© DMTI (2017) | Resource Materials

© DMTI (2017) | Resource Materials
The Developing Mathematical Thinking Institute (DMTI) is dedicated to enhancing students’ learning of mathematics by supporting educators in the implementation of research-based instructional strategies through high-quality professional development. For more information contact Dr. Brendefur at © DMTI (2017) | Resource Materials

Multiplication Fact Fluency
Supplemental Module © DMTI (2017) | Resource Materials

About this Supplemental Module
This module can be used by teachers at many different grade levels to support students’ fluency with basic multiplication facts. The process for developing fact fluency used in the DMTI modules focuses on practicing number relationships and making use of visual models and guided language structures. Students who lack fact fluency are often attempting to memorize facts individually and are not able to connect related facts in ways that help them recall the facts fluently. The DMTI approach to fact fluency has students progress gradually through a series of tasks that will not have students working particularly quickly in the early part of the module. Over time, though, the tasks accelerate students’ experiences and will support their recall of basic facts. The tasks in this module should be used for 5-15 minutes per day, 3-5 days per week for a period of weeks before a short break is taken. Then, coming back to tasks that students need more work with on a similar schedule will ultimately increase fact fluency. © DMTI (2017) | Resource Materials

Module Sequence Note to Teachers: This supplemental module may be used with a variety of grade levels and in many different instructional settings. It may be unnecessary to proceed from lesson to lesson depending on students’ prior knowledge or the intended purpose of using the module. Parts 2, 3 and 4 develop fluency concepts in a gradual progression from very informal to more complex and explicit understandings. You may find it useful to start at different points within the lessons depending on the needs of students. Part 1: Anchor Facts Part 2: Using 5 Part 3: Halving and Doubling Part 4: Using Squares Part 5: Using 10 Part 6: Practice © DMTI (2017) | Resource Materials

Part 1 Multiplication Anchor Facts © DMTI (2017) | Resource Materials

Note to Teachers Part 1 of this module serves as an introduction to the concept of Anchor Facts strategies which will be used throughout to support students’ development of basic fact fluency. For younger students who may not have much background with fact fluency, it may be best to begin with Part 2 and proceed through the module sequentially. The information in Part 1 may not be helpful to students who have never practiced facts in this way. However, the information in Part 1 can be useful for older students and teachers in order to focus and bring coherence to future learning and the DMTI approach to building fact fluency. © DMTI (2017) | Resource Materials

Part 1: Anchor Facts Let’s count by 5’s to solve 6 x 5. We will use a table to show these six multiples of 5. We made 6 iterations of a unit of 5. To iterate means to copy something over and over. 1 5 2 10 3 15 4 20 5 25 6 30 Students should copy the table in their notebooks/math journals. Define multiples as the, “…the numbers you create by multiplying.” Use the example of 2 x 5 = 10 and explain that 10 is a multiple of both 2 and 5. It is important that students learn to describe multiplication not only as “groups of” but also “iterations of a unit.” In later grades, students will solve contextual problems in which there are no whole groups. For example, ¼ x 5 describes situations in which you find the measure of ¼ of a unit of 5. This means you do not have any whole groups. If students arrive at this point in their learning understanding multiplication as a process of iterating, it is easier for them to adjust to this new situation and realize that they are now just doing the opposite of iterating which is partitioning. However, at the early stages of learning, a vision of multiplication as “groups of” is also helpful. 6 iterations of a unit of 5 is written as 6 x 5. We can also say, “6 groups of 5.” So if 6 x 5 = 30, what would 7 x 5 be and how does 7 x 5 relate to 6 x 5? © DMTI (2017) | Resource Materials

Part 1 So if 6 x 5 = 30, what would 7 x 5 be and how does 7 x 5 relate to 6 x 5? 1 5 2 10 3 15 4 20 5 25 6 30 7 35 Let’s use a bar model to show how 6 x 5 and 7 x 5 are related. 6 x 5 = 30 1 x 5 = 5 7 x 5 = 35 Students should share their ideas prior to being given the bar model example. Have students copy the bar model through the first few iterations and then see if they can complete it independently. Students should be asked to describe where 6 x 5 and 1 x 5 are in the model before the examples are given. 1 5 2 10 5 3 15 5 4 20 5 25 5 6 30 5 7 35 5 5 © DMTI (2017) | Resource Materials

