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2nd Nine Weeks: Number and Operations in Base Ten: Addition Strategies

Use place value understanding and properties of operations to add and subtract.Standard: 2.NBT.5 Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Support Document:• Numbers with no regrouping:

o Start with concrete; move to pictorial; then move to numbers only• With regrouping:

o Problem-solving situations only (algorithm is not taught)• Students need strong understanding of base-10 concepts to be able to generate

strategies (of their own choosing) for solving problems• Students should be asked to:

• Show their strategies with concrete objects and/or pictures• Justify the reasonableness of their answers

o Part-part-whole model should be emphasized

Caution: Base-10 pieces should not be used too soon.

Children who are 7 and 8 years old may have conservation of small numbers (they realize that 7 objects remain the same, even when rearranged into different groupings), but they may not have conservation of larger numbers. "To such children, it is not obvious that 26 cubes grouped into 2 tens with 6 left over is still 26. The simple action of rearranging, or regrouping, may seem like an operation such as addition or subtraction, and so children may think that after regrouping they end up with a number that is different from the one they started with (Kathy Richardson, Developing Number Concepts)."

o Start with loose materials, such as beans and cups or connecting materialso Give children many experiences in reorganizing connecting cubes into longs and units before the

idea of trading is introduced. (This gives the children the opportunity to count objects again to verify that the "new" arrangement is the same as the one they started with.)

o When children are ready to deal with large numbers, the idea of trading then becomes useful.

Sample LEQs-change the word addition to subtraction, as needed:• How do I show addition with manipulatives?• How do my drawings show addition?• How many different strategies can I use to show addition?• How can I use part-part-whole relationships to prove that my addition/subtraction is correct?• How do I know that my answer makes sense?

Part-part-whole relationships using large base-10 pieces; finding the missing part:• How far am I from 100?• How far am I from _? (a number smaller than 100)• How far am I from 200? From 500?

Adding On (using base-10 pieces)•What do I need to add to this construction to make 50? To make 300? To make 500?

Sample LEQs for Estimation-standard 2-2.9: •What do I think my answer is going to be? (Will it be closer to _or to_?)•What do I think that the answer will be? How do I know that this might be a good estimate?•What strategies can I use to estimate the answer before I rearrange the manipulatives?• How do I use a different estimation strategy to help me decide if I need to change my first estimate?'" How can finding out how close my estimates are to tr.:: C1r.tual answers h31p me make better estimates?

2nd Nine WeeksNumber and Operations in Base Ten: Addition Strategies

Standard: 2.NBT.5 Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Number Relationships:

Materials:Developing Number Concepts, Book Two: pages 111-172 Developing Number Concepts, Book Three: pages 99-131 Van de Walle, pages 145-150

General Guidelines for working with each of the 4 number relationships:

a Introduce one strategy at a time.

a Model using the strategy a few times for the children.

a Have children start at the concrete level with manipulatives.

oOrally, give the children a simple word problem to solve with the manipulatives each time you ask them to use the strategy.

oRepeat the above process over several days.

oHave the children explain how to use the strategy to a partner.

oHave them draw/write how they use the strategy.

oOnce some children are comfortable with the strategy-which means that these children can usethe strategy without having to count the manipulatives to get the answer- use dice games and "Van de Walle" type flashcards with the children.

oWhile some children are playing games and doing activities at the "representational level" continue doing similar activities using manipulatives with children who are not yet comfortable with the strategy.

oAs strategies are introduced, add each one to an anchor chart (a list of each strategy-possibly with picture cues-posted in a way that allows children to refer to as they work)

Second Nine Weeks: Number and Operations in Base Ten: Addition/Subtraction Strategies


Number Relationships-Numbers 1-10:STEP ONE:Begin working on each relationship using numbers 1-10.

o Van de Walle, pages 43-54o Developing Number Concepts: Book 2-Chapter 3-page 99)

• Activities such as those on pages 132-150 can be used. These activities can be differentiated by using larger/smaller numbers or different types of dice, as needed.

