Partner Investigation • O ne student makes an amount up to 5 with counters and the partner makes an image that looks exactly the same. Together, they figure out the total number of counters. Further Talk • Discuss the strategies that students used and make an anchor chart. • Briefly show one of the fast images, showing identical sides, and then take it away. How many dots did you see and how do you know? What is a matching equation? (e.g., 3 plus 3 equals 6) • Repeat these activities another day with other fast images of doubles (see BLM 17: More Dice Doubles). • This activity can also be done with ten-frames using BLM 18: Ten-Frame Doubles (1–5) or Digital Slides 106–110, 10 counters per student, and BLM 5: Blank Ten-Frames. • It can also be done showing identical numbers of beads on the top and bottom rows of the arithmetic rack. Further Investigation • H ave students create all of the doubles up to 5 and 5 using connecting cubes of two colours, as double-decker trains (with one row on top of the other rather than as one long train). • Students can keep the double-decker trains and flash them to each other like fast images to practise recognizing doubles. They can respond, “I saw 3 plus 3 equals 6.” Addition and Subtraction to 10 153
12Lesson Whole-Part-Part: Decomposing 10 Math Curricular Competencies Learning Standards • Reasoning and analyzing: Use reasoning to explore and make Teacher connections; develop mental math strategies and abilities to make sense of Look-Fors quantities Previous Experience • U nderstanding and solving: Develop, demonstrate, and apply with Concepts: Students have had mathematical understanding through play, inquiry, and problem solving; opportunities to create visualize to explore mathematical concepts various combinations of the same quantity using • Communicating and representing: Explain and justify mathematical concrete materials. ideas and decisions; represent mathematical ideas in concrete, pictorial, and symbolic forms • Connecting and reflecting: Connect mathematical concepts to each other and to other areas and personal interests Content • W ays to Make 10: Decomposing 10 into parts; benchmarks of 10 and 20 • Addition and subtraction to 20 (understanding of operation and process): Mental math strategies: counting on, making 10, doubles; addition and subtraction are related • Change in quantity to 20, concretely and verbally: Verbally describing a change in quantity (e.g., I can build 7 and make it 10 by adding 3) • M eaning of equality and inequality: Demonstrating and explaining the meaning of equality and inequality; recording equations symbolically using = and ≠ Possible Learning Goals • Decomposes 10 using concrete materials and various strategies • Connects addition and subtraction as operations that ‘undo’ each other • Finds most or all of the combinations for subtracting from 10 • Explains or shows how the equations match the concrete representations • Recognizes that the same number of pairs are involved in both the addition and subtraction questions for 10 154 Number and Operations
McmbdsoaaeaumtlctbhacbotnhrVimcnaioneacpcg,ttoai,cosebtonequams,ulka,eapretoyaiq:oswuneaa,syt,ions, About the This activity gives students more experience with finding all combinations of 10, except from a ‘whole into parts’ context by solving a separating subtraction problem. This offers a good opportunity to link the operations of addition and subtraction and show how they ‘undo’ each other. It also highlights that, unlike addition, the commutative property does not apply for subtraction since the order of subtracting does matter (e.g., 2 brownies can be taken away from 5 brownies, but 5 brownies can’t be taken away from 2 brownies). Materials: Minds On (10 minutes) concrete materials, chart paper, markers • P ropose the following problem: There are 5 birds in a tree. Some fly away. Time: 45 minutes How many birds could be left in the tree? • Have students turn and talk to a partner to figure out what the solutions might be. Ask whether they think they have found all of the possibilities and how they know. • Ask students what subtraction equations would match their solutions. Record these equations on chart paper. Working On It (15 minutes) • Students work in pairs. Have students solve the following problem: There are 10 birds in a tree. Some fly away. How many could be left? Have students find all of the possibilities using any of the materials they used in the past lessons, and record their results (see Differentiation). Differentiation • Students can record their combinations by drawing simple pictures (e.g., differently coloured dots), by recording number combinations (e.g., 9 and 1, 8 and 2), or by creating subtraction equations (e.g., 10 – 9 = 1). Assessment Opportunities Observations: Are students applying any of the strategies that they used for composing combinations to decompose 10? (e.g., decompose by 1 more each time) Conversations: Probe deeper if students are using guess and check, and not using one combination to form another: Teacher: What did you do here? Student: I had 10, and I took 1 away and there are 9 left. Teacher: Can you visualize what would happen if you took 1 more away? Try it. Student: Now I have 8 and 2. continued on next page Addition and Subtraction to 10 155
