. Now, reach up with your right hand and, using just the fingertips, feel gently along the front of the neck until you locate the long band of vertical muscle stretching from your collar bone up to the side of your head, called the sternocleidomastoid muscle, which is illustrated in Figure 4-2.

3. "Walk" your fingertips up and down this muscle, feeling for where it connects near the center of your collar bone (the origin) and up along the base of your skull (the insertion).

Do certain parts feel tighter than others? Is part of the muscle thinner than others?

4. Grasp the muscle between your fingertips, as if it were a guitar string and you were going to pluck it.

Be careful not to dig your fingers into the sensitive front part of your neck.

5. Still grasping the muscle, slowly bring your head back to center, feeling the softening in the muscle between your fingers as you do so.

Repeat the process several times, back and forth.

6. Walk your fingers down to the base of this muscle and then onto the collar bone, following it along out toward the shoulder.

How does the bone feel different than the muscle? In what ways is it the same?

7. Now walk your fingers away from the collar bone up over toward your back until you reach the top of your shoulder.

Use a little firmer pressure to feel along the length of this muscle. Where does it feel harder? Where does it feel softer? Are there any "knots" or "bands" of harder tissue within the more pliable surrounding area?

Notice whether there are any points that feel more tender when you touch them, and whether these tender points correspond to the "knots".

Take several minutes to do this. Get a feeling for feeling. Let your fingers become familiar with all the permutations of texture, density, and tone that you can find just below the surface of the skin.

Bony landmarks

If you attend massage school yourself one day, youTl learn all kinds of intimidating anatomical terms with which to impress your friends and loved ones, such as "Boy, Cheryl, that’s one exquisite medial malleolus you have there."

Cheryl probably won’t know that you’re talking about her inner ankle bone. And there’s a very good chance she won’t care either. Therefore, I’m not going to bore you or her by loading you down with all kinds of Latinate words and phrases. Instead, I’m going to do something fun, in plain English, that’s going to help give you an idea where things are located anatomically.

Your medial malleolus, or inner ankle bone, is one of at least forty seven "bony landmarks" throughout your body. Now, before you go making any crude comments about bony landmarks, let me assure you that this is indeed what they are called by professionals everywhere. They have compelling names, such as Xiphoid process, occipital protuberance, And Greater tuberosity of the humerus. I’m going to use laymen’s terms, though, and expose you to a few of these landmarks as part of a game. That’s right, it’s time to play____

The bony landmark game

It can really be a lot of fun getting to know what’s where beneath your skin, and, in fact, for many centuries (before the invention of TV) people the world over would sit around the campfire playing the bony landmark game. This was a great way to pass the time between wolverine attacks, and it’s an effective method to teach anatomy to the young at the same time.

The game is simple: I describe a particular landmark (see Figure 4-3) for you in terms that you can understand and give you directions on how to locate it through Palpation. Then, all you have to do is supply me with the common, everyday term we use to describe this landmark. It’s important that you actually do the palpation, not just read the words, because that is what will familiarize you with the terrain you massage in future chapters.

Ready?

1. You can find this landmark by holding one hand out in front of you, with your palm facing you. Feeling with the fingertips of your other hand, notice that you have two bones in the forearm, one on the pinkie finger side (the ulna), and one on the thumb side (the radius). Follow the bone on the pinkie finger side all the way from your wrist to its extreme other end. You’ll find a bump there, called the Olecranon process, Otherwise known as the.

2. Cross one of your feet up and rest it on the other knee so you can examine it. Then feel with both hands along the shin bone (tibia) in the front of your lower leg. Follow it down all the way to your foot and see what happens to it. Feel how it curves back toward you and ends up in a bump at the top of your foot? This is the Medial malleolus, Or_.

3. Trace your fingers down onto your foot and then back in the opposite direction from your toes onto the Calcaneus, Or__bone.

4. Now sit up straight. Reach down along one side of your body until your hand almost slips underneath you. Right at that point you should feel a bone called the Greater trochanter of the femur, Which is otherwise known as the bony knob at the top of the longest bone in the body, the _bone.

