Foundations of Algebraic Thinking: Place Value K-2 Name Counte glass rs blobs Numb Date er Wo rd Tens Single button s s Tens square Single tile s Tens Single toothp icks s Tens lima bea ns Single s Tens Single s Building Instructional Leadership Across Oregon “Developing Algebraic Thinking” Session 3 March 2010 K-2 Goal: Students use number sense, place value, and patterning to construct concrete models for thinking algebraically. I can describe how students begin to represent their thinking K2 and how this builds more abstract thinking, models, and representations through grade 8. Foundations of Algebraic Thinking: Place Value K-2 March Follow Up K-8 Math Institute Foundations for Algebraic Thinking Vocabulary 3 Activities Activity Descriptions, Note Pages 5 Participant Activity Planner 9 Number Talk: Ten Frame Flash A. Compatible Pairs B. Groups of Ten C. Three Other Ways D. The Other Part of 100 E. Close to 100, Close to 1000 F. Solve Two Ways 10 11 20 24 29 31 34 2 Addition, Subtraction, Patterning K-2 Vocabulary Properties Associative Property : (Addition, Multiplication) For a given number sentence that combines three quantities (2 at a time); the initial pairing of the quantities is arbitrary. The way quantities associate will not change the result of the operation. (a + b) + c = a + (b + c) and (a x b) x c = a x (b x c). Commutative Property: (Addition, multiplication) The order of factors or multiples doesn’t change the sum or product. Identity Property: (Addition: 0, Multiplication: 1) Adding 0 to another number (or multiplying by 1) cannot change the final quantity. Strategies: Addition & Subtraction Compensation: Turning a number into a ‘friendly’ number to perform an operation, then adjusting for this change. Students often use compensation intuitively once they have a good understanding of quantity and number order. Example: Compose/decompose: Numbers can be broken down into their smaller parts to allow children to have greater facility with numbers and operations. Students may use place value or landmark (friendly, benchmark) numbers to break numbers apart, depending on the numbers and context. This will transfer to understandings with larger numbers in subsequent grades. 3 Comparison (subtraction): Some contexts do not lend themselves to modeling with a “take‐away” model. How many more? How many less? Both of these call for comparing to arrive at a difference. The following comparison reflects the original question as no one is taking away or removing cars. Begin with young students using concrete models. In this example, young students would be using small toy cars, line them up, compare and count how many to see the difference. As children develop mathematically, we move them to more abstract models for understanding operations (example below with number lines). The ultimate goal is to move toward fluency with abstract numbers. Sam has 12 cars. Josie has 16 cars. How many more does Josie Have? Take Away (subtraction): Some contexts will involve ‘taking away’ a specified amount and ask about the remaining amount. A Note on Fluency: Fluency does not relate to a specific number of math facts correctly performed in a specific amount of time. (In fact, research continues to show the practice of ‘timed tests’ most often detrimental to student learning and understanding!) Students need to have conceptual understanding of operations. Once they have this understanding teachers can help them learn various ways of remembering these basic facts. Why does it work that way? What does the student already know that will help them remember new facts? Building a connection between concrete models and representations to more abstract numbers provides the foundation for conceptual understanding and computational fluency. Look for strategies embedded in teaching materials to help identify how students work toward fluency. 4 Number Talk: Ten Frame Flash: Teaching Student‐Centered Mathematics; Grades K‐3 (2006) Van de Walle & Lovi, p.47 Materials • Mini Ten Frames • Chart Paper • Download blacklines at www.ablongman.com/vandewalleseries Activity This activity helps children see groups of ten. This Math Talk provides a wonderful setting for checking for understanding, sharing ideas, and multiple strategies. Recording different ways of “seeing” and thinking (as we do in class) facilitates students’ understanding of a wider range of ideas and makes individual thinking available to others. When we don’t take the time to record the ideas, we lose out on these wonderful opportunities. Keeping the Math Talk posters will also give teachers ways to connect new knowledge and concepts to prior learning. Pull out a previous poster, “Remember when we…?” __________________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ ____________________________________________________________________________________ Activity A: Compatible Pairs Teaching Student‐Centered Mathematics; Grades K‐3 (2006) Van de Walle & Lovi, p.148 Materials • Direction Cards • Compatible Pairs Record Sheet • Compatible Pairs Number Sheet Activity Students work with a partner, take turns, and find a combination of numbers that equals 50, 100, 1000, or some other choice. Students justify their choice to their partner, and then cover the two numbers with a cube if they’re correct. Players continue until they’ve matched all