Part 1 Before we look at the four most helpful anchor facts for multiplication, let’s make sure we understand a very important concept called the Commutative Property. You already have a bar model for 7 x 5. Think about what a bar model for 5 x 7 (5 iterations of a unit of 7) would look like and try to draw the model for 5 x 7. The Commutative Property says that you can put the numbers in a multiplication problem in any order and the product will be the same. 7 x 5 = 5 x 7 Define product as, “the result of multiplying.” © DMTI (2017) | Resource Materials

Part 1 What is the same and what is different between 7 x 5 and 5 x 7? The product of 7 x 5 and 5 x 7 is the same. The product is 35. 7 x 5 and 5 x 7 are different because we are iterating different units. There are many situations where the order of the numbers in a multiplication problem are very important. But, when we are trying to increase our fact fluency, we can use the Commutative Property to order the numbers in a way that makes the problem easier to think about. It will be helpful for students to be comfortable using the commutative property when applying anchor facts strategies. If students’ thinking is not flexible in this way, they will likely struggle to find ways to use anchor facts in which the first fact is decomposed as in using 5 x 7 as an anchor fact to solve 6 x 7. If students can only think of 5 x 7 as “5 iterations of a unit of 7” they will have to count by 7’s to solve the anchor fact if they do not already know it. This is a valuable skill to explore in situations in which models and contexts are the focus of the lesson. However, when fluency development is the focus of the lesson it is helpful for students to think of the facts in the most accessible way. It is likely much easier for students to think of 5 x 7 as iterating 5’s a total of 7 times if they have not memorized the fact already. Therefore, the Commutative Property is extremely useful when developing fact fluency. © DMTI (2017) | Resource Materials

Part 1 Using 6 x 5 to solve 7 x 5 is an example of an anchor fact. Anchors are heavy metal hooks boats use to keep them in place in the water. The anchor is hooked to the ground under the water. Anchor facts are used in the same way. We connect facts that are difficult to remember to the easier anchor fact. We solved 7 x 5 by thinking of the problem as 6 x 5 and 1 x 5. We also used a model to show our answer was correct. 6 x 5 is an anchor fact because 5’s are easy numbers to multiply. © DMTI (2017) | Resource Materials

Part 1 There are four anchor facts we can use to solve multiplication facts. The four anchor facts for multiplication fluency are: Using 5 Halving and Doubling Using Squares Using 10 We previously used bar models to model multiplication facts. Now we will use special types of arrays called area models to give examples of each of the anchor facts. Area models are helpful when decomposing multiplication facts because they can be decomposed in more ways than bar models. If students need clarification on the difference between arrays and area models, explain that all area models are arrays and that area models are just special arrays composed of square units that are all connected. Drawing a 3 x 4 array of dots and a 3 x 4 array of connected square units and discuss how the models show the same product but the area models are covering a continuous area and the arrays are individual dots. There are more examples of this topic in the Grade 3 and Grade 4 modules and materials. © DMTI (2017) | Resource Materials

Part 1 Let’s look at each of the anchor facts using this multiplication fact: 6 x 8 Materials needed: students need at least one sheet of 1 cm grid paper (available from the link below) © DMTI (2017) | Resource Materials

Part 1 8 6 x 8 Let’s practice Using 5 to solve 6 x 8. Where do you see 6 x 5 in the model? How does 6 x 5 help you solve 6 x 8? 6 Students should draw the area model on their grid paper. The orientation of the area model (6 rows of 8) was determined to be the more common presentation in the first grades students tend to encounter multiplication fact fluency practice (Grades 3 and 4). In later grades, the orientation will likely rotate so that the 6 is the horizontal length in accordance with the conventional order of ordered pairs in the coordinate plane. Students should ideally be flexible in the way they think about area models, but for this module the models will be consistently presented like the example shown above. © DMTI (2017) | Resource Materials

Part 1 8 6 x 8 Let’s practice Using 5 to solve 6 x 8. Where do you see 6 x 5 in the model? How does 6 x 5 help you solve 6 x 8? 6x5=30 6 Students should draw the area model on their grid paper. © DMTI (2017) | Resource Materials

