a r X i v :a s t r o -p h /0505330v 1 16 M a y 2005
How many dark energy parameters?
Eric V.Linder 1and Dragan Huterer 2
1
Physics Division,Lawrence Berkeley Laboratory,Berkeley,CA 947202
Kavli Institute for Cosmological Physics,and Astronomy and Astrophysics Department,University of Chicago,Chicago,IL 60637
(Dated:February 2,2008)
For exploring the physics behind the accelerating universe a crucial question is how much we can learn about the dynamics through next generation cosmological experiments.For example,in de?ning the dark energy behavior through an e?ective equation of state,how many parameters can we realistically expect to tightly constrain?Through both general and speci?c examples (including new parametrizations and principal component analysis)we argue that the answer is 42–no,wait,two.Cosmological parameter analyses involving a measure of the equation of state value at some epoch (e.g.w 0)and a measure of the change in equation of state (e.g.w ′)are therefore realistic in projecting dark energy parameter constraints.More elaborate parametrizations could have some uses (e.g.testing for bias or comparison with model features),but do not lead to accurately measured dark energy parameters.
I.
INTRODUCTION
The discovery of the acceleration of the cosmic expan-sion has thrown physics and astronomy research into a ferment of activity,from a search for fundamental theo-ries to investigation of parametrizations relating models and the cosmological dynamics to development of astro-physical surveys yielding improved measurements.The question of the nature of the accelerating mechanism (“dark energy”)impacts the composition of some 70%of the energy density of the universe,the evolution of large scale structure,and the fate of the universe.Learn-ing the physics responsible will advance the fundamental frontiers of science,in high energy physics,extra dimen-sions,gravity beyond Einstein relativity,or possibly the uni?cation of gravity and quantum physics.
Given the open landscape of proposed theories,it is di?cult to assess the prospects for a de?nitive measure-ment of the physics.We consider here how much in-formation the next generation of cosmological measure-ments is likely to realistically provide on the nature of dark energy.By “realistically”,we mean several things.Knocking down theories one by one is unlikely to be useful,given theorists’fecundity and ?ckleness.Direct reconstruction,whether of the dark energy equation of state,density,or potential,while formally possible (in a limited range at least),is stymied by statistical and systematic errors in the observations.So the question becomes,while describing dark energy in as model inde-pendent fashion as possible,what is the greatest degree of informative parametrization justi?ed by the future ob-servations.
In §II we examine various parametrizations,including designing some new ones useful for studying aspects of the dark energy dynamics,utilizing one to four parame-ters.In §III we de?ne success criteria for a parameter es-timation and project what constraints future supernova distance,CMB,and weak lensing shear measurements can impose,analyzing the resulting maximum number of parameters tightly ?t.Discussion of trade o?s between
success and model independence,such as in the mini-mum variance approach,is another issue addressed.We consider in §IV uncorrelated and non-parametric char-acterizations.§V examines what the data requirements would be to obtain more information on the nature of dark energy and summarizes the conclusions on the vi-able number of dark energy parameters.
II.MODELING DARK ENERGY
Dark energy appears in the Friedmann equations of cosmological dynamics through its e?ective energy den-sity and pressure.We emphasize that these need only be e?ective quantities and not necessarily correspond to characteristics of a physical scalar ?eld,say.The ratio of the energy density to pressure,known as the equation of state ratio w ,has importance within the equations re-gardless of its physical origin.This has been emphasized by [1](see also [2]),in that any deviation of the Hubble parameter H =˙a/a from the matter dominated behavior can be written in terms of an e?ective equation of state.So a function w (a ),where a is the scale factor of the universe,is key to understanding the dynamics of the cos-mological expansion,in particular its acceleration.With only imperfect knowledge of the expansion history a (t ),through noisy distance measures over a ?nite redshift range,one cannot derive the full function w (a ).The minimum characteristics one would seek to provide in-sight on the origin of the acceleration and nature of the dark energy would be a measure of w at some epoch,e.g.today,and a measure of its dynamics,i.e.its change over time (see [3]for a demonstration of the strength of this approach;see our §V for a discussion of the role of sound speed).In particular these could yield consistency or con?ict with the two key properties of the cosmologi-cal constant as the explanation:w unchanging and equal to ?1.
Within a realistic scenario of next generation observa-tions,how much can we expect to learn about the dark
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energy function w (a )?We consider three approaches to answering this question.After reviewing the basis for the current
two
parameter model,in §II A we attempt to generalize it with a phenomenological approach,calling out a set of four characteristics of an evolving equation of state,and then examining restrictions of the phase space.In §II B we instead extend the two parameter model with physically motivated third https://www.doczj.com/doc/c37789674.html,ter,for a more dark energy model independent tack,in §IV we investi-gate a principal component approach.
A.
Describing Dark Energy Evolution
The two basic characteristics of a dynamical function mentioned above –a value and a change in value –can be implemented in many ways.The one in perhaps most common use presently is employing the present value w 0and the logarithmic derivative with respect to scale fac-tor,w ′≡dw/d ln a ,at redshift z =1(a =1/2):
w (a )=w 0?2w ′
(1?a )
=w 0+w a (1?a ).
[Model 2.0](1)
This possesses many useful properties,including a close
approximation to a wide variety of dark energy dynamics,boundedness at early times (so CMB observations can be readily treated),analyticity of the Hubble parameter expression (involving an integral over w ),and ease of physical interpretation [4,5].
To go beyond this description of dark energy dynam-ics,one might characterize the physics in a broadly phe-nomenological manner as having some equation of state value far in the past,w p ,(deep in the matter dominated epoch,near the time of CMB last scattering,say),some value far in the future,w f ,(deep in the dark energy dominated epoch),and treat the transition between the two as occurring at some scale factor a t and with some rapidity τ.
We choose to treat the transition in terms of the e-fold variable N =ln a ,since this is the characteristic scale of the cosmological background evolution.That is,any dynamics driven purely by the expansion will have a transition scale length of order one in this variable.If the equation of state changes according to ˙w ～H ,then w ′
=dw/dN ～1.
In the neighborhood of the transition we can write
w (N )≈w t 1+(N ?N t )1
dN t
.(2)
There are many possible ways to extend this to a full function describing the past and future evolution of the equation of state.We would like this to be bounded in both directions (i.e.the past N ?N t and the future N ?N t ).We choose to view Eq.(2)as the ?rst order term in an exponential,and applying our boundedness criteria we adopt
w (N )=w f +
?w
1?e 1/?
1?e (1?a )/?
1+(a/a t )1/τ
,[Model 4.0](5)
and so
H 2/H 2
0=?m a ?3+(1??m )(1+a 1/τ
t )?3τ′
×
a ?3(1+w f )[1+(a/a t )?1/τ]3τ′
,
(6)
where ?m is the dimensionless matter density,τ′=τ?w ,and we assume a spatially ?at universe (as throughout this article).
Our complete set of cosmological parameters for dis-tance data then becomes {M ,?m ,w f ,?w,a t ,τ},where M is a nuisance parameter combining supernova abso-lute magnitude and the Hubble constant H 0.The fourth of the dark energy parameters,measuring the rapidity,can be chosen to be dw/dN t rather τif desired.This is one new equation of state model we will consider;one unfortunate aspect is that Model 4.0does not reduce to Model 2.0for any choice of parameters,so it cannot be viewed as an extension of the “baseline”Model 2.0.
B.
Extending Two Parameter Dynamics
Given the successful properties of the two parameter model,and that extensive estimation of constraints on the two model parameters exists for present and next generation cosmological probes,let us attempt to extend Model 2.0to a third parameter,while keeping the virtues as intact as possible.Studies of many dark energy models in the w ′?w phase plane reveal interesting properties of the dynamics [3]and show the success –and limitations –of Model 2.0.
This model keeps the derivative dw/da constant,or the logarithmic derivative w ′≡dw/dN ≡dw/d ln a ～a .One might conjecture that the logarithmic derivative,giving the change in dark energy equation of state per e-fold of cosmic expansion,is an important characteristic of the dynamics [8];so let us venture a generalization by enlarging its scope of behavior.(A di?erent approach in terms of changing the transition epoch is considered by [9].)We consider w ′～a b :
w (a )=w 0+w a (1?a b ).
[Model 3.1]
(7)
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Since Model2.0is a fairly successful approximation for many dark energy models we might expect that b does not deviate greatly from unity.In terms of the phase space dynamics of[3],their“thawing”models are a spe-cial case of Model3.1,with w a=?(1+w0)and their slope equal to our b.
Of course for a model with w a=0(such as taking the cosmological constant as the?ducial model),obser-vations will place no constraints on the third parameter b.Therefore,this is not a general extension of the two parameter model.So we also consider a di?erent way of changing the dynamics w′.We introduce some scale factor dependence to dw/da by adding a term of higher order in1?a:
w(a)=w0+w a(1?a)+w3(1?a)3.[Model3.2](8) Using the cubic power rather than a quadratic preserves the trend of the logarithmic derivative w′at the present, a=1.For example if w a=0then with this cubic the sign of w′does not?ip around the present day,unlike with a quadratic.We make no claims,however,that the cubic form is of great signi?cance;it merely represents a possible three parameter form with reasonable proper-ties,extending the standard two parameter model.
It is instructive to also investigate a more extreme evo-lution of the equation of state,such as
w(a)=w0+w a(1?a)+w e[(1+z)e z?1],[Model3.3]
(9) to see the in?uence of runaway behavior at high redshifts. This will not be suitable for treating CMB or growth related data,and we only consider it brie?y.
