Bridging Thinking: Designing Effective Mathematics Intervention Experiences PROFESSIONAL LEARNING www.origoeducation.com Rob Nickerson M Ed
Bridging Thinking: Designing Effective Mathematics Intervention Experiences Copyright 2011 ORIGO Education For more information visit www.origoeducation.com All rights reserved. Unless specifically stated, no part of this publication may be reproduced, or copied into, or stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of ORIGO Education. No resale of this material is permitted.
What mathematics do your students struggle with? don t understand? can t do? 3
1. Record the top 3 ideas that your students struggle with in the spaces provided in column 1. Struggles Disconnects Causes 2. For each item in column 1, record the disconnects or related misunderstandings. 3. In column 3, record the possible causes of these misunderstandings. What prior experiences might have led to a misunderstanding? 4
In column 1, record the misconceptions related to the previously identifi ed struggles. Misconceptions (What is it the students don t understand?) Needed Conceptual Understandings (If a student really understood they would understand ) For each misconception listed in column 1, record the related conceptual understanding needed for students to be successful in mathematics. 5
An overview of the process Identifying the problem Identify common problem areas Pinpoint the disconnect Identify possible causes prior experiences that led to the disconnect Identify the misconception Identify the needed conceptual understanding Identify key experiences to develop the needed conceptual understanding 6
What do you think? A student was solving some simple math fact problems in her mathematics journal. This is what she wrote. 4 + 5 = 5 4 + 6 = 6 4 + 7 = 7 4 + 8 = 8 4 + 9 = 9 What is the student s error? What is the disconnect? How would you intervene? 7
Becoming a Detective The art of researching the problem Ask questions to fi nd out what they are really thinking. Listen to what they say instead of what you hope to hear. Question to determine depth of understanding. Investigate before you correct. Make sure you are correcting the actual problem. Build on what they do understand. Don t correct too soon. Remember the value of informal assessment Formal assessments tell you there is a problem. Informal assessments help determine what the problem is. 8
Developing the number facts Introduce Use key visual models Reinforce Link concrete, pictorial, verbal, and symbolic representations Practice Build fl uency through fun meaningful activities Extend Apply to numbers beyond the basic number facts 9
The Bridge to Ten Strategy Pre-activities 10
The Bridge to Ten Strategy Pre-activities Activity 1 Dominoes Matching Quantities Materials: One set of double-nine five frame dominoes per group of 3 or 4 How to play: Place all of the dominoes upside-down on the table. Each player selects 7 dominoes. The person with the largest double begins by placing the double on the table. Continuing clockwise, each player takes a turn matching their domino to one end of the domino train. If a player is unable to attach a domino to the train, he or she must draw from the pile until they can. The first player to play all of his or her dominoes wins. 11
The Bridge to Ten Strategy Pre-activities Activity 2 Dominoes Who has more? Materials: One set of double-nine five frame dominoes per pair How to play: Turn all of the dominoes upside-down. Distribute all of the dominoes so that each player has half of the set. Each player places all of the dominoes in their hand upsidedown in a stack on the table. Each player turns over the top card in the deck. The player with the greatest total describes how she knows she her domino has a greater total than her opponent s. That player takes both dominoes and places them at the bottom of the deck. If there is a tie, each player places 2 additional dominoes face down on his or her initial play. They then place a third domino face up on top of the stack. The player with the greatest total claims both stacks by describing how they determined they had the greatest total. Continue play until a player has captured all of the dominoes. 12
The Bridge to Ten Strategy Pre-activities 13