NOTE: Children must first be secure in their understanding of the number relationships referred to in Van de Walle-pages 42-43-subitizing, 1 or 2 more/less, anchors to 5 and to 10, part-part-whole. Remember that part-part-whole is to be emphasized in 2nd grade; however, children who do not have the other relationships will need work on each one.

SubitizingUse 10-Frames and Dot Cards frequently so that children will be able to mentally "see" numbers that they will be asked to work with.

Sample Games:War and Variations of War: Basic War:Use the dot cards. Each person places a card face up on the table at the same time. The person who has the largest number of dots wins the cards. (If the numbers are equal, each person turns over a new card.)

Double War-emphasis on explaining strategies: Use the dot cards.Play as basic War-except each person places two cards face up at the same time. The person with the"higher number" should explain or show why his/her card has more dots.

War with a Twist: ProbabilityPlay War or Double War as usual, except the winner might not be the person who has collected the most cards. At the end of the game, flip a coin. If the coin shows "heads" the person with the most cards wins. If the coin shows "tails" the person with the smallest number of cards wins.

Difference WarPlay in pairs. Put about 50 of some type of counter in a cup-dried beans, buttons, etc. Each person turns over the top card in his/her set of cards. Decide what the difference is between the two numbers. The person who has the larger card takes that number of counters from the container.

-+ Example: Person A turns over a card that has 9 dots. Person B turns over a card with 6 dots. Person A has 3 more dots than Person B, so Person A wins 3 counters.

-+ When you run out of cards, shuffle the cards again and keep playing.-+ The game ends when all of the counters in the cup are gone. . The person who has the most counters at the end of the game wins.

More Addition and Subtraction Activities: Basic AdditionTurn over two dot cards at the same time. Add the two numbers together. Explain how you used the arrangements of the dots to help you get the answer. (You may decide that you are ready to add the numbers on 3 cards together.)

Basic SubtractionTurn over 2 dot cards at the same time. Subtract the smaller number from the larger.

Week 1 Addition Strategy of the week:


1 or 2 more I 1 or 2 less

Around the World with Subitizing and 1 More/1 Less; 2 More/2 Less Materials: Dot Cards from Van de Walle

Round One (Subitizing): (Note: In this round, one child answers; two children do not compete.)Tell the class that you are going to show a card for 3 seconds. When you put the card down, the child will tell:

o The total # of dots on the cardo How they were

grouped.• On card A, the child might say, "9...there was a group of 6 and a group of 3"• On card B, the child might say, "9...there were two groups of 4 and 1 by itself'• On card C, the child might say, "5...there was one row of 5 dots"

o Show the card again for the child to see if he/she was right. Allow the child to correct his/her answer, if necessary.

Move to the next child, and repeat until each child has had a turn.To keep all children engaged, occasionally ask, "Who saw this card a different way?"

Round Two: (In this round, two children may compete-or you may continue with one child at a time.)Show a card for 3 seconds. The child is supposed to tell you one more than is shown on the card.

o On card A, the child would say 10. o On card B, the child would say 10. o On card C, the child would say 6.

Repeat until each child has had a turn.

Round Three: (In this round, two children may compete-or you may continue with one child at a time.)Show a card for 3 seconds. The child will tell what is one less than the number shown.

o On card A, the child would say 8. o On card B, the child would say 8. o On card C, the child would say 4.

Round Four I Round Five: Repeat the above process, but in round 4 the child will give 2 more; in round 5 the child will give 2 less.

NOTE:Only devote a few minutes to this game-perhaps as a quick warm-up activity. If the children are slow giving answers-or you have a large math class, do one of the following:

o Do Round One on Day One, Round Two on Day Two, Round Three on Day Three, etc. o Randomly call on a few children instead of every child.o If children are unable to give correct

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Week 8 Addition Word Problems: Missing addend/Counting On


Part-part-whole activities (recommended order of presentation)

Concrete:

o Question: How many different combinations can you make using two parts? (Giveeach child a target number that is appropriate for his/her level of understanding. He/she should record all the combinations found.)