Below we give the expressions for the modi?cations to the Hubble expansionδH2=H2/H20??m a?3,divided by(1??m),for the models considered:
[Model3.1]:a?3(1+w0+w a)e?3w a(1?a b)/b(10) [Model3.2]:a?3(1+w0+w a+w3)×(11)
e?3w a?11w3/2+3(w a+3w3)a?9w3a2/2+w3a3 [Model3.3]:a?3(1+w0+w a?w e)e?3w a(1?a)+3w e(e z?1)(12)
III.SEARCHING FOR SUCCESSFUL
CONSTRAINTS
One question to address before further investigation is what is our criteria for success in constraining dark energy parameters.We can obviously?t a model with many parameters,each poorly,but what we mean by “how many dark energy parameters?”is how many can we determine to an accuracy that provides real physical insight.What the criteria for success should be is not a solved problem,though recent progress has been made by[3,10].
We adopt the following unrigorous but reasonable seeming criteria:1)a parameter describing the equa-tion of state at some epoch should be determined to a fractional error of<10%,relative to its?ducial value, e.g.the future equation of state value should be found to within0.1,and2)a parameter describing the change in equation of state,or a derivative,should be estimated to within0.2(i.e.20%of the Hubble time scale).It is hard to see that much looser constraints would be of substan-tial use in unraveling the physics.
We carry out a full Fisher analysis of the cosmolog-ical parameter uncertainties to test how many equation of state parameters can be reasonably constrained,in ad-dition to the matter density?m,and other parameters relevant to the cosmological measurements.Three types of cosmological probes are considered:Type Ia supernova distances(SN),weak lensing shear(WL),and the cosmic microwave background(CMB)angular power spectrum. We employ supernova distance data of the quality ex-pected from the next generation experiment SNAP[11], with2000supernovae from z=0.1?1.7plus300local su-pernovae from z=0.03?0.08,with intrinsic magnitude dispersion0.15magnitudes(7%in luminosity distance) plus an irreducible systematic of0.02(1+z)/2.7mag per 0.1redshift bin.This method carries with it the param-eter M to be marginalized over.
For the?ducial weak lensing survey we assume sky coverage of1000sq.deg.with100galaxies per arcmin2, which roughly corresponds to coverage and depth ex-pected from SNAP[12].We also brie?y consider a LSST-type survey with15000sq.deg.and30galaxies per arcmin2.Throughout we consider lensing tomogra-phy with10redshift bins equally spaced between z=0 and z=3and use the lensing power spectra on scales 50≤?≤3000.Intrinsic shape noise of each galaxy is ?xed toσγ=0.22.In addition to the dark energy param-eters(those describing the equation of state,plus?m) we vary the the amplitude of mass?uctuations with the ?ducial valueσ8=0.9,the physical matter and baryon densities?m h2=0.147and?b h2=0.023,and the scalar spectral index n=1.0.The sum of the neutrino masses is held?xed at mν=0.1eV.We compute the linear power spectrum using the?tting formulae of Hu&Eisenstein [14].We generalize the formulae to w=?1by appropri-ately modifying the growth function of density perturba-tions.To complete the calculation of the full nonlinear power spectrum we use the?tting formulae of Smith et al.[15].We emphasize that the calculations here are overoptimistic since we do not consider any systematics in the weak lensing survey.(On the other hand,it is con-ceivable that additional information,not captured by the two-point correlation function,can and will be extracted from weak lensing maps.)Understanding of weak lensing systematics is at an early stage and we hope to include a more realistic treatment in the future;for preliminary requirements on WL systematics see,e.g.,[16].
The third piece of information we(optionally)add is the full Fisher matrix corresponding to the expected con-straints from the Planck CMB experiment with tempera-ture and polarization information[13].In the SN+CMB case(i.e.when WL information is not considered),the
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Planck information is captured by the determination of the CMB peak locations,or measurement of the angu-lar diameter distance to the last scattering surface with a fractional precision of0.7%.As shown in[17],this nicely complements SN measurements,typically being more powerful than an independent prior on?M com-ing from large-scale structure surveys.
A.Fitting the Four Parameter Model
To provide a strong chance for tight parameter con-straints on all four parameters in Model4.0,we take a sensitive?ducial cosmological model with w f=?1,?w=0.8(w p=?0.2),a t=0.5,andτ=0.2.Note that the last value corresponds to dw/dN t=?1.So this model has a strong transition,at an epoch accessible to the data,over a time scale also visible to the data.
To get a?rst look at how di?cult the task is,we look at the unmarginalized uncertainties,i.e.the pa-rameter estimations if all other parameters(for both the equation of state and?m,M)are?xed.In this case,SN+CMB data provideσ(?m,w f,?w,a t,τ)= (0.0043,0.010,0.043,0.014,0.018),or0.09for σ(dw/dN t).These all satisfy the success criteria (note the constraint onτor dw/dN t would be borderline if we had applied blindly the10%fractional precision rule;for consistency,if we apply the0.2rule to?w and dw/dN t then the precision requirement onτshould be ～20%).
Other than the rapidity,the equation of state parameter estimations succeed by factors of4-10. Since this is not a huge margin,we have our?rst indication that it may be di?cult to characterize ac-curately the equation of state with several parameters, given the loss in precision that marginalization would cause.However we should also add in the WL data (though this is already marginalized over the large scale structure parameters)and now the(mostly) unmarginalized estimations areσ(?m,w f,?w,a t,τ)= (0.0030,0.011,0.065,0.015,0.016)for SN+WL and (0.0029,0.0099,0.038,0.13,0.015)for SN+WL+CMB. We now examine this model in detail,gradually re-stricting the four parameter model to fewer parameters by?xing the others,one at a time,in a physically moti-vated manner.Full results appear in Table I.For the full four parameter model even all three probes together only allow one parameter(a t)to be satisfactorily?t.This is despite the clear complementarity:for example on a t, SN+CMB gives an uncertainty of0.25,SN+WL gives 0.17,SN+WL+CMB provides0.045.We also note that moving away from the?ducial model can cause drastic increase in the uncertainties;for example withτ=1or w p=?0.8the parameter estimation errors exceed unity. So even with next generation measurements of dis-tances to better than1%,wide area weak lensing shear information,and accurate determination of the distance to the CMB last scattering surface we will be unable to accurately characterize the dark energy equation of state with four parameters.In the next subsection we explore the situation for reduced parameter sets.
Modelσ(?w)σ(τ)
0.10.05
0.150.045
3par:?x w f0.0950.050
0.0470.043
3par:?x a t0.290.13
0.0570.044
2par:?x a t,τ0.13–
0.0410.043
2par:?x?w,a t–0.056
–0.037
2par:?x w f,a t0.0530.041
–0.025
TABLE I:Cosmological parameter sensitivities for the four parameter equation of state model are here estimated for the combination of next generation supernovae,weak lensing,and CMB data.Fixed parameters are denoted with–.The sec-ond row shows the criteria for successful determination of each parameter(see text for details).The matter density?m esti-mation is always better than0.01.
1.Reduction to three parameters
Fixing parameters corresponds to using theoretical prejudice to pre-determine characteristics of the equation of state.Ideally,this is physically motivated prejudice, but sometimes the choice is not clear and one should scan the phase space to see how the?ducial choice of the?xed parameter a?ects the results.
First we address the variable w f describing the asymp-totic future value of the equation of state.Since obser-vational data only exist for the past,we have no direct constraint on the future.If the transition from past to future values is not complete,then the data should not be sensitive to the exact choice of w f.We?x w f=?1, corresponding to an asymptotic deSitter state,arguably better motivated than some other value.For the resulting three free parameter equation of state the uncertainties using SN+WL+CMB meet the criteria for success for only two parameters(?w and a t).Again,changing the ?ducial values tends to increase the errors dramatically. Next we?x the transition epoch a t=0.5.This could be roughly justi?ed by the expectation that dark energy should start to show increased dynamics as its energy density begins to overtake the matter density.Note that the standard Model2.0evaluates w′at a=0.5for the same reason,and proves successful in matching many scalar?eld and extended gravity model behaviors.Now
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the three free parameter equation of state model fails to ?t any parameter satisfactorily(recall that a t was the one successful?t in the full four parameter case,and now we have rendered that moot).Moving the transition epoch further in the past allows better determination of w f,e.g. taking a t=0.3yieldsσ(w f)=0.053rather than0.15, but raises the uncertainty on?w andτ.Conversely, making the transition more recent improves estimation of the rapidityτ(and of the high redshift value w p)but worsens other parameter errors.
Fixing the rapidityτor dw/dN t has less strong moti-vation,but one could argue that dw/dN t=?1,i.e.evo-lution on the Hubble time scale,is a reasonable choice. For the?ducial parameters this corresponds toτ=0.2. With?xedτ=0.2,the three free parameter equation of state allows successful?t of all three parameters.But as we say,it is less obvious that we can justify holding the rapidity?xed,and if the?ducial is changed toτ=1, then none of the parameters can be reasonably estimated. Finally,taking?xed the degree of change of the equa-tion of state from the far past to far future,?w= w p?w f,again less physically justi?ed,allows success-ful estimation of two of the three remaining equation of state parameters(for the extreme case?w=0.8).
One would need to carry out a comprehensive scan of phase space(e.g.along the lines of the Markov chain Monte Carlo[18]applied to the four parameter model Eq.4in terms of scale factor a)to be sure that the re-duced three parameter equation of state model cannot satisfy the characterization criteria in any particular case. However we have shown that we cannot generically ex-pect satisfactory three parameter?ts,and indeed cannot obtain them(except in the?xed rapidity case)in a?du-cial model designed to be especially favorable to see the behavior of the equation of state(we have checked that τ=0.2gives near optimal sensitivity).Moreover,since we do not know which?ducial case represents the true universe,we cannot claim three equation of state param-eters are justi?ed unless such a model is successfully?t over a broad region of phase space.