The Bridge to Ten Strategy Pre-activities Activity 3: Dominoes Total Ends to the 20Ends How How to play: to play: Challenge Each student each student selects to four select dominoes four dominoes at random at random and take turns to and take arrange turns the to tiles arrange as shown the tiles below, as shown so the below, total number so the of dots on total the number outer of ends dots is on close the outer to the ends sum is of 20 a multiple of fi ve. The number made is that player s score. No points (A variation are scored to this if a game multiple is to of allow fi ve sums cannot larger be made. to 20). Several Player s scores score are possible is the difference with the to tiles 20. above, For example: but the the numbers greatest on the possible ends are score 4 + is 620 + as 3 + shown. 2 = 15. The The fi rst difference player to between 15 reach and 10020 points is 5, the wins score the game. for that player is 5. Variation Repeat and play 5 rounds. The player with the lowest score is declared the winner. If one player cannot make a scoring arrangement with their tiles and another player can, the second player scores the relevant number of points. 5 + 6 + 0 + 9 = 20 Adapted from Number Concept Activities in A Little Book of Big Ideas: Dominoes, Calvin Irons (ORIGO Education, 2007). 14
The Bridge to Ten Strategy Introduce 15
The Bridge to Ten Strategy Introduce The bridge-to-ten strategy can be used to introduce and develop these facts. + 1 2 3 4 5 6 7 8 9 1 2 3 4 5 6 7 8 9 10 2 3 4 5 6 7 8 9 10 11 3 4 5 6 7 8 9 10 11 12 4 5 6 7 8 9 10 11 12 13 5 6 7 8 9 10 11 12 13 14 6 7 8 9 10 11 12 13 14 15 7 8 9 10 11 12 13 14 15 16 8 9 10 11 12 13 14 15 16 17 9 10 11 12 13 14 15 16 17 18 Directions 1. Hold one card as shown below. Ask the students to tell you what they see and how they can figure out the total number of dots. 2. Close the flap to demonstrate the idea of bridging to ten. Nine add four is the same as ten add three. 3. Ask the students to write the two number facts to help them see that the answers are the same. 9 + 4 = 13 10 + 3 = 13 4. Repeat Steps 1 to 3 with other cards from the set. 5. Use the same cards to develop the turnaround facts. The addend that is closer to ten should be represented on the right. 16
The Bridge to Ten Strategy Reinforce 11 Name: Jade had 9 girls and 6 boys at her party. How many guests in all? 1. a. These ten-frames show 9 counters. Draw 6 more counters. Start here b. Write the total. What did you notice? c. Complete the sentence. 9 + 6 is the same as 10 + 2. a. These ten-frames show 8. Draw 4 more counters. b. Write the total. c. Complete the sentence. 8 + 4 is the same as 10 + Start here ORIGO Publications Permission is given to instructors to reproduce this page for classroom use. 30 Make a Ten 17
The Bridge to Ten Strategy Reinforce Name: 11 1. Draw more counters then complete the sentence. a. b. Draw 7 more Draw 5 more + is the same as c. d. Draw 5 more + is the same as Draw 3 more ORIGO Publications Permission is given to instructors to reproduce this page for classroom use. + + is the same as is the same as 2. For each of these, draw an arrow to a number sentence below that has the same answer. Write the answer. a. 9 + 8 b. 8 + 6 c. 9 + 3 10 + 2 = 10 + 7 = 10 + 4 = Make a Ten 31 18
The Bridge to Ten Strategy Extend Name: 12 1. For each of these, draw more counters then complete the sentence. a. b. Draw 7 more Draw 6 more 20 19 + is the same as c. d. Draw 5 more 29 Draw 8 more + is the same as ORIGO Publications Permission is given to instructors to reproduce this page for classroom use. 2. For each of these, draw an arrow to a number sentence below that has the same answer. Write the answer. a. 29 + 8. 30 + 5 =. + is the same as b. 58 + 3 c. 28 + 7 d. 59 + 5 30 + 7 =. 60 + 4 =. + is the same as 60 + 1 = Make a Ten 33 19
The Bridge to Ten Strategy Extend Name: 6 1. For each of these, draw more counters then complete the sentence. a. b. Draw 16 more Draw 17 more 29 + is the same as 27 + is the same as 30 30 c. d. Draw 27 more Draw 17 more ORIGO Publications Permission is given to instructors to reproduce this page for classroom use. 2. For each of these, draw an arrow to a number sentence below that has the same answer. Write the answer. a. 47 + 18 + is the same as b. 68 + 27 c. 69 + 33 d. 48 + 26. 50 + 24 =. 50 + 15 =. 70 + 25 =. + is the same as 70 + 32 = Make a Ten 19 20
