Connecting Academics & Parents Academic seminars to sharpen skills and build understanding in Multiplication & Division Basic Fact Fluency Critical Point: Welcome parents to the session. This training was intended to take an hour, but can be adjusted to take more/less time as needed. Step-by-Step Directions: Welcome parents and introduce trainers. Give purpose for training, which is to build the parent’s understanding of what their children are learning and strategies they will be using in the classroom, which align with the Math Florida Standards. Explain that parents will also be leaving with “purposeful practice,” games and strategies to practice at home and will reinforce what they are learning in school. Materials: counters/centimeter cubes/square tiles, Up Up and Array cards and game sheet, speakers, paper clips Copies in packet: graph paper, Up Up and Array, The Product Game, Facts Triangles (copied on cardstock if possible), Article on delaying memorized facts

Let’s Solve It!!! Solve the problem using 2 different ways. Record an equation that matches your problem. A touchdown is worth 7 points. The Buccaneers scored 9 touchdowns in their first 3 games. How many points did the Buccaneers score in touchdowns? Critical Point: To engage parents in a real-world problem that is similar to what students will be doing in the classroom. Step-by-Step Directions: Pose problem. Give parents 2 minutes to solve on their own using any strategy they choose. As parents are solving, walk around and monitor different strategies being used. You will want to select 2 or 3 strategies to share. Pose questions to get parents to think deeper about their strategy and to justify their solution: What do the cubes represent? What does each group represent? Focus on getting parents to use precise vocabulary when discussing their model/strategy. Select the parents that are sharing. (Consider some one that used a direct model, number line, quick picture, mental math, etc). If none of these strategies were used, have a “Fake participants” strategy shared that is prepared ahead of time by you. Have parents share and make connections between the different strategies. The purpose of this slide is for parents to get an idea of what kind of problem solving is expected in the classroom to demonstrate understanding of multiplication/division. It is essential that understanding is achieved before students begin using more advanced fluency/basic fact strategies. Explain that by the end of the training, they will be familiar with different basic facts strategies that help students become more efficient and fluent in basic facts. Also share that this is typically how your child’s math class is started. Students are expected to share thinking with words/pictures/or numbers to either a partner/group or the whole class. Discuss that it’s important to think about the actions or what’s happening in the multiplication/division problems, not the key words such as: “total, more, separated, groups of, etc.” to determine which operation to use. Trainer Note – Solving and sharing should take no longer than 5 minutes for this slide. Encourage parents to think about the way their children may solve the problem with this new learning.

Problem: A touchdown is worth 7 points Problem: A touchdown is worth 7 points. The Buccaneers scored 9 touchdowns in their first 3 games. How many points did the Buccaneers score in touchdowns? 9 x 7 =63 Critical Point: For parents to see examples of strategies that demonstrate understanding of multiplication. Step-by-Step Directions: Focus on the fact that before beginning fluency strategies, students should be able to solve one problem in multiple ways and be able to explain how their model/strategy relates to the problem. When a child demonstrates understanding like shown in this slide with the examples, it’s a signal for the teachers and parents that fluency is on its way. Share that they will see examples of how students that have “understanding” may solve the problem they just solved. Click once and the array that directly models 9 x 7 will appear. Click again and the equal groups picture that directly models 9 x 7 will appear. Click again and the repeated addition and multiplication equations will appear. You can also make a connection to skip counting here. Click again and the basic fact 9 x 7 appears. Discuss that even though fluency is the focus of this training, students need a deep understanding of multiplication/division models and strategies in order for that to happen. Trainer Notes – This should be a quick overview of strategies shared by parents on slide 2. This slide is hear in case parents did not demonstrate some of the “understanding” strategies on the previous slide. 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 + 7 = 63

Mathematics Florida Standards Focus Fluency is … … quickly and efficiently computing accurately. Grade 3 MAFS.3.OA.2.5 Apply properties of operations as strategies to multiply and divide. MAFS.3.OA.3.7 multiply and divide within 100, using strategies such as the relationship between multiplication and division or properties of operations. By the end of Grade 3, know from memory all products of two one-digit numbers. MAFS.3.OA.4.8 Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding. Fluently Fluency is not … … instant recall of basic facts. Critical Point: Share the standards that are being learned throughout building understanding of multiplication and division. Step-by-Step Directions: Share the standards that will be covered. Discuss that these are not the only multiplication/division standards in 3rd grade, but these will be the focus of this training. Click and the word “fluently” in the second standard will grow. Click again and the definition of what fluency is and is not will appear. Share that though fluency of multiplication and division is a standard and expectation for students by the end of 3rd grade, students are expected to have a deep understanding through various strategies before the fluency occurs. We do not rush to flashcards/basic facts. Rather we put multiplication/division in context of real-world story problems and build understanding by moving from concrete hands-on models to pictures to abstract thinking. Staying with this process is essential for a deep understanding of multiplication and division. If the parents want more information on the standards, encourage them to visit flstandards.org