2.Reduction to two parameters
Of course we know that it is possible to achieve a suc-cessful two parameter characterization of the dark en-ergy equation of state:that is precisely what the stan-dard Model2.0does.For the data simulation used here, Model2.0deliversσ(w0,w′)=(0.066,0.11)for a?ducial model of the cosmological constant.
Reducing Model4.0to a two parameter case by?xing two parameters at a time gives six permutations.These are presented in Table I.Brie?y,all cases whereτis one of the parameters?xed can successfully?t two param-eters(note that we also can?nd the derived parameter w p,the high redshift value of the equation of state to better than0.1).However whenτis one of the two free parameters,we fail to estimate it well and so only attain a one parameter?t in the equation of state.
Note that with only one of either supernova or weak
lensing data,the only one of the six permutations of the doubly reduced four parameter model that even
marginally succeeds in?tting two parameters is where
we?x a t andτ–the major part of the dynamics we seek to?nd!This further demonstrates that it is remarkably
di?cult to characterize detailed dynamics of dark energy, even using next generation data,and complementarity is
essential.
B.Results from Three Parameter Extensions
If the full four parameter dynamics generalization fails, perhaps a more limited extension of the standard two
parameter(value and derivative)approach would work, as proposed in§II B.
1.Power law extension
In Model3.1,we allowed dw/da to depend on a,as a
power law.Recall that no constraint can be placed on the third parameter b when the?ducial model has w a=0.
Also,a?ducial value near b=1(i.e.looking like the stan-dard Model2.0)does not change the constraints on the
other equation of state parameters very much.Taking b to di?er from unity by10%changes the uncertainties
in w0and w a by roughly10%and7%.However,even when using a?ducial model with w a=0it is not possible to obtain a reasonable constraint on b.In the SUGRA model with w0=?0.82,w a=0.58,the third parame-
ter can only be estimated withσ(b)≈0.8(without all
three probes this becomes worse than2).So this model does not allow characterization of dark energy behavior beyond two parameters.
2.Cubic extension
In the cubic Model3.2,we do not have the problem with no determination of the third parameter for?ducial w a=0.However now covariance with the additional parameter signi?cantly degrades estimation of the?rst two parameters,especially w a,unless tightly controlled by inclusion of all three probes.With all three sources of information,for the?ducial cosmological constant case the uncertaintyσ(w3)=0.64.While all three parame-ters can be satisfactorily?t for the SUGRA case(0.074 uncertainty in w3),we have no guarantee that the uni-verse lives in such a favorable case,and so we cannot generically gain a third parameter to teach us about dark energy.
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3.Misleading sensitivity
To attempt to constrain a third parameter tightly enough to be useful,we are led to models with high sen-sitivity to the equation of state,e.g.by rapid variation. Indeed this is basically what allows Model3.2to succeed as well as it does:the cubic behavior,though bounded, gives strong evolution.To carry the sensitivity to an extreme,we consider the exponential Model3.3.Even using only the single probe of supernovae distances,we ?nd that indeed we can obtain successful constraints of ～0.2on w e.By contrast,with supernovae only,the third parameters in Models3.1and3.2can only be determined to～10.
This enhanced sensitivity is a bug,not a feature.The equation of state model is pathological,blowing up for high redshifts.Indeed we cannot straightforwardly em-ploy CMB data or weak lensing data(because it depends on the growth function from high redshift).The same situation arises,though less extremely,in the pioneering, but now obsolete,parametrization?rst considered in[19]: w(z)=w0+w1z.[Model2.1](13) Even cutting this o?at some moderate redshift and stitching on a more well-behaved parametrization can cause misestimation of cosmological parameter determi-nation.
For example,spurious estimations can result from us-ing Model2.1for baryon oscillation surveys.These data typically extend to z=1.5?3.5and sensitivities for baryon oscillations alone can appear quite promising.Pa-rameter estimations in terms of w1are factors of1.9-2.7 times better than those in terms of w′≡w a/2;however, as stated above this is a bug not a feature.This hyper-sensitivity arises from the extreme evolution forced upon the equation of state(even when the?ducial model is taken to be w1=0,the form still enters in the Fisher derivatives).One can test this by considering only low redshift(z<1)data,where Model2.1is not too egre-gious;indeed then the sensitivities in terms of w1and w′=w a/2agree.This is not to say that baryon oscil-lations do not provide a possible cosmological probe–they do in complementarity with other methods.Rather it cautions against applying a parametrization outside its realm of validity.Since we are now in the fortunate sit-uation of employing data sets extending beyond z≈1, Model2.1has become unsuitable and should be retired.
C.Success vs.Model Independence
It appears that obtaining a three parameter character-ization of the dark energy equation of state is nontrivial. We simultaneously need high sensitivity of the dark en-ergy to the third parameter,and small covariance with the other parameters,so as not to increase their uncer-tainties.We use this idea in§IV to make one more at-tempt at describing the equation of state in more detail.
First we note that the equation of state parameter un-certainties calculated need not be the minimum errors. For a given form of w(a)one can look for a change of vari-ables that minimizes the error on some variable,or some other quantity(such as volume of the N-dimensional un-certainty ellipsoid).However the price to pay for this improvement is breakdown of model independence.The new variables formed will no longer have the same phys-ical meaning as one changes the experiment(e.g.maxi-mum redshift),cosmological probe,or?ducial model.
To emphasize this point more strongly,we consider the case of minimum variance,or the“pivot redshift”(see [20]and also Appendix A of the omnivaluable[13]).For the standard Model2.0,say,we can ask what variable characterizing the value of the equation of state shows the minimum uncertainty.The uncertainty on w(a)is
σ2(w(a))=σ2(w0)+(1?a)2σ2(w a)+2(1?a)COV[w0,w a],
(14) where COV represents an entry in the covariance matrix. Minimizing this with respect to a,to?nd the smallest uncertainty on the value of the equation of state at any epoch,we?nd that at a min=1+COV[w0,w a]/σ2(w a) the minimum uncertainty is obtained,on the new vari-able w min=w0+w a(1?a min):
σ2(w min)=σ2(w0)?COV2[w0,w a]/σ2(w a).(15) This is manifestly smaller thanσ(w0).
But w min has no absolute meaning independent of probe,survey characteristics,priors,or?ducial model, and so it cannot be used to compare experiments or mod-els.For example,for the supernovae plus CMB data set,z min=0.28andσ(w min)=0.038.But that was for a?ducial cosmological constant;if a?ducial SUGRA model is used then w min has a di?erent meaning,and z min=0.52andσ(w min)=0.016.If no CMB data is used,instead a0.03prior on the matter density,then the cosmological constant case has z min=0.10and σ(w min)=0.072.
Since the main point of a phenomenological equation of state rather than an exact solution for a speci?c scalar ?eld dynamics is the utility of model independence(not to mention survey independence and experiment inde-pendence),such a variable rede?nition is not so useful in seeking a general answer to the question of how many dark energy parameters can be characterized.
IV.HOW MANY PRINCIPAL COMPONENTS? Consider the simplest case of a single cosmological pa-rameter extended to two parameters.If the?ducial value of the new parameter is such that it does not a?ect the previous parameter sensitivity,then we can treat this as adding a row and column to the previous Fisher matrix. The uncertainty on the?rst variable then grows from
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σ(v1)=(F11)?1/2to
σ(v1)=[F22/(F11F22?F212)]1/2=(F11)?1/211?r,
(16) where r=F212/(F11F22)is the covariance between the parameters.Covariance blows up parameter errors un-less the additional parameter is rather uncorrelated with the previous parameter.This leads to the idea of inves-tigating how many uncorrelated parameters can be used to characterize the dark energy equation of state.
The function describing the background behavior of dark energy–such as the equation of state or energy density–can be decomposed into its principal compo-nents(PC).Note that we seek the number of components directly informative about the nature of dark energy,so while we might obtain,say,17PCs to1%for distances to z=0.1?1.7or8PCs of the Hubble parameter,these would require di?erentiation of noisy data to teach us about the dark energy dynamics directly.Therefore we concentrate on the dark energy equation of state from the start.
Principal components o?er several nice features.They are uncorrelated,orthogonal,and widely separated by how accurately they can be measured.No functional form is assumed,but on the other hand,the eigenmodes have no meaning independent of model,survey,or tech-nique(cf.§III C).The?rst principal component gives the quantity most accurately measured by a particular survey.Here we extend the work of[21]applied to SN measurements by computing the principal components measured by weak lensing surveys,and also adding the CMB power spectrum information.