Part 1 What happened to the 8 in 6 x 8? We decomposed 8 into 5 and 3. 8 6 x 8 Let’s practice Using 5 to solve 6 x 8. Where do you see 6 x 5 in the model? How does 6 x 5 help you solve 6 x 8? 6x5=30 6x3=18 6 Students should draw the area model on their grid paper. Students should count rows or columns of the grid squares if they do not know the anchor fact. Over time, they will remember these anchor facts more easily than the more difficult related facts. 6 x 5 = 30 6 x 3 = 18 6 x 8 = 48 © DMTI (2017) | Resource Materials

Part 1 8 6 x 8 Let’s practice Using 5 to solve 6 x 8. Where do you see 5 x 8 in the model? How does 5 x 8 help you solve 6 x 8? 6 Students should draw the area model on their grid paper. © DMTI (2017) | Resource Materials

Part 1 8 6 x 8 1x8=8 Let’s practice Using 5 to solve 6 x 8. Where do you see 5 x 8 in the model? How does 5 x 8 help you solve 6 x 8? 6 5x8=40 Students should draw the area model on their grid paper. Students should count rows or columns of the grid squares if they do not know the anchor fact. Over time, they will remember these anchor facts more easily than the more difficult related facts. 5 x 8 = 40 1 x 8 = 8 6 x 8 = 48 © DMTI (2017) | Resource Materials

Part 1 8 6 x 8 Let’s practice Halving and Doubling to solve 6 x 8. How could you partition 6 in half? How does half of 6 help you solve 6 x 8? 6 Students should draw the area model on their grid paper. Define partition as “splitting into equal parts.” © DMTI (2017) | Resource Materials

Part 1 This strategy is called Halving and Doubling because you partition one part of the multiplication fact in half and then double that amount to find the total product. 8 6 x 8 Let’s practice Halving and Doubling to solve 6 x 8. How could you partition 6 in half? How does half of 6 help you solve 6 x 8? 3x8=24 6 3x8=24 Students should draw the area model on their grid paper. Define partition as “splitting into equal parts.” 3 x 8 = 24 6 x 8 = 48 © DMTI (2017) | Resource Materials

Part 1 8 6 x 8 Let’s practice Halving and Doubling to solve 6 x 8. How could you partition 8 in half? How does half of 8 help you solve 6 x 8? 6 Students should draw the area model on their grid paper. Define partition as “splitting into equal parts.” © DMTI (2017) | Resource Materials

Part 1 8 6 x 8 Let’s practice Halving and Doubling to solve 6 x 8. How could you partition 8 in half? How does half of 8 help you solve 6 x 8? 6x4=24 6x4=24 6 Students should draw the area model on their grid paper. Define partition as “splitting into equal parts.” 6 x 4 = 24 6 x 8 = 48 © DMTI (2017) | Resource Materials

Part 1 8 6 x 8 Let’s practice Using Squares to solve 6 x 8. Where do you see 6 x 6 in the model? How does 6 x 6 help you solve 6 x 8? 6 Students should draw the area model on their grid paper. Remind students that any number multiplied by itself (e.g. 6 x 6) is called a “square” because the associated area model is the shape of a square. © DMTI (2017) | Resource Materials

Part 1 8 6 x 8 6x2=12 Let’s practice Using Squares to solve 6 x 8. Where do you see 6 x 6 in the model? How does 6 x 6 help you solve 6 x 8? 6 6x6=36 Students should draw the area model on their grid paper. Students should count rows or columns of the grid squares if they do not know the anchor fact. Over time, they will remember these anchor facts more easily than the more difficult related facts. 6 x 6 = 36 6 x 2 = 12 6 x 8 = 48 © DMTI (2017) | Resource Materials

Part 1 10 6 x 8 Let’s practice Using 10 to solve 6 x 8. Where do you see 6 x 8 in the model? How does 6 x 10 help you solve 6 x 8? 6 Students should draw the area model on their grid paper. © DMTI (2017) | Resource Materials

Part 1 10 6 x 8 Let’s practice Using 10 to solve 6 x 8. Where do you see 6 x 8 in the model? How does 6 x 10 help you solve 6 x 8? 6x8=48 6x2=12 6 Students should draw the area model on their grid paper. Students should count rows or columns of the grid squares if they do not know the anchor fact. Over time, they will remember these anchor facts more easily than the more difficult related facts. Students may be familiar with the concept of “compensation” in which you change numbers in a problem to make the problem easier and then adjust the answer to account for the way in which the numbers were changed. This is an example of compensation with multiplication. Students are more likely to have used compensation for problems such as = in prior grades. 6 x 10 = 60 -(6 x 2 = 12) 6 x 8 = 48 © DMTI (2017) | Resource Materials