MAFS Learning Progression: Multiplication & Division Basic Fact Fluency Critical Point: To show where building understanding of multiplication and division standards are and how they are connected between the different grade levels. Step-by-Step Directions: Share the standards briefly that address multiplication/division. Show how learning of multiplication and division basic fact fluency progresses from earlier grades to future grades. Copyright 2009

Children’s Strategies for Solving Multiplication & Division Problems Stage 3: Derived Facts Stage 2: Counting & Adding/ Subtracting Strategies Stage 1: Direct Modeling Strategies Critical Point: Children progress through 3 stages of development when solving problems. Step-by-Step Directions: Share the three stages of development children go through when solving problems. Click and the third stage will get larger. This is the stage we will focus on in this training. Stages 1 and 2 are addressed in the Understand Multiplication and Division Training. Discuss how it’s important to go in order of these 3 stages of development so students “understand” instead of just memorizing basic facts. Using this order, children will derive their basic facts with understanding and achieve fact fluency. (Fluency is getting a correct answer quickly – they can use mental math strategies based on experiences in Stage 1 and Stage 2 to help gain fluency).

Strategies for Building Basic Fact Fluency of Multiplication & Division Multiplying by 0 & 1 Multiplying by 2 & 4 Multiplying by 5 & 10 Distributive Property to Multiply by 3, 6, 7, 8 & 9 Division Basic Fact Fluency using Fact Families Critical Point: These are the strategies we will focus on in this training. Step-by-Step Directions: Share the five strategies that will be focused on in the training and that you will go into more detail on them throughout. Trainer Notes – This slide should take only about a minute, purpose is to give them the agenda for the rest of the day. Copyright 2009

Step-by-Step Directions: Critical Point: To show how the multiplication chart can be used for basic facts that are mastered. Step-by-Step Directions: Share the sample multiplication chart and that there is a blank copy in the participant packet. Discuss that as their child masters a basic fact conceptually, they can highlight the section they mastered with a crayon. Children can then just focus on the facts they have not mastered. After each basic fact factor is discussed in the training, a multiplication table will pop-up showing the facts we have mastered. Copyright 2009

Multiply & Divide by 0s & 1s Multiplying by one and zero often seems like a simple rule, but it is complex for students to understand conceptually. Through modeling and exploration your child will discover the patterns and rules to master their 0 and 1s facts. Khalib had six cups. He put one marble in each cup. How many marbles did he use? Critical Point: To share strategies for mastering basic facts with factors of 0 and 1. Step-by-Step Directions: Read the details about multiplying by 0 and 1. Discuss that though we are moving towards basic fact fluency, many times models need to be brought back to understand patterns/rules for multiplying by 0 and 1. It’s assumed that these are easy facts to achieve, but students sometimes need to model to really understand and represent the patterns. Pose the problem. Click again to reveal the model of equal groups. Click to reveal the array that models the problem. Click to reveal the equation. Ask parents what they notice about the answer. Ask parents if 1 x 6 would also relate to this problem. The answer is no. Though it would give the same product, it doesn’t directly model the problem based on the fact the first number represents how many groups and second number is how many in each group. Trainer Note – You may want to discuss with parents that multiple problems with multiplying by 1 should occur until students observe that the other factor is always the answer. 1 2 3 4 5 6 6 x 1 = 6

Multiply & Divide by 0s & 1s Khalib had 6 marbles and he put them in one cup. How many marbles does he have? 6 4 1 3 2 5 Critical Point: To share strategies for mastering basic facts with factors of 0 and 1. Step-by-Step Directions: Read the details about multiplying by 0 and 1. Discuss that though we are moving towards basic fact fluency, many times models need to be brought back to understand patterns/rules for multiplying by 0 and 1. Pose the problem. Click again to reveal the model of equal groups. Click to reveal the array that models the problem. Click to reveal the equation. Ask parents what they notice about the answer. Ask parents how this problem relates to the problem on slide 10. Trainer Note – You may want to discuss with parents that multiple problems with multiplying by 1 should occur until students observe that the other factor is always the answer. 1 x 6 = 6

