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Multiplication and the Common Core Standards 5th and 6th Grade Teachers
Welcome, Please make a Name Tent • First name in the middle, • Favorite desert in top left corner, • School in bottom left corner, • # years teaching in top right corner, and • Grade level teaching in the bottom right corner.
Share your information with your table • start with the person who had the longest drive and • going clockwise within the group.
person with the shortest drive • introduce yourself and • your group to the rest of the participants in your group giving • their name and • one thing you learned about that person today
Goal • Increasing your understanding of the Common Core Standards; focusing on Multiplication • Intertwined within this framework will be strands of the Mathematics Process Standards • increasing your knowledge of pedagogical strategies for questioning.
You will take on two roles throughout the course of the workshop: • At times, I will ask you to take the role of a student, where you will be doing the mathematics. • At other times, I will ask that you take the role of a teacher and think about how you will teach the concepts, what challenges your students may face, etc.
Norms Come up with ideas (norms) for things to make the workshops meaningful and productive Think about past workshops that you have been to; what helped to make the workshop meaningful and relevant to you? For example: honor the clock Honor the clock goes both ways, --I will honor the clock by making sure we end of the day on time --Participants please be on time after break
Agenda • Common Core Shifts • Questioning • Activities involving multiplication
Common Core Shifts for Mathematics In order for students to be successful, educators must effectively implement the new changes to the standards. There are three key shifts associated with Arizona’s Common Core Standards in mathematics. 1. Focus — The goal is to focus strongly where the standards focus. The curriculum significantly narrows and deepens the way time and energy are spent in the mathematics classroom. The standards focus deeply on the major work of each grade so that students can gain strong foundations: solid conceptual understanding, a high degree of procedural skill and fluency, and the ability to apply the mathematics they know to solve problems inside and outside the mathematics classroom.
Common Core Shifts for Mathematics 2. Coherence — Coherence is connecting ideas across grades, and linking to major topics within grades. The standards are designed around coherent progressions from grade to grade. Students build new understanding onto foundations built in previous years. Each standard is not a new event, but an extension of previous learning. Instead of allowing additional or supporting topics to detract from the focus of the grade, these topics can serve the grade-level focus.
Common Core Shifts for Mathematics 3. Rigor — In major topics, conceptual understanding, procedural skill and fluency, and application are pursued with equal intensity.
Common Core Shifts for Mathematics Emphasis is placed on conceptual understanding of key concepts, such as place value and ratios. Teachers support students’ ability to access concepts from a number of perspectives so that students are able to see math as more than a set of mnemonics or discrete procedures. Students build speed and accuracy in calculation. Teachers structure class time and/or homework time for students to practice core functions, such as single-digit multiplication, so that they have access to more complex concepts and procedures. Students use math flexibly for applications. Teachers provide opportunities for students to apply math in context. Teachers in content areas outside of math, particularly science, ensure that students are using math to make meaning of and access content.
Discuss: What challenges have you experienced in implementing cognitively challenging tasks that promote thinking, reasoning, and problem solving?
Work through this problem: How could you break apart each factor to make 46 x 58 an easier problem? • What problem could you start with? • How could you make sure you had completed all the multiplication required by this problem? • Will this work for every two-digit multiplication problem?
Solution Share solution strategies Are there other ways your students might go about solving the problem?
Questioning • What questions can you ask students as they are working through this problem as you are monitoring? • Did you come up with questions for the different needs of levels of your students? Extension questions? Intervention questions?
Framework for Mathematical Task: • Selecting and Setting Up a Mathematical Task • Supporting Students’ Exploration of the task • Sharing and Discussing the task
Brainstorm: • How can you use this process within your 90 minute math class? • What do you need to do to facilitate this process with your students?
Mathematical Practices Look through the Mathematical Practices Standards for your grade level Which did we address through the problem that we went through?
