Catalog Information
Title
Elementary Linear Algebra.
(Credit Hours:Lecture Hours:Lab Hours)
Offered
Prerequisite
Minimal learning outcomes
Upon completion of this course, the successful student will be able to:
- Use Gaussian elimination to do all of the following: solve a linear system with reduced row echelon form, solve a linear system with row echelon form and backward substitution, find the inverse of a given matrix, and find the determinant of a given matrix.
- Demonstrate proficiency at matrix algebra. For matrix multiplication demonstrate understanding of the associative law, the reverse order law for inverses and transposes, and the failure of the commutative law and the cancellation law.
- Use Cramer's rule to solve a linear system.
- Use cofactors to find the inverse of a given matrix and the determinant of a given matrix.
- Determine whether a set with a given notion of addition and scalar multiplication is a vector space. Here, and in relevant numbers below, be familiar with both finite and infinite dimensional examples.
- Determine whether a given subset of a vector space is a subspace.
- Determine whether a given set of vectors is linearly independent, spans, or is a basis.
- Determine the dimension of a given vector space or of a given subspace.
- Find bases for the null space, row space, and column space of a given matrix, and determine its rank.
- Demonstrate understanding of the Rank-Nullity Theorem and its applications.
- Given a description of a linear transformation, find its matrix representation relative to given bases.
- Demonstrate understanding of the relationship between similarity and change of basis.
- Find the norm of a vector and the angle between two vectors in an inner product space.
- Use the inner product to express a vector in an inner product space as a linear combination of an orthogonal set of vectors.
- Find the orthogonal complement of a given subspace.
- Demonstrate understanding of the relationship of the row space, column space, and nullspace of a matrix (and its transpose) via orthogonal complements.
- Demonstrate understanding of the Cauchy-Schwartz inequality and its applications.
- Determine whether a vector space with a (sesquilinear) form is an inner product space.
- Use the Gram-Schmidt process to find an orthonormal basis of an inner product space. Be capable of doing this both in R n and in function spaces that are inner product spaces.
- Use least squares to fit a line ( y = ax + b) to a table of data, plot the line and data points, and explain the meaning of least squares in terms of orthogonal projection.
- Use the idea of least squares to find orthogonal projections onto subspaces and for polynomial curve fitting.
- Find (real and complex) eigenvalues and eigenvectors of 2 × 2 or 3 × 3 matrices.
- Determine whether a given matrix is diagonalizable. If so, find a matrix that diagonalizes it via similarity.
- Demonstrate understanding of the relationship between eigenvalues of a square matrix and its determinant, its trace, and its invertibility/singularity.
- Identify symmetric matrices and orthogonal matrices.
- Find a matrix that orthogonally diagonalizes a given symmetric matrix.
- Know and be able to apply the spectral theorem for symmetric matrices.
- Know and be able to apply the Singular Value Decomposition.
- Correctly define terms and give examples relating to the above concepts.
- Prove basic theorems about the above concepts.
- Prove or disprove statements relating to the above concepts.
- Be adept at hand computation for row reduction, matrix inversion and similar problems; also, use MATLAB or a similar program for linear algebra problems.
Lesia M. Ohnivchuk
Abstract
The article considers way to extend the functionality of LMS Moodle when creating e-learning courses for the mathematical sciences, in particular e-learning courses "Elementary Mathematics" by using flash technology and Java-applets. There are examples of the use of flash-applications and Java-applets in the course "Elementary Mathematics".
Keywords
LMS Moodle; e-learning courses; technology flash; Java applet, GeoGebra
References
Brandão, L. O., "iGeom: a free software for dynamic geometry into the web", International Conference on Sciences and Mathematics Education, Rio de Janeiro, Brazil, 2002.
Brandão, L. O. and Eisnmann, A. L. K. “Work in Progress: iComb Project - a mathematical widget for teaching and learning combinatorics through exercises” Proceedings of the 39th ASEE/IEEE Frontiers in Education Conference, 2009, T4G_1–2
Kamiya, R. H and Brandão, L. O. “iVProg – a system for introductory programming through a Visual Model on the Internet. Proceedings of the XX Simpósio Brasileiro de Informática na Educação, 2009 (in Portuguese).
Moodle.org: open-source community-based tools for learning [Electronic resource]. – Access mode: http://www.moodle.org.
MoodleDocs [Electronic resource]. – Access mode: http://docs.moodle.org.
Interactive technologies: theory, practice, evidence: methodical guide to auto-installation: O. Pometun, L. Pirozhenko. – K.: APN; 2004. – 136 p.
Dmitry Pupinin. Question Type: Flash [Electronic resource]. – Access mode: https://moodle.org/mod/data/view.php?d=13&rid=2493&filter=1 – 02/26/14.
Andreev A.V., Gerasimenko P.S.. Using Flash and SCORM to create final control tasks [Electronic resource]. – Access mode: http://cdp.tti.sfedu.ru/index.php?option=com_content&task=view&id=1071&Itemid=363 –02.26.14.
GeoGebra. Materials [Electronic resource]. – Access mode: http://tube.geogebra.org.