We assume w(z)to be described by20piecewise con-stant values in redshift uniformly distributed in redshift between z=0and z=2.We form another,21st bin at2 The top left panel in Fig.1shows the three best measured principal components of w(z)for weak lens-ing SNAP and LSST surveys.Note that the best mea-sured PC peaks at z～0.4–as expected,the sensitivity of weak lensing surveys to dark energy is at higher red-shifts than that of SN,which are shown in the top right panel.We also see that while adding the WL+CMB in-formation does not drastically change the shape of the best SN PC,the second best-measured PC,and higher ones,are moved to higher redshifts. The bottom panel of Fig.1shows the principal result: the inverse signal-to-noise in measurements of each PC for SN alone,SN+CMB,SN+WL,WL+CMB,and the three probes combined.More precisely,we computed the coe?cient of each principal component entering the?du-cial w(z)and its error,and de?neσP C=error/coe?cient. [Note that it isσPC,and not the raw error,that matters and determines at how many“sigma”each PC is de-tected1].To guide the eye we also plot two horizontal lines corresponding to two criteria for parameter mea-surement accuracy:the weak criterionσPC=1(below which we have a“1-σdetection”of a given PC)and the strong criterionσPC=0.1which is precisely equivalent to our10%criterion from Sec.III. We see that SN measurements alone,with systemat-ics,measure three PCs to S/N>1,and they are further signi?cantly helped by adding the Planck measurement of d A(z lss)(cf.[17]).Weak lensing power spectra alone, without systematics however,measures four PCs to pre-cision satisfying the weak criterion(not shown).Finally, combining all three probes leads to?ve(nearly six)PCs measured to S/N>1.This is quite encouraging and shows in a slightly di?erent light that all three surveys contribute nontrivially to the information content.How-ever,we also?nd that all three surveys combined lead to only two PCs passing the strong criterion,and two more if we dilute it to25%precision(σPC=0.25).Our results are in rough agreement with those of Knox et al. [22]who also consider the error in measurement of the PCs with both SN and WL.(Note that taking instead LSST-like WL data does not change the numbers of PCs quoted above.) These results indicate that it is not just a matter of ?nding a“better”parametrization.Even with allowing w(a)to take on whatever form the data best?t,next generation data can only generically provide tight con-straints on two dark energy parameters. V.CONCLUSION We have examined various characterizations of the dark energy in both parametric and non-parametric forms.Even with observational data corresponding to next generation probes of supernova distances,weak lens-ing shear,and the CMB,we?nd it is extremely challeng-ing to give a model independent description of the equa-tion of state with more than two parameters:the value at some epoch and a measure of its variation.The good news is that the two parameter model in current use is well understood,physically motivated,and accurate. To some extent this should not be surprising.If the FIG.1:Top left:three best measured principal components of w(z)for the upcoming weak lensing surveys;higher depth and galaxy density(SNAP)and wider area(LSST)each contribute well.Note that the best measured mode peaks at z～0.4.Top right:Two of the best measured PCs of w(z)expected for a SNAP-type survey for SN alone,and SN combined with weak lensing and CMB(Planck d A(z lss)).The best PC from SN alone peaks at z～0;this becomes z～0.2,if?M is independently known to better than0.01(not shown).Note that the second best measured PC is pushed out to higher redshift once the WL and CMB information is added in.Bottom:inverse signal-to-noise in measuring the PCs of w(z);that is,accuracies in measuring the PC coe?cients divided by their?ducial values.We show the cases of SN alone,SN+CMB,SN+WL,WL+CMB, and the three probes combined.Throughout,we include the systematics in SN constraints,but not in WL estimates as the systematics contributions to the latter are highly uncertain. underlying microphysics,e.g.potential of the dark energy ?eld,is reasonably smooth,described by a few parame-ters,then we should not expect more parameters to be evident in the equation of state.However we note that other microphysics,such as couplings to matter or non-canonical sound speed[23,24],have not been addressed here(although if w≈?1then we do not expect the sound speed to have much observable e?ect[25,26,27,28]). A more fundamental question to ask is what physics would we learn anyway with?tting a third parameter; the physical interpretation of basic dynamical character-istics such as an equation of state value and its variation is clear,but an extra parameter for the sake of having more may not hold much revelation,i.e.no real quali-tative di?erence.Furthermore,if we concatenate all our cosmological probes together in the quest for one more piece of information then we run the risk of losing the ability to test a more general framework.For example, by contrasting the expansion history with the growth his-tory[5,29,30,31],e.g.supernovae and weak lensing separately,we can seek whether the general concept of an e?ective equation of state w(a)corresponds to the speci?c origin of a high energy physics scalar?eld or a modi?cation of Einstein gravity. More elaborate parametrizations have some uses,e.g. testing for bias in the recovered w(z)(e.g.