Problem Sets • Choose a problem from the handout* • Work out the problem • What standards are being addressed? • Develop questions to meet learning needs of your students – along with questions that address the mathematical practices of the CCS *Questions came from two books: “Good Questions for Math Teaching: Why Ask Them and What to Ask Grades 5 – 8” and Good Questions for Math Teaching: Why Ask Them and What to Ask Grades K – 6”
Brainstorm Where can you get these types of questions to use in your classroom? Write some questions based on upcoming math standards along with questions you can use with the task
BREAK 15 Minutes
Multiplication Three in a Row Step 1: Prior to your turn, choose one number from Box A and one number from Box B. Multiply these numbers on your scratch paper. Be prepared with your answer when your turn comes. Step 2: On your turn, announce your numbers and the product of your numbers. Explain your strategy for finding the answer. Step 3: Another player will check your answer with a calculator after you have announced your product. If your answer is correct, place your counter on the appropriate space on the board. If the answer is incorrect, you may not place your counter on the board and your turn ends. Step 4: Your goal is to be the first one to make “three-in-a-row,” horizontally, vertically, or diagonally.
FORMATIVE ASSESSMENT QUESTIONS •Who is winning the game? How do you know? •(To the winner) What was your strategy? •Is there any way to predict which factors would be best to use without having to multiply them all? Explain. •How are you using estimation to help determine which factors to use? •How many moves do you think the shortest game of this type would be if no other player blocked your move? Why?
Extension •A variation of the game above is to require each player to place a paper clip on the numbers they use to multiply. The next player may move only one paper clip either the one in Box A or the one in Box B. This limits the products that can be found and adds a layer of strategy to the game. •Another variation is for students to play “Six in a Row” where students need to make six products in a row horizontally, vertically, or diagonally in order to win. •Eventually, you will want to challenge your students with game boards that contain simple 3-digit numbers (e.g. numbers ending with a 0 or numbers like 301) in Box A or multiples of 10 (i.e., 10, 20, ... 90) in Box B. As their competency develops, you can expect them to be able to do any 3-digit by 2-digit multiplication problem you choose.
Intervention •Allow students time to view the game boards and work out two or three of the problems ahead of time to check their readiness for this activity. •Use benchmark numbers in Box A, such as 25, 50, 100, etc.
STANDARDS FOR MATHEMATICAL PRACTICE What questions could you ask as you monitor your students during this game? Which Mathematical Practice Standards does this activity address? • SMP 1. Make sense of problems and persevere in solving them. • SMP 3. Construct viable arguments and critique the reasoning of others. • SMP 4. Model with mathematics. • SMP 6. Attend to precision. • SMP 7. Look for and make use of structure. • SMP 8. Look for and express regularity in repeated reasoning.
Multiplication Teaser Activity For each problem, find the missing factor. Do not solve problems by dividing. Instead, use your calculator and the guess and check problem solving strategy. See how many guesses it takes for you to solve each one! For example, to solve 4 x___= 87, you might start with 23 and then adjust. See the solution steps on the next slide as example of how to record your guesses.
4 x _____ = 87 4 x 23 = 92 4 x 22 = 88 4 x 21 = 84 4 x 21.5 = 86 4 x 21.6 = 86.4 4 x 21.7 = 86.8 4 x 21.8 = 87.2 4 x 21.74 = 86.96 4 x 21.75 = 87 It took 9 guesses!
STANDARDS FOR MATHEMATICAL PRACTICE What questions could you ask as you monitor your students during this game? Which Mathematical Practice Standards does this activity address? • SMP1. Make sense of problems and persevere in solving them. • SMP 2. Reason abstractly and quantitatively. • SMP 3. Construct viable arguments and critique the reasoning of others. • SMP 6. Attend to precision. • SMP 7. Look for and make use of structure. • SMP 8. Look for and express regularity in repeated reasoning.
ESSENTIAL QUESTIONS How can we efficiently solve multiplication and division problems with decimals? How can we multiply and divide decimals fluently? What strategies are effective for finding a missing factor or divisor? How can we use estimation to assist in solving problems with decimal operations?
FORMATIVE ASSESSMENT QUESTIONS How did you get your answer? How do you know your answer is correct? What patterns are you noticing?
Extension Provide students with problems that incorporate factors to the thousands and repeating decimals to challenge their thinking.
Intervention Work with small groups of struggling students to model the problem solving process by doing a think-aloud to work through an example problem. Incorporate the use of manipulatives (grid paper, base ten blocks, etc.) to assist with student guesses.
Closure The Teaching Principle (PSSM, 2000) states, “Worthwhile tasks alone are not sufficient for effective teaching. Teachers must also decide what aspects of a task to highlight, how to organize and orchestrate the work of students, what questions to ask to challenge those with varied levels of expertise, and how to support students without taking over the process of thinking for them and thus eliminate the challenge” p.19. Based on your own teaching experiences, reflect on this statement