The Bridge to Ten Strategy Extend 6 Name: Aden bought a camera for $27 and a lm pack for $18. How much did he spend in all? 1. a. These ten-frames show 27 counters. Draw 18 more counters. Start here b. Write the total. c. Complete this sentence to match the picture above. 27 + 18 is the same as 30 + What do you notice? 2. Use the same method to gure out 148 + 27 in your head. Try making a ten. Complete this sentence to help you. 148 + 27 is the same as 150 + ORIGO Publications Permission is given to instructors to reproduce this page for classroom use. 18 Make a Ten 21
The Bridge to Ten Strategy Extend Name: 6 1. For each of these, draw an arrow to a number sentence below that has the same answer. Write the answer. a. 97 + 38 b. 128 + 47 c. 119 + 26 a. 130 + 45 = 100 + 35 = 120 + 25 = 2. Complete each sentence. a. 68 + 27 b. 247 + 36 c. is the same as is the same as 70 + 250 + 169 + 26 is the same as 170 + ORIGO Publications Permission is given to instructors to reproduce this page for classroom use. d. 38 + 127 e. 26 + 139 f. is the same as is the same as + 130 + 140 3. For each of these, draw an arrow to the number sentence you could use to gure it out. Write the answer. a. 138 + 47 120 + 65 = b. 66 + 119 170 + 15 = c. 28 + 157 140 + 45 = d. 168 + 17 160 + 25 = 47 + 329 is the same as + 330 Make a Ten 19 22
The Bridge to Ten Strategy Extend Name: 5 1. For each of these, write an easier number sentence that will help you gure out the problem below. Write the answer. a. b. c. so 4.6 + 3.8 = so 8.9 + 4.3 = so 7.8 + 5.4 = d. e. f. so so 1.9 + 6.5 = 2.8 + 3.6 = so 9.9 + 6.7 = 2. Write the answers. Place a above the numbers you adjusted. a. 6.9 + 8.4 = b. 5.3 + 7.8 = c. 8.9 + 4.4 = d. 7.7 + 8.8 = e. 6.5 + 3.9 = f. 4.8 + 4.9 = 3. Add the numbers on the spokes to the number in the center. Write the answers around the outside. a. b. 8.7 5.3 3.6 7.3 8.6 6.5 4.9 4.2 2.7 3.8 9.5 5.3 1.8 7.5 6.4 2.8 4.7 7.9 23
The Bridge to Ten Strategy Extend Name: 5 1. Adjust each of these to make a new sentence that is easier to gure out. Write the answer. a. 23.9 + 15.6 b. is the same as 42.7 + 14.8 is the same as c. 31.8 + 16.9 d. is the same as 53.8 + 24.7 is the same as 2. Write the answers. Place a above the numbers you adjusted. a. 24.3 + 13.9 = b. 16.8 + 31.7 = c. 45.9 + 11.6 = d. 32.7 + 15.8 = e. 53.9 + 34.8 = f. 26.7 + 42.9 = 3. Add the numbers on the spokes to the number in the center. Write the answers around the outside. a. b. 26.8 23.6 34.8 23.7 13.9 35.7 12.9 31.9 25.5 22.8 14.6 16.7 56.7 32.3 16.9 44.9 51.8 25.9 24
Fundamentals The Bridge to Ten Strategy 2 players Reinforcing the make-a-ten strategy Purpose This game is designed to help students to see ten as one ten as well as ten ones. This realization will encourage students to move beyond countall strategies to more efficient strategies, such as bridging to ten to solve addition and subtraction problems. For example, a student who bridges to ten to solve 7 + 5 would think 7 + 3 = 10 and 10 + 2 = 12. Although this game involves the addition of one- and two-digit addends, the students will also be using the thinking associated with missing-addend subtraction, for example, I have 19. How many more do I need to make 40? Materials Each pair of players will need One (1) standard number cube showing numerals or dot patterns 1-6. Each player will need A First to Forty game board (page 54) as shown below. Forty (40) counters (a different color for each player). How to Play The aim is to fill four ten-frames with counters. Players must start at the top row and fill from left to right in each ten-frame. The first player rolls the number cube. The player then places that number of counters in the first ten-frame on his or her game board. The other player has a turn. The first player to fill all of his or her ten-frames is the winner. It is not necessary to roll an exact number to finish. Reading the Research Ten-frames are good for developing part-whole understandings involving the landmark numbers 5 and 10. These understandings are especially useful in addition- and subtraction-fact work (Isaacs & Carroll, 1999; Van de Walle, 2001). 54 ORIGO Publications Permission is given to instructors to reproduce this page for classroom use. 52 25
The Bridge to Ten Strategy 26