[32])or search- ing for a particular property(e.g.rapid evolution),but this must be weighed against the fact that extra param-eters can only be weakly constrained.In fact,even if we reduce a higher parameter model down to two parame-ters by a physically motivated?xing of some parameters, in most cases the complementarity of all three cosmolog-ical probes is essential.The standard parametrization in terms of w0and w a is one of the few successful enough to manage a generic?t with only a subset of probes;an-other is the new four parameter model introduced,with ?xed transition epoch and rapidity. The principal component decomposition of dark energy equation of state is a powerful approach that explicitly protects against biases in parameter determination and separates the accurately measured parameters from the poorly measured ones.But even the PC method has weaknesses–the recovered principal components depend on the survey speci?cations,and are not related to funda-mental physics of dark energy in any obvious way.Most likely the most powerful approach of characterizing dark energy will be testing the data in multiple ways,through both a parametrized function and PCA to check for con-sistency. The conclusion is that more than two parameters is not viable for a general?t with next generation data if we are interested in parameters directly informative about the nature and dynamics of dark energy that do not require further di?erentiation with respect to redshift.We can then ask how good do we need–what are the require-ments on the data such that we would be successful in characterizing the dark energy dynamics with a third pa-rameter.For the power law extension,Model3.1,to esti-mate b to0.2(with a?ducial SUGRA model),we require next-next generation improvements in all three probes together.Next-next generation experiments are de?ned as those that determine the distances for z=0?1.7 to0.05%,averaged over redshift bins?z=0.1,weak lensing shear power spectrum at the level equivalent to that of a completely systematics free survey with10000 square degrees and100galaxies per square arcminute, and the CMB distance to last scattering to0.1%.For the cubic case,Model3.2,the requirement is next-next generation accuracy on either SN+CMB or WL+CMB. For the four parameter dynamics,one can attain suc-cessful?ts to all four parameters with either next-next generation improvements to SN or to WL.However,this was for the most sensitive?ducial model and does not hold generally.For example,next-next accuracy on all three probes still fails to?t even three parameters if the true model instead possesses,say,a t=0.3orτ=1. So while it is not clear what we would gain in physics insight from going beyond two parameters,it is apparent that it would be extraordinarily di?cult to achieve such characterization.The two basic dynamical properties of the dark energy equation of state value and variation seem a reasonable and realistic goal for the next genera-tion. Acknowledgments This work has been supported in part by the Di-rector,O?ce of Science,Department of Energy under grant DE-AC03-76SF00098and by NSF Astronomy and Astrophysics Postdoctoral Fellowship under Grant No. 0401066. [1]E.V.Linder& A.Jenkins,MNRAS346,573(2003) [astro-ph/0305286] [2]U.Alam,V.Sahni,T.D.Saini,A.Starobinsky,MNRAS 344,1057(2003)[astro-ph/0303009] [3]R.R.Caldwell&E.V.Linder,astro-ph/0505xxx [4]E.V.Linder,Phys.Rev.Lett.90,091301(2003) [astro-ph/0208512] [5]E.V.Linder,Phys.Rev.D70,023511(2004) [astro-ph/0402503] [6]P.-S.Corasaniti& E.J.Copeland,Phys.Rev.D67, 063521(2003)[astro-ph/0205544] [7]B.A.Bassett,M.Kunz,J.Silk,C.Ungarelli,MNRAS 336,1217(2002)[astro-ph/0203383] [8]E.V.Linder&R.Miquel,Phys.Rev.D70,123516(2004) [astro-ph/0409411] [9]D.Rapetti,S.W.Allen,J.Weller,MNRAS accepted [astro-ph/0409574] [10]E.V.Linder,in draft(2005) [11]SNAP–https://www.doczj.com/doc/c37789674.html,; Aldering,G.et al.2004,PASP,submitted [astro-ph/0405232] [12]A.Refregier et al.,Astron.J.127,3102(2004) [astro-ph/0304419][13]D.J.Eisenstein,W.Hu and M.Tegmark,Astrophys.J., 518,2(1999) [14]D.J.Eisenstein&W.Hu,Astrophys.J.,511,5(1999) [15]R.E.Smith et al.,MNRAS,341,1311(2003) [16]D.Huterer,M.Takada,G.Bernstein&B.Jain,in draft (2005) [17]J.A.Frieman,D.Huterer,E.V.Linder and M.Turner, Phys.Rev.D,67,083505(2003) [18]P.S.Corasaniti,M.Kunz,D.Parkinson,E.J.Copeland and B.Bassett,Phys.Rev.D70,093006(2004) [19]A.Cooray&D.Huterer,Astrophys.J.,513,L95(1999) [20]D.Huterer&M.S.Turner,Phys.Rev.D64,123527 (2001) [21]D.Huterer&G.Starkman,Phys.Rev.Lett.,90,031301 (2002) [22]L.Knox, A.Albrecht and Y.-S.Song,Proceed- ings of NOAO Workshop on Observing Dark Energy [astro-ph/0408141] [23]W.Hu,Astrophys.J.,506,485(1998) [24]S.DeDeo,R.R.Caldwell and P.J.Steinhardt,Phys.Rev. 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D,69,124015(2004) [30]E.V.Linder,in DARK2004,Proceedings of5th Interna- tional Heidelberg Conference(2004)[astro-ph/0501057] [31]L.Knox,Y.-S.Song and J.A.Tyson,astro-ph/0503644 [32]B.Bassett,P.S.Corasaniti and M.Kunz,Astrophys.J., 617,L1(2004) 三年级上册 Module 1 I我 am (I'm=I am)是（我是） aah啊 hello (hi)你好 ooh嗬 goodbye (bye-bye)再见 are是 How are you? 你好么？ 你好！ good好的 morning早晨，上午 fine健康的 thank谢谢 you你；你们 Module 2 Ms女士 boy男孩 girl女孩 whoops哎哟 and那么；和 too也 ha ha哈哈 what什么 is (what's= what is) 是（是什么）your你的，你们的 name名字 please请 afternoon下午 Mr 先生 Module 3 point指 to向…… the这（那）个，这（那）些door门 sit坐 down向下 stand站 up向上 window窗户 blackboard黑板 bird鸟 tweet（鸟）啾啾的叫声desk书桌 chair椅子 Module 4 it它 it's= it is它是 red红色的 look看 wow呀，哇 yellow黄色的 blue蓝色的 a(an) 一个，一chameleon变色龙 my我的 panda熊猫 now现在 green绿色的 black黑色的 dog狗 cat猫 cap帽子 小学英语单词总表（供外研社新标准英语三年级起始用） Module 5 how many多少 one 一 two二 three三 four四 five五 six六 seven七 eight八 nine九 oh噢，哦 ten十 eleven十一 twelve十二 Module 6 happy快乐的 birthday生日 here (here's=here is)这里（这是）present礼物 this这个 pencil铅笔 pen钢笔 cake蛋糕 old ……岁的 how old多大 yes是的 you're=you are你是 Module 7 teacher教师 pupil小学生school学校 classroom教室 English英语 that那个 say说 again再一次 schoolbag书包 ball球 book书 Module 8 monster怪物 new新的 kite风筝 or或者 don't=do not不 know知道 no不 not (isn't=is not)不（不是） help救命（呼救用语） where (where's=where is)哪里（在哪 里） in在……里 bag书包 Module 9 mother母亲 father父亲 sister姐妹 brother兄弟 she (she's=she is)她（她是） grandpa祖父，外祖父 grandma祖母，外祖母 that's=that is那是 me我 he(he's=he is)他（他是） doctor医生 policeman警察 nurse护士 driver司机 farmer农民 Module 10 his他的 head头 leg腿 foot脚 on在……上 arm胳膊 hand手 her她的 nose鼻子 eye眼睛 mouth嘴 ear耳朵 三年级下册 Module 1 song歌曲 TV电视 favourite特别喜欢的 colour颜色 Here you are.