Teacher: Do you still have 10 altogether? Student: (Student may answer, yes, immediately or may need to count.) Teacher: (if they counted again) How many birds should there be altogether? How do you know? Teacher: What strategy did you just use? Student: I took 1 more away. Teacher: Will that work every time? Student: I’m not sure. Teacher: Why don’t you try, and see what happens? Consolidation (20 minutes – 10 minutes to meet with a partner and 10 minutes to meet with the whole class) • Have each pair of students meet with another pair to compare their solutions and to determine if they have all the combinations. • As a class, discuss students’ strategies and record all of the subtraction equations. Ask students how they could put the combinations in a list so they show a pattern. • Show students the chart made in Lesson 11 with all of the adding combinations for 10. Ask what they notice is the same and what is different (e.g., in both operations, 9 and 1 go together when adding to or subtracting from 10; they are different because you can have 9 + 1 = 10, which gives the same sum as 1 + 9 = 10, and you can have 10 – 9 = 1 but not 9 – 10 = 1). • Link addition and subtraction. Addition has part with part to make a whole, while subtraction has the whole broken into parts. Further Practice • Independent Problem Solving in Math Journals: Pose one of the following prompts: – Show how 8 + 2 and 10 – 2 are related. – Make many subtraction stories about 7 cookies. 156 Number and Operations
13Lesson Subtraction as ‘Think Addition’ Math Curricular Competencies Learning Standards • Reasoning and analyzing: Use reasoning to explore and make Teacher connections; develop mental math strategies and abilities to make sense of Look-Fors quantities Previous Experience • U nderstanding and solving: Develop, demonstrate, and apply with Concepts: Students have had mathematical understanding through play, inquiry, and problem solving; experience with addition visualize to explore mathematical concepts and subtraction activities and have created related • Communicating and representing: Explain and justify mathematical equations. ideas and decisions; represent mathematical ideas in concrete, pictorial, and symbolic forms • Connecting and reflecting: Connect mathematical concepts to each other and to other areas and personal interests Content • W ays to Make 10: Decomposing 10 into parts; benchmarks of 10 and 20 • Addition and subtraction to 20 (understanding of operation and process): Mental math strategies: counting on, making 10, doubles; addition and subtraction are related • Change in quantity to 20, concretely and verbally: Verbally describing a change in quantity (e.g., I can build 7 and make it 10 by adding 3.) • M eaning of equality and inequality: Demonstrating and explaining the meaning of equality and inequality; recording equations symbolically using = and ≠ Possible Learning Goals • Solves part-part-whole problems using a variety of strategies • Connects addition and subtraction as operations that ‘undo’ each other • Creates a screened activity that can be presented to peers • S elects an appropriate strategy to solve screened activities • E xplains or shows how they solved the problem • C ounts on from total under screen when using ‘think addition’ strategy • E xplains how addition and subtraction are related Addition and Subtraction to 10 157
Msseeuiaqqgbtunuhta,raaVtt‘iictooohtcnn,ianssmbk,,uimbnalaaudarlsdtayc,i:nthmicoinening,,’us About the cdoemcopmospeo,se As Marian Small points out, “any addition situation can also be viewed as a subtraction one, and vice versa” since the operations ‘undo’ each other (Small, 2009, p. 107). While addition and subtraction are often taught separately, considerable time needs to be spent on linking the inverse operations. As Van de Walle emphasizes, screened activities that use part-part-whole models can reveal both operations and help students view subtraction as ‘think addition’ (Van de Walle & Lovin, 2006, p. 74). In a screened activity, the whole is shown and then a part of the whole is visibly removed, while the remaining part is covered. Students determine the part that is covered. These activities encourage a ‘think addition’ strategy such as, “What goes with the part shown to make the whole?” rather than a ‘count what is left’ strategy. Making this connection explicit not only links the operations, but will help students as they learn their number facts. The following Minds On activity was developed by Van de Walle (Van de Walle & Lovin, 2006, p. 74). Materials: Minds On (15 minutes) paper bag, • Pose the following problems: counters or connecting cubes, chart − There are 9 cookies in the bag. (Put 9 objects in the bag to represent the paper, markers cookies.) 3 children came and ate 1 cookie each. (Remove 3 of the cookies while students watch.) How many cookies are left in the bag? Time: 40–45 minutes Ask students how they can be sure that their response is correct. (This is a partially screened task—the total is hidden and students see how many are removed.) − There are 9 cookies in the bag. (Put 9 objects in a bag.) Some children come and take 1 cookie each. (Remove 4 cookies, but do not show students how many were removed.) There are 5 cookies left. How many cookies were taken out of the bag? (This is a fully screened task—the total is hidden and the amount removed is not known.) Have students discuss their solution and the strategy that they used. • Tell students that they will be solving similar problems with a partner. Model the activity with a student. Put out 7 counters on a piece of paper so the student can see. Then have the student cover his/her eyes while you break the 7 counters into two groups (e.g., 3 and 4) and then cover one of the groups. The student uncovers his/her eyes and figures out how many counters are covered, saying, “7 minus 3 equals 4.” Or they may say, “3 plus 4 equals 7.” The covered part can then be revealed to see whether the student is correct. Model how they can record their thinking with matching addition and subtraction equations. 158 Number and Operations