the pairs. __________________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ 5 Activity B: Groups of Ten Teaching Student‐Centered Mathematics; Grades K‐3 (2006) Van de Walle & Lovi, p.130 Materials • Containers of Counters • Number word sheet • Recording sheet • Counting Mat Activity Students get a recording sheet and choose a container of counters. The student dumps out the contents, records the amount as a number word, and groups the objects in as many tens as possible. The final count is then recorded on the recording sheet. Children collect their objects and choose another container. __________________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ ____________________________________________________________________________________ Activity C: Three Other Ways Teaching Student‐Centered Mathematics; Grades K‐3 (2006) Van de Walle & Lovi, p.133 Materials • Number strips with written numbers • glue • Recording sheets • Items to represent hundreds, tens & ones (toothpicks, dots, color tiles, etc.) Activity Students, working in groups or pairs, get a recording sheet and choose a number strip. They glue the number strip on their recording sheet. Using standard representations (dots for ones, toothpicks for tens, & squares for hundreds), students record their number one way. Then they find three other ways to record them. __________________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ ____________________________________________________________________________________ 6 Activity D: The Other Part of 100 Teaching Student‐Centered Mathematics; Grades K‐3 (2006) Van de Walle & Lovi, p.147 Materials • Mini ten‐frame cards Activity Two students work together with min ten‐frame cards. One student makes a two‐digit number. Then both students work mentally to determine what would go with the first number to make 100. Students justify their answer. Each person takes turns choosing the first number. __________________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ ____________________________________________________________________________________ Activity E: Close to 100, Close to 1000 Grade 3, Grade 4 Investigations, 1st Edition (Pearson) Materials • Game sheet • Numeral Cards 0‐9 Activity Six cards are dealt to each player. Each player uses any four of these cards to make two numbers, that when added, come as close as possible to a total of 100. __________________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ ____________________________________________________________________________________ 7 Activity F: Solve Two Ways Materials • Sample problems • glue • Blank paper Activity Choose a problem. Glue it down (top) of a blank page. (Add name/date!) Estimate an answer. Jot this down – students should have an idea of a reasonable answer. Solve ‐ Is this a reasonable answer? Solve another way – reinforces multiple strategies and gives students another way to solve to verify. Write a story (context) to match the original __________________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ __________________________________________________________________________________________ ____________________________________________________________________________________ 8 NAME DATE Work with some of the activities. Try to get to at least 3. Identify the skill targeted (can you match this to a grade level or standard(s)?) Identify ways the activity can be modified (for both accelerated learning and depth). The Other Part of 100 Three Other Ways Compatible Pairs Groups of Ten Close to 100, Close to 1000 Name Counters glass blobs Date Number Word Tens Singles buttons Solve Two Ways! Tens Singles square tile Tens Singles toothpicks Tens Singles lima beans Tens Singles print 9 Number Talk: Ten Frame Flash What number properties and strategies are evident in this activity? 10 A. Compatible Pairs This activity can be done as a worksheet activity or with the full class using the overhead projector. Prepare a transparency or duplicate a page with a search task. For today you will select a mat to work with and follow the rules below: Markers of any type can be used to cover number pairs… 11 Name Date Compatible Pairs Record Sheet Make __________ Represent your thinking = Represent your thinking = Represent your thinking = Represent your thinking = 12 Name Date Make __________ Compatible Pairs Record Sheet Represent your thinking + = Represent your thinking + = Represent your thinking + = Represent your thinking + = 13 Compatible Pairs! Make 50 14 Compatible Pairs! Make 1000 15 - Compatible Pairs! Make ? 16 Compatible Pairs! Make ? 17 18 19 B. Groups of Ten Choose a container of counters. Get a record sheet and write your name on it Dump out and count the contents Number Words Record the amount as a number word eleven ten one twelve twenty two thirteen thirty three fourteen forth four fifteen fifty five sixteen sixty six seventeen seventy seven eighteen eighty eight nineteen ninety nine Group the counters in as many tens as possible (use a counting mat to help you!) Name Counter s glass blob s Date Number Word Tens Sin gles buttons Record this on your paper. Tens Sin gles square tile Tens Sin gles toothpic ks Tens lima bea ns Sin gles Tens Sin gles Return the counters to the container. 