给你。 Module 2 they他（她／它）们 they're=they are他（她／它）们是 monkey猴子 baby幼兽，幼畜 all每个；全体 zoo动物园 tiger老虎， lion狮子 elephant大象 fat胖 man人，男人 short矮的 tall高的 small小的 thin瘦的 big大的 Module 3 like喜欢 football足球 them他（她／它）们 ouch哎哟 basketball球 table tennis乒乓球 morning exercises早操 ride骑 bike自行车 swim游泳 skip跳绳 Module 4 meat肉 pass传递 rice米饭 mum妈妈 儿童歌曲大全 01.外婆的澎湖湾15.小毛驴29.红星歌 02.世上只有妈妈好16.我上幼儿园30.爷爷为我打月饼 03.爱我你就抱抱我17.虫儿飞31.少先队队歌 04.小兔乖乖18.采蘑菇的小姑娘32.好爸爸坏爸爸 05.打电话19.三只熊33.红星闪闪 06.蓝精灵20.爸爸妈妈听我说34.大家一起喜羊羊 07.拔萝卜21.我的好妈妈35. 小星星中文版 08.让我们荡起双浆22.大头儿子和小头爸爸36.泥娃娃 09.两只老虎23.小螺号37.爸爸妈妈最爱我 10.小燕子24.春天在哪里38.学习雷锋好榜样 11.蜗牛与黄鹂鸟25.葫芦兄弟39.快乐的节日 12.数鸭子26.白龙马40.粉刷匠 13.健康歌27.我爱洗澡 14.生日快乐28.字母歌 41.茉莉花55.儿童歌曲串烧50 首69.一分钱 42.小龙人56.小二郎70.猪之歌 43.丢手绢57.捉泥鳅71. 拆拆拆(喜羊羊与 灰太狼 44.幸福拍手歌58.我爱北京天安门之兔年顶呱呱插曲) 45.每当我走过老师窗前59.拍拍手72.最美的画 46.分果果排排坐60.黑猫警长73.我不上你的当 47.娃哈哈61.劳动最光荣74.少年先锋队队歌 48.26 个字母62.三个和尚75.我有一只小毛驴 49.闪闪的红星63.花仙子之歌76.校园的早晨 50.卖报歌64.小宝贝快快睡77.鲁冰花 51.妈妈的吻65.宝宝睡觉了78.蓝精灵之歌 52.葫芦娃66.歌声与微笑79.歌唱二小放牛郎 53.铃儿响叮当67.聪明的一休80.西游记 54.童年68.少年少年祖国的春天 81.嘚啵嘚啵嘚93.哈啰歌105.礼貌歌 82.一闪一闪亮晶晶94.小红帽106.铃儿叮当响 83.小白船95.我在马路边捡到一分 钱107.小兔子乖乖 84.伦敦铁桥垮下来96.拨浪鼓108.读唐诗 85.司马光砸缸97.大苹果109.两只老虎独唱版 86.中国少年先锋队队歌98.小鸭子110.北京北京祝福你 87.祖国祖国多美丽99.七色光之歌111.小手拍拍 88.小花猫100.找朋友112.说唱脸谱 89.歌唱祖国101.走在乡间的小路上113.祝你生日快乐 90.八月十五月儿圆102.吉祥三宝114.春晓 91.长亭外古道边103.友谊地久天长115.小星星英文版 92.烛光里的妈妈104.凤阳花鼓116.小松树 人教版小学三年级英语下册Unit3Howmany单元测试卷2带 答案 〔时间：40分钟〕 Listening Part 〔50分〕 一、听音.写出你所听到的大写或小写字母.〔4分〕 二、听音.选出录音所读的单词、字母或数字.在〔〕内打“√”〔8分〕 1. monkey〔〕 2. eleven〔〕 3. twenty〔〕 4. jeep 〔〕 mouse 〔〕 seven 〔〕 twelve〔〕 jump 〔〕 5. 17 〔〕 6. Coke 〔〕 7. N J 〔〕 8. hak 〔〕 27 〔〕 coffee〔〕 N G 〔〕 hka 〔〕 三、听音.按照录音所读的顺序.用大写字母给下列单词标号.〔10分〕 1. doll ball balloon 〔〕〔〕〔〕 2. mom dad man 〔〕〔〕〔〕 3. green great black bird 〔〕〔〕〔〕〔〕 四、听音.用阿拉伯数字写出下面图形的数量.〔8分〕 〔〕〔〕〔〕〔〕五、听音.判断下列各图是否与听到的内容一致.是的画.否则画.〔12分〕 〔〕〔〕〔〕〔〕〔〕 〔〕 六、听音.根据录音所读的内容.连成一个完整的句子.〔8分〕 1. How many pencils can you see? 2. How many cats do you have? 3. Look at my my mother. 4. She is new crayons. Writing Part 〔50分〕 一、按字母表顺序.写出大小写字母.从Ee—Nn.〔8分〕 E e N n 二、把相应的大小写字母连起来.〔6分〕 三、看图.将相对应的单词和图片连线.〔12分〕 kite ice-cream egg girl kangaroo man 四、根据栏中的问句.在栏中选出正确答语.〔10分〕 〔〕1. How many crayons do you have? A. Nice to meet you. 〔〕2. Who's that girl? B. OK! 〔〕3. How many birds can you see? C. I can see eleven. 小学英语分册词汇(外 研版三年级起点) （最新版） 第一册 MOUDLE 1 I 我 am 是 hello（hi）你好goodbye（bye-bye）再见how 怎样 are 是 you 你 good 好的 morning 早上 fine 很好 thank 谢谢 MODULE 2 Ms 女士 too 也 and 那么、和 boy 男孩 girl 女孩 what 什么 is 是 your 你的 name 名字 afternoon 下午 Mr 先生 MODULE 3 the 这个，那个，这些，那些 door 门 please 请 window 窗户 blackboard 黑板 bird 鸟 desk 桌子 chair 鸟MODULE4 my 我的 it 它 red 红 a（an）一个，一 panda 熊猫 blue 蓝 yellow 黄 green 绿 black 黑 dog 狗 cat 猫 cap 帽子 MODULE 5 many 许多 how many 有多少 one 一 two 二 three 三 four 四 five 五 six 六 seven 七 eight 八 nine 九 ten 十 eleven 十一 twelve 十二 hat 帽子 MODULE6 this 这个 school 学校 pupil 小学生 classroom 教室 English 英语 teacher 教师 that 那个 bag 书包 pencil 铅笔 pen 钢笔 book 书 MODULE 7 happy 快乐 Birthday 生日 here 这里，这是 cake 蛋糕 old 年岁 how old 多大 loot 看 MODULE 8 no 不 not 不，不是 yes 是的 help 帮助 kite 风筝 where 哪里，在哪里 in 在什么里面 bag 袋子 MODULE 9 mother 母亲 father 父亲 doctor 医生 grandpa 祖父，外祖父 grandma 祖母，外祖母 sister 姐妹 me 我（宾格） brother 兄弟 driver 司机 policeman 警察 farmer 农民 nurse 护士 he 他 she 她 MODULE 10 DVD1 01、采蘑菇的小姑娘 02、拔萝卜 03、打电话 04、一分钱 05、小白兔乖乖 06、小司机 07、两只老虎 08、世上只有妈妈好 09、小燕子 10、泥娃娃 11、大公鸡 12、小星星 13、小号手之歌 14、生日歌 15、跳舞歌 16、小喇叭 17、竹叶船 18、捕鱼歌 19、做做做 20、洋娃娃之歌 21、过马路 22、摇啊摇 23、小天使 DVD2 01、小老鼠上灯台 02、小兔子乖乖 03、心爱的小马车 04、小放牛 05、粗心的小画家 06、跳飞机（粤语） 07、麻雀与小孩 08、我是一个大苹果 09、小红花 10、花蛤蟆 11、胡子掉下来 12、猫儿歌 13、劳动最光荣 14、东南西北 15、小熊请客 16、爱清洁 17、看花灯 18、啄木鸟 19、可爱的小白鹅 20、小露珠 21、小白船 22、小杜鹃 23、新年好 24、落雨大（粤语） 25、鹿中的王子 26、小小粉刷匠 27、小羊 28、青蛙妈妈睡醒了 29、对鲜花 30、妈妈你歇歇吧 31、小鸟醒来了 32、我给小鸟搭歌台 33、牧童之歌 34、小手帕 35、捉泥鳅 36、拾豆豆 37、七只小乌鸦 38、我是一粒米 39、小拜年 40、小小音乐家 41、妈妈再见 42、交朋友 43、洋娃娃玛苏卡 44、紫竹调 45、这是什么 46、青蛙和小姑娘 47、小懒猫 48、丢手绢 49、马兰花 50、我是小叮当 DVD3 01、卖汤圆 02、老黄牛 03、烧饼油条 04、雪花 05、齐跳跳（粤语） 06、猫咪别淘气 07、玩具进行曲 08、妈妈好 09、猎人之歌 10、小山羊小绵羊 Unit6: How many The first lesson 一、教学目标 1、能够听、说、读、写单词 kite kites eleven twelve 2、句子：How many….句型的应用掌握 二、教具准备：扑克牌 三、Warm –up 1、sing a song 问这些问题以后请一个学生上台做“小老师”，进行问答。 五、presentation 1、Look at the picture Kite [kait]学生自己拼出 再展示Kites 复数形式 2、T: “How many kites do you see ?” S: “one two … six ” Six kites . 3、Look at me . a、This… <要伸出5个手指> b、PPT .eleven 出示 4、做游戏加强练习“eleven”. （1）分组依次报数，从1开始到7，报7的学生不报数只报手，报到11全组齐读，顺序报完即为胜利。 （2）看哪组获的花多 5、接着看图片PPT 用“How many….”造句，提出“12”，展示“12”单词，twelve 6、用以上游戏报数练习12. 六、practice 1、分组练习，看哪组获胜 Teaching Design 一、Title （题目）：How many 二、Less on type （课型）：conv ersati on . 2. New teaching 三、教学目标： 1、知识目标 ①、掌握A部分Let ‘‘ s le中的单词。 ②、能够听、说、认读A部分Let ‘‘ s le中的单词。 2、能力目标 ①、能听、说、认读1—15的数字单词，并能在日常生活中正确使用。 ②、听懂有关数字的指令性语言并作出相应的反应。 3、过程与方法 本单元以How many为主题，分为三个部分。以A,B部分中的1—20的数字单词为线索，Let ‘‘ s为桥梁，C部分中的任务活动为中心，贯穿整个单元。 4、情感态度与价值观 ①、养成热爱学习，懂得学习重在积累的优良品质。 ②、注重小组合作学习，培养相互沟通忽然交流的能力。 5、技能目标： 能将所学的交际用语用于日常交际。 四、教学重点： 1. 在巩固数字1 —10 的基础上，学习数字eleven, twelve, thirteen, fourteen, fifteen 。 2. 能熟练掌握11-15这些数字和“ How many…can you see及其回答“lean see …” 3. 听懂有关数字的指令性语言并作出相应的反应。 五、教学难点： 1. 能区别数字eleven, twelve, thirteen, fourteen, fifteen 并正确发音。 