5. Walk your fingers back up along the side of your body about 6-8 inches until you hit the next bony landmark. It should be a sharp ridge that sticks out and that you can follow along toward the front of your body for a few inches. This is the Iliac crest, Also known as the_bone.

6. Reaching your hands up to your face, locate your chin and then feel back along the lower ridge toward your ear. It curves up here, forming the Ramus of the mandible, Otherwise known as the point at the angle of the_bone.

7. You’ll need a partner for this one. Have her lie face-down on a comfortable surface with her back exposed, and then gently lift her arm, bend it at the elbow, and place her hand on her lower back. Let her upper arm rest down along her side. You’ll notice that by doing this you cause a big bump to appear on her upper back. Feel along the edges of this triangular-shaped bone, otherwise known as the_.

Answers: 1. elbow, 2. inner ankle bone, 3. heel, 4. thigh, 5. hip, 6. jaw, 7. shoulder blade.

These are just a few of the many landmarks you can palpate, and this game is meant to get you comfortable with the fact that you can actually feel and affect the structures of anatomy without being a scientist or doctor. When you practice hands-on massage, remember this and use your knowledge to guide you through your partner’s body.

Soft tissues

Now that you know how to familiarize yourself with bony landmarks, you’re probably wondering about all the other parts of your body that are Not Bony landmarks. After all, you’re not going to be massaging bones. It’s the Soft tissue That you’ll have in your hands most of the time, and by soft tissue I don’t mean Kleenex brand facial tissues. I mean muscles, mostly, and a little bit of connective tissue as well.

Muscles comprise 40 to 60 percent of your total body weight, depending upon your gender and physical condition, and you have over 600 of them, large and small. Each one is compartmentalized in a sheath of Fascia, Which sets it apart and helps it function as a distinct unit, although the truth is that you never use just one muscle to perform any given action. As Mark Beck says in Theory and Practice of Therapeutic Massage, "Muscles have anatomic individuality, but they do not have functional individuality." They are always working in groups to create movement. That’s their whole purpose for being.

The larger obvious muscles you can see simply through observing a body in motion are called Skeletal muscles. There are also two other types of muscle tissue: Cardiac And Smooth. Cardiac, as the name implies, is the special muscle tissue that makes up the heart. Smooth muscle lines the stomach, intestines, and blood vessels.

The slightest movement of the most mundane part of your anatomy (your left knee, for example) requires the precisely timed and perfectly executed synchronization of many muscles, and there’s absolutely no way that you could consciously coordinate all that without going nuts. Imagine Michael Jordan driving in for a lay-up and having to fire off messages to every single one of his separate muscles to do so. It would look something like this:

"Okay, contract the quadriceps, especially the rectus femoris, and simultaneously pull in the psoas, push off the soleus, shorten the gastrocnemius, and extend the web of flexors and the tibialis anterior. Now compensate for the lifted foot by tightening the opposite gluteus maximus and bracing all the muscles in the lower back, too numerable to mention here. Whoops, that threw me off, and… whoa!"

And down he’d go, before even moving one step. In fact, it’s much more complex than that for even the simplest of maneuvers, and we’d all be helpless to try and stand up, sit down, or walk to the refrigerator if we had to think about it.

So how do we do it? Basically, we learn to move one little piece at a time as we develop during infancy and childhood, laying each chunk of the pattern down in a movement-memory groove, and then building upon it with the next movement. That’s why you see babies experimenting with things like kicking their legs out, bobbing their heads around, and bringing small electrical appliances toward their mouths for examination. Every time they do something successfully and then master it through repetition, they file it away, and that’s one less thing they have to consciously think about next time. Of course, this is the same procedure that athletes use later in life through their practice as they gradually layer all the perfect little movements they need one upon the next until they no longer have to think about it but rather, "Just do it."

By all of this explanation, I mean to say that muscles don’t just flex and contract — they Learn. What you’re holding in your hands when you massage someone is conscious matter. In fact, it’s your muscles that tell you where you are in space and time, through special nerve endings embedded in your muscles known as Proprioceptors. I don’t want to freak you out with bizarre-sounding anatomical terms, but there are two of these proprioceptors that are particularly interesting and important, and I want to share them with you.