Multiply & Divide by 0s & 1s Khalib had zero marbles and he decides to put them in six cups. How many marbles is in each cup? 0 ÷ 6 = 0 Critical Point: To share strategies for mastering basic facts with factors of 0 and 1. Step-by-Step Directions: Discuss that though we are moving towards basic fact fluency, many times models need to be brought back to understand patterns/rules for multiplying/dividing by 0 and 1. Pose the problem. Click again to reveal the model of equal groups. Click to reveal the equation. Since I have no marbles to separate, there will be zero marbles in each cup. Ask parents what they notice about the answer. Trainer Note – You may want to discuss with parents that multiple problems with multiplying by 1 should occur until students observe that the other factor is always the answer.

Multiply & Divide by 0s & 1s Khalib had 6 marbles and he decides to put them in 0 cups. How many marbles is in each cup? 6 ÷ 0 = can’t be done You can never divide by zero! Critical Point: To share strategies for mastering basic facts with factors of 0 and 1. Step-by-Step Directions: Discuss that though we are moving towards basic fact fluency, many times models need to be brought back to understand patterns/rules for multiplying/dividing by 0 and 1. Pose the problem. Click to reveal the six marbles. Click again to reveal that we have no cups to put them in. This is a difficult model for students to understand. If we do not have any cups to put them in this action is impossible. You cannot divide by 0. There is no answer to this problem. Click again and the multiplication chart showing we have mastered our 0 and 1s basic facts will appear. Trainer Note – You may want to discuss with parents that multiple problems with multiplying by 1 should occur until students observe that the other factor is always the answer. Copyright 2009

2 x 9 9 + 9 Multiply by 2 Double the number to multiply by 2 Repeated addition is a natural strategy to show doubling Children begin to learn about doubling in 1st grade 2 x 9 Critical Point: To share strategies for developing basic fluency when 2 is a factor by doubling and relating multiplication to addition. Step-by-Step Directions: Share the details about multiplying by 2. You want to focus on doubling and using repeated addition to solve problems when multiplying by 2. Discuss that you are going to show a “doubles” model for 2 x 9. Click and the equal groups model for 2 x 9 will appear. Click again and the repeated addition sentence will appear. Discuss how using “precise’ vocabulary is essential. The problem is not “2 times 9” it’s “2 groups of 9.” Discuss the importance of seeing the multiplication symbol as instead the “equal groups” symbol. Trainer Notes – Let parents know that students have been working with doubling numbers since 1st grade. 9 + 9

doubled 4 x 9 = (2 x 9) + (2 x 9) 9 + 9 + 9 + 9 or 18 + 18 Multiply by 4 The 4s basic facts are mastered by knowing to double the number that’s not four and then double again to multiply by 4 (double double) doubled 4 x 9 = (2 x 9) + (2 x 9) Critical Point: To relate multiplying by 4 to doubling and doubling again. Step-by-Step Directions: Share the details about multiplying by 4. First you double the number that’s not a four. Then you double it again to find the product. (Could refer to it as “double double”) Click again and the equal groups model with the matching expression will appear. Click again and the repeated addition sentence and doubling number sentence will appear. Ask parents to make comparisons to the equal groups picture and the original problem. Trainer Notes – Let parents know that students have been working with doubling numbers since 1st grade. 9 + 9 + 9 + 9 or 18 + 18 Copyright 2009

Multiply by 4 A fun sized bag of Skittles has 8 Skittles in it. How many Skittles are in 4 bags? Use your what you know about doubling to solve this problem. Critical Point: To have parents apply what they learned about using doubles twice to solve multiplication with factors of 4. Step-by-Step Directions: Pose the problem to the parents. Encourage them to use what they know about doubling and multiplying by 4 to solve the problem. Monitor parents as they are solving to select a parent to share that has a good description of doubling and doubling again. Have the parent share and ask questions to get parents to think deeper about the strategy. Discuss how at this point when children are developing their basic fact fluency, models are still needed to make sense of the patterns and facts. Click again and the multiplication chart showing that the 0s, 1s, 2s, and 4s facts are mastered (or have been covered). Copyright 2009