2. 数字11-15的灵活运用及单词twelve , thirteen中“ve,” “th,” “teen的发音。 六、教材分析： 本节课是人教版小学英语三年级下册第三单元第一课时。本单元主要内容是学 习数字11—20的英语表达法和与数字相关的一些句型的运用及游戏。本课是第一课时，是一节数字教学，主要是在复习1-10的基础上学习11—15的数字表达。它是展开本单元how many话题和后面16-20学习的基础。在学习本课时之前，学生已学习了1到10的数字单词，学生对数字单词1到10的读音、拼写已经掌握。这样比较易于对新课的深入与扩展，这样的安排，既体现了教材循序渐进、由易到难的编排意思，又符合学生的知识水平和认知水平。 七、Teachi ng methods 教学方法： 1. Situational teaching 2. Heuristic method of teaching 3. Students are center，the teacher is a helper，guide，supervisor ，prompter and monitor 4. WLL CA TBL TPR 八.Length of lesson：40 minutes 九.Age of learners: around 10 十：Teaching Material 教具准备： 数字11-15的单词卡片、录音机、磁带、课件、黑板上画好的四线三格书写格式、跳绳、男女生竞赛表等教具。 （将学生分成男女生组，竞赛回答课堂中的问题，教师给予单独奖励小贴画并为所在组赢得五角星。此表贴于黑板右上角。如下：） 十^一- .Teachi ng procedures:（教学过程） Step 1:Warm-up and Revisio n 1. 复习数字1-10：T: How many （教师竖起手指）How many times （教师拍几下手）. 活动：T：“show me 1 and 2, show me 3 and 4 ..... ”. a song:《Ten little can dies dance〉课件播放歌曲。 儿童歌曲大全 01.外婆的澎湖湾 02.世上只有妈妈好 03.爱我你就抱抱我 04.小兔乖乖 05.打电话 06.蓝精灵 07.拔萝卜 08.让我们荡起双浆 09.两只老虎 10.小燕子 11.蜗牛与黄鹂鸟 12.数鸭子 13.健康歌 14.生日快乐15.小毛驴 16.我上幼儿园 17.虫儿飞 18.采蘑菇的小姑娘 19.三只熊 20.爸爸妈妈听我说 21.我的好妈妈 22.大头儿子和小头爸爸 23.小螺号 24.春天在哪里 25.葫芦兄弟 26.白龙马 27.我爱洗澡 28.字母歌 29.红星歌 30.爷爷为我打月饼 31.少先队队歌 32.好爸爸坏爸爸 33.红星闪闪 34.大家一起喜羊羊 35.小星星中文版 36.泥娃娃 37.爸爸妈妈最爱我 38.学习雷锋好榜样 39.快乐的节日 40.粉刷匠 …………………………………………………………………………………… 41.茉莉花 42.小龙人 43.丢手绢 44.幸福拍手歌 45.每当我走过老师窗前 46.分果果排排坐 47.娃哈哈 48.26个字母 49.闪闪的红星 50.卖报歌 51.妈妈的吻 52.葫芦娃 53.铃儿响叮当 54.童年55.儿童歌曲串烧50首 56.小二郎 57.捉泥鳅 58.我爱北京天安门 59.拍拍手 60.黑猫警长 61.劳动最光荣 62.三个和尚 63.花仙子之歌 64.小宝贝快快睡 65.宝宝睡觉了 66.歌声与微笑 67.聪明的一休 68.少年少年祖国的春天 69.一分钱 70.猪之歌 71.拆拆拆(喜羊羊与灰太狼 之兔年顶呱呱插曲) 72.最美的画 73.我不上你的当 74.少年先锋队队歌 75.我有一只小毛驴 76.校园的早晨 77.鲁冰花 78.蓝精灵之歌 79.歌唱二小放牛郎 80.西游记 …………………………………………………………………………………… 81.嘚啵嘚啵嘚 82.一闪一闪亮晶晶 83.小白船 84.伦敦铁桥垮下来 85.司马光砸缸 86.中国少年先锋队队歌 87.祖国祖国多美丽 88.小花猫 89.歌唱祖国 90.八月十五月儿圆 91.长亭外古道边 92.烛光里的妈妈93.哈啰歌 94.小红帽 95.我在马路边捡到一分钱 96.拨浪鼓 97.大苹果 98.小鸭子 99.七色光之歌 100.找朋友 101.走在乡间的小路上 102.吉祥三宝 103.友谊地久天长 104.凤阳花鼓 105.礼貌歌 106.铃儿叮当响 107.小兔子乖乖 108.读唐诗 109.两只老虎独唱版 110.北京北京祝福你 111.小手拍拍 112.说唱脸谱 113.祝你生日快乐 114.春晓 115.小星星英文版 116.小松树 外研版小学三年级英语教学 计划 一、课程总目标： 基础教育阶段英语课程的任务是：激发和培养学生学习英语的兴趣，使学生树立自信心，养成良好的学习习惯和形成有效的学习策略，发展自主学习的能力和合作精神；使学生掌握一定的英语基础知识和听、说、读、写技能，形成一定的综合语言运用能力；培养学生的观察、记忆、思维、想象能力和创新精神；帮助学生了解世界和中西方文化的差异，拓展视野，培养爱国主义精神，形成健康的人生观，为他们的终身学习和发展打下良好的基础。 二、本学期课程目标： 进一步激发学生学习英语的兴趣，培养他们学习英语的积极态度，使他们建立初步的学习英语的自信心；培养学生一定的语感和良好的语音、语调基础，使他们形成初步运用英语进行简单日常交流和书写，为进一步学习打下基础，同时，培养学生简单的阅读理解能力。 三、学情分析： 本班共有学生ｘｘ人，除了个别同学因种种原因学习基础较差以外，大部分同学他们聪明活泼、勤奋好学，这些学生曾在一年级的时候就初步接触了英语，对英语有着浓厚的兴趣。 四、教材和其他课程资源分析： 《新标准英语第六册》供三年级使用，是北京外研室编写的一套全新的中下学衔接的英语教材。这套教材是根据教育部制定的《国家英语课程标准》和《小学英语教学基本要求》编写而成的。本册供以小学一年级为起点、开设英语的学校第三学年第二学期使用。 《新标准英语》的设计和编写体现了外语教学思想的继承和发展。在分析、研究许多种国内外小学英语教材的基础上，取其精华，博采众长，形成了本套教材特有的编写体系。同时，有吸收了当今国内外英语作为外语教学的理论和成功经验，把这些教学理论和实践经验同我国的小学外语教学实际相结合，以形成我国小学英语的外语教学模式和教学方法。 本册教材具有以下几个特点： 1、注重学生语言运用能力的培养，突出语言的实践性和交际性，同时也突出语言的真实性和实用性。 2、注重学生自学能力和学习策略的培养，为学生的进一步学习或终身学习奠定基础。 3、注重中外文化的双向交流，使学生通过学习，培养未来跨文化交际所需要的能力。 作文我想对爸爸妈妈说大全 我想对爸爸妈妈说作文1：作者：顾可茵 亲爱的爸爸妈妈，我有几句话一直想对你们说。今天终于趁着这个机会，鼓足勇气说出了口。 妈妈，您每天都要做菜、洗碗、洗衣服什么的，很劳累。从小到大，我知道妈妈都很照顾我，我也知道您养我很不容易，所以我想对妈妈说：“妈妈，谢谢您!” 爸爸，您是家里的顶梁柱，每天风里来，雨里去的，很累，我知道，家里的经济大都由您承担，所以您的压力很大，您每天都要考虑很多的事情，并且把它们都考虑得很周到，偶尔，我也有不会做的题目要向您请教，而你也会很有耐心地告诉我，所以我也想对爸爸您说：“爸爸，您辛苦了!” 看到爸爸妈妈这么辛劳，我想，我应该替他们做一些事，在爸爸妈妈回来的时候在他们面前放一双拖鞋，倒一杯热茶，做些力所能及的事，让劳累了一天的爸爸妈妈好好休息。 亲爱的爸爸妈妈，女儿一定不会辜负你们的期望，努力学习，将来做个有用的人，来报答你们对我的养育之恩的! 我想对爸爸妈妈说作文2：作者：毛岚 很多人都是快乐幸福的，我也不例外，爸爸妈妈为我挡风遮雨，而我们却从来没有说过一声“谢谢”。 我要对爸爸说：你辛苦了，您为了我，在外面努力的工作， 回家后还要辅导我功课。妈妈，我想对您说，“感谢您，为我做饭，洗衣服，从不报怨”。父母就是我最温暖的家，最坚强的保护神。 爸爸妈妈我想对你说，谢谢你们对我无微不致的照顾，小时候，我身体不好，是你们的细心照料，使我的病情很快的好转。为了让我的身体更健康，体力更充足，我参加了学校田径队，是你们早早起床为我准备丰富的早餐。送我到学校训练。 爸爸妈妈我想对你说，谢谢你们对我的精心培养。为了让我手脚协调能力更强，您给我报了篮球班;为了让我的头脑更聪明，您让我去学习围棋;为了让我写一手漂亮的好字，每周还领我去学习硬笔书法。在你们的精心培养下，我正在向德、智、体方向全面发展。 我觉得我很幸福，因为爸爸妈妈教会了我做人，我生活在一个充满爱的家庭里，我想对你们说“我爱你们”! 我想对爸爸妈妈说作文3：作者：张瑾 爸爸妈妈，我有一肚子的心里话想对你们说，今天我就借此机会跟你们说一说。 妈妈，你还记得去年的一件事吗?那天，我上完兴趣班，等着您来接我时，您却迟迟也没来，同学们一个接一个被家长接走了，我徘徊在门口，却连您的影子也没看见，当作是我是多么孤单又痛苦呀，眼泪在我的眼眶里打转，差点哭了。 过了足足半个小时，可是来接我的是舅舅，不是您!也许，这半个小时对你来看是微不足道的，可是对我而言，简直是度日如年!我气呼呼的回到家，看见您在悠闲自得的打麻将时，我的心都快碎了，对您说，您为什么不来接我时，您说，她要打麻将。我 Howmany第三课时 三年级英语教案 课题Unit Three How many?教学重点1.学习英文字母Jj,Kk,能够按正确笔顺对字母进行描红。2.学习单词jeep, jump, kangaroo, key, 能听懂会说。教学难点1.印刷体与手写体的区别2.Jj在单词中的发音。教具预备1.图片jeep, jump, kangaroo, key2. 课题Unit Three How many? 教学重点 1.学习英文字母Jj,Kk,能够按正确笔顺对字母进行描红。 2.学习单词jeep, jump, kangaroo, key, 能听懂会说。 教学难点 1.印刷体与手写体的区别 2.Jj在单词中的发音。 教具预备 1.图片jeep, jump, kangaroo, key 2.写有大小写Jj的字母卡 3.教材相配套的教学课件和录音带 教学过程 (一)热身、复习(Warmup/Revision) 复习字母Aa-Ii 1.做游戏:抢读字母 这是一个练习学生认读字母的游戏,教师将全班分成若干小组,然后逐个出示字母卡片,学生们举手抢答,教师让最先举手的学生读出该字母,读对的给该组记10分,最后得分最多的组为优胜。 2.做游戏:看谁快 这是一个练习学生听字母的游戏,将全班分成两组,一组学生持大写字母,另一组学生持小写字母,教师快速念字母,要求持有该字母的学生迅速站起来,最先站起来的人得两分,后站起来的得一分,没站出来的得零分,得分多的组获胜。 (二)呈现新课(Presentation)课题 1.学习字母Jj和单词jeep, jump。 1)教师播放Let’s say部分动画,学生观看。 外研版小学英语（三年级起点）四年级英语教案 （第四册） 四年级英语教学计划 一、学生知识能力习惯态度分析 本班共有28人，其中女生15人，男生13人。他们来自不同的11个自然村。由于各种原因，他们的英语水平也不同。大部分学生对英语有着较浓厚的学习兴趣，但也有少数学生由于遇到困难，学习兴趣会随之减弱。尤其从这学期开始，对学生又提出了新的要求：培养听、说、读、写的技能。所以教师应该面向全体学生，以学生的发展为宗旨，始终把激发学生的学习兴趣放在首位，引导学生端正学习态度，掌握良好的学习方法，培养学生良好的学习习惯。 二、教材内容分析： 本册教材——New Standard English三册是供小学四年级下学期使用的。全书共分11个模块，内含一个期末复习模块。每个模块分为两个单元。一般情况下，第一单元呈现本模块所要学习的语言内容，第二单元提供若干任务性练习，包括一首歌谣或小诗。通过歌谣或小诗，培养学生的语感和节奏感，提高发音的正确性，介绍一定的西方文化。在这一册，我们将进一步描述人、物品、动物等的基础上重点学习对动作的描述。此外，我们还要学习如何表达时间以及如何描述地点方位等内容。 三、教学目的任务 1、能按四会、三会的要求掌握所学单词。 2、能按四会要求掌握所学句型。 3、能使用日常交际用语，活用四会句型，进行简单的交流，做到大胆开口，发音正确。 4、能在图片、手势、情境等非语言提示的帮助下，听懂清晰的话语和录音。 5、初步培养良好的书写习惯，能做到书写整洁、规范。 6、养成响亮清晰读英语、说英语的习惯，认真模仿语音、语调，以培养语感。 7、能在完成某个任务（如涂色，小制作）的过程中学会相关的词句，并且培养动手能力。 8、能演唱已学过的英语歌曲，诵读已学过的歌谣。 三、教材重点难点 1、能按四会、三会的要求掌握所学单词。 2、能按四会要求掌握所学句型。 3、能使用日常交际用语，活用四会句型，进行简单的交流，做到大胆开口，发音正确。 4、能在图片、手势、情境等非语言提示的帮助下，听懂清晰的话语和录音。 5、初步培养良好的书写习惯，能做到书写整洁、规范。 四、主要措施 １、以活动为课堂教学的主要形式，设计丰富多彩的教学活动，让学生在乐中学、学中用，从而保证学生英语学习的可持续性发展。 ２、通过听、说、读、写、唱、游、演、画、做等形式，进行大量的语言操练和练习。 ３、将直观教具和电教手段，多媒体课件相结合，培养学生良好的朗读习惯，打下良好的语音语调基础。 ４、设计全面、高效的课外作业，培养学生良好的书写习惯，做到整洁、规范、正确地书写。 HowMany 三年级英语教案 unit 3 lesson 2 part a let’s talk ; let’s practise teaching aims : 1. be able to understand and say: how many … can you see? i can see…and use them in real situation. 2. .be able to understand and use the phrase which can praise other ones: “ it’s beautiful! how beautiful!” and use them in the proper situation. focus points & difficult points : be able to use the new structures as following in the real life: how many …can you see? i can see… *be able to understand and say “ the black one is a bird.” teaching preparation: vcd, a kite, powerpoint. designing for the blackboard: how many kites can you see? i can see 12 teaching steps: step1. warming –up 1. sing a song 《one two three four five》. 2. greetings. 3. slow action to review the words. 4. t claps her hands and ss say the times step2.presentation and practice. 1. 1) t shows her hand: how many fingers can you see? s: i can see ten fingers. t writes down: i can see…… t teaches this sentence. t continue to ask: how many fingers can you see? 第一册 MOUDLE 1 hello (hi) 你好 I am (I’m) 我是goodbye(bye-bye) 再见good morning 早上好 How are you? 你好吗 fine（身体）很好 thank you 谢谢 and 和 MODULE 2 too 也 boy 男孩 girl 女孩 Ms 女士 good afternoon 下午好 Mr 先生 what 什么 your 你的 name 名字 MODULE 3 a 一个 panda 熊猫 now 现在 it is（it’s ）它是 red 红的 blue 蓝的 yellow 黄的green 绿的 black 黑的 dog 狗 desk 书桌 chair 椅子 orange 橘黄色的MODULE4 how many 多少个？ one 一 two 二 three 三 four 四 five 五 six 六 seven 七 eight 八 nine 九 ten 十 cap 帽子 hand 手 cat 猫 MODULE 5 stand up 起立 point to 指向 door 门 window 窗户 sit down 坐下 bird 鸟 eleven 十一 twelve 十二 MODULE6 pupil 小学生 this 这个 my 我的 school 学校 classroom 教室 English 英语 teacher 教师 that 那个 bag 书包 pencil 铅笔 pen 钢笔 book 书 MODULE 7 happy birthday生日快乐 head 头to 给…… on 在……上面 under 在……下面 hat 帽子 where 那里 in 在……里面 cake 蛋糕 here 这里 present 礼物 MODULE 8 today 今天 how old 多大 know 知道 no 不，不是 dragon 龙 yes 是的 help 帮助 kite 风筝 look 看 MODULE 9 mother 母亲 father 父亲 doctor 医生 grandpa 祖父，外祖父grandma 祖母，外祖母sister 姐妹 me 我（宾格）brother 兄弟 driver 司机policeman 警察farmer 农民 nurse 护士 he 他 she 她 MODULE 10 arm 手臂 leg 腿 foot(feet) 脚 全是宝宝喜欢听的儿歌，我经常放给她听，她现在两岁半，差不多都能唱啦，与大家分享一下，点击相关图片就可以播放啦 爸爸好-中文儿歌FLASH 爸爸好爸爸好爸爸好爸爸的力气真不小做的多说的少脏活累活他全包了爸爸好爸爸好爸爸好爸爸的力气真不小挣的多说的少剩菜剩饭他全包了... 中文儿歌FLASH喜羊羊与灰太狼主题歌 喜羊羊，美羊羊，懒羊羊，沸羊羊，慢羊羊，软绵绵，红太狼，灰太狼。别看我只是一只羊，绿草因为我变得更香，天空因为我变得更蓝，白云因为我变得柔软。别看我只是一只羊，羊儿的聪明... [中文儿歌FLASH]蜗牛和黄鹂鸟 阿门阿前一棵葡萄树阿嫩阿嫩绿地刚发芽蜗牛背著那重重的壳呀一步一步地往上爬阿树阿上两只黄鹂鸟阿嘻阿嘻哈哈在笑它葡萄成熟还早地很哪现在上来干什么阿黄阿黄鹂儿不要笑... 中文儿歌FLASH 爸爸妈妈听我说 童星段丽阳推出北京电视台公益广告歌曲《爸爸妈妈听我说》flash完整版庆“六一”... flash儿歌花花我 妈妈总是对我说爸爸妈妈最爱我我却总是不明白爱是什么爸爸总是对我说爸爸妈妈最爱我我却总是不明白爱是什么如果真的爱我就陪陪我如果真的爱我就亲亲我如果真的爱我就夸夸我如果真的爱我就抱抱我... 儿歌FLASH好孩子要诚实 好孩子要诚实歌词：小花猫喵喵叫，喵喵叫是谁把花瓶打碎了？爸爸没看见，妈妈不知道小花猫对我叫，喵！喵！喵！... flash儿歌劳动最光荣 flash儿歌劳动最光荣歌词：太阳光金亮亮雄鸡唱三唱花儿醒来了鸟儿忙梳妆小喜鹊造新房小蜜蜂采蜜忙... 儿歌FLASH大公鸡 儿歌FLASH大公鸡大公鸡, 每天早起, 对着我窗口高给高给的, 高给高给催促我们快快上学去... 1《小宝贝快快睡》 小宝贝快快睡， 梦中会有我相随， 陪你笑陪你累，有我相依偎， 小宝贝快快睡， 你会梦到我几回，有我在梦最美，梦醒也安慰，花儿随流水， 日头抱春归， 粉面含笑微不露， 嘴角衔颗相思泪， 山间鸟徘徊，彩霞伴双飞， 惊鸿一蔑莫后退， 离开也让春风醉。 看蒙蒙的睡眼， 有谁值得你留恋， 同林鸟分飞雁， 一切是梦魇， 传说中神话里， 梦中的我在梦你， 神仙说梦会醒， 可是我不听， 流水葬落花，更凭添牵挂， 尝过相思百味苦， 从此对情更邋遢， 寒风催五谷， 遥风到天涯， 枯木也能发新芽， 馨香播种摇篮下。2.《左手右手》 在困难来临的时候，请你举起你的左手，左手代表着方向， 他不会向困难低头，当遇到挫折的时候，请你举起你的右手，右手代表着希望， 他不会为挫折发愁，当左手攀向右手， 我们的步伐就有节奏，当右手攀向左手， 我们的力量就有源头。3《虫儿飞》 黑黑的天空低垂， 亮亮的繁星相随， 虫儿飞虫儿飞， 你在思念谁， 天上的星星流泪，地上的玫瑰枯萎，冷风吹冷风吹， 只要有你陪， 虫儿飞花儿睡， 一双又一对才美，不怕天黑只怕心碎，不管累不累， 也不管东南西北 4《时钟在说话》听谁在说话， 叮咚叮咚 时钟在说话， 叮咚叮叮咚 时钟啊提醒我， 时间不停在溜走 时钟啊我知道， 时间走了不回头 听谁在说话， 叮咚叮咚 时钟在说话，叮咚叮叮咚 时钟啊提醒我， 时间不停在溜走 时钟啊我知道， 时间走了不回头 所以现在要努力， 功课要温习 等到考试才拼命， 一定来不及 听谁在说话， 叮咚叮咚 时钟在说话， 叮咚叮叮咚 谢谢它提醒我做事不要慢慢拖谢谢它提醒我用功才会有收获5《家族歌》 爸爸的爸爸叫什么？ 爸爸的爸爸叫爷爷； 爸爸的妈妈叫什么？ 爸爸的妈妈叫奶奶； 爸爸的哥哥叫什么？ 爸爸的哥哥叫伯伯； 爸爸的弟弟叫什么？ Howmany教案2 三年级英语教案 教 学 目 标 掌握句型：How many … can you see? I can see…It’s beautiful.理解名词复数及读音 教学重点句型：How many … can you see? I can see… It’s beautiful. 教学难点名词复数的读音。 教 具教材相配套的教学课教材相配套的教学录音带 教学 方法 教学过程：二次备课 （一）热身／复习（Warm-up/Revision） 1.Listen and do. Show me 1 and 2. …… 2.Guess（复习数字1－10） 教师举起右手做出各种表示数字的手势，让学生看好后，教师立即把手放到身后，让学生说出来。（速度由慢到快）也可小组竞赛，看哪组正确率高，在黑板上记分。 （二）呈现新课（Presentation） 1.教师分别出示画有苹果、香蕉、橘子、桃和梨的图片给学生：I have many fruits here. Do you want to know how many they are? OK，Let’s count. 2.数完后，教师问学生：How many apples/bananas/oranges/peaches/pears can you see? 让学生回答出：I can see eleven apples. I can see twelve bananas. I can see thirteen oranges. I can see fourteen peaches. I can see fifteen pears.（教师要注意及时纠正复数的错误读音。） 3.教师把图片贴到黑板上，指着图片教学生正确读出数字11－15。 4.教师拿出一个风筝问学生：What can you see? 让学生回答：I can see a kite. 教师接着问：Is it beautiful? 让学生回答：Yes, it’s beautiful. __________________________________________________ 小学英语三年级上下册单词表(外研版三年级 起点) 第一册MOUDLE 1I 我 am 是 hello（hi）你好goodbye（bye-bye）再见 how 怎样 are 是 you 你 good 好的morning 早上 fine 很好 thank 谢谢MODULE 2 Ms 女士too 也and 那么、和boy 男孩girl 女孩what 什么 is 是 your 你的 name 名字afternoon 下午Mr 先生 MODULE 3the 这个，那个，这些，那些door 门please 请 window 窗户 blackboard 黑板 bird 鸟 desk 桌子 chair 椅子 MODULE4 my 我的it 它 red 红 a（an）一个，一 panda 熊猫 blue 蓝 yellow 黄 green 绿 black 黑 dog 狗cat 猫 cap 帽子 MODULE 5many 许多 how many 有多少 one 一two 二three 三four 四five 五 six 六seven 七 eight 八nine 九ten 十 eleven 十一 twelve 十二 hat 帽子 MODULE6 this 这个school 学 校 pupil 小学生 classroom 教室 English 英语 teacher 教师that 那个bag 书包 pencil 铅笔pen 钢 笔book 书 MODULE 7happy 快乐 Birthday 生日 here 这里，这是 cake 蛋糕 old 年岁 how old 多大 look看 MODULE 8no 不 not 不，不是yes 是 的help 帮助kite 风筝where 哪里， 在哪里 in 在什么里面 bag 袋子 MODULE 9mother 母亲father 父亲 doctor 医生 grandpa 祖父，外祖 父grandma 祖母， 外祖母sister 姐妹 me 我（宾格） brother 兄弟driver 司机policeman 警 察farmer 农民 nurse 护士he 他 she 她 MODULE 10 body 身体 his 他的 head 头arm 手臂 leg 腿foot 脚her 她的eye 眼睛ear 耳朵mouth 嘴nose 鼻子 句子 Module 1 1、Hello... 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