Number Facts Teaching Facts Using a Number Sense Approach Addition/Subtraction Addition Count On 0, 1, 2 Use Doubles Doubles + 1 Subtraction Count On/Count Back Use Doubles Use Ten Doubles + 2 Link above to addition... Use Ten + 0 1 2 3 4 5 6 7 8 9 0 0 1 2 3 4 5 6 7 8 9 1 1 2 3 4 5 6 7 8 9 10 2 2 3 4 5 6 7 8 9 10 11 3 3 4 5 6 7 8 9 10 11 12 4 4 5 6 7 8 9 10 11 12 13 5 5 6 7 8 9 10 11 12 13 14 6 6 7 8 9 10 11 12 13 14 15 7 7 8 9 10 11 12 13 14 15 16 8 8 9 10 11 12 13 14 15 16 17 9 9 10 11 12 13 14 15 16 17 18 27
Summarizing the process Planning for Intervention Identify common misconceptions Identify needed conceptual understandings Determine how far back to plug the hole Identify key visual models Link concrete, visual/pictorial, verbal, and symbolic representations Plan to bridge to grade level using similar models Be purposeful in material use and model selection Spiral Classroom Protocols Allow for mistakes Don t panic! Ask versus tell (whenever possible) Use Think-pair-share and other informal interview techniques to determine understanding Ask questions to determine understandings versus reaching an instructional goal Require students to always explain thinking Build connections 28
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The ORIGO Range ORIGO Education has a broad and exciting range of high-quality educational resources. From the independent challenges of The Think Tank cards to an instructional series like Algebra for All, or a new teaching resource like DecaCards, our focus is on helping you make a difference in the classroom. Visit www.origoeducation.com to find out more. ORIGOmath A Step-by-Step Approach to Computation ORIGOmath is a quality supplemental program that provides a developmentally appropriate sequence to teach and assess students mathematical understanding and computational skill. The program can also be used as a grade-level intervention for elementary students who need a new approach. Teacher Sourcebook ation y-step A Step-b Comput ch to Approa The ORIGOmath program has been added to the list of approved intervention programs in Montgomery County, Maryland, and both Fort Bend ISD in Texas and Denver Public Schools have approved the program for Tier II Intervention. ORIGOmath s balance of concept-based and process development is just one of the features leading to the program s approval. The Box of Facts Developing Number Fact Strategies The Box of Facts kits are packed with visual aids for introducing and developing the basic number facts and number fact strategies. Each contains over 200 sturdy cards organized by strategy and designed for use with the activities in The Book of Facts series as well as lessons in ORIGOmath. In Washington s OSPI Instructional Materials Review of over 40 supplemental programs, The Box of Facts and The Book of Facts ranked the highest for content quality. They were the only resources with an average score of over 80%. These are must-have materials for your classroom! www.origoeducation.com 30
Fundamentals The 2s, 4s, and 8s Facts 4 players Doubling single-digit numbers Purpose From a very young age, many students are able to double numbers quite easily. Doubling is a powerful mental strategy for addition that begins with number facts and is continued beyond the number fact range. In this game, students double single-digit numbers. Materials Each group of players will need A Double Up game board (page 58) as shown below. One (1) number cube showing the numerals 3-8. This can be made from a blank wooden cube. Each player will need Four (4) counters (a different color for each player). How to Play The aim is to place four counters on the game board. The first player rolls the number cube and doubles the number rolled. Example: Carla rolls 7 and thinks double 7 is 14. The player claims the answer on the game by covering it with a counter. If an answer is unavailable, the player misses a turn. The other players have a turn. The first player to place four counters on the game board is the winner. Reading the Research The acquisition of basic facts needs to focus on the systematic use of strategies to encourage increasingly more sophisticated number fact knowledge (Bobis, Mulligan, Lowrie & Taplin, 1999). 58 ORIGO Publications Permission is given to instructors to reproduce this page for classroom use. 56 Fundamentals 31
The 2s, 4s, and 8s Facts 58 Fundamentals Permission is given to instructors to reproduce this page for classroom use. 32