Multiply by 5s & 10s To master the 5s and 10s basic facts students apply what they know about patterns and skip counting. A number line is a great tool to use to model the patterns. Now Faster!!! Critical Point: Discuss strategies for basic fact fluency when dealing with 5s and 10s multiplication problems. Step-by-Step Directions: Share the details about multiplying by 5s and 10s. Discuss how patterns are important in achieving fact fluency. (Example – What do you notice about all the products of 5? They end in 5 or 0). Click and it will reveal the number line that shows the action of skip counting by 5. Discuss how this strategy can also be used with multiplying by 10s. Click again and the multiplication chart will appear showing the facts for 0s, 1s, 2s, 4s, 5s and 10s have been covered. Trainer Notes – Let parents know that students have been working skip counting by 5 since 2nd grade and 10s since kindergarten. 5, 5, 10, 10, 15, 15, 20, 20, 25, 25, 30, 30, 35, 35, 40, 40, 45, 45, 50, 50,

3 x 8 (2 x 8) + (1 x 8) Distributive Property to Multiply by 3s and 6s The Distributive Property refers to the idea that one of two factors in a product can be split into two or more parts and each part multiplied separately and then added. The results is the same as when the original factors are multiplied. Van De Walle Critical Point: Model the distributive property with arrays to develop understanding of the property. This strategy will help develop fact fluency for 3s, 6s, 7s, 8s and 9s. Step-by-Step Directions: Discuss what multiplication problem is modeled with the array. Click and the expression 3 x 8 will appear. Click again to show how the array separates into 2 arrays. Click again to show how the original expression turns into (2x8) + (1x8). Ask parents to relate the two arrays and expressions. The important things for children when learning abstract properties is to make them hands-on by using models. Continuously pose questions to have students relate their new models/expressions to the original problem will help with this transition. Fluency will not happen immediately and we want the children to have understanding before using the abstract property. Share a definition of the distributive property. Discuss how using what children know about breaking apart numbers flexibly, makes it quicker more efficient to multiply with factors of 3, 6, 7, 8, and 9. You start off by modeling the process, like shown in the slide, but then move to quicker mental math strategies for deriving the basic facts. Ask parents, “How was the distributive property definition just modeled?” Have a discussion connecting the definition and the model.

(5 x 7) + (1 x 7) 6 x 7 Critical Point: Model the distributive with arrays to achieve 6s facts fluency. Step-by-Step Directions: Show the array on the slide. Ask parents what problem is being modeled with the array. Click and the multiplication problem 6 x 7 will appear. Discuss ways for finding the product by counting each square, skip counting the number in each row, etc. Ask parents, “Is that the most efficient strategy to find the product of 6 and 7?” They should come to the conclusion that though it is a strategy that could be used, it’s not the most efficient/effective way. Click and it will show how the array could be broken into two problems 5 x 7 and 1 x 7. Click again to show the new expression modeled by the broken apart arrays. Ask parents if this would be a quicker more efficient way to solve the problem. Trainer Note – The next slide will show another way to break apart the same problem. Then parents will be asked to find another way and the most efficient way.

What’s another way to break apart the problem 6 x 7? (3 x 7) + (3 x 7) 6 x 7 What’s another way to break apart the problem 6 x 7? Critical Point: Model the distributive with arrays to achieve 6s facts fluency. Step-by-Step Directions: Show the array and multiplication problem 6 x 7 on the slide. Discuss ways for finding the product by counting each square, skip counting the number in each row, etc. Click and it will show how the array could be broken into two problems 3 x 7 and 3 x 7. Click again to show the new expression modeled by the broken apart arrays. Ask parents if this would be a quicker more efficient way than the previous slide of 1 x 7 and 5 x 7 to solve the problem. Click again and the multiplication chart will appear with 0s, 1s, 2s, 3s, 4s, 5s, 6s colored in. Discuss the importance of this strategy in determining easy ways to decompose the 6 so there are two easier problems for solving. Discuss how though you are looking for the easiest/most efficient strategy, the most efficient/easiest strategy may not be the same for all children. Trainer Note – This slide should take no longer than 2 minutes.