Hohenvator M. Introduction to GeoGebra / M. Hohenvator / trans. T. S. Ryabova. – 2012. – 153 p.
REFERENCES (TRANSLATED AND TRANSLITERATED)
Brandão, L. O. "iGeom: a free software for dynamic geometry into the web", International Conference on Sciences and Mathematics Education, Rio de Janeiro, Brazil, 2002 (in English).
Brandão, L. O. and Eisnmann, A. L. K. “Work in Progress: iComb Project - a mathematical widget for teaching and learning combinatorics through exercises” Proceedings of the 39th ASEE/IEEE Frontiers in Education Conference, 2009, T4G_1–2 (in English).
Kamiya, R. H and Brandão, L. O. “iVProg – a system for introductory programming through a Visual Model on the Internet. Proceedings of the XX Simpósio Brasileiro de Informática na Educação, 2009 (in English)..
Moodle.org: open-source community-based tools for learning. – Available from: http://www.moodle.org (in English).
MoodleDocs. – Available from: http://docs.moodle.org (in English).
Pometun O. I., Pirozhenko L. V. Modern lesson, Kiev, ASK Publ., 2004, 192 p. (in Ukrainian).
Dmitry Pupinin. Question Type: Flash . – Available from: https://moodle.org/mod/data/view.php?d=13&rid=2493&filter=1 – 02/26/14 (in English).
Andreev A., Gerasimenko R. Using Flash and SCORM to create tasks final control. – Available from: http://cdp.tti.sfedu.ru/index.php?option=com_content&task=view&id=1071&Itemid=363 – 02.26.14 (in Russian).
GeoGebra Wiki. – Available from: http://www.geogebra.org (in English).
Hohenwarter M. Introduction to GeoGebra / M. Hohenwarter. – 2012. – 153 s. (in English).
DOI: https://doi.org/10.33407/itlt.v48i4.1249
Copyright (c) 2015 Lesia M. Ohnivchuk
In the traveling salesman problem, to form an optimal route around n cities, you need to choose the best one from (n-1)! options based on time, cost or route length. This problem involves determining a Hamiltonian cycle of minimum length. In such cases, the set of all possible solutions should be represented in the form of a tree - a connected graph that does not contain cycles or loops. The root of the tree unites the entire set of options, and the tops of the tree are subsets of partially ordered solution options.
Purpose of the service. Using the service, you can check your solution or get a new solution to the traveling salesman problem using two methods: the branch and bound method and the Hungarian method.
Mathematical model of the traveling salesman problem
The formulated problem is an integer problem. Let x ij =1 if the traveler moves from the i-th city to the j-th and x ij =0 if this is not the case.Formally, we introduce (n+1) a city located in the same place as the first city, i.e. the distances from (n+1) cities to any other city other than the first are equal to the distances from the first city. Moreover, if you can only leave the first city, then you can only come to the (n+1) city.
Let's introduce additional integer variables equal to the number of visits to this city along the way. u 1 =0, u n +1 =n. In order to avoid closed paths, leave the first city and return to (n+1), we introduce additional restrictions connecting the variables x ij and the variables u i (u i are non-negative integers).
U i -u j +nx ij ≤ n-1, j=2..n+1, i=1..n, i≠j, with i=1 j≠n+1
0≤u i ≤n, x in+1 =x i1 , i=2..n
Methods for solving the traveling salesman problem
- branch and bound method (Little's algorithm or subcycle elimination). An example of a branch and bound solution;
- Hungarian method. An example of a solution using the Hungarian method.
Little's algorithm or subcycle elimination
- Reduction operation along rows: in each row of the matrix, the minimum element d min is found and subtracted from all elements of the corresponding row. Lower limit: H=∑d min .
- Reduction operation by columns: in each column of the matrix, select the minimum element d min and subtract it from all elements of the corresponding column. Lower limit: H=H+∑d min .
- The reduction constant H is the lower bound of the set of all admissible Hamiltonian contours.
- Finding powers of zeros for a matrix given by rows and columns. To do this, temporarily replace the zeros in the matrix with the sign “∞” and find the sum of the minimum elements of the row and column corresponding to this zero.
- Select the arc (i,j) for which the degree of the zero element reaches the maximum value.
- The set of all Hamiltonian contours is divided into two subsets: the subset of Hamiltonian contours containing the arc (i,j) and those not containing it (i*,j*). To obtain a matrix of contours including arc (i,j), cross out row i and column j in the matrix. To prevent the formation of a non-Hamiltonian contour, replace the symmetric element (j,i) with the sign “∞”. Arc elimination is achieved by replacing the element in the matrix with ∞.
- The matrix of Hamiltonian contours is reduced with a search for the reduction constants H(i,j) and H(i*,j*) .
- The lower bounds of the subset of Hamiltonian contours H(i,j) and H(i*,j*) are compared. If H(i,j)
- If, as a result of branching, a (2x2) matrix is obtained, then the Hamiltonian contour obtained by branching and its length are determined.
