Multiply and Conquer

This site provides a lesson plan that has students decompose 2-digit numbers, model area representations using the distributive property and partial product arrays, and align paper-and-pencil calculations with the arrays. The lesson provides conceptual understanding of what occurs in a 2-digit multiplication problem. Partial product models serve as transitions to understanding the standard multiplication algorithm. This lesson will allow for a deeper understanding of the two-digit multiplication process beyond the algorithm itself. This lesson also helps to reinforce the relationship between area and multiplication. Several options for assessment are also included. This lesson also includes enrichment activities to further the learning.

Weblink address (URL):

Standards & Objectives

Academic standards

CCSS.Math.Content.4.MD.A.2: Use the four operations to solve word problems involving distances, intervals of time, liquid volumes, masses of objects, and money, including...
CCSS.Math.Content.4.MD.B.4: Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Solve problems involving addition and subtraction of...
CCSS.Math.Content.4.NBT.A.3: Use place value understanding to round multi-digit whole numbers to any place.
CCSS.Math.Content.4.NBT.B.5: Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place...
CCSS.Math.Content.4.NBT.B.6: Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the...
CCSS.Math.Content.4.NF.B.3: Understand a fraction a/b with a > 1 as a sum of fractions 1/b.
CCSS.Math.Content.4.NF.B.4: Apply and extend previous understandings of multiplication to multiply a fraction by a whole number.
CCSS.Math.Content.4.OA.A.3: Solve multistep word problems posed with whole numbers and having whole-number answers using the four operations, including problems in which...
CCSS.Math.Content.4.OA.B.4: Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether...
GLE 0406.1.2: Apply and adapt a variety of appropriate strategies to problem solving, including estimation, and reasonableness of the solution.
GLE 0406.2.2: Develop fluency with multiplication and single-digit division.
GLE 0406.2.3: Identify prime and composite numbers.
GLE 0406.2.5: Add and subtract fractions with like and unlike denominators.
GLE 0406.2.6: Solve problems involving whole numbers, fractions, and/or decimals using all four arithmetic operations.
GLE 0406.4.2: Understand and use measures of length, area, capacity, and weight.
GLE 0406.4.3: Solve problems that involve estimating and measuring length, area, capacity and weight.
GLE 0406.5.1: Collect, record, arrange, present, and interpret data using tables and various representations.
SPI 0406.2.10: Solve contextual problems using whole numbers, fractions, and decimals.
SPI 0406.2.11: Solve problems using whole number multi-digit multiplication.
SPI 0406.2.12: Solve problems using whole number division with one- or two-digit divisors.
SPI 0406.2.3: Identify the place value of a specified digit in a number and the quantity it represents.
SPI 0406.2.4: Find factors, common factors, multiples, and common multiples of two numbers.
SPI 0406.2.8: Add and subtract proper fractions with like and unlike denominators and simplify the answer.
SPI 0406.3.1: Use letters and symbols to represent an unknown quantity and write a simple mathematical expression
SPI 0406.4.7: Determine appropriate size of unit of measurement in problem situations involving length, capacity or weight.
SPI 0406.4.9: Solve problems involving area and/or perimeter of rectangular figures.
SPI 0406.5.1: Depict data using various representations (e.g., tables, pictographs, line graphs, bar graphs).
SPI 0406.5.2: Solve problems using estimation and comparison within a single set of data.
TSS.Math.4.MD.A.2: Solve one- or two-step real-world problems involving whole number measurements with all four operations within a single system of measurement including...
TSS.Math.4.MD.B.4: Make a line plot to display a data set of measurements in fractions of a unit (1/2, 1/4, 1/8). Use operations on fractions for this grade to solve problems...
TSS.Math.4.NBT.A.3: Round multi-digit whole numbers to any place (up to and including the hundred-thousand place) using understanding of place value.
TSS.Math.4.NBT.B.5: Multiply a whole number of up to four digits by a one-digit whole number and multiply two two-digit numbers, using strategies based on place value and the...
TSS.Math.4.NBT.B.6: Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of...
TSS.Math.4.NF.B.3: Understand a fraction a/b with a > 1 as a sum of fractions 1/b. For example, 4/5 = 1/5 + 1/5 + 1/5 + 1/5.
TSS.Math.4.NF.B.4: Apply and extend previous understandings of multiplication as repeated addition to multiply a whole number by a fraction.
TSS.Math.4.OA.A.3: Solve multi-step contextual problems posed with whole numbers and having whole-number answers using the four operations, including problems in which...
TSS.Math.4.OA.B.4: Find all factor pairs for a whole number in the range 1-100. Recognize that a whole number is a multiple of each of its factors. Determine whether a given...

Alignment of this item to academic standards is based on recommendations from content creators, resource curators, and visitors to this website. It is the responsibility of each educator to verify that the materials are appropriate for your content area, aligned to current academic standards, and will be beneficial to your specific students.

Learning objectives:

Learning Objectives

Students will:

Create and solve multiplication stories.
Practice selected multiplication facts.

Essential and guiding questions:

Questions for Students:

How does your model show the answer to your multiplication problem?
Why do we decompose a 2-digit number?
Is this method easier than the traditional way or harder?
What do the partial products show?

Lesson Variations

Blooms taxonomy level:

Applying

Extension suggestions:

Extensions:

This model of multiplication can be extended to 3-digit numbers although it becomes less efficient. Some students may enjoy the challenge of being given larger numbers to distribute and multiply. Three-digit numbers should be decomposed into hundreds, tens, and ones. Since three dimensions are required, they must be modeled with base ten blocks rather than paper grids.
Ask students to write journal entries comparing array models with the traditional algorithm. Ask them to determine what is happening when they use a regrouping method and how it works.
To practice the array models, play a game whose object is to color in as many squares as possible on a sheet of graph paper. Roll 3 dice, 2 of one color and 1 of another color. The same-color dice represent a 2-digit number and the other die represents a single-digit number. Students color in the partial product arrays. The first to correctly color in as much of their paper as possible is the winner. For an extra challenge, have students play in pairs, coloring the same grid. This adds strategy to the game.
Instead of giving students the factors to use in a model, have them discover the best strategy on their own. Students need to think flexibly about the factors as they develop strategies for multiplicative thinking. For example, ask students to compute 14 × 6, but let them discover other factors that work, such as 6 × (10 + 4), 6 × (9 + 5), 6 × (2 + 12), etc.Use questioning, discussions, and mathematical communication throughout the process.

Helpful Hints

Materials:

Base 10 blocks
2 × 1 Multiplication Activity Sheet
2 × 2 Multiplication Activity Sheet
Computers or tablets with internet access (optional)

References

Contributors:

Angela Collins

Originator:

Ayers Institute for Learning & Innovation