Golgi tendon organs Are nerve endings found, strangely enough, in your tendons. They measure how far any particular tendon has stretched, how much pressure it’s putting on the nearby bone, and if the tendon’s in danger of snapping. It’s through these little organs that you are saved from ripping yourself to shreds and pulling all your muscles and tendons right off the bone.

^ Muscle spindle cells Are found in the center of muscles, what’s known as the "belly," where they perform the important task of constantly communicating the state of the muscle’s contraction and movement back to the central nervous system. They are basically scouts on the outpost of your active physical self. Without them, you wouldn’t be able to tell where you were going, how fast, or if you were going at all.

Cartilage, ligaments, and tendons

Many people find themselves confusedly referring to various connective tissue structures between the muscles and bones as "tendons" or "ligaments" or "cartilage" without really knowing what the heck they’re talking about. Now, I know you’re not one of those people, but just in case you have a friend who’s guilty of such anatomical faux pas, here’s the skinny to set you straight:

V Cartilage gives shape to external features like the nose and ears, and it is also found between bones as a cushion at the joints. (Vertebral discs are made from cartilage, for example.)

Ligaments connect bone to bones. ^ Tendons connect muscle to bone.

Muscle tissue itself is largely insensate, meaning if someone were to cut, jab, or even burn you directly on an exposed muscle, you quite likely wouldn’t feel much at all. Your muscles don’t so much Feel Massage as they Experience Massage as it retrains them how to Be More relaxed in stillness, and fluid in movement.

Muscles learn, and massage teaches.

Name that muscle

What often intimidates people when they’re first learning massage is the anonymity and invisibility of muscles. Skin is plainly visible: You can clearly watch your own hands make contact with somebody else’s body, skin to skin. But muscles? How can you really tell what muscle you’re touching when that muscle is covered by the skin? And besides, the muscles are all jumbled together and not that clearly defined, except in professional body builders, whose muscles are so hard and tight that they’re almost impossible to massage anyway. So how are you expected to really know what the heck you’re touching when you’re touching a body?

Glad you asked. This brings us directly to our next little exercise: Name That Muscle. This game is a bit tougher because you need to know the names of the muscles, which you might have forgotten if you weren’t paying close attention in high school anatomy class. So, to be fair, I’m going to give you the answers first. Can’t complain about that, right? All you have to do is match the right muscle in Figure 4-4 to its description and action. If you’re not sure about one, see whether you can use common sense and deductive reasoning to figure out which the best choice might be. It will help if you actually perform the action described in the questions so that you FEEL the muscles you’re looking for. After all, feeling is what this book is all about.

First, here are the answers:

Pectorals V Deltoid

Spinalis & Rectus abdominis \^ Gastrocnemius & Hamstrings I> Triceps W Gluteus maximus ^ Trapezius ^ Biceps

1. Standing with your back against a wall, push against the wall with one heel and reach around to the back of your thigh to feel a tightening in your_.

2. Place the back of your hand flat on the edge of a desk or table in front of you and then push that hand firmly down onto the desk. With the other hand, feel the underside of your upper arm and discover a tightening in your_.

3. Standing up tall, lift your right leg out behind you as far as it will go comfortably, keeping it straight. Place your hand on your right buttock to feel a tightening in the_.

4. Reach across with your left hand and place it on top of your right shoulder, right next to your neck. Now shrug your right shoulder as far up toward your ear as possible to feel a contraction of your_.

5. Standing up, push onto your tip toes to feel a contraction of the _in the rear of your lower legs.

6. Sitting in front of a desk or table, place one hand palm-up beneath the

Table and lift up, creating a contraction in the_muscle of your

Upper arm.

7. When you lie down to perform a sit-up, the muscles in the front of your body that you’re trying to tone through contraction are the_.

8. Stand in front of a wall and push forward against it with your right hand while touching the right upper portion of your chest with the left hand. The muscles you feel bulging beneath your fingers are the_.