Working On It (15 minutes) • Students work in pairs playing the modelled game, switching roles and creating their own problems. After each turn, have students record both the addition and subtraction equations on chart paper. Differentiation • For some students who may need support with the language, you can create sentence starters that are linked to pictures and/or equations (e.g., minus (–) equals (=) or plus (+) equals (=) ) to help them verbally formulate their ideas. Model how to use them as you act out a problem. • If students have difficulty recording matching addition and subtraction equations, they can record the number combinations (e.g., 3, 4, 7). You can help them set up a chart with columns labelled Part/Part/Whole. Assessment Opportunities Observations: As students play the game, observe the strategies that they are using. This can help to assess students’ number sense and whether students see numbers as related to each other or as random quantities. – Are they counting on, are they counting three times, or do they just ‘know’ how many there are? – Do they use their fingers to create both sets or do they use their fingers to track as they count on? – Do they adjust their strategies according to the difficulty of the numbers presented? (e.g., If the difference is 1, they may automatically know, while they may need to create sets with concrete materials for numbers that differ by larger amounts.) Conversations: Make note of the observations which can be addressed in one-to-one interviews with certain students. For example, if students are still counting three times, make two sets and cover one with a piece of paper so they cannot see how many are underneath. Print the numeral on the top. Tell them, “There are 4 counters under here, so how many are there altogether?” Consolidation (10–15 minutes) • Discuss the strategies that students used to play the game. • Have a discussion of whether these questions are addition or subtraction problems. Highlight the idea that they can be solved using either operation. • Ask how addition and subtraction are related. (e.g., They are the opposite of each other.) Addition and Subtraction to 10 159
14Lesson Compare Problems: Differences Math Curricular Competencies Learning Standards • Reasoning and analyzing: Use reasoning to explore and make Previous Experience connections; develop mental math strategies and abilities to make sense of with Concepts: quantities Students have had several experiences with adding • U nderstanding and solving: Develop, demonstrate, and apply and subtracting with concrete materials. mathematical understanding through play, inquiry, and problem solving; Students can create visualize to explore mathematical concepts matching equations for models that represent • Communicating and representing: Explain and justify mathematical addition or subtraction. ideas and decisions; represent mathematical ideas in concrete, pictorial, and Teacher symbolic forms Look-Fors • Connecting and reflecting: Connect mathematical concepts to each other and to other areas and personal interests Content • W ays to make 10: Decomposing 10 into parts; benchmarks of 10 and 20 • Addition and subtraction to 20 (understanding of operation and process): Mental math strategies: counting on, making 10, doubles; addition and subtraction are related • Change in quantity to 20, concretely and verbally: Verbally describing a change in quantity (e.g., I can build 7 and make it 10 by adding 3.) • M eaning of equality and inequality: Demonstrating and explaining the meaning of equality and inequality; recording equations symbolically using = and ≠ Possible Learning Goal • Compares quantities using a variety of strategies • Accurately finds the difference between two quantities • Explains or shows how they found the difference • Explains how the problem can be solved with either addition or subtraction 160 Number and Operations
Math Vocabulary: About the eec‘qtqohuumiaanpttkiiaooarnnedss,d,,dimtbiifoafaenltar’cenhncinecg,e, In comparing problems, the quantities of two sets are being compared, yet they Materials: are not subsets of each other like in part-part-whole problems. Instead, the connecting cubes, focus is on the difference between the quantities of the two sets. For example, chart paper, markers the problem might be, “Jake has 6 cookies. Anna has 4 cookies. How many Time: 45 minutes more does Jake have?” Students can use a variety of strategies using concrete materials to figure out the difference. They may: • C reate a set of 6 cubes and a set of 4 cubes, put them side by side, matching the cubes from each set using one-to-one correspondence, and then count the cubes that are left over. • C reate a set of 6 cubes and a set of 4 cubes, and then count on from 4 to 6, tracking the count on their fingers. • C reate a set of 4 cubes in one colour, add on different coloured cubes until a set of 6 is created, and count the cubes in the second colour. • C reate a set of 6 cubes, remove cubes until a set of 4 is left, and count the cubes that were removed. It is important to vary the problems. For example: • A nna has 4 cookies. Jake has 2 more cookies than Anna. How many cookies does Jake have? (compare quantity unknown) • Jake has 6 cookies. He has 2 more cookies than Anna. How many cookies does Anna have? (referent unknown) Minds On (15 minutes) • H ave two students stand. Give one student a train made from 5 connecting cubes. Give the second student a train made from 9 connecting cubes. • Ask which train has more cubes and how they know. Ask how they could figure out how many more cubes the second student has than the first student (the strategy is more important than the answer). Students can demonstrate their thinking using the connecting cubes. Through questioning, try to elicit at least two or three strategies. • Pose another problem using a different pair of students. Give one student a train that is 4 cubes long. Give the second student a train that is 7 cubes long, but put it in a bag so the rest of the students cannot see it. Tell them that the second train is 3 cubes longer than the first train. Ask students how long the second train is. Addition and Subtraction to 10 161