Note to Teachers Several of the following sections of this module provide experiences for students to develop proficiency with using specific anchor facts to increase multiplication fact fluency. The lessons are often presented with the assumption students will be using graph paper to visually model the composing and decomposing of multiplication facts using area models. There are notes indicating the possible need for the use of enactive (physical) models such as square tiles or cubes. However, not all students need to enactively (physically) model their strategies. Teachers are encouraged to use their best judgement about whether to use the tiles or counters along with the visual (iconic) models or whether to just use the visual models. © DMTI (2017) | Resource Materials

Part 2 Materials needed: Students need access to several sheets of 1 cm graph paper (or a single sheet placed inside a transparent sleeve to be used with a dry erase marker). Students may need square tiles or cubes to enactively (physically) model the anchor facts strategies prior to drawing them visually. Multiplication Using 5 © DMTI (2017) | Resource Materials

Part 2: Using 5 Because it is often easy to count by 5’s, we are going to use multiples of 5 to help solve multiplication facts. Using 5 will be our first anchor fact for multiplication fluency. For each of the given facts, try to think of a x5 multiplication fact that could be used to help you solve the given fact. Then, model the x5 fact and use the sentence frame to describe your strategy. We will go through this process together for the first fact. © DMTI (2017) | Resource Materials

Part 2 What are some x5 facts that could help us solve this problem? We could decompose 6 in to 5 and 1 and use 5 x 8. We could also decompose 8 in to 5 and 3 and use 6 x 5. 6 x 8 8 I know _____ x _____ = _____. 6 I also know _____ x _____ = _____. Remind students that decomposing is “breaking apart.” Have students try to model and describe their strategies before presenting the examples. If students do not know 6 x 5 or 5 x 8, have them count rows or columns of 5 in their models to determine these partial products. Over time, the multiples of 5 will be committed to memory if students are connecting them to models and using x5 facts to solve more difficult facts. _____ + _____ = _____ So, that means _____ x _____ = _____. © DMTI (2017) | Resource Materials

Part 2 6 x 8 8 I know _____ x _____ = _____. 6 x 5 = 30 6 x 3 = 18 6 I also know _____ x _____ = _____. Have students read through the sentence frame and make sure they can describe the strategy using the model. To explain the use of parentheses in the equation below the area model, describe parentheses as the symbol used to “group” different operations so that we know to complete the operations inside the parentheses before any other part of the equation. _____ + _____ = _____ 6 x 5 = 30 6 x 3 = 18 6 x 8 = 48 (6x5) + (6x3) = 6 x 8 = 48 So, that means _____ x _____ = _____. © DMTI (2017) | Resource Materials

Part 2 6 x 8 8 1 x 8 = 8 I know _____ x _____ = _____. 6 I also know _____ x _____ = _____. 5 x 8 = 40 Have students read through the sentence frame and make sure they can describe the strategy using the model. To explain the use of parentheses in the equation below the area model, describe parentheses as the symbol used to “group” different operations so that we know to complete the operations inside the parentheses before any other part of the equation. _____ + _____ = _____ 5 x 8 = 40 1 x 8 = 8 6 x 8 = 48 (5x8) + (1x8) = 6 x 8 = 48 So, that means _____ x _____ = _____. © DMTI (2017) | Resource Materials

Part 2 Now we will practice Using 5 for many more facts. Use models and the sentence frame to show and describe your Using 5 strategy for each fact. Students can work through each example individually, in pairs, in small groups or as a whole class. Students may use different multiples of 5 (e.g. 5 x 8 or 6 x 5 for 6 x 8). Encourage them to explore more than one x5 fact and then decide which they prefer. Some facts will not have two different x5 facts that are sensible, however others may use x5 as a compensation strategy. For example, 7x4 = (7x5) – (7x1). The following slides could also be printed and used as “practice packet.” © DMTI (2017) | Resource Materials

Part 2 6 x 7 7 I know _____ x _____ = _____. 6 I also know _____ x _____ = _____. _____ + _____ = _____ So, that means _____ x _____ = _____. © DMTI (2017) | Resource Materials