9. Sitting up straight, reach across with your right hand and place it on the left shoulder, out by the arm. Now lift the left arm straight out to the side

Until it’s at a ninety degree angle from your body, engaging the_

Muscle beneath your hand.

10. You’ll need a partner for this one. Have him lie on his stomach, with no shirt on, and then ask him to lift his head and shoulders off the floor with no help from his arms. The two long cords of muscle down along either side of his spine are part of the_group.

Answers: 1. hamstrings, 2. triceps, 3. gluteus maximus, 4. trapezius, 5. gastrocnemius, 6. biceps, 7. rectus abdominis, 8. pectorals, 9. deltoid, 10. spinalis.

These three extra credit muscles are more obscure and I don’t blame you if you don’t know them offhand, but it may be fun to see whether you can decipher which is which.

Iu* Rhomboid ^ Latissimus dorsi Psoas

1. If you stand up and lift one leg in front of you with the knee bent, you engage a deep muscle that connects your leg bone to your backbone called the_.

2. You’ll need to observe a partner for this one. Have her sit facing away from you with her back exposed and then gently reach one of her hands up as far as possible along her spine. Her shoulder blade will lift and you’ll be able to feel between it and the spine for the_muscle.

3. Lift your right hand over your head and reach across with the left hand to grasp your right side below the armpit. The large muscle you feel there is the_.

Extra credit answers: 1. psoas, 2. rhomboid, 3. latissimus dorsi.

This is not a test. Repeat, this is not a test. It’s just a way for you to become familiar with locating muscles. I refer you to several of these landmarks, bony and otherwise, when you move through the how-to massage chapters in Part III.

Other Body Systems

Don’t get the idea that it’s just the skin, muscles, and bones that count when it comes time to massage somebody. You also deal with a few other anatomical systems that are strongly affected by your touch as well. These include the…

W Circulatory system

Nervous system W Endocrine system

Digestive system ^ Respiratory system

The next few sections take a brief look at these systems and discuss how they’re important when you give or receive a massage.

Circulatory system

The heart is constantly pumping your blood (about 11 pints of it in a 160-pound adult) out through your arteries and into each and every tiny little cell of your body, carrying the nutrients and oxygen that make it possible for you to stay alive. Then the blood travels back to your heart through the veins. On this return trip, the blood has to pass through a series of one-way valves that keep it from accidentally heading back in the wrong direction.

Massage strokes have a direct effect on the flow of blood in the veins, so keep in mind that when you massage someone, your strokes should always be in the direction of venous flow. You wouldn’t want to accidentally push the blood back through these valves and therefore weaken them. When a number of the valves weaken and stop working efficiently, blood can pool up visibly and form varicose veins.

As much as half of all your blood is in your skin at any given moment, which accounts for that rosy glow certain people have, and also for the less healthy appearance of varicose veins and other problems. Massage works powerfully on your circulatory system, and for this reason you should always be aware of how your hands are affecting it.

Massage also affects that other circulating fluid referred to in Chapter 1, the lymph. In fact, there is an entire system of massage called manual lymphatic drainage meant to assist the movement of the lymph because, as you may know from Chapter 1, lymph has no heart of its own to pump it along.

NerVous system

J^ALty/ As a busy person in the twenty-first century, you don’t have any time to /^i^v fiddle around reminding your heart to beat, your lungs to breathe, and so on. pfe0 I Luckily, your Autonomic nervous system Takes care of all that for you. This ^Jf^jL^ System is further broken down into the Sympathetic And Parasympathetic ner-vous systems. The sympathetic nerves prepare your body for action, and the parasympathetic nerves calm you down. Massage is a great way to stimulate the parasympathetic nervous system, thereby lowering the pulse, slowing breathing, and in general, chilling you out.

The largest and longest nerve in the body is the Sciatic, Which many people are painfully familiar with. It runs from the base of your spine down the back of your leg, and when any of its length becomes pinched or trapped between muscles, bones, and connective tissues, it can cause the condition known as Sciatica. That’s the way all nerves work; you don’t want to get in their way or piss them off. Massage can help soften the muscles and other soft tissue that surround nerves and sometimes entrap them.

