However, this extension is problematic for many students, who start struggling with mathematics when fractions are introduced. Concerning these issues, mathematics educators have debated whether student difficulties with fractions and irrational numbers are exacerbated by viewing multiplication as repeated addition for a long time before these numbers are introduced, and relatedly whether it is acceptable to significantly modify rigorous mathematics for elementary education, leading children to believe statements that later turn out to be incorrect. From scaling to multiplication[ edit ] Multiplication can also be thought of as scaling.

Quantum equation predicts universe has no beginning February 9, by Lisa Zyga, Phys.

Note on the left the dramatic expansion not to scale occurring in the inflationary epoch, and at the center the expansion acceleration. The scheme is decorated with WMAP images on the left and with the representation write a repeated addition equation stars at the appropriate level of development.

The model may also account for dark matter and dark energy, resolving multiple problems at once. The widely accepted age of the universeas estimated by general relativityis In the beginning, everything in existence is thought to have occupied a single infinitely dense point, or singularity.

Only after this point began to expand in a "Big Bang" did the universe officially begin. Although the Big Bang singularity arises directly and unavoidably from the mathematics of general relativity, some scientists see it as problematic because the math can explain only what happened immediately after—not at or before—the singularity.

Ali and coauthor Saurya Das at the University of Lethbridge in Alberta, Canada, have shown in a paper published in Physics Letters B that the Big Bang singularity can be resolved by their new model in which the universe has no beginning and no end.

Old ideas revisited The physicists emphasize that their quantum correction terms are not applied ad hoc in an attempt to specifically eliminate the Big Bang singularity. Their work is based on ideas by the theoretical physicist David Bohm, who is also known for his contributions to the philosophy of physics.

Starting in the s, Bohm explored replacing classical geodesics the shortest path between two points on a curved surface with quantum trajectories. Raychaudhuri was also Das's teacher when he was an undergraduate student of that institution in the '90s.

Using the quantum-corrected Raychaudhuri equation, Ali and Das derived quantum-corrected Friedmann equations, which describe the expansion and evolution of universe including the Big Bang within the context of general relativity.

Although it's not a true theory of quantum gravitythe model does contain elements from both quantum theory and general relativity. Ali and Das also expect their results to hold even if and when a full theory of quantum gravity is formulated.

The student writes a word problem as well as the corresponding repeated addition and multiplication equations. The student clearly explains the relationship between repeated addition and multiplication. Division as Repeated Subtraction. This is a complete lesson with teaching and exercises, showing how division can be seen as repeated subtraction. Browse arrays and repeated addition resources on Teachers Pay Teachers, a marketplace trusted by millions of teachers for original educational resources. Students may have a scavenger hunt, play SCOOT or other fun yunusemremert.com will write an equation to express the total array as a sum of equal adden. Subjects: Arithmetic, Basic .

No singularities nor dark stuff In addition to not predicting a Big Bang singularity, the new model does not predict a "big crunch" singularity, either.

In general relativity, one possible fate of the universe is that it starts to shrink until it collapses in on itself in a big crunch and becomes an infinitely dense point once again.

Ali and Das explain in their paper that their model avoids singularities because of a key difference between classical geodesics and Bohmian trajectories. Classical geodesics eventually cross each other, and the points at which they converge are singularities.

In contrast, Bohmian trajectories never cross each other, so singularities do not appear in the equations. In cosmological terms, the scientists explain that the quantum corrections can be thought of as a cosmological constant term without the need for dark energy and a radiation term. These terms keep the universe at a finite size, and therefore give it an infinite age.

The terms also make predictions that agree closely with current observations of the cosmological constant and density of the universe. New gravity particle In physical terms, the model describes the universe as being filled with a quantum fluid.

The scientists propose that this fluid might be composed of gravitons—hypothetical massless particles that mediate the force of gravity. If they exist, gravitons are thought to play a key role in a theory of quantum gravity. In a related paper, Das and another collaborator, Rajat Bhaduri of McMaster University, Canada, have lent further credence to this model.

They show that gravitons can form a Bose-Einstein condensate named after Einstein and another Indian physicist, Satyendranath Bose at temperatures that were present in the universe at all epochs.

Motivated by the model's potential to resolve the Big Bang singularity and account for dark matter and dark energythe physicists plan to analyze their model more rigorously in the future. Their future work includes redoing their study while taking into account small inhomogeneous and anisotropic perturbations, but they do not expect small perturbations to significantly affect the results.

Ahmed Farag Ali and Saurya Das. Volume4 FebruaryPages — Saurya Das and Rajat K. Bhaduri, "Dark matter and dark energy from Bose-Einstein condensate", preprint:This is a complete lesson with teaching and exercises, showing how division can be seen as repeated subtraction.

It is meant for third grade. Students solve divisions by 'subtracting' or crossing out equal-size groups from the total in the visual model, until there is nothing left. Examples show how divisions can be solved by repeatedly subtracting . Division as Repeated Subtraction.

This is a complete lesson with teaching and exercises, showing how division can be seen as repeated subtraction. After understanding the exponential function, our next target is the natural logarithm..

Given how the natural log is described in math books, there’s little “natural” about it: it’s defined as the inverse of e^x, a strange enough exponent already.

Waves are everywhere. Whether we recognize it or not, we encounter waves on a daily basis. Sound waves, visible light waves, radio waves, microwaves, water waves, sine waves, cosine waves, stadium waves, earthquake waves, waves on a string, and slinky waves and are just a few of the examples of our daily encounters with waves.

In addition to waves, there are a variety of phenomena in our. Practice solving word problems by using one of the following multiplication strategies: create an array, skip counting, repeated addition, or writing a multiplication sentence.

Part two of three. Multiplication is defined as meaning that you have a certain number of groups of the same size. Then, it can be solved by repeated addition. Students fill in the missing parts in multiplication and addition sentences to match the given visual models (pictures).

They also draw pictures to .

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