Part 2 6 x 9 9 I know _____ x _____ = _____. I also know _____ x _____ = _____. 6 _____ + _____ = _____ So, that means _____ x _____ = _____. © DMTI (2017) | Resource Materials

Part 2 7 9 x 7 I know _____ x _____ = _____. I also know _____ x _____ = _____. 9 _____ + _____ = _____ So, that means _____ x _____ = _____. © DMTI (2017) | Resource Materials

Part 3 Materials needed: Students need access to several sheets of 1 cm graph paper (or a single sheet placed inside a transparent sleeve to be used with a dry erase marker). Students may need square tiles or cubes to enactively (physically) model the anchor facts strategies prior to drawing them visually. Multiplication Halving and Doubling © DMTI (2017) | Resource Materials

Part 3: Halving and Doubling
When multiplication facts have large numbers in them, sometimes it is easier to partition one of the large numbers in half and them double that product to find the answer to the original problem. Doubling is something that is usually pretty easy to do. What is 3 doubled? 2 x 3 = = 6 So if 2 x 3 = 6, what will the product of 4 x 6 be? © DMTI (2017) | Resource Materials

Part 3 4 x 6 = (2 x 6) + (2 x 6) = = 24 2 x 6 = 12 = 24 4 x 6 = 24 6 2x6=12 4 2x6=12 Students should model the area model and describe how either notation example matches the model and the strategy. Younger students should focus on the series of equations in blue while more experienced (or older) students should focus on the equation using the parentheses. What would the area model look like if we used 4 x 3 to solve 4 x 6? © DMTI (2017) | Resource Materials

Part 3 4 x 6 = (4 x 3) + (4 x 3) = = 24 4 x 3 = 12 = 24 4 x 6 = 24 6 4 4x3=12 4x3=12 Students should model the area model and describe how either notation example matches the model and the strategy. Younger students should focus on the series of equations in blue while more experienced (or older) students should focus on the equation using the parentheses. © DMTI (2017) | Resource Materials

Part 3 6 Both of these examples show a Halving and Doubling strategy. Some multiplication facts are easy to partition in half and then double to find the total product. This won’t work for all facts, but can make some very difficult facts much easier. Let’s practice this strategy. 2x6=12 4 2x6=12 6 Define partition as “splitting into equal parts.” 4 4x3=12 4x3=12 © DMTI (2017) | Resource Materials

Part 3 How could we partition one of the numbers in half and then double that product? We could partition 6 into 3 and 3 or 8 into 4 and 4. 6 x 8 8 I know _____ x _____ = _____. 6 Remind students that partitioning is “splitting into equal parts.” Have students try to model and describe their strategies before presenting the examples. If students do not know 6 x 4 or 3 x 8, have them count rows or columns in their models to determine these partial products. Over time, these multiples will be committed to memory if students are connecting them to models and using the facts to solve more difficult facts. _____ + _____ = _____ So, that means _____ x _____ = _____. © DMTI (2017) | Resource Materials

Part 3 6 x 8 8 I know _____ x _____ = _____. 6 x 4 = 24 6 x 4 = 24 6 Have students read through the sentence frame and make sure they can describe the strategy using the model. To explain the use of parentheses in the equation below the area model, describe parentheses as the symbol used to “group” different operations so that we know to complete the operations inside the parentheses before any other part of the equation. _____ + _____ = _____ 6 x 4 = 24 6 x 8 = 48 (6x4) + (6x4) = 6 x 8 = 48 So, that means _____ x _____ = _____. © DMTI (2017) | Resource Materials

Part 3 6 x 8 8 I know _____ x _____ = _____. 3 x 8 = 24 6 3 x 8 = 24 Have students read through the sentence frame and make sure they can describe the strategy using the model. To explain the use of parentheses in the equation below the area model, describe parentheses as the symbol used to “group” different operations so that we know to complete the operations inside the parentheses before any other part of the equation. _____ + _____ = _____ 3 x 8 = 24 6 x 8 = 48 (3x8) + (3x8) = 6 x 8 = 48 So, that means _____ x _____ = _____. © DMTI (2017) | Resource Materials

Part 3 Now we will practice using Halving and Doubling for many more facts. Use models and the sentence frame to show and describe your Halving and Doubling strategy for each fact. Students can work through each example individually, in pairs, in small groups or as a whole class. Students may use different halving and doubling strategies in the case of both factors being even. Some students may need to explore why using a halving and doubling strategy with odd factors is not a particularly helpful strategy to build fluency (e.g. 6x9 is easier to solve if the 6 is partitioned in half but not if the 9 is partitioned in half). © DMTI (2017) | Resource Materials