As I mention earlier in this chapter, you also have specialized nerves called proprioceptors that tell you where your body is in space, giving you your sense of depth, position, and movement. Without them, you’d be internally blind, and by making you more aware of them, massage can help you "see" yourself in a new way from the inside out. Chapter 7 has some exercises to get you in touch with your proprioceptors. Look for "The limp arm experiment."

The mind-body connection

Did you ever wonder what the heck people were talking about when they used the term mind-body connection? Is it part of the nervous system you weren’t told about in school? Did you think maybe there was a tube or special cable of some kind near the base of your neck that linked your mind and your body, and that you were the only one who hadn’t been shown where it was? Well, don’t worry; you’re not alone. In a far-reaching survey conducted by my wife one day at her restaurant, it was ascertained that only 2.4 percent of normal people understand what the term mind-body connection really means, and those people are new-age geeks.

Typical incorrect responses about what the mind-body connection is included the following:

\* That sinking feeling you get when your mind realizes your body did something it shouldn’t have

^ Nerves

The neck

Actually, the mind-body connection is simply Awareness. It’s an awareness that permeates way down into every cell of your body, as compared to the awareness of your brain alone. It’s the entire You Consciously affecting every other part of you.

This whole mind/body split problem developed gradually over many centuries and was not really caused by any one individual, but many scholars have pointed to the French philosopher and mathematician Rene Descartes as having had the greatest influence. He’s the one who coined that famous Latin phrase, "Cogito, ergo sum" Which means "I think, therefore I am." That was in 1637. Well, pretty much ever since then people have been assuming that it was only specific types of electrical activity inside their skulls that proved they indeed existed. What’s glossed over in the history books is that Descartes never received a great massage from an expert holistically oriented practitioner. If he had, he certainly would have modified his statement a bit, to something like, "Cogito etsentio, ergo sum" I think And I feel, Therefore I am."

Endocrine system

Heard of hormones? The glands that make up the endocrine system are what secrete hormones into your body. It’s been shown that massage stimulates the release of human growth hormone (HGH), among others, thereby affecting the healthy maturation of your entire body.

Digestive system

Your digestive system is a tube approximately five times as long as you are tall (see Figure 4-5). This tube, along with several digestive organs, has the magical ability to transform whatever enters it into a very special substance known as "you." Massage can beneficially stimulate this process if you’re familiar with the various twists and turns this tube follows through your body, especially over the large intestine.

Cleaning the pipes

If you visit a health food store and search through the herbal potions that line the shelves, you’ll find some strange-looking mixtures that promise "internal cleansing." The ingredients in these products have two major actions: absorption and expulsion, and they act primarily in your large intestine, also called the colon. First, certain ingredients (often psyllium husks) absorb much of the matter that tends to get lodged in the many folds of the colon, and then a mixture

Of herbs comes along to "sweep" it all out. Ifs similar to the technique favored by many garage mechanics of throwing sawdust on dirty grease before pushing it away with a broom.

This type of cleansing is highly advisable, and some extra massage at the same time may aid the elimination process by stimulating lymph

One landmark along the digestive pathway that many people are able to palpate is the Cecum, Which is a little pouch at the beginning of the large intestine, or Colon. You can locate it by first touching bony landmark number 5, your right hip bone, and then walking your fingers in toward your belly button a couple of inches. Sometimes this spot makes a liquid-squishing noise, especially after meals. In Part III, I discuss how to use this landmark as a starting point for some abdominal massage techniques.

Respiratory system

Breathing is an extremely important activity for human beings, as can be attested to by the millions of people around the world who have stopped breathing and suffered serious side-effects, even death. Massage is an excellent opportunity to engage in some full deep breathing, as described in Chapter 7. This reconnects you with the source of life. It also fills your blood with fresh oxygen because the first place your blood goes when it leaves your heart is the lungs.

Surprising facts about your stomach

Most of us think of the stomach as a large roundish ball in the center of our abdomens, but actually it’s a smallish oval sack up and to the left, tucked mostly under the ribs on the side of your body by your heart. Like the entire intesti-

Nal wall, it’s lined with smooth muscle. What people are really referring to when they point at the center of their abdomen and say "Look how flat my stomach is" is actually the intestines.