Working On It (15 minutes) • Students work in pairs and take turns creating trains and asking the questions. One student looks away, while the other student makes two trains that are different in length. He/she hides one of the trains. The second student looks at the train and the first student poses a question, such as, “The second train is 4 cubes longer. How long is it?” Students can check their answers by revealing the second train. Differentiation • For students who cannot mentally figure out the length of the second train, encourage them to use other cubes to build and support their thinking. • For students who master the activity and need more of a challenge, pose some of the following problems: – The difference between 2 numbers is 7. What might the numbers be? – The difference between 2 numbers is at least 5. What might the numbers be? Assessment Opportunities Observations: Pay attention to how students are figuring out the length of the hidden train. – If using mental strategies, are they subitizing the original amount of cubes in the train? Do they need to touch and count each cube? – Can they provide a solution for 1 more/less than problems without counting the length of the train, but by knowing the number sequence? – Are they mentally counting on or back? How do they track the size of the new train? (e.g., on their fingers, nodding their head for each count) – Are they counting on when using concrete materials, or are they creating the original train, adding or removing the difference, and then counting the cubes in the train again? (Ask probing questions highlighted in earlier lessons to get students to count on.) Conversations: • To probe further, ask how students are finding out the difference. Ask what they are counting on their fingers. • If students find 1 more/less than problems difficult, you can carry out some related Math Talks (e.g., 1 more/less than) with small groups or the entire class to reinforce how the differences are related to the number sequence. 162 Number and Operations
Consolidation (15 minutes) • Inside/Outside Circle: Have each student create two trains of different sizes. Split the class in half and make two circles, one within the other (the inside and outside circles). Students on the inside and outside face each other, with each person in front of a partner. The partners put one of their trains behind their backs so the other students cannot see. They take turns showing their one train, and then posing prompts that compare it to the train behind their backs. After a couple of minutes, have the students on the outside move one person to the right. (You can signal the time to change with a bell or by playing music.) Students now have a new partner and can pose the same or different questions. Repeat this for two or three rotations. • Discuss the mental strategies that students used to figure out the differences between the two trains. • Ask if these are addition or subtraction problems. Through discussion, students should realize that the problems can be solved with either operation. Further Practice • Independent Problem Solving in Math Journals: Verbally pose one of the following prompts: – The difference between two numbers is 3. What might the two numbers be? − Show that there are 4 more cookies on one plate than on another plate. Materials: Math Talk: large arithmetic Math Focus: Mental strategies for comparing problems rack, small arithmetic racks (or BLM 5: Blank Let’s Talk (10–15 minutes) Ten-Frames and counters, or connecting Select the prompts that best meet the needs of your students. cubes) • W ith your partner, solve this problem in more than one way on your arithmetic rack. There are 9 birds in the tree and 3 birds on the ground. How many more birds are in the tree? continued on next page Addition and Subtraction to 10 163
Teaching Tip • Possible solutions to discuss: Integrate the math Removal: talk moves (see page 6) throughout − Slide across 9 beads on the top row (either all at once or 5 red beads and Math Talks to then the 4 white beads); maximize student participation and − Slide back 6, one at a time, counting 1, 2, 3, 4, 5, 6, until they can see active listening. that there are only 3 left; − Slide back all beads except the 3 (students can subitize this) and then count how many they slid back; − Slide back one at a time counting back from 9, 8, 7, 6, 5, 4, 3, and then counting the beads they slid back. Adding On: − Slide across 3 beads to the right on the bottom row (or they can work on the top row); − Slide across 2 more red beads on the bottom row to create a group of 5, and then 4 white beads all at once to make a group of 9 (2 moves). Then count the beads that they slid over 1, 2, 3, 4, 5, 6; − Mentally count on from 3 to 9, tracking each number on their fingers, and then sliding across a group of 6 counters on the bottom row, since it is easy to subitize the 5 red beads and 1 white bead (1 move). Comparing: − Build 9 on the top and 3 on the bottom. Visually match corresponding beads on the top and bottom and count the ‘extra’ beads on the top, 1, 2, 3, 4, 5, 6. • G ive students time to solve the problem. How did you solve this problem? Put your thumb up if you solved it the same way. How did you start? What do those beads represent? What action are you doing? Who can explain in their own words how this strategy works? How is this strategy different from Jon’s strategy? Partner Investigation • G ive students another problem to solve to try out some of the strategies that were discussed. Further Talk • M ake an anchor chart of students’ strategies, illustrating them with red and white dots for beads, and annotating the movement. You can name the strategies after the students who explained them. 164 Number and Operations