Multiply by 7, 8 & 9 Solve the problem below using the distributive property. Be prepared to share your thinking. John is baking cookies with his mom. They baked 6 rows of 9 cookies. How could they separate the cookies onto two different trays to make it easier to find how many cookies they are baking? Critical Point: To have parents apply their understanding of the distributive property in real world context. Step-by-Step Directions: Pose the problem and task to parents. Give them 2 minutes to solve using the distributive property. Monitor and pose HOT questions to get parents to think deeper about their strategy/model. Select a couple parents to share looking for someone who used an array model, record the distributive expression, etc. Have the parents share. Have a discussion on which number they chose to break apart. May want to pose the question, “Why did you break up the number you did? Could you have broken up the other number?” Pose questions to get parents to make connections between the different strategies shared. Click again and the multiplication chart with all facts will be shaded in.

Division Basic Fact Fluency using Fact Families How can you use multiplication to solve this problem quickly and efficiently? We need to arrange 42 chocolate chip cookies onto trays to take to a class party. Only 6 cookies fit onto each row of the tray. How many rows will I have to make to fit all of the cookies? Critical Point: To have students use their multiplication fluency to develop division fluency. Step-by-Step Directions: Pose the question, “How can you use multiplication to solve this problem quickly and efficiently?” Have parents discuss with their group. Pose the problem and encourage parents to use multiplication to solve the problem quickly. Have a parent share how they used multiplication to solve the division problem. Focus on the equation 6 x ? = 42, focusing on the unknown factor. Discuss how using fact families and multiplication fluency to solve division problems is a quick and efficient strategy.

Fact Families 2 x 4 = 8 8 4 x 2 = 8 8 ÷ 4 = 2 8 ÷ 2 = 4 2 4 X / ÷ 2 4 X / ÷ Critical Point: To have students apply the distributive property with a game they can play at home with their children. Step-by-Step Directions: Share the fact triangle picture with the parents. Ask parents how they think the card works and what equations are modeled with the one triangular card. Encourage parents to have their children put the equations in context of a story and draw models to show the actions in the equation. Click to share all the equations related to this triangular card with the numbers 8, 2 and 4. Discuss how they can have their child go through the same process of looking at one card and determining the four equations that relate to the card. Click again and the 8 will be covered up. Discuss which equations are being modeled with the 8 missing, which are the two multiplication equations. Click again and the 4 will be covered up. Discuss that this models the 8 divided by 2 equation. Click again and the 2 will be covered up. Discuss that this models the 8 divided by 4 equation. Trainer Notes –You may also want to spend time on discussing how to use the Fact Triangle cards effectively. Fact Triangles are in their handout packet.

Step-by-Step Directions Critical Point: To have parents apply the different strategies discussed, using the unknown factor and relating division and multiplication. Step-by-Step Directions Read the directions for the Product Game to the parents. Pass out counters and 2 paper clips to each set of parents. Give parents about 4-5 minutes to play and think about strategies that are helping them get four squares in a row. After the first few minutes, bring parents back together to discuss strategies/observations. In the discussion you want parents to realize they are using division to acquire desired products. They are also using the strategy of “unknown factors” to determine where to move their paper clip. Copyright 2009

Timed Tests? What do the experts say? http://goo.gl/oMpBhU Flash cards are best used after students have learned strategies of multiplication and have an understanding of multiplication. Flashcards in isolation are not going to build understanding. Timed Tests? What do the experts say? http://goo.gl/oMpBhU Critical Point: To share the importance of having students understand multiplication and division rather than just memorizing facts. Step-by-Step Directions: Click once and the flashcard graphic will appear. Click again and the red slash with explanation will appear. Discuss how understanding and basic fact fluency is achieved with the strategies covered in this training and that it doesn’t happen in one day, week, month. Sometimes it takes repeated problems with similar facts and exploration of patterns for students to make connections and understand. Trainer Notes – Let parents know there is an article in their packet that goes into detail about avoiding memorization of facts.

Find real world examples of multiplication! DO TRY THIS AT HOME! Take it Home and Try It! Warning: Implementing this engaging activity will result in an increase in motivation and long-lasting learning. Purposeful Practice: The Product Game Product War Up, Up and Array Distributive Property Matching Game Fact Triangles Find real world examples of multiplication! Critical Point: To give parents purposeful practice and resources to use at home to reinforce understanding multiplication and division. Step-by-Step Directions: When the slide appears, there are “purposeful practice” tasks. Go through the different details and encourage parents to look in their packet at the resources. Encourage them to try it at home and use those as a guide to develop their own meaningful problems and tasks. Click again and focus on making real world connections at home. Pictures of real world arrays and multiplication scenarios will appear. Discuss how multiplication story problems can be created just from a picture. Trainer Notes – If time permits, you may want to play one of the games in the training. Copyright 2009