- The length of the Hamiltonian contour is compared with the lower boundaries of the dangling branches. If the length of the contour does not exceed their lower boundaries, then the problem is solved. Otherwise, branches of subsets with a lower bound less than the resulting contour are developed until a route with a shorter length is obtained.
Example. Solve the traveling salesman problem with a matrix using Little's algorithm
| 1 | 2 | 3 | 4 | |
| 1 | - | 5 | 8 | 7 |
| 2 | 5 | - | 6 | 15 |
| 3 | 8 | 6 | - | 10 |
| 4 | 7 | 15 | 10 | - |
Solution. Let's take as an arbitrary route: X 0 = (1,2);(2,3);(3,4);(4,5);(5,1). Then F(X 0) = 20 + 14 + 6 + 12 + 5 = 57
To determine the lower bound of the set, we use reduction operation or reducing the matrix row by row, for which it is necessary to find the minimum element in each row of matrix D: d i = min(j) d ij
| i j | 1 | 2 | 3 | 4 | 5 | d i |
| 1 | M | 20 | 18 | 12 | 8 | 8 |
| 2 | 5 | M | 14 | 7 | 11 | 5 |
| 3 | 12 | 18 | M | 6 | 11 | 6 |
| 4 | 11 | 17 | 11 | M | 12 | 11 |
| 5 | 5 | 5 | 5 | 5 | M | 5 |
| i j | 1 | 2 | 3 | 4 | 5 |
| 1 | M | 12 | 10 | 4 | 0 |
| 2 | 0 | M | 9 | 2 | 6 |
| 3 | 6 | 12 | M | 0 | 5 |
| 4 | 0 | 6 | 0 | M | 1 |
| 5 | 0 | 0 | 0 | 0 | M |
d j = min(i) d ij
| i j | 1 | 2 | 3 | 4 | 5 |
| 1 | M | 12 | 10 | 4 | 0 |
| 2 | 0 | M | 9 | 2 | 6 |
| 3 | 6 | 12 | M | 0 | 5 |
| 4 | 0 | 6 | 0 | M | 1 |
| 5 | 0 | 0 | 0 | 0 | M |
| d j | 0 | 0 | 0 | 0 | 0 |
| i j | 1 | 2 | 3 | 4 | 5 |
| 1 | M | 12 | 10 | 4 | 0 |
| 2 | 0 | M | 9 | 2 | 6 |
| 3 | 6 | 12 | M | 0 | 5 |
| 4 | 0 | 6 | 0 | M | 1 |
| 5 | 0 | 0 | 0 | 0 | M |
The elements of the matrix d ij correspond to the distance from point i to point j.
Since there are n cities in the matrix, then D is an nxn matrix with non-negative elements d ij ≥ 0
Each valid route represents a cycle in which the traveling salesman visits the city only once and returns to the original city.
The route length is determined by the expression: F(M k) = ∑d ij
Moreover, each row and column is included in the route only once with the element d ij .
Step #1.
Determining the branching edge
| i j | 1 | 2 | 3 | 4 | 5 | d i |
| 1 | M | 12 | 10 | 4 | 0(5) | 4 |
| 2 | 0(2) | M | 9 | 2 | 6 | 2 |
| 3 | 6 | 12 | M | 0(5) | 5 | 5 |
| 4 | 0(0) | 6 | 0(0) | M | 1 | 0 |
| 5 | 0(0) | 0(6) | 0(0) | 0(0) | M | 0 |
| d j | 0 | 6 | 0 | 0 | 1 | 0 |
The largest sum of reduction constants is (0 + 6) = 6 for the edge (5,2), therefore, the set is divided into two subsets (5,2) and (5*,2*).
Edge exclusion(5.2) is carried out by replacing the element d 52 = 0 with M, after which we carry out the next reduction of the distance matrix for the resulting subset (5*,2*), as a result we obtain a reduced matrix.
| i j | 1 | 2 | 3 | 4 | 5 | d i |
| 1 | M | 12 | 10 | 4 | 0 | 0 |
| 2 | 0 | M | 9 | 2 | 6 | 0 |
| 3 | 6 | 12 | M | 0 | 5 | 0 |
| 4 | 0 | 6 | 0 | M | 1 | 0 |
| 5 | 0 | M | 0 | 0 | M | 0 |
| d j | 0 | 6 | 0 | 0 | 0 | 6 |
Enabling an edge(5.2) is carried out by eliminating all elements of the 5th row and 2nd column, in which the element d 25 is replaced by M to eliminate the formation of a non-Hamiltonian cycle.
| i j | 1 | 3 | 4 | 5 | d i |
| 1 | M | 10 | 4 | 0 | 0 |
| 2 | 0 | 9 | 2 | M | 0 |
| 3 | 6 | M | 0 | 5 | 0 |
| 4 | 0 | 0 | M | 1 | 0 |
| d j | 0 | 0 | 0 | 0 | 0 |
The lower bound of the subset (5,2) is equal to: H(5,2) = 35 + 0 = 35 ≤ 41
Since the lower boundary of this subset (5,2) is less than the subset (5*,2*), we include edge (5,2) in the route with a new boundary H = 35
Step #2.