20 Number Words eleven ten one twelve twenty two thirteen thirty three fourteen forth four fifteen fifty five sixteen sixty six seventeen seventy seven eighteen eighty eight nineteen ninety nine Groups of Ten Reference Page 21 Name Counters Date Number Word glass blobs Tens Singles buttons Tens Singles square tile Tens Singles toothpicks Tens Singles lima beans Tens Singles Groups of Ten Record Sheet 22 10-frame check mats 23 C. Three Other Ways Students work in groups or pairs. or 1. Draw out a strip with a written number. Read it with your partner(s). Five hund venty-five sed sevent red re y-five hund Five 2. Glue the strip of paper down on your record sheet. Five hundred seventy-five 3. Build the number using standard representation (with your model) 4. Find and record three more ways to show this number. 24 Name Date Three Other Ways Record Sheet (Tens, Units) glue written number here Tens ______ Ones ______ Tens ______ Ones ______ Tens ______ Ones ______ Tens ______ Ones ______ Build the number with standard representation. Find three more ways! glue written number here Tens ______ Ones ______ Tens ______ Ones ______ Tens ______ Ones ______ Tens ______ Ones ______ glue written number here Tens ______ Ones ______ Tens ______ Ones ______ Tens ______ Ones ______ Tens ______ Ones ______ 25 Name Date Three Other Ways Record Sheet (Hundreds, Tens, Units) Write Numeral ____________ + _____________ + ______________ = _______________ Use standard Notation and build to match. Next, Find at least three other ways to build the same number. Hundreds _______ Tens ______ Hundreds _______ Tens ______ Hundreds _______ Tens ______ Hundreds _______ Tens ______ Ones ______ Ones ______ Ones ______ Ones ______ Write Numeral ____________ + _____________ + ______________ = _______________ Use standard Notation and build to match. Next, Find at least three other ways to build the same number. Hundreds _______ Tens ______ Ones ______ Hundreds _______ Tens ______ Ones ______ Hundreds _______ Tens ______ Ones ______ Hundreds _______ Tens ______ Ones ______ Write Numeral ____________ + _____________ + ______________ = _______________ Use standard Notation and build to match. Next, Find at least three other ways to build the same number. Hundreds _______ Tens ______ Ones ______ Hundreds _______ Tens ______ Ones ______ Hundreds _______ Tens ______ Ones ______ Hundreds _______ Tens ______ Ones ______ 26 one hundred forty-four eight hundred twenty-two five hundred seven three hundred eleven two hundred eighty-nine six hundred nineteen nine hundred twenty-one five hundred forty-seven seven hundred fifty-one six hundred thirty-four two hundred sixty-three three hundred twenty-eight two hundred seventy-six six hundred thirty-four one hundred eighty-nine one hundred forty-six two hundred sixty-two four hundred thirty-five one hundred six three hundred sixteen one hundred ten five hundred sixty eight hundred three hundred eighty-two three hundred fifteen two hundred three hundred forty six hundred fifty-two six hundred ninety-six four hundred seventy seven hundred fourteen seven hundred seventy-eight five hundred thirty-nine five hundred twenty-four five hundred seventy-seven seven hundred twenty-nine four hundred ninety-four one hundred eighty-six seven hundred fifty-five three hundred fifty-eight 27 forty-four twenty-two seven eleven eighty-nine nineteen twenty-one forty-seven fifty-one thirty-four sixty-three twenty-eight seventy-six thirty-four eighty-nine forty-six sixty-two thirty-five six sixteen one hundred ten sixty eight hundred eighty-two fifteen forty-eight forty fifty-two ninety-six seventy fourteen seventy-eight thirty-nine twenty-four d seventy-seven twenty-nine ninety-four eighty-six fifty-five fifty-eight 28 D. The Other Part of 100 Two students work together with a set of little ten-frame cards. 1. One student makes a two-digit number. 2. Then both students work mentally to goes with the ten-frame to make 100. determine what 3. They write their solutions on paper and then check by making the other part with the cards to see if the total is 100. 4. Students take turns making the starting number. 29 Name Date The Other Part of 100 Record Sheet 1 Represent first number What’s the other part of 100? Write the numeral Write the numeral Represent first number What’s the other part of 100? Write the numeral Write the numeral Represent first number What’s the other part of 100? Write the numeral Write the numeral Represent first number What’s the other part of 100? Write the numeral Write the numeral Justify 2 Justify 3 Justify 4 Justify 30 Investigations, 1st Edition, Grade 3 (Pearson) + + = ______ = ______ 32 + + = ______ = ______ 33 F. Solve Two Ways! Choose a problem and stick it to the top of your work paper. Estimate an answer… write it down Solve 2 different ways Hmmm…is my answer “reasonable?” Write a story (give the problem a context) to match the original problem. Note: Print number sentences on return address labels or cut apart (to be glued on to blank paper) 34 354 + 537 = 307 - 289 = 1025 - ? = 375 789 + 1038 = 6521 - 387 = 73 + ? = 1000 1578 + 199 = 1000 – 271 = 100 – ? = 33.75 325+ 327+ 324= 1001 – 552 = 6.75 – ? =3.21 2957 + 123 = 352 - 175 10.25 – ? = 6.82 55 + 175 + 350= 210 – 35 = 9974 – ? = 55 724 + 1268 = 149 – 94 = ?- 1,581 = 16,785 562+625+256= 333 – 39 = ? + 526 = 2583 253+ 257+ 255= 17,458 – 873= ? – 73.2 = 2052 642 + 658 = 2952- 766 = ? – 0.671 = 372.6 35