Part 3 6 x 7 7 I know _____ x _____ = _____. 6 _____ + _____ = _____ So, that means _____ x _____ = _____. © DMTI (2017) | Resource Materials

Part 3 8 9 x 8 I know _____ x _____ = _____. 9 _____ + _____ = _____ So, that means _____ x _____ = _____. © DMTI (2017) | Resource Materials

Part 4 Multiplication Using squares © DMTI (2017) | Resource Materials

Part 4: Using Squares When you multiply a number by itself, you compose an area that is the shape of a square. Draw an area model of 6 x 6. Notice that the area is a square. Square facts tend to be easy to remember. We will practice using squares to solve facts. 6 6 Materials needed: Students need access to several sheets of 1 cm graph paper (or a single sheet placed inside a transparent sleeve to be used with a dry erase marker). Students may need square tiles or cubes to enactively (physically) model the anchor facts strategies prior to drawing them visually. © DMTI (2017) | Resource Materials

Part 4 What is a square fact that could help us solve this problem? We could decompose 8 into 6 and 2 and then use 6 x 6 to solve 6 x 8. 6 x 8 8 I know _____ x _____ = _____. 6 I also know _____ x _____ = _____. Remind students that decomposing is “breaking apart.” Have students try to model and describe their strategies before presenting the examples. If students do not know 6 x 6, have them count rows or columns of 6 in their models to determine this partial product. Over time, the squares will be committed to memory if students are connecting them to models and using square facts to solve more difficult facts. _____ + _____ = _____ So, that means _____ x _____ = _____. © DMTI (2017) | Resource Materials

Part 4 6 x 8 8 I know _____ x _____ = _____. 6 x 6 = 36 6 x 2 = 12 6 I also know _____ x _____ = _____. Have students read through the sentence frame and make sure they can describe the strategy using the model. To explain the use of parentheses in the equation below the area model, describe parentheses as the symbol used to “group” different operations so that we know to complete the operations inside the parentheses before any other part of the equation. _____ + _____ = _____ 6 x 6 = 36 6 x 2 = 12 6 x 8 = 48 (6x6) + (6x2) = 6 x 8 = 48 So, that means _____ x _____ = _____. © DMTI (2017) | Resource Materials

Part 4 Now we will practice using Squares for many more facts. Use models and the sentence frame to show and describe how you used a Square fact to solve the more difficult fact. Students can work through each example individually, in pairs, in small groups or as a whole class. Students may need to model the facts with tiles, cubes or counters. If students are unfamiliar with the square facts, encourage them to solve the square fact by counting rows or columns in their models. Over time they will commit the squares to memory by following this process and using squares to solve more difficult facts. © DMTI (2017) | Resource Materials

Part 4 6 x 5 5 I know _____ x _____ = _____. I also know _____ x _____ = _____. 6 _____ + _____ = _____ So, that means _____ x _____ = _____. © DMTI (2017) | Resource Materials

Part 4 9 10 x 9 I know _____ x _____ = _____. I also know _____ x _____ = _____. 10 Students may use a compensation strategy for this fact. (10x10) – (10x1) = 10x9 _____ + _____ = _____ So, that means _____ x _____ = _____. © DMTI (2017) | Resource Materials

Part 5 Materials needed: Students need access to several sheets of 1 cm graph paper (or a single sheet placed inside a transparent sleeve to be used with a dry erase marker). Students may need square tiles or cubes to enactively (physically) model the anchor facts strategies prior to drawing them visually. Multiplication Using 10 © DMTI (2017) | Resource Materials

Part 5: Using 10 Multiplying any number by 10 increases the number by one place value. This is called the 10 Rule. Let’s see how to model this pattern. Create a model for 3 units of one. Now iterate 3 units of one a total of 10 times. Iterate vertically which means up and down. Define iterate as “copying without any gaps or overlaps.” Students may accidentally iterate the 3 units of one 10 more times for a total of 11 x 3. Remind them that the first model of 3 units of one was 1 x 3. If 1 x 3 = 3 and 10 x 3 = 30, where do you see the 3 units of ten in the model? The 3 units one became 3 units of ten. So, 10 x 3 increased 3 by one place value. 10 x 3 = 30 © DMTI (2017) | Resource Materials