Determining the branching edge and divide the entire set of routes relative to this edge into two subsets (i,j) and (i*,j*).
For this purpose, for all cells of the matrix with zero elements, we replace the zeros one by one with M (infinity) and determine for them the sum of the resulting reduction constants, they are given in parentheses.
| i j | 1 | 3 | 4 | 5 | d i |
| 1 | M | 10 | 4 | 0(5) | 4 |
| 2 | 0(2) | 9 | 2 | M | 2 |
| 3 | 6 | M | 0(7) | 5 | 5 |
| 4 | 0(0) | 0(9) | M | 1 | 0 |
| d j | 0 | 9 | 2 | 1 | 0 |
The largest sum of reduction constants is (0 + 9) = 9 for the edge (4,3), therefore, the set is divided into two subsets (4,3) and (4*,3*).
Edge exclusion(4.3) is carried out by replacing the element d 43 = 0 with M, after which we carry out the next reduction of the distance matrix for the resulting subset (4*,3*), as a result we obtain a reduced matrix.
| i j | 1 | 3 | 4 | 5 | d i |
| 1 | M | 10 | 4 | 0 | 0 |
| 2 | 0 | 9 | 2 | M | 0 |
| 3 | 6 | M | 0 | 5 | 0 |
| 4 | 0 | M | M | 1 | 0 |
| d j | 0 | 9 | 0 | 0 | 9 |
Enabling an edge(4.3) is carried out by eliminating all elements of the 4th row and 3rd column, in which the element d 34 is replaced by M to eliminate the formation of a non-Hamiltonian cycle.
After the reduction operation, the reduced matrix will look like:
| i j | 1 | 4 | 5 | d i |
| 1 | M | 4 | 0 | 0 |
| 2 | 0 | 2 | M | 0 |
| 3 | 6 | M | 5 | 5 |
| d j | 0 | 2 | 0 | 7 |
The lower bound of the subset (4,3) is equal to: H(4,3) = 35 + 7 = 42 ≤ 44
Since 42 > 41, we exclude the subset (5,2) for further branching.
We return to the previous plan X 1.
Plan X 1.
| i j | 1 | 2 | 3 | 4 | 5 |
| 1 | M | 12 | 10 | 4 | 0 |
| 2 | 0 | M | 9 | 2 | 6 |
| 3 | 6 | 12 | M | 0 | 5 |
| 4 | 0 | 6 | 0 | M | 1 |
| 5 | 0 | M | 0 | 0 | M |
| i j | 1 | 2 | 3 | 4 | 5 |
| 1 | M | 6 | 10 | 4 | 0 |
| 2 | 0 | M | 9 | 2 | 6 |
| 3 | 6 | 6 | M | 0 | 5 |
| 4 | 0 | 0 | 0 | M | 1 |
| 5 | 0 | M | 0 | 0 | M |
Determining the branching edge and divide the entire set of routes relative to this edge into two subsets (i,j) and (i*,j*).
For this purpose, for all cells of the matrix with zero elements, we replace the zeros one by one with M (infinity) and determine for them the sum of the resulting reduction constants, they are given in parentheses.
| i j | 1 | 2 | 3 | 4 | 5 | d i |
| 1 | M | 6 | 10 | 4 | 0(5) | 4 |
| 2 | 0(2) | M | 9 | 2 | 6 | 2 |
| 3 | 6 | 6 | M | 0(5) | 5 | 5 |
| 4 | 0(0) | 0(6) | 0(0) | M | 1 | 0 |
| 5 | 0(0) | M | 0(0) | 0(0) | M | 0 |
| d j | 0 | 6 | 0 | 0 | 1 | 0 |
The largest sum of reduction constants is (0 + 6) = 6 for the edge (4,2), therefore, the set is divided into two subsets (4,2) and (4*,2*).
Edge exclusion(4.2) is carried out by replacing the element d 42 = 0 with M, after which we carry out the next reduction of the distance matrix for the resulting subset (4*,2*), as a result we obtain a reduced matrix.
| i j | 1 | 2 | 3 | 4 | 5 | d i |
| 1 | M | 6 | 10 | 4 | 0 | 0 |
| 2 | 0 | M | 9 | 2 | 6 | 0 |
| 3 | 6 | 6 | M | 0 | 5 | 0 |
| 4 | 0 | M | 0 | M | 1 | 0 |
| 5 | 0 | M | 0 | 0 | M | 0 |
| d j | 0 | 6 | 0 | 0 | 0 | 6 |
Enabling an edge(4.2) is carried out by eliminating all elements of the 4th row and 2nd column, in which the element d 24 is replaced by M to eliminate the formation of a non-Hamiltonian cycle.
The result is another reduced matrix (4 x 4), which is subject to the reduction operation.
After the reduction operation, the reduced matrix will look like:
| i j | 1 | 3 | 4 | 5 | d i |
| 1 | M | 10 | 4 | 0 | 0 |
| 2 | 0 | 9 | M | 6 | 0 |
| 3 | 6 | M | 0 | 5 | 0 |
| 5 | 0 | 0 | 0 | M | 0 |
| d j | 0 | 0 | 0 | 0 | 0 |
The lower bound of the subset (4,2) is equal to: H(4,2) = 41 + 0 = 41 ≤ 47
Since the lower boundary of this subset (4,2) is less than the subset (4*,2*), we include edge (4,2) in the route with a new boundary H = 41
Step #2.