15Lesson Creating Using Benchmarks Mental Strategies Math Curricular Competencies Learning Standards • Reasoning and analyzing: Use reasoning to explore and make connections; Previous Experience develop mental math strategies and abilities to make sense of quantities with Concepts: Students have had • U nderstanding and solving: Visualize to explore mathematical concepts; opportunities to add and subtract with concrete develop and use multiple strategies to engage in problem solving materials such as ten- frames, and can read and • Communicating and representing: Explain and justify mathematical create matching equations for the concrete ideas and decisions representations. • Connecting and reflecting: Reflect on mathematical thinking Teacher Look-Fors Content • W ays to Make 10: Decomposing 10 into parts; benchmarks of 10 and 20 • Addition and subtraction to 10 (understanding of operation and process): Mental math strategies: counting on, making 10, doubles Possible Learning Goals • Solves problems by applying various mental strategies • Investigates benchmarks of 5 and 10 when solving addition and subtraction problems • Uses mental strategies to find sums presented on ten-frames • Explains or shows strategies and how they mentally rearranged the dots on the ten-frame • Accurately recreates the dot images using tens-frames and counters, and rearranges them to match their thinking • R eflects on strategy and explains how their rearrangement helped them solve the problem McrmeoacmatohtpcmVhopoisnocegas, beed,uqeleucaqaoryutmi:aoptnioossnes,, About the Students have had many opportunities to add and subtract with concrete materials, which has helped them to discover the advantages of using 5 and 10 as benchmark numbers. These experiences help students create mental images of these concrete representations, so they can be retrieved later and used to visualize and solve other problems. Addition and Subtraction to 10 165
BLM 28: Shake and Spill Game Combinations for 9 Minds On (20 minutes) BLM 29: Fast Images (Set 2) © 2018 Scholastic Canada Ltd. MATH PLACE GRADE 1: Number Sense (Early in the Year) ISBN 978-0-0000-0000-0 Image 1 Materials: • Using BLM 19 or Digital Slides 111–116, briefly show Image 1 and then take it away. Have students explain how they know BLM 19: Fast Images (Set 1) and Digital how many dots they saw. Image 2 Slides 111–116; BLM 20: Fast Images (Set 2) 95 and Digital Slides 117–127; chart paper; • Repeat, showing some of the following fast images of ten-frames. After each BLM 5: Blank Ten- one, ask students for their strategy and an equation that would match their Frames; counters thinking. 96 © 2018 Scholastic Canada Ltd. MATH PLACE GRADE 1: Number Sense (Early in the Year) ISBN 978-0-0000-0000-0 BLM 29: Fast Images (Set 2) BLM 29: Fast Images (Set 2) 98 © 2018 Scholastic Canada Ltd. MATH PLACE GRADE 1: Number Sense (Early in the Year) ISBN 978-0-0000-0000-0 BLM 29: Fast Images (Set 2) Time: 50 minutes Image 4 − Image 2: (5 + 2 = 7) Image2 © 2018 Scholastic Canada Ltd. MATH PLACE GRADE 1: Number Sense (Early in the Year) ISBN 978-0-0000-0000-0 Image 3 Image 4 − Image 3: (5 + 4 = 9) − Image 4: (5 + 5 = 10) 97 − Image 5: Ask students what they did in their minds to BLM 29: Fast Images (Set 2) find out how many counters there were on the 2 ten- frames combined. (e.g., Move the sixth counter in the © 2018 Scholastic Canada Ltd. MATH PLACE GRADE 1: Number Sense (Early in the Year) ISBN 978-0-0000-0000-0 Image 5 second frame to finish the row of 4 in the first frame.) 99 Optional: Write the expression for the original configuration (4 + 6) and the recomposed configuration (5 + 5). Ask how the first was changed into the second. Ask what the sum is for both and how they know. 100 © 2018 Scholastic Canada Ltd. MATH PLACE GRADE 1: Number Sense (Early in the Year) ISBN 978-0-0000-0000-0 BLM 29: Fast Images (Set 2) − Image 6: Ask students how they figured out the total. Image 6 (e.g., mentally moved 3 counters on the second ten-frame to make a row of 5 with the 2 counters on the first ten- frame; mentally moved the 2 counters on the first frame to the second ten-frame with 8 counters to make a full ten-frame) Optional: Write the expression for the original configuration (2 + 8) and the recomposed configuration (5 + 5). Ask how the first was changed into the second. Ask what the sum is for both and how they know. Working On It (15 minutes) • Provide each student with a few ten-frame images from BLM 20. Have students work in pairs, taking turns to show a fast image of two ten-frames and then taking it away. The other student determines the amount, explains his/her thinking, and then recreates the original image using ten-frames and counters. He/she then shows how the arrangement can be recomposed to solve the problem (e.g., decomposing and recomposing the amounts). The two students then compare to see if the recreation matches the fast image and whether the two sums are the same. 166 Differentiation • For students who need more work with establishing a benchmark for 5, give them simpler fast images with quantities up to 6 or 7, and images showing 1 or 2 more or less than a group of 5. • For students who need more of a challenge, they can create matching equations for the original fast image and the recomposed image. In Number and Operations