Part 5 Use the 10 Rule to solve the following problems. Remember, every unit of place value will increase by one place value when you multiply by 10. 4 x 10 7 x 10 9 x 10 10 x 10 12 x 10 Describe how you found the product for each example by explaining the increasing place values. It is important for students to avoid describing the pattern as, “…adding a zero.” This is not only mathematically incorrect (e.g =12, not 120) but the 10 Rule is a more accurate description for future problems involving decimals. Students who only focus on adding zeroes often incorrectly believe 1.2 x 10 = Students who understand the 10 Rule will recognize the increasing place values and correctly solve 1.2 x 10 = 12. © DMTI (2017) | Resource Materials

Part 5 Use the 10 Rule to solve the following problems. Remember, every unit of place value will increase by one place value when you multiply by 10. 4 x 10 = 40 7 x 10 = 70 9 x 10 = 90 10 x 10 = 100 12 x 10 = 120 © DMTI (2017) | Resource Materials

Part 5 Now let’s use the 10 Rule to solve multiplication facts that are a little more or a little less than multiplying by 10. We have used 6 x 8 before so let’s use that as a fact to practice the Using 10 strategy. © DMTI (2017) | Resource Materials

Part 5 10 6 x 8 Let’s practice Using 10 to solve 6 x 8. Where do you see 6 x 8 in the model? How does 6 x 10 help you solve 6 x 8? 6 Students should draw the area model on their grid paper. If needed students can model the compensation strategy using tiles or cubes and then transfer the representation to grid paper. © DMTI (2017) | Resource Materials

Part 5 10 6 x 8 Let’s practice Using 10 to solve 6 x 8. Where do you see 6 x 8 in the model? How does 6 x 10 help you solve 6 x 8? 6x8=48 6x2=12 6 Students should draw the area model on their grid paper. Students should count rows or columns of the grid squares if they do not know the anchor fact. Over time, they will remember these anchor facts more easily than the more difficult related facts. Students may be familiar with the concept of “compensation” in which you change numbers in a problem to make the problem easier and then adjust the answer to account for the way in which the numbers were changed. This is an example of compensation with multiplication. Students are more likely to have used compensation for problems such as = in prior grades. 6 x 10 = 60 -(6 x 2 = 12) 6 x 8 = 48 © DMTI (2017) | Resource Materials

Part 5 10 6 x 8 I know _____ x _____ = _____. 6x8=48 6x2=12 6 I also know _____ x _____ = _____. _____ - _____ = _____ 6 x 10 = 60 -(6 x 2 = 12) 6 x 8 = 48 So, that means _____ x _____ = _____. © DMTI (2017) | Resource Materials

Part 5 12 6 x 12 How could 6 x 10 help you solve 6 x 12? Where do you see 6 x 10 in the model? 6 Students should draw the area model on their grid paper. If needed students can model the decomposing strategy using tiles or cubes and then transfer the representation to grid paper. © DMTI (2017) | Resource Materials

Part 5 12 6 x 12 I know _____ x _____ = _____. 6 6x10=60 6x2=12 I also know _____ x _____ = _____. Students should draw the area model on their grid paper. If needed students can model the decomposing strategy using tiles or cubes and then transfer the representation to grid paper. _____ + _____ = _____ So, that means _____ x _____ = _____. © DMTI (2017) | Resource Materials

Part 5 Now we will practice Using 10 for many more facts. Use models and the sentence frame to show and describe how you used a x10 fact to solve the more difficult fact. You will be given the area model for the multiplication fact you need to solve. You will want to think about how you can change the model to show a x10 fact. Students can work through each example individually, in pairs, in small groups or as a whole class. Students may need to model the facts with tiles, cubes or counters. If students are unfamiliar with the x10 facts, encourage them to solve the x10 fact by counting rows or columns in their models and focus on the 10 Rule. Over time they will commit the x10 facts to memory by following this process and using x10 to solve more difficult facts. © DMTI (2017) | Resource Materials

Part 5 5 x 9 9 I know _____ x _____ = _____. 5 I also know _____ x _____ = _____. _____ - _____ = _____ So, that means _____ x _____ = _____. © DMTI (2017) | Resource Materials