Determining the branching edge and divide the entire set of routes relative to this edge into two subsets (i,j) and (i*,j*).
For this purpose, for all cells of the matrix with zero elements, we replace the zeros one by one with M (infinity) and determine for them the sum of the resulting reduction constants, they are given in parentheses.
| i j | 1 | 3 | 4 | 5 | d i |
| 1 | M | 10 | 4 | 0(9) | 4 |
| 2 | 0(6) | 9 | M | 6 | 6 |
| 3 | 6 | M | 0(5) | 5 | 5 |
| 5 | 0(0) | 0(9) | 0(0) | M | 0 |
| d j | 0 | 9 | 0 | 5 | 0 |
The largest sum of reduction constants is (4 + 5) = 9 for the edge (1,5), therefore, the set is divided into two subsets (1,5) and (1*,5*).
Edge exclusion(1.5) is carried out by replacing the element d 15 = 0 with M, after which we carry out the next reduction of the distance matrix for the resulting subset (1*,5*), as a result we obtain a reduced matrix.
| i j | 1 | 3 | 4 | 5 | d i |
| 1 | M | 10 | 4 | M | 4 |
| 2 | 0 | 9 | M | 6 | 0 |
| 3 | 6 | M | 0 | 5 | 0 |
| 5 | 0 | 0 | 0 | M | 0 |
| d j | 0 | 0 | 0 | 5 | 9 |
Enabling an edge(1.5) is carried out by eliminating all elements of the 1st row and 5th column, in which the element d 51 is replaced by M to eliminate the formation of a non-Hamiltonian cycle.
As a result, we obtain another reduced matrix (3 x 3), which is subject to the reduction operation.
After the reduction operation, the reduced matrix will look like:
| i j | 1 | 3 | 4 | d i |
| 2 | 0 | 9 | M | 0 |
| 3 | 6 | M | 0 | 0 |
| 5 | M | 0 | 0 | 0 |
| d j | 0 | 0 | 0 | 0 |
The lower bound of the subset (1,5) is equal to: H(1,5) = 41 + 0 = 41 ≤ 50
Since the lower boundary of this subset (1,5) is less than the subset (1*,5*), we include edge (1,5) in the route with a new boundary H = 41
Step #3.
Determining the branching edge and divide the entire set of routes relative to this edge into two subsets (i,j) and (i*,j*).
For this purpose, for all cells of the matrix with zero elements, we replace the zeros one by one with M (infinity) and determine for them the sum of the resulting reduction constants, they are given in parentheses.
| i j | 1 | 3 | 4 | d i |
| 2 | 0(15) | 9 | M | 9 |
| 3 | 6 | M | 0(6) | 6 |
| 5 | M | 0(9) | 0(0) | 0 |
| d j | 6 | 9 | 0 | 0 |
The largest sum of reduction constants is (9 + 6) = 15 for the edge (2,1), therefore, the set is divided into two subsets (2,1) and (2*,1*).
Edge exclusion(2.1) is carried out by replacing the element d 21 = 0 with M, after which we carry out the next reduction of the distance matrix for the resulting subset (2*,1*), as a result we obtain a reduced matrix.
| i j | 1 | 3 | 4 | d i |
| 2 | M | 9 | M | 9 |
| 3 | 6 | M | 0 | 0 |
| 5 | M | 0 | 0 | 0 |
| d j | 6 | 0 | 0 | 15 |
Enabling an edge(2.1) is carried out by eliminating all elements of the 2nd row and 1st column, in which the element d 12 is replaced by M to eliminate the formation of a non-Hamiltonian cycle.
As a result, we obtain another reduced matrix (2 x 2), which is subject to the reduction operation.
After the reduction operation, the reduced matrix will look like:
| i j | 3 | 4 | d i |
| 3 | M | 0 | 0 |
| 5 | 0 | 0 | 0 |
| d j | 0 | 0 | 0 |
∑d i + ∑d j = 0
The lower bound of the subset (2,1) is equal to: H(2,1) = 41 + 0 = 41 ≤ 56
Since the lower boundary of this subset (2,1) is less than the subset (2*,1*), we include the edge (2,1) in the route with a new boundary H = 41.
In accordance with this matrix, we include edges (3,4) and (5,3) in the Hamiltonian route.
As a result, along the branching tree of the Hamiltonian cycle, the edges form:
(4,2), (2,1), (1,5), (5,3), (3,4). The route length is F(Mk) = 41
Decision tree.
| 1 | |||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| (5*,2*), H=41 | (5,2) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| (4*,2*), H=47 | (4,2) | (4*,3*), H=44 | (4,3) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| (1*,5*), H=50 | (1,5) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| (2*,1*), H=56 | (2,1) | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| (3,4) | (3*,4*), H=41 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
| (5,3) | (5*,3*), H=41 | ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
The SAT Math Test covers a range of mathematical methods, with an emphasis on problem solving, mathematical models, and the strategic use of mathematical knowledge.