In This Chapter

^ Going step-by-step through a problem using suggested techniques

^ Using both sketch and table to illustrate the problem

^ Solving a problem using Pythagoras’s theorem and a quadratic equation

Solving a math word problem may seem daunting at first, but it doesn’t have to be if you have a plan and the proper mindset. You don’t have to use all the steps and procedures in this chapter for every math problem, but here you see how different techniques are useful when attempting to solve a problem.

In this chapter, I tell you what to look for in terms of the question and information needed. I fill you in on all the steps used to solve an equation — and the proper order to do them in. And I show you the need for checking your answer at the end.

Laying Out the Steps to a Solution

A math word problem presents challenges in understanding, organization, and launching the mathematical problem to be solved. To illustrate all these steps (and more), consider a problem involving two friends and their walking adventure. They both leave the same place at the same time; one walks north and the other walks east. One walks faster than the other. And, for some reason known only to them, they can determine how far apart they are after a period of time.

The Problem: Shelly and Shirley leave their dorm at 8 a. m. and start walking in different directions. Shelly starts walking due north, and Shirley walks due east. Shirley takes her time to smell the flowers and Shelly walks at a pretty steady pace. So, at noon, when they stop walking, Shelly has gone 1 mile less than twice as far as Shirley. At noon, they are 17 miles apart. How far did Shelly walk?

How old was the wife?

At the time a man and woman married, the man wife’s age was now % of his age. How old was determined that his wife’s age was % of his age. his wife when they were married? After 12 years of marriage, he found that his

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.VLAiy Words, words, words. Somewhere in the description of the problem, you find the question that needs to be answered, information that needs to be organized, other information that can be ignored, an appropriate drawing that can be made, and an algebra problem begging to be written. In the following sections, I walk you through the various steps — steps you can apply to any math word problem.

Step 1: Determine the question

The main thing that empowers you when you’re attempting a math word problem is determining the question. You wade through all that information and wonder what in the world you’re going to do with everything. So, find the question, and you’ll have more direction. The question is usually at the end of the problem description.

In this problem, the question is, indeed, at the end. How far did Shelly walk? The only unit mentioned is Miles — because they’re 17 miles apart — so your answer should be something like: "Shelly walked_miles."

Step 2: Organize the information

After you’ve determined what the question is (see the preceding section), you can go back to the problem and decide what information is needed and in what order you want to do the various steps. If you have an answer guess or goal in mind, it’ll help you determine how the steps get arranged.

Eliminating the unneeded

The problem mentions the distance that Shelly and Shirley walked and the distance they’re apart. Also, in the problem description you see times listed. You don’t need the time they started and finished, because there’s no mention of rates — requiring you to use D = Rt. So just eliminate or ignore the mention of the times. It’s just fluff. It also doesn’t matter why Shirley walked slower; it could have been that her legs are shorter.

Shelly and Shirley leave their dorm at 8 a. m. and start walking in different directions. Shelly starts walking due north, and Shirley walks due east. Shirley takes her time to smell the flowers, and Shelly walks at a pretty steady pace. So, at noon, when they stop walking, Shelly has gone 1 mile less than twice as far as Shirley. At noon, they are 17 miles apart. How far did Shelly walk?

Making an educated guess

The problem involves two people walking in different directions and ending up 17 miles apart. If they walked in opposite directions — one north and one south — you’d expect the sum of the distances they walked to be 17. Neither could have walked more than 17 miles. In this case, they walked at right angles from one another, so the sum of the distances that they walked has to be less than 17. Keeping in mind a Reasonable Answer, you’re more likely to set up the process (equations and operations) properly. So some possible answers are that Shelly walked 10 miles and Shirley walked 5 miles. The numbers don’t really need to fit or be the right answer. You just have a relationship in mind — that Shelly walked farther and the sum of their distances is less than 17.