Doubles and Near Doubles 33
50 Fundamentals The 2s, 4s, and 8s Facts 4 players Doubling to multiply by two and four Purpose Doubling is powerful mental strategy for multiplication involving 2, 4, and 8. In this game, the students practice the fours facts by doubling and doubling again. Although, the main game focuses solely on number facts, the extension requires the students to think beyond the number fact range. Both games also involve the twos facts. Materials Each group of players will need A Double Double game board (page 50) as shown below. One (1) doubling cube. Using a blank wooden cube, write double on three faces and double double on the remaining three faces. One (1) number cube showing numerals 3-8. This can also be made from a blank wooden cube. Each player will need Four (4) counters (a different color for each player). How to Play The aim is to arrange all four counters on the game board. The first player rolls the two cubes. The player follows the instruction, doubling the number or doubling and doubling it again. Example: Tristan rolls 6 and double double. He thinks double 6 is 12 and double 12 is 24. Six multiplied by 4 is 24. The player claims the answer on the game board by covering it with a counter. If an answer is unavailable, the player misses a turn. Each of the other players has a turn. The first player to place all four counters on the game board is the winner. Reading the Research Thinking strategies for basic facts and other forms of mental computation must build directly on children s understanding of number and not be taught as mindless procedures (Van de Walle & Bowman Watkins, 1993). 16 12 14 48 32 28 6 16 10 24 8 12 20 ORIGO Publications Permission is given to instructors to reproduce this page for classroom use. Fundamentals 34
The 2s, 4s, and 8s Facts 50 12 16 12 14 32 28 6 16 10 24 8 20 Fundamentals Permission is given to instructors to reproduce this page for classroom use. 35
Fundamentals The 2s, 4s, and 8s Facts 4 players Doubling to multiply by four and eight Purpose This game uses a doubling strategy to practice the fours and eights facts. It extends the thinking that was introduced in Double Double on pages 48-51. The main game focuses solely on number facts. The extension requires the students to think beyond the number fact range. Both games also involve the fours facts. How to Play Materials Each group of players will need A Do the Ds game board (page 54) as shown below. One (1) doubling cube. Using a blank wooden cube, write double double on three faces of the cube and double double double on the remaining three faces. One (1) number cube showing numerals 3-8. This can also be made from a blank wooden cube. Each player will need The aim is to arrange all four counters on the game board. The first player rolls the two cubes. The player follows the instruction, doubling the number two or three times. Example: Lily rolls 4 and double double double. She thinks double 4 is 8, double 8 is 16, double 16 is 32. Four multiplied by 8 is 32. The player claims the answer on the game board by covering it with a counter. If an answer is unavailable, the player misses a turn. Each of the other players has a turn. The first player to place all four counters on the game board is the winner. Five (5) counters (a different color for each player). Reading the Research Games involving number cubes can provide an incentive 52 54 64 40 16 24 12 20 48 56 24 28 32 20 ORIGO Publications Permission is given to instructors to reproduce this page for classroom use. for inventing more efficient strategies as well as hours of interesting practice (Baroody & Standifer, 1993). Fundamentals 36
The 2s, 4s, and 8s Facts 54 64 40 16 24 12 20 48 56 28 32 20 24 Fundamentals Permission is given to instructors to reproduce this page for classroom use. 37
Number Facts Multiplication Chart x 0 1 2 3 4 5 6 7 8 9 0 0 0 0 0 0 0 0 0 0 0 1 0 1 2 3 4 5 6 7 8 9 2 0 2 4 6 8 10 12 14 16 18 3 0 3 6 9 12 15 18 21 24 27 4 0 4 8 12 16 20 24 28 32 36 5 0 5 10 15 20 25 30 35 40 45 6 0 6 12 18 24 30 36 42 48 54 7 0 7 14 21 28 35 42 49 56 63 8 0 8 16 24 32 40 48 56 64 72 9 0 9 18 27 36 45 54 63 72 81 38
Key models for dividing numbers Quotitive Division 27 dots 3 in each row 18 dots 6 equal rows 39
Key models for dividing numbers Quotitive Division 1. 9 257 2. 15 1325 3. 22 2435 4. 44 2435 40