SAT Math Test: just like in the real world
Instead of testing you on every math topic, the new SAT tests your ability to use the math you'll rely on most times and in many different situations. Math test questions are designed to reflect problem solving and models that you will be dealing with in
University studies, directly studying mathematics, as well as natural and social sciences;
- Your daily professional activities;
- Your daily life.
For example, to answer some questions, you will need to use several steps - because in the real world, situations where one simple step is enough to find a solution are extremely rare.
SAT Math Format
SAT Math Test: Basic Facts
The SAT Math section focuses on three areas of mathematics that play a leading role in most academic subjects in higher education and professional careers:
- Heart of Algebra: Fundamentals of algebra, which focuses on solving linear equations and systems;
- Problem Solving and Data Analysis: Problem solving and data analysis essential to general mathematical literacy;
- Passport to Advanced Math: Fundamentals of advanced mathematics, which asks questions that require manipulating complex equations.
The math test also draws on additional topics in mathematics, including geometry and trigonometry, which are most important for university studies and professional careers.
SAT Math Test: video
Basics of algebra
Heart of Algebra
This section of SAT Math focuses on algebra and the key concepts that are most important for success in college and career. It assesses students' ability to analyze, solve and construct linear equations and inequalities freely. Students will also be required to analyze and fluently solve equations and systems of equations using multiple methods. To fully assess knowledge of this material, problems will vary significantly in type and content. They can be quite simple or require strategic thinking and understanding, such as interpreting the interaction between graphical and algebraic expressions or presenting a solution as a reasoning process. Test takers must demonstrate not only knowledge of solution techniques, but also a deeper understanding of the concepts that underlie linear equations and functions. SAT Math Fundamentals of Algebra is scored on a scale of 1 to 15.
This section will contain tasks for which the answer is presented in multiple choice or independently calculated by the student. The use of a calculator is sometimes permitted, but not always necessary or recommended.
1. Construct, solve or interpret a linear expression or equation with one variable, in the context of some specific conditions. An expression or equation may have rational coefficients, and several steps may be required to simplify the expression or solve the equation.
2. Construct, solve or interpret linear inequalities with one variable, in the context of some specific conditions. An inequality may have rational coefficients and may require several steps to simplify or solve.
3. Construct a linear function that models a linear relationship between two quantities. The test taker must describe a linear relationship that expresses certain conditions using either an equation with two variables or a function. The equation or function will have rational coefficients, and several steps may be required to construct and simplify the equation or function.
4. Construct, solve and interpret systems of linear inequalities with two variables. The examinee will analyze one or more conditions existing between two variables by constructing, solving, or interpreting a two-variable inequality or system of two-variable inequalities, within certain specified conditions. Constructing an inequality or system of inequalities may require several steps or definitions.
5. Construct, solve and interpret systems of two linear equations in two variables. The examinee will analyze one or more conditions that exist between two variables by constructing, solving, or analyzing a system of linear equations, within certain specified conditions. The equations will have rational coefficients, and several steps may be required to simplify or solve the system.
6. Solve linear equations (or inequalities) with one variable. The equation (or inequality) will have rational coefficients and may require several steps to solve. Equations may have no solution, one solution, or an infinite number of solutions. The examinee may also be asked to determine the value or coefficient of an equation that has no solution or has an infinite number of solutions.
7. Solve systems of two linear equations with two variables. The equations will have rational coefficients, and the system may have no solution, one solution, or an infinite number of solutions. The examinee may also be asked to determine the value or coefficient of an equation in which the system may have no solution, one solution, or an infinite number of solutions.
8. Explain the relationship between algebraic and graphical expressions. Identify the graph described by a given linear equation or the linear equation that describes a given graph, determine the equation of a line given by verbally describing its graph, identify key features of the graph of a linear function from its equation, determine how a graph might be affected by changing its equation.
Problem solving and data analysis
Problem Solving and Data Analysis
This section of SAT Math reflects research that has identified what is important for success in college or university. Tests require problem solving and data analysis: the ability to mathematically describe a certain situation, taking into account the elements involved, to know and use various properties of mathematical operations and numbers. Problems in this category will require significant experience in logical reasoning.
Examinees will be required to know the calculation of average values of indicators, general patterns and deviations from the general picture and distribution in sets.
All problem solving and data analysis questions test examinees' ability to use their mathematical understanding and skills to solve problems they might encounter in the real world. Many of these issues are asked in academic and professional contexts and are likely to be related to science and sociology.
Problem Solving and Data Analysis is one of three subsections of SAT Math that are scored from 1 to 15.
This section will contain questions with multiple choice or self-calculated answers. Using a calculator here is always permitted, but not always necessary or recommended.
In this part of SAT Math, you may encounter the following questions:
1. Use ratios, rates, proportions, and scale drawings to solve single- and multi-step problems. Test takers will use a proportional relationship between two variables to solve a multi-step problem to determine a ratio or rate; Calculate the ratio or rate and then solve the multi-step problem using the given ratio or ratio to solve the multi-step problem.