Getting organized

The question asks for the distance that Shelly walked. So you want some algebraic expression involving that distance. You could let the distance be represented by X. But look back at the problem to see what other distances are involved. You have the number 17 representing the distance between them, so you don’t need a symbol for that distance. The only other distance is the distance that Shirley walked. You could let the distance that Shirley walked be represented by Y, But then you’d have two different variables — X And Y — to worry about. You want to keep to one variable, if possible.

The last bit of information that hasn’t been used is that Shelly has gone 1 mile less than twice as far as Shirley. The two distances are compared to one another. Letting the distance that Shirley traveled be represented by Y, You can let Shelly’s distance be written in terms of Y. Shelly went 2y – 1 miles: twice Shirley’s less 1. Now you have Y For Shirley’s distance and 2y – 1 for Shelly’s distance.

Step 3: Draw a picture or make a chart

This problem just begs to have a picture drawn. But a chart or table may be helpful, too. In the following sections, I show you both options and let you decide which works best.

Providing some artwork

Shelly walks north and Shirley walks east. You have to assume that it’s possible for them to stay exactly on track and walk due north and due east, respectively, and not have to veer off. Math problems are usually about optimal situations, not the reality of how roads are laid out.

Figure 4-1 shows you a sketch of the paths taken by Shelly and Shirley. The representations of their distances — Y For Shirley and 2Y - 1 for Shelley — are shown, as is their distance apart.

Shelley’s path 2y – 1

Figure 4-1:

The paths are at right angles.

\ 17

Y

Shirley’s path

The picture of their paths seems to suggest a right triangle. And a right triangle comes with that most famous formula, the Pythagorean theorem. If you hadn’t thought of that theorem before you saw the picture, you probably did after seeing it.

Figuring out possible values with a table

This problem has Shelly traveling one less than twice as far as Shirley. You could make a table of possible values for the distances they traveled. You’d probably be interested only in whole-number values, which won’t solve the problem if the answer is a fraction, but you may get fairly close to the answer. Table 4-1 has some possible numbers or distances, starting with Shirley going 1 mile and ending to keep the total distance from getting larger than 17.

Counting cars

On a warm, sunny day, Clark went to the car races to watch an automobile race. As the cars sped around the track and Clark tried to watch them, he got dizzy. So he decided to keep his eyes on one particular car — the bright green one.

Clark then decided to count how many cars were in the race. He noticed that the total number of cars was equal to one-third of the cars in front of the green car plus three-quarters of the cars behind the green car plus the green car. How many cars were in the race?

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This table doesn’t really get you any closer to the answer, but it shows you how the distances are related. You know that the total distance has to be less than 17, so several of the entries in the table give possible solutions to the problem.

Step 4: Align the units

Many problems come with two or more different units. You may have information on time in terms of minutes and seconds, information on distance in terms of feet and yards, or some combination of all these. You can’t really

Change minutes and feet into the same unit, but feet and yards can be changed to feet, and minutes and seconds can be changed into minutes (or seconds). (Turn to Chapter 3 for a full coverage of how to deal with units.)

In this problem, because the only unit mentioned (after the time was eliminated) is miles, you just leave the unit as is. You’re finished in terms of the units.

Step 5: Set Up the operations or tasks

This is where all the previous steps should be coming together. You have the question in mind and a picture of what’s happening. You have an estimate or guess of how the answer should come out. You have variable expressions representing the distances traveled. Now you need to set up a process or equation. Sometimes the process is no more than multiplying a number by 2 or 3. That’s the best-case scenario. But it’s much more fun when you set up an equation to be solved.

The previous work on this problem suggests an equation. You have two different distances represented by expressions involving a Y. Write an equation and solve for Y. What equation? Why Pythagoras’s, of course!

In a right triangle whose shorter sides are A And B In length and whose longest side is C Long, the following is always true: A2 + B2 = C2.

The distances that Shirley and Shelly walked are the A And B Of a right triangle. The distance that they’re apart, 17 miles, is the C Value. Substituting into the Pythagorean theorem, the equation A2 + B2 = C2 Becomes (y)2 + (2y – 1)2 =172.

Finding a concrete solution

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■03 = 6 + 11 41!* Win

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Solving the Problem

A math word problem is different from other arithmetic and algebra problems, because you first have to translate from the words to the symbols before you do the operations or solve the equation for the answer.

