### MATH

**Unit 10: Deepening Reasoning of Multiplication and Division Relationships and Extending Multiplication Fact Strategies**

Students need to work with area models of real base ten blocks and use pictorial representations of those models area models first before any other written strategies are shared. Once students understand those partial products, connections can be made to algorithms including the standard algorithm for recording those partial products.

Computation

Computation

Students multiply a two-digit number with a one-digit number using mental math strategies, using partial products and area models, and the commutative, associative, and distributive properties. All of these types of thinking involve students seeing ahead of time how large the solution might be (estimation and knowledge of place value.) Time spent with invented strategies will pay off in students understanding other algorithms. When students work with algorithms, they may progress through a series of ways to record their solutions. The standard algorithm is just a recording strategy that may or may not save time, may or may not produce efficiency or accuracy. The standard algorithm utilizes single digit thinking and has a specific recording style. Students need to see how these models of recording work in order to determine flexibly what the best strategy is for them concerning the numbers used in the problem.

Problem Solving - Multiplying with Larger Numbers

Problem Solving - Multiplying with Larger Numbers

Students solve multiplication problems including multiplicative comparisons with larger numbers. The CGI structures for multiplication will help students to determine what the situation is about. Students will use representations to help them understand the problem type. Students will solve one-step and two-step problems based on objects, pictorial models, arrays, area models, and equal groups, properties of operations, or recall of facts. Problem situations including bringing back up work with categorical data - solving problems that involve using a frequency table, dot plot, pictograph, or bar graph with scaled intervals. Students also use multiplicative comparison expressions to solve problems involving multiplication of a number and a comparative factor.

Representations of Multiplication with Larger Numbers

Representations of Multiplication with Larger Numbers

In working with multiplication problems, students use representations to help them understand what is happening in the situation. These include arrays, strip diagrams, and equations. Students need to use their understandings from the problem situation to develop these models that show those relationships and may help with comprehension.

Multiplication and Division Relationships

Multiplication and Division Relationships

Students solve problems where understanding the relationships between multiplication and division help support the multiplicative thinking necessary for these types of problems. STAAR has included solving problems given a contextual situation and having strip diagrams that display the relationship of a dividend with a given number of groups, but the amount in each group is missing as the resulting choices for answers. The strip diagram is similar to the one shown below:

In the above diagram, students use the relationship between multiplication and division when they find a missing factor that involves 18 groups x ___ = 54. Although students do not solve problems like 54 divided by 18 (just a one-digit divisor in grade 3), they can use problem solving and reasoning to figure this out as a multiplication with missing factor situation.

They also represent real-world relationships using number pairs in a table and verbal descriptions. Many of these problems are easier to solve if students used their understandings of the relationship between multiplication and division in figuring out the pattern in the table. Students may be given the verbal description then has to find or create the table of values that would work with that situation. They may also be given a table of values that models a relationship given in a problem situation and then has to make decisions about what can be true about the relationships having to find or create verbal descriptions that match the relationships in the table.

**Connecting Big Ideas****Operation Meanings & Relationships -**The same number sentence (e.g. 12-4=8) can be associated with different concrete or real-world situations, AND different number sentences can be associated with the same concrete or real-world situation. (TEKS 3.4K, 3.5B-C)

**Properties -**For a given set of numbers there are relationships that are always true, and these are the rules that govern arithmetic and algebra. (TEKS 3.4G, 3.4K)

**Basic Facts & Algorithms -**Basic facts and algorithms for operations with rational numbers use notions of equivalence to transform calculations into simpler ones. (TEKS 3.4G)

**Mathematical situations and structures can be translated and represented abstractly using variables, expressions, and equations. (TEKS 3.5B, 3.5D)**

Variable -

Variable -

**Relations & Functions -**Mathematical rules (relations) can be used to assign members of one set to members of another set. A special rule (function) assigns each member of one set to a unique member of the other set. (TEKS 3.5E)

**Equations & Inequalities -**

Rules of arithmetic and algebra can be used together with notions of equivalence to transform equations and inequalities so solutions can be found. (TEKS 3.5B, 3.5D)

__Essential Questions:__

Variable -

Variable -

Mathematical situations and structures can be translated and represented abstractly using variables, expressions, and equations. (TEKS 3.5B, 3.5D)

Relations & Functions -

Relations & Functions -

Mathematical rules (relations) can be used to assign members of one set to members of another set. A special rule (function) assigns each member of one set to a unique member of the other set. (TEKS 3.5E)

Equations & Inequalities -

Equations & Inequalities -

Rules of arithmetic and algebra can be used together with notions of equivalence to transform equations and inequalities so solutions can be found. (TEKS 3.5B, 3.5D)

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