2. Solve single and multi-step problems with percentages. The examinee will solve a multi-level problem to determine percentage. Calculate the percentage of a number and then solve a multi-level problem. Using a given percentage, solve a multi-level problem.
3. Solve single- and multi-step calculation problems. The examinee will solve a multi-level problem to determine the rate unit; Calculate a unit of measurement and then solve a multi-step problem; Solve a multi-level problem to complete the unit conversion; Solve a multi-stage density calculation problem; Or use the concept of density to solve a multi-step problem.
4. Using scatter diagrams, solve linear, quadratic, or exponential models to describe how variables are related. Given the scatterplot, select the equation of the line or curve of fit; Interpret the line in the context of the situation; Or use the line or curve that best suits the prediction.
5. Using the relationship between two variables, explore the key functions of the graph. The examinee will make connections between the graphical expression of data and the properties of the graph by selecting a graph that represents the described properties or using a graph to determine values or sets of values.
6. Compare linear growth with exponential growth. The examinee will need to match two variables to determine which type of model is optimal.
7. Using tables, calculate data for various categories of quantities, relative frequencies and conditional probabilities. The examinee uses data from various categories to calculate conditional frequencies, conditional probabilities, association of variables, or independence of events.
8. Draw conclusions about population parameters based on sample data. The examinee estimates the population parameter, taking into account the results of a random sample of the population. Sample statistics can provide confidence intervals and measurement error that the student must understand and use without having to calculate them.
9. Use statistical methods to calculate averages and distributions. Test takers will calculate the mean and/or distribution for a given set of data or use statistics to compare two separate sets of data.
10. Evaluate reports, draw conclusions, justify conclusions, and determine the appropriateness of data collection methods. Reports can consist of tables, graphs, or text summaries.
Fundamentals of Higher Mathematics
Passport to Advanced Math
This section of SAT Math includes topics that are especially important for students to master before moving on to advanced math. The key here is understanding the structure of expressions and the ability to analyze, manipulate and simplify those expressions. This also includes the ability to analyze more complex equations and functions.
Like the previous two sections of SAT Math, questions here are scored from 1 to 15.
This section will contain questions with multiple choice or self-calculated answers. The use of a calculator is sometimes permitted, but is not always necessary or recommended.
In this part of SAT Math, you may encounter the following questions:
1. Create a quadratic or exponential function or equation that models the given conditions. The equation will have rational coefficients and may require several steps to simplify or solve.
2. Determine the most appropriate form of expression or equation to identify a particular attribute, given the given conditions.
3. Construct equivalent expressions involving rational exponents and radicals, including simplification or conversion to another form.
4. Construct an equivalent form of the algebraic expression.
5. Solve a quadratic equation that has rational coefficients. The equation can be represented in a wide range of forms.
6. Add, subtract and multiply polynomials and simplify the result. The expressions will have rational coefficients.
7. Solve an equation in one variable that contains radicals or contains a variable in the denominator of the fraction. The equation will have rational coefficients.
8. Solve a system of linear or quadratic equations. The equations will have rational coefficients.
9. Simplify simple rational expressions. Test takers will add, subtract, multiply or divide two rational expressions or divide two polynomials and simplify them. The expressions will have rational coefficients.
10. Interpret parts of nonlinear expressions in terms of their terms. Test takers must relate given conditions to a nonlinear equation that models those conditions.
11. Understand the relationship between zeros and factors in polynomials and use this knowledge to construct graphs. Test takers will use the properties of polynomials to solve problems involving zeros, such as determining whether an expression is a factor of a polynomial, given the information provided.
12. Understand the relationship between two variables by establishing connections between their algebraic and graphical expressions. The examinee must be able to select a graph corresponding to a given nonlinear equation; interpret graphs in the context of solving systems of equations; select a nonlinear equation corresponding to the given graph; determine the equation of the curve taking into account the verbal description of the graph; identify key features of the graph of a linear function from its equation; determine the effect on the graph of changing the governing equation.
What does the SAT math section test?
General mastery of discipline
A math test is a chance to show that you:
Perform mathematical tasks flexibly, accurately, efficiently and using solution strategies;
- Solve problems quickly by identifying and using the most effective approaches to solution. This may include solving problems by
performing substitutions, shortcuts, or reorganization of information you provide;
Conceptual understanding
You will demonstrate your understanding of mathematical concepts, operations, and relationships. For example, you may be asked to make connections between the properties of linear equations, their graphs, and the terms they express.
Application of subject knowledge
Many SAT Math questions are taken from real-life problems and ask you to analyze the problem, identify the basic elements needed to solve it, express the problem mathematically, and find a solution.
Using the calculator
Calculators are important tools for performing mathematical calculations. To successfully study at a university, you need to know how and when to use them. In the Math Test-Calculator part of the test, you will be able to focus on finding the solution and analysis itself, because your calculator will help save your time.