Step 6: Perform the operations or solving the equation

The problem involving Shelly and Shirley boils down to an equation to solve. The equation is quadratic — requiring either factoring (coupled with the multiplication property of zero) or the quadratic formula. Some operations are performed on the equation first — squaring the binomial and simplifying terms, and the result of the simplifying is a quadratic equation set equal to 0. The equation (y)2 + (2y – 1)2 =172 can actually be solved by factoring, but I’ll show it both that way and with the quadratic formula, because the factoring may not be entirely obvious to you.

Solving the equation using factoring

The equation (y)2 + (2y – 1)2 =172 is quadratic. You first square each of the terms, including the binomial, and then simplify the terms by combining what you can. Then move all the terms to the left to set it equal to 0.

(Y )2 + (2y – 1)2 = 172 Y2 – 4y2 – 4y + 1 = 289 5y2 – 4y – 288 = 0

The factorization of this quadratic is the product of two binomials. The first terms in the binomials have to be 5y and Y. There’s no other choice. It’s the second numbers that will be the challenge. You have to find two numbers whose product is 288 — that’s challenge enough. But then you have to figure out how to arrange the factors so that the difference between the outer and inner products is 4Y.

5y2 – 4y – 288 = 0 (5y )((y ) = 0

The two factors that work are 36 and 8. Their product is 288, and, when you multiply the 8 by 5 and the 36 by 1 you get 40 and 36, respectively. The difference between the two products is 4. The product 40Y Has to be negative for

The difference 4y to come out negative. So the factor 8 is negative and the factor 36 is positive in the factorization.

(5y + 36)(y – = 0

Setting the two factors equal to 0, you get Y =— Or Y = 8. The negative

Fraction doesn’t make any sense if this is supposed to represent distance, so you go with the solution Y = 8, only.

Solving the equation using the quadratic formula

Not all quadratic equations can be solved by factoring. And sometimes those that can be solved by factoring are more easily solved using the quadratic formula. On the other hand, All Quadratic equations can be solved using the formula. It’s just that factoring is usually quicker, easier, and more accurate (not as many opportunities for error).

The quadratic formula says that if a quadratic equation is written in the form

Ax2 + Bx + C = 0, then its solutions are found with X -

B ±Jb2 — 4ac

2a

Solving the equation 5y2 – 4y – 288 = 0 with the quadratic formula, you get

-(— 4) ±7(— 4)2 — 4(5)(—288)

Y =-oTF\-

2(5)

4 ± /16 + 5,760

= -

10

4 ± 5,776

= -

10

= 4 ± 76 = 80 or —72 10 10 10

This gives you the same two answers that you get with the factoring method: 8 and — -55-.

Step 7: Answer the question

After working hard at solving the quadratic equation, it’s tempting to just sit back, relax, and think that your work is done. Not so. The solution of the equation that you think will work is that Y = 8. What is Y? Does its value answer the question?

First, Y Represents the distance that Shirley walked. The question asks how far Shelly walked, so the number 8 is not the answer to the question. To determine how far Shelly walked, use 2y – 1 and replace the Y With 8. You get that 2(8) – 1 = 16 – 1 = 15. So Shelly walked 15 miles.

Step S: Check for accuracy and common sense

Accuracy and common sense go hand in hand. You can’t have one without the other. The accuracy part goes to the arithmetic or algebra involved in solving the problem and whether the relationships hold. The common sense refers to whether the answer — even though it’s the solution of an equation — really fits the problem and the real world.

In the case of Shirley and Shelly who walked away from each other in directions that are right angles from one another, you find that Shirley walked 8 miles and Shelly walked 15 miles. They’re only 17 miles apart — even though they walked a total of 23 miles — because their journeys and the distance between them formed a right triangle. They didn’t walk in opposite directions. As far as the accuracy of the solution, check out the Pythagorean theorem with the distances:

82 + 152 = 172 64 + 225 = 289

289 = 289

The equation checks, the answer makes sense, and the problem is solved with the question answered.

Part II