conversation, ask students how the two equations can still represent the same amount. Ask students to explain how the first equation was changed into the second equation by showing their concrete representations. Assessment Opportunities Observations: Observe what students are doing by paying attention to their actions with the materials. Are they moving counters 1 at a time to find what may complete a row, or do they move 2 or 3 counters at a time with an intended plan in mind? Conversations: For students who tend to use guess and check, ask them to visualize how to rearrange the counters and then make a prediction. This helps them develop visual images and mental strategies. Consolidation (15 minutes) • Using BLM 20 or Digital Slides 117–127, show one or two of the fast images to the class and discuss the strategies that students used to solve them. • Have ten-frames and counters available for each pair of students. Challenge them to create the original fast image and then recompose it to make an easier representation. • For one of the discussed examples, show expressions that represent the original and the recomposed arrangements (e.g., 3 + 7 and 5 + 5, or 9 + 1 and 10 + 0). Ask students to explain how the first arrangement was changed into the second, using concrete representations to support their explanations. • Ask how knowing 5 and 10 can help them figure out other numbers. Further Practice • Independent Problem Solving in Math Journals: Give each student one fast image of 2 ten-frames. Have them use counters and ten-frames to rearrange the counters and find the sum. They can use a template of empty ten-frames to show what was on the ten-frame and how it was rearranged. They can annotate what they did using arrows. For students who need a further challenge, have them create a matching equation for the original and recomposed images. Addition and Subtraction to 10 167
16Lesson Linking Addition and Subtraction Math Curricular Competencies Learning Standards • Reasoning and analyzing: Use reasoning to explore and make connections; Previous Experience develop mental math strategies and abilities to make sense of quantities with Concepts: Students have worked • U derstanding and solving: Develop, demonstrate, and apply mathematical with various problem structures (e.g., joining, understanding through play, inquiry, and problem solving; visualize to separating, part-part- explore mathematical concepts whole, compare problems) as they apply • C ommunicating and representing: Explain and justify mathematical their understanding of addition and subtraction. ideas and decisions; represent mathematical ideas in concrete, pictorial, and They have had symbolic forms experiences with connecting equations to • Connecting and reflecting: Connect mathematical concepts to each other their matching stories. and to other areas and personal interests Teacher Look-Fors Content • W ays to Make 10: Decomposing 10 into parts; benchmarks of 10 and 20 • Addition and subtraction to 20 (understanding of operation and process): Addition and subtraction are related Possible Learning Goals • Creates various addition and subtraction problems for a given answer • Investigates the connection between addition and subtraction • Creates a story that makes sense and matches the given challenges • Explains how the story either represents addition or subtraction • Explains how a given matching equation represents their story problem • Explains or shows how addition and subtraction are related Math Vocabulary: About the Lesson eesqquuubaatrttaiioocnntsisn,,gmadadtcinhgin, g Students have had several experiences working with different structures of addition and subtraction problems using a variety of strategies. This activity reinforces understanding of these structures by offering practice in creating addition and subtraction stories. The key concept is to connect the operations of addition and subtraction as operations that ‘undo’ each other. Time can be spent in the Consolidation making these links, through the stories and also through creating and comparing the matching equations. 168 Number and Operations
Materials: Minds On (15 minutes) chart paper, markers, concrete materials • T ell students that today they are going to make adding and subtracting Time: 50 minutes problems rather than solve them. • Tell them that the answer to the problem is “5 cookies.” Have them turn and talk with a partner to discuss what the problem might be. Encourage them to be creative and find at least two different problems. They can use concrete materials to help them devise their problems. • Have two or three pairs share one of their problems with the class (selectively choosing so that at least one represents addition and one represents subtraction). Have the rest of the class put their thumbs up if they had a similar problem. • Ask why the problems can be both adding and subtracting, yet still have the same answer. Optional: Ask what a matching equation would be for each of the problems. Have students explain how each part of the equation relates to the context of the problem. Working On It (15 minutes) • Students can work in pairs or individually. Pose one of the following open-ended questions to students. They are to create at least two problem stories and include both addition and subtraction. − The answer is 7 