However, a calculator, like any tool, is only as smart as the person using it. There are some questions on the Math Test where it is best not to use a calculator, even if you are allowed to do so. In these situations, test takers who can think and reason are likely to arrive at the answer before those who blindly use a calculator.
The Math Test-No Calculator portion makes it easy to evaluate your general knowledge of the subject and your understanding of certain math concepts. It also tests familiarity with computational techniques and understanding of number concepts.
Questions with answers entered into a table
Although most questions on the math test are multiple choice, 22 percent are questions where the answers are the result of the test taker's calculations - called grid-ins. Instead of choosing the correct answer from a list, you need to solve the problems and enter your answers into the grids provided on the answer sheet.
Answers entered into a table
Mark no more than one circle in any column;
- Only answers indicated by completing the circle will be counted (You will not receive points for everything written in the fields located above
circles).
- It doesn't matter in which column you start entering your answers; It is important that the answers are written inside the grid, then you will receive points;
- The grid can only contain four decimal places and can only accept positive numbers and zero.
- Unless otherwise specified in the task, answers can be entered into the grid as decimal or fractional;
- Fractions such as 3/24 do not need to be reduced to minimum values;
- All mixed numbers must be converted to improper fractions before being written into the grid;
- If the answer is a repeating decimal number, students must determine the most accurate values that will
consider.
Below is a sample of the instructions test takers will see on the SAT Math exam:
An elementary math curriculum for supplementary or home school should teach much more than the “how to” of simple arithmetic. A good math curriculum should have elementary math activities that build a solid foundation which is both deep and broad, conceptual and “how to”.
Time4Learning teaches a comprehensive math curriculum that correlates to state standards. Using a combination of multimedia lessons, printable worksheets, and assessments, the elementary math activities are designed to build a solid math foundation. It can be used as a , an , or as a for enrichment.
Time4Learning has no hidden fees, offers a 14-day money-back guarantee for brand new members, and allows members to start, stop, or pause at anytime. Try the interactive or view our to see what’s available.
Teaching Elementary Math Strategies
Children should acquire math skills using elementary math activities that teach a curriculum in a proper sequence designed to build a solid foundation for success. Let’s start with what appears to be a simple math fact: 3 + 5 = 8
This fact seems like a good math lesson to teach, once a child can count. But the ability to appreciate the concept “3 + 5 = 8” requires an understanding of these elementary math concepts:
- Quantity– realizing that numbers of items can be counted. Quantity is a common concept whether we are counting fingers, dogs or trees.
- Number recognition– knowing numbers by name, numeral, pictorial representation, or a quantity of the items.
- Number meaning– resolving the confusion between numbers referring to a quantity or to the position in a sequence (cardinal vs. ordinal numbers.
- Operations– Understanding that quantities can be added and that this process can be depicted with pictures, words, or numerals.
To paint a more extreme picture, trying to teach addition with “carrying over” prior to having a solid understanding of place value is a recipe for confusion. Only after mastering basic math concepts should a child try more advanced elementary math activities, like addition. Trying to teach elementary math strategies prior to mastering basic math concepts cause confusion, creating a sense of being lost or of being weak at math. A child can end up developing a poor self image or a negative view of math all because of a poor math curriculum.
It’s important to implement an elementary math curriculum that teaches math in a sequence, using elementary math activities that allow children to progressively build understanding, skills, and confidence. Quality teaching and curriculum follows a quality sequence.
Time4Learning teaches a personalized elementary math curriculum geared to your child’s current skill level. This helps to ensure that your child has a solid math foundation before introducing harder, more complex elementary math strategies. , included in the curriculum, provides practice in foundation skill areas that is necessary for success during elementary school. Get your child on the right path, about Time4Learning’s strategies for teaching elementary math.
Time4Learning's Elementary Math Curriculum
Time4Learning’s math curriculum contains a wide range of elementary math activities, which cover more than just arithmetic, math facts, and operations. Our elementary math curriculum teaches these five math strands.*
- Number Sense and Operations– Knowing how to represent numbers, recognizing ‘how many’ are in a group, and using numbers to compare and represent paves the way for grasping number theory, place value and the meaning of operations and how they relate to one another.
- Algebra– The ability to sort and order objects or numbers and recognizing and building on simple patterns are examples of ways children begin to experience algebra. This elementary math concept sets the groundwork for working with algebraic variables as a child’s math experience grows.
- Geometry and Spatial Sense– Children build on their knowledge of basic shapes to identify more complex 2-D and 3-D shapes by drawing and sorting. They then learn to reason spatially, read maps, visualize objects in space, and use geometric modeling to solve problems. children will be able to use coordinate geometry to eventually specify locations, give directions and describe spatial relationships.
- Measurement– Learning how to measure and compare involves concepts of length, weight, temperature, capacity and money. Telling the time and using money links to an understanding of the number system and represents an important life skill.
- Data Analysis and Probability– As children collect information about the world around them, they will find it useful to display and represent their knowledge. Using charts, tables, graphs will help them learn to share and organize data.
Elementary math curriculums that cover just one or two of these five math strands are narrow and lead to a weak understanding of math. Help your child build a strong, broad math foundation.
