- 3 By 1 Multiplication Worksheet
- Multiplication 1 10
- Multiplication 0 1 2 Worksheet
- Disk Graph 2 1 15 Multiplication Worksheets
- Disk Graph 2 1 15 Multiplication Worksheet
When it comes to explaining the basics of multiplication, many kids are thrown into the vat of multiplication facts and asked to memorize them, having absolutely no idea of what multiplication really is all about.
3 By 1 Multiplication Worksheet
The definition is a great place to start and with this Graph Grid Multiplication Game, your students will be understanding what it means in no time at all!
Grab these few supplies, and you are on your way:
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1. First off, place a sheet of graph paper between two or more kids. You can download our graph paper here, or use your own (see below for more information).
2. One after another, students roll the pair of dice. (These 10-sided dice are super cool! I love this set of 100 multi-sided dice.)
3. Students look at the two numbers rolled and create a grid. For example, if the student rolls a 3 and a 4, they draw a grid that is 3 high and 4 wide, or 4 high and 3 wide. The box is then labeled with the number and the answer. If students do not know the answer, they can count the squares. This is just fine as this action helps kids understand that multiplication is area and repeated addition!
4. Students color in their grid and/or place their initials inside. Players take turns rolling the dice and making grids.
5. They continue playing until a player rolls numbers that equal a grid that cannot be drawn on paper. If there are more than two players, play continues until each player rolls two numbers that do not fit.
6. Students count the number of squares in the grid to determine the winner. The winner has the most squares!
Want to see the game played?
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Notes: You can use the more common 6-sided dice, or you can invest in some 9 and 12-sided dice. They are perfect for this game. You can find them here.
You may use the graph paper you have at a home, but beware: The number of squares in your graph paper could significantly change the length of your game! We suggest using this free PDF 25X25 graph paper.
For beginners, you might want to consider this 20X20 PDF graph paper or even this 15X15 PDF graph paper. Sitesucker 2 11 6 – automatically downloads complete web sites.
![Disk Graph 2 1 15 Multiplication Disk Graph 2 1 15 Multiplication](https://media.cheggcdn.com/study/384/3841c8a9-c573-4800-9c42-af3ca614080d/image.png)
And for extended learning, we present to you:
This Array Math Art Activity will give students more time to practice multiplication. Students roll the dice and list all the different grids (or fact families) that can be formed using those two numbers. Repeat 15 times, then trace all the different grids on the paper, overlapping fact families. Color in your grids, and you have yourself some art worthy of the fridge! For a more complete explanation: Multiplication Array Art Activity.
If multiplication understanding is already mastered, then it's time to move on to this super fun competitive version of Multiplication War! Your students will forget they are practicing their times tables in no time!
Multiplication 1 10
Have fun!
The Cartesian product of graphs.
In graph theory, the Cartesian productGH of graphs G and H is a graph such that
- the vertex set of GH is the Cartesian productV(G) × V(H); and
- two vertices (u,u' ) and (v,v' ) are adjacent in GH if and only if either
- u = v and u' is adjacent to v' in H, or
- u' = v' and u is adjacent to v in G.
The Cartesian product of graphs is sometimes called the box product of graphs [Harary 1969].
The operation is associative, as the graphs (FG) H and F (GH) are naturally isomorphic.The operation is commutative as an operation on isomorphismclasses of graphs, and more strongly the graphs GH and HG are naturally isomorphic, but it is not commutative as an operation on labeled graphs.
The notation G × H has often been used for Cartesian products of graphs, but is now more commonly used for another construction known as the tensor product of graphs. The square symbol is an intuitive and unambiguous notation for the Cartesian product, since it shows visually the four edges resulting from the Cartesian product of two edges.[1]
Examples[edit]
- The Cartesian product of two edges is a cycle on four vertices: K2 K2 = C4.
- The Cartesian product of K2 and a path graph is a ladder graph.
- The Cartesian product of two path graphs is a grid graph.
- The Cartesian product of n edges is a hypercube:
- Thus, the Cartesian product of two hypercube graphs is another hypercube: Qi Qj = Qi+j.
- The Cartesian product of two median graphs is another median graph.
- The graph of vertices and edges of an n-prism is the Cartesian product graph K2 Cn.
- The rook's graph is the Cartesian product of two complete graphs.
Properties[edit]
Multiplication 0 1 2 Worksheet
If a connected graph is a Cartesian product, it can be factorized uniquely as a product of prime factors, graphs that cannot themselves be decomposed as products of graphs.[2] However, Imrich & Klavžar (2000) describe a disconnected graph that can be expressed in two different ways as a Cartesian product of prime graphs:
- (K1 + K2 + K22) (K1 + K23) = (K1 + K22 + K24) (K1 + K2),
where the plus sign denotes disjoint union and the superscripts denote exponentiation over Cartesian products.