penguins. What might be the story? Create as many stories as you can. Optional: Make matching equations for each story. − Create many different stories that have 5 and 4 in them. Optional: Make matching equations for each story. • Observe whether students are only making addition questions. Ask if they can create a story that includes another operation. Differentiation • For some students, the context of penguins, for example, may be unfamiliar. Leaving the question too open can also be overwhelming, since it is difficult to know where to start. Help the students think of a familiar context that makes sense to them. Using real objects, rather than concrete materials representing other objects, can help to give the context more direct meaning. • For students who need more of a challenge, open up the problems. For example, “The answer is 7 penguins” can be opened up by posing “The answer is 7.” Open the second problem by letting students choose the two numbers they wish to use. Addition and Subtraction to 10 169
Assessment Opportunities Observations: This is a good time to check whether students see the connection between addition and subtraction. Conversations: • If students are creating only addition problems, ask whether it is possible to make a subtraction problem with an answer of 7. Ask what action can take place in a subtraction problem. (e.g., taking away or removing) Ask what problem they can create that has a ‘leftover’ of 7. • If students are creating both addition and subtraction problems, ask if they can turn one of their addition stories into a subtraction story. • To help them understand the relationship between addition and subtraction, ask students to investigate the following problem. Record the expressions 5 + 2, and 7 – 2. Ask what would happen if they continually did the first expression, using counters, followed by the second expression, using the same counters, followed again by the first expression. Allow students to investigate and visit them later to ask what they discovered. Consolidation (20 minutes – 10 minutes with another student or pair and 10 minutes with the whole class) • Have individuals or pairs of students meet with another student or pair and pose their problems to each other. Have the other students decide whether it can be solved with adding, subtracting, or both. Optional: Together, they can make equations that match the stories. • As a class, have students share some of their solutions. Ask them to explain how addition and subtraction make their stories different. • Show an addition equation that matches one of the students’ stories. Ask how the equation and question could be altered so they represent subtraction. • Ask how addition and subtraction are related. (e.g., One operation undoes the other operation.) Further Practice • Independent Problem Solving in Math Journals: Verbally pose the following prompt: − Create adding and subtracting problems that have an answer of 4. Use pictures, numbers, and/or words to explain. 170 Number and Operations
Materials: Math Talk: BLM 5: Blank Ten- Frames, counters Math Focus: Doubles plus/minus 1 strategies (or arithmetic racks) Let’s Talk Teaching Tip Select the prompts that best meet the needs of your students. Integrate the math talk moves (see • B riefly show 3 on the top row and 3 on the bottom row of a ten-frame or page 6) throughout Math Talks to arithmetic rack. How many did you see and how do you know? (e.g., I saw 6 maximize student because I know 3 and 3 are 6; I saw 6 and counted 2, 4, 6; I saw 6 because participation and they look like the 6 on a die.) active listening. • B riefly show 3 on the top row and 4 on the bottom row. How many did you see and how did you know? (e.g., I saw 7 because it is 1 more than 6; I saw 7, 3 and 3 are 6 so 1 more is 7) Where did you see 3 and 3? So knowing the double helped you, is that what you are saying? Did anyone see it differently? (e.g., I saw 4 and 4 is 8, but 1 was missing so it was 7.) Can someone show us where Raj visualized 4 and 4? How did knowing a double help Raj? • Continue showing ten-frames with 1 more on the top or the bottom. Dot configurations that show 1 more or less dot can also be used. Partner Investigation • S tudents can add 1 more cube to the top or bottom of their double-decker trains that they previously created (see Lesson 11). They can take turns flashing them to each other. The partners say how many there are, and then state the closest double that they could see. Follow-Up Talk • Discuss how the doubles and near doubles trains differed. Have a discussion about odd and even numbers. Math Talk: Math Focus: Doubles and near doubles with number sentence sequences NOTE: Once students have had experiences working with doubles and near doubles with concrete and visual representations, they can work through number sentence sequences. Let’s Talk Select the prompts that best meet the needs of your students. • Show some of the following expressions one at a time. For each, ask some of the following prompts. Incorporate the math talk moves to maximize student participation and active listening. What do you visualize? What is the sum? How do you know? How did an earlier problem help you solve this one? How is this problem like the one before it? continued on next page Addition and Subtraction to 10 171