Part 5 5 9 x 5 I know _____ x _____ = _____. I also know _____ x _____ = _____. 9 _____ - _____ = _____ So, that means _____ x _____ = _____. © DMTI (2017) | Resource Materials

Part 6: Problem Strings You will be given a string of multiplication problems. Your goal is to use the products to solve the final fact you are given. Here is an example: 3 x 5 = 15 2 x 5 = 10 5 x 5 = ? © DMTI (2017) | Resource Materials

Part 6: Problem Strings You will be given a string of multiplication problems. Your goal is to use the products to solve the final fact you are given. Here is an example: 3 x 5 = 15 2 x 5 = 10 5 x 5 = 25 Students should be able to make connections between the given problems. If needed, have them model the problems with tiles or on grid paper. © DMTI (2017) | Resource Materials

Strategy Cards Print the following slides for individual or pairs of students. Students may not need to practice all of these facts. Provide them with cards that match the facts they are the least fluent with. There is a blank template as well that can be used to create additional facts students need to practice. 1. Students cutout the cards (horizontally), fold in half and then shuffle the cards. 2. On the back of the card, students draw models and/or write equations that match their preferred strategy. 3. If desired, students can include a second strategy or model that could be useful for the same fact. The space provided could be any of the following combinations: 2 equations, 1 equation and 1 model, 2 models, 2 equations and 2 models Note: students may have strategies for specific facts that do not fit any of the practiced Anchor Facts. At this time in their learning, encourage students to use their favorite strategy provided it is accurate and increases fluency. Students should use area models and equations to communicate their thinking. Their area models should be “open” area models so that students do not spend too much time drawing the square units. If students need to show all of the individual square units because they are still becoming comfortable with the anchor facts, provide graph paper to guide their drawings. Encourage them to commit open area models to their strategy cards. The Strategy Cards can be used by pairs of students or can be taken home for additional practice. If the cards are sent home, provide the above directions and possibly copies of some of the informational slides in this module so parents are informed about how to best use the cards as well as the purpose for developing fact fluency with the DMTI approach. © DMTI (2017) | Resource Materials

Strategy Card Example 7 x 6 Strategy or Model 7 x 5 = 35 7 x 1 = 7 = 42 7 x 6 = 42 Additional strategy cards can be made by using the following slides as a template. Fold here. Students should be presented with the front of the card (showing the fact) with the back facing their partner. © DMTI (2017) | Resource Materials

Part 6: Anchor Facts Practice
Now that we have good strategies to solve basic multiplication facts, let’s practice our facts to see if we can become more fluent. The word fluent means that you know or find the product very quickly and accurately. Because we have been practicing our multiplication facts, you will likely have started to memorize some facts. Try to see if you can remember the product for each given fact as quickly as possible. Then, describe your favorite strategy you could use to solve the fact, even if you just have it memorized. You will be given a Strategy Menu to select from. Make sure students try to recall the fact as quickly as possible but avoid giving the impression that recalling facts is always better than using a strategy. Students will ultimately become fluent with their facts regardless of whether they have the facts memorized or need a brief moment to use a strategy. If they feel comfortable using strategies, over time students will memorize a much larger number of facts than if they intentionally try to memorize them. An over-emphasis on memorization can inadvertently have a negative effect on fact fluency for some students, particularly those who find math difficult or do not memorize well (e.g. names, directions, phone numbers). © DMTI (2017) | Resource Materials

Part 6 Strategy Menu Using Halving and Doubling Using a Square Using 10 I know 6 x 4 = ____. My strategy was __________________. 6 x 4 © DMTI (2017) | Resource Materials

Practice For this section students should first use the sentence frame to state the entire equation including the missing factor. You can then ask them to use a strategy to show they are correct. The notes on the first slide provide guidance on this process. Missing factors © DMTI (2017) | Resource Materials

Part 6 I know 48 =____x 8. 48 =__x 8 Students should say, “I know 48 = 6 x 8.” If needed, they can then be asked, “How do you know 6 x 8 = 48?” Students can then use a strategy and describe their thinking just as they have done before. © DMTI (2017) | Resource Materials

© DMTI (2017) | Resource Materials | www.dmtinstitute.com
Brendefur and Strother (2017). DMTI Inc. © DMTI (2017) | Resource Materials |

© DMTI (2017) | Resource Materials

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