A Cartesian product is vertex transitive if and only if each of its factors is.[3]
A Cartesian product is bipartite if and only if each of its factors is. More generally, the chromatic number of the Cartesian product satisfies the equation
- χ(G H) = max {χ(G), χ(H)}.[4]
The Hedetniemi conjecture states a related equality for the tensor product of graphs. The independence number of a Cartesian product is not so easily calculated, but as Vizing (1963) showed it satisfies the inequalities
- α(G)α(H) + min{|V(G)|-α(G),|V(H)|-α(H)} ≤ α(GH) ≤ min{α(G) |V(H)|, α(H) |V(G)|}.
The Vizing conjecture states that the domination number of a Cartesian product satisfies the inequality
Disk Graph 2 1 15 Multiplication Worksheets
- γ(GH) ≥ γ(G)γ(H).
The Cartesian product of unit distance graphs is another unit distance graph.[5]
Cartesian product graphs can be recognized efficiently, in linear time.[6]
Algebraic graph theory[edit]
Algebraic graph theory can be used to analyse the Cartesian graph product.If the graph has vertices and the adjacency matrix , and the graph has vertices and the adjacency matrix , then the adjacency matrix of the Cartesian product of both graphs is given by
- ,
where denotes the Kronecker product of matrices and denotes the identity matrix.[7] The adjacency matrix of the Cartesian graph product is therefore the Kronecker sum of the adjacency matrices of the factors.
Category theory[edit]
Viewing a graph as a category whose objects are the vertices and whose morphisms are the paths in the graph, the cartesian product of graphs corresponds to the funny tensor product of categories. The cartesian product of graphs is one of two graph products that turn the category of graphs and graph homomorphisms into a symmetricclosed monoidal category (as opposed to merely symmetric monoidal), the other being the tensor product of graphs.[8] The internal hom for the cartesian product of graphs has graph homomorphisms from to as vertices and 'unnatural transformations' between them as edges.[8]
History[edit]
According to Imrich & Klavžar (2000), Cartesian products of graphs were defined in 1912 by Whitehead and Russell. They were repeatedly rediscovered later, notably by Gert Sabidussi (1960).
Notes[edit]
- ^Hahn & Sabidussi (1997).
- ^Sabidussi (1960); Vizing (1963).
- ^Imrich & Klavžar (2000), Theorem 4.19.
- ^Sabidussi (1957).
- ^Horvat & Pisanski (2010).
- ^Imrich & Peterin (2007). For earlier polynomial time algorithms see Feigenbaum, Hershberger & Schäffer (1985) and Aurenhammer, Hagauer & Imrich (1992).
- ^Kaveh & Rahami (2005).
- ^ abWeber 2013.
References[edit]
Disk Graph 2 1 15 Multiplication Worksheet
- Aurenhammer, F.; Hagauer, J.; Imrich, W. (1992), 'Cartesian graph factorization at logarithmic cost per edge', Computational Complexity, 2 (4): 331–349, doi:10.1007/BF01200428, MR1215316.
- Feigenbaum, Joan; Hershberger, John; Schäffer, Alejandro A. (1985), 'A polynomial time algorithm for finding the prime factors of Cartesian-product graphs', Discrete Applied Mathematics, 12 (2): 123–138, doi:10.1016/0166-218X(85)90066-6, MR0808453.
- Hahn, Geňa; Sabidussi, Gert (1997), Graph symmetry: algebraic methods and applications, NATO Advanced Science Institutes Series, 497, Springer, p. 116, ISBN978-0-7923-4668-5.
- Horvat, Boris; Pisanski, Tomaž (2010), 'Products of unit distance graphs', Discrete Mathematics, 310 (12): 1783–1792, doi:10.1016/j.disc.2009.11.035, MR2610282.
- Imrich, Wilfried; Klavžar, Sandi (2000), Product Graphs: Structure and Recognition, Wiley, ISBN0-471-37039-8.
- Imrich, Wilfried; Klavžar, Sandi; Rall, Douglas F. (2008), Graphs and their Cartesian Products, A. K. Peters, ISBN1-56881-429-1.
- Imrich, Wilfried; Peterin, Iztok (2007), 'Recognizing Cartesian products in linear time', Discrete Mathematics, 307 (3–5): 472–483, doi:10.1016/j.disc.2005.09.038, MR2287488.
- Kaveh, A.; Rahami, H. (2005), 'A unified method for eigendecomposition of graph products', Communications in Numerical Methods in Engineering with Biomedical Applications, 21 (7): 377–388, doi:10.1002/cnm.753, MR2151527.
- Sabidussi, G. (1957), 'Graphs with given group and given graph-theoretical properties', Canadian Journal of Mathematics, 9: 515–525, doi:10.4153/CJM-1957-060-7, MR0094810.
- Sabidussi, G. (1960), 'Graph multiplication', Mathematische Zeitschrift, 72: 446–457, doi:10.1007/BF01162967, hdl:10338.dmlcz/102459, MR0209177.
- Vizing, V. G. (1963), 'The Cartesian product of graphs', Vycisl. Sistemy, 9: 30–43, MR0209178.
- Weber, Mark (2013), 'Free products of higher operad algebras', TAC, 28 (2): 24–65.
External links[edit]
- Weisstein, Eric W.'Graph Cartesian Product'. MathWorld.
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