− 3 + 3 (establishing a double) − 3 + 4 (1 more than a double) How does this connect to the first problem? − 4 + 3 (addends in reversed position) How does this connect to the first and second problems? − 4 + 4 (establishing a double) − 4 + 3 (1 less than 4 + 4 or 1 more than 3 + 3) − 5 + 5 (establishing a double) − 5 + 4 (1 less than 5 + 5 or 1 more than 4 + 4) What other problems helped you solve this? How? − 4 + 5 (the same sum as above) How is this like the problem above? Why? Partner Investigation • S tudents can take turns showing each other similar expressions on cards while their partner explains their strategy for finding the sum. Further Talk • Make an anchor chart of students’ strategies. 172 Number and Operations
17Lesson Reinforcement Activities Math • All of the Learning Standards identified for this unit Learning Standards Teacher • Solves word problems using appropriate addition or subtraction strategies Look-Fors • Explains or shows strategies and justifies how they solve the problem • Explains the meaning of the addition, subtraction, and equal symbols • Explains or shows how addition and subtraction are related • Explains or shows how a given equation can match a story problem Materials: About the Lesson different colours The following activities can be carried out by the whole class in small of connecting groups, or as centres through which students rotate over a few days. They cubes or tiles can also be used throughout the unit any time you decide to offer guided math lessons. For example, you may want to meet with small groups over Time: 20–25 minutes a few days and tailor the lessons to meet the needs of the students. While per activity over a few this can be done while the rest of the students solve the same problem in days small groups, you may wish to observe how each group works through the same concept. In this case, one group meets with you each day, while the other groups rotate through some of the following activities. See the Overview Guide for more information on how to manage guided math lessons. Centre 1: Combination Challenges • Students find all of the combinations for 5, using more than 2 groups to make the total (e.g., 2 red connecting cubes, 2 green connecting cubes, and 1 yellow connecting cube). Have them create equations for each combination. Differentiation • For students who need more of a challenge, have them find all the combinations for any other numbers between 6 and 10, using 3 or more addends. They can also choose their own concrete materials to solve the problem. Addition and Subtraction to 10 173
Materials: Centre 2: Creating Compare Problems on the Farm “What Do You See?” • Have students use the “What Do You See?” picture to solve ‘How many (pages 6–7 in the Number Sense little more or less?’ problems. Students create their own problems and verbally books), chart pose them to one another. For example, a student might ask his/her partner paper how many more rabbits there are than cats. Materials: Centre 3: Grab Bag Subtraction (from Marilyn Burns, 2000, p. 170) connecting cubes or tiles, • Students work in pairs. They select a number between 5 and 10 and put that paper bag many items in the bag. One student reaches in and removes some of the Materials: items, showing how many have been removed. Both predict how many items connecting cubes, they think are left in the bag. Then they check their predictions, and record chart paper, the matching equation(s). markers Centre 4: Snap It! (from Marilyn Burns, 2000, p. 170) Materials: • Students work in pairs. Both students decide on a length of train (up to 10), “What Do You See?” (pages 6–7 in the and each build one using connecting cubes. Both students put their trains Number Sense behind their backs. One student says, “Snap it!” Both students break their big book), problem trains into two pieces and continue to hold them behind their backs. They written on card, take turns showing the cubes in one hand, while the other student figures chart paper, out how many cubes are behind their back. On the next turn, the other markers student says, “Snap it!” Have students record some of their equations. Centre 5: Who Is in the Barnyard? • Pose the following problem: You see 10 legs in the barnyard. What animals might you be seeing? Have students draw a picture and make matching equations. Students can look at the “What Do You See?” picture to get ideas about the different kinds of animals on the farm and how many legs they have. Encourage students to be creative. For example, ask what types of insects (e.g., honeybees) might be on the farm and whether they could be represented in one of the examples. 174 Number and Operations
Materials: Centre 6: What’s the Difference? playing cards without • Students play in pairs or in small groups. Each player chooses two cards. the face cards, but including the aces They take turns figuring out the difference between the two cards and take as 1s; counters; BLM 5: that many counters. They keep playing until one player has 25 counters. Blank Ten-Frames; They can keep track of their counters using five- or ten-frames. BLM 6: Blank Five- Frames; dice Variation: Students roll two dice and figure out the difference between the (optional) two amounts showing. They take counters to equal the difference. The first to have 25 counters wins the game. Building Growth Mindsets: Reflect back on the lessons and pose some of the following prompts to reinforce growth mindset messages: – A sk what students’ favourite activities were. (Math is interesting to investigate.) – A sk what they found challenging. (Hard tasks are good and if we keep trying, we can overcome them.) – A sk what they have learned. (Celebrate the accomplishments.) – A sk what they still have to learn. (We may not know it YET, but we will with time.) – A sk how mistakes can help them learn. (Mistakes help us to try new strategies and learn new ways of trying so we can do things better.) Addition and Subtraction to 10 175
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