• Bi-colored counters

• Unifix cubes

• Color tiles

Pictorial:

oGames with Dot Cards

o"Flash Cards" with dots and ten frames

Missing Parts:

Concrete:

o Use a target number appropriate for each child. Count out that number of counters.Hide "part" of the counters. Show the child what is not hidden. Ask: What part is hidden?

Pictorial:

o Missing part cards: Refer to the picture on page 51 in Van de Walle. Make cards similar to these for independent practice. (Remember, independent practice comes after the child has the concept.

o "I Wish I Had": Choose a target number (such as 9). Show the child a dot plate or a 10- frame card with 4 dots. Say, I have 4. I wish I had 9. How many do I need?

Use different target numbers according to the level of the children.

A. I Wish I Had 10

B. I Wish I Had 12

c. I Wish I Had 16

*D. I Wish I Had _ (This strategy will be used later when working withmuch larger numbers, and with making change in money.)

2nd Nine Weeks: Addition Strategies


Anchoring Numbers to 5 and to 10

IMPORTANT: USE 10-FRAMES FREQUENTLY SO CHILDREN WILL "SEE" HOW CLOSE TO 5 AND 10 THE NUMBERS ARE AND HOW FAR FROM 5 AND 10 THE NUMBERS ARE.

•When modeling at the overhead, make statements such as, "I know that this number has to be 8 because I see 2 empty boxes on the 10-frame." "I know that this number has to be 7 because one row of 5 is full and I see 2 dots in the other row."

• Encourage children to make similar statements as they explain how they got their answers whengiving the total number of dots on dot cards as well as when they work counters on 10-frames.

Make a 10 on a 10-Frame:(Based on part-part-whole combinations of 10 (9 + 1, 8 + 2, 7 + 3, 6 + 4)

• 8 + 6 = (I have 8 on the 10-frame and 6 counters in my hand. If I use 2 of the counters to fill the ten-frame, I will have 4 counters in my hand. 10 + 4 = 14)

• 7 + 6 = (I have 7 on the 10-frame and 6 counters in my hand. If I use 3 more counters to fill the 10-frame, I will have 3 counters left in my hand.)

Games with Making Ten (after children have had much practice with 10-frames):•Pyramid 10 (attached)

•Tens Go Fish (playing cards with face cards removed or with several sets of Van de Walle's dot cards-tens removed)

o Each player receives 5 cards; the remaining cards are placed face down in themiddle.

o Players put aside cards that equal 1O; play continues until all the cards in the deck are used up, at which point players record the combinations that they made.

o Before beginning play, the players replace each pair that they are able to put aside by picking up 2 new cards from the deck (for each pair put aside).

o When all combinations are put aside, player A then asks player B for a specific card. o If player B cannot give player A the card that he asked for, player A will "Go Fish" by

taking the top card from the deck.o Each time during the game that a player puts down a combination that equals 10, he

picks up one card from the deck.

•Sweet 10 (The actual name of this game is Sweet 13, but different target numbers can be used. The directions for Sweet 13 are below.)

•Individual books of 10 (Make list of all equations with 2 addends on one day (using counters); with 3 addends on the next day; with 4 addends the next ...until 10 addends. Compile in a book

NOTE: These games and activities can be repeated as needed using different target numbers.

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Pyramid 10You need:

1 deck of cards, with face cards removed

RulesTwo partners play against the deck. The object of the game is to pick up the entire pyramid as you go through the deck once.

1. Shuffle the deck, and arrange twenty-one cards in a pyramid, as shown below:

2. Put the rest of the deck face down.

3. Definition of a free card: A free card is one that has no other card overlapping it. When the game begins, only the six cards on the bottom row of the pyramid qualify as free cards.

4. Remove all free 1Os and pairs of cards that add up to ten in th ls bottom row. Set these aside. These cards are now completely out of play.

5. If additional 1Os and pairs that add up to ten have now become free cards, remove th em as well, and set them aside, out of play. Continue this process until there are no more possibilities to be picked up.

6. Turn over one card from your deck. If it's a 10 or can combine to make ten with a free card from the pyramid, set these cards aside. If you can't use the card that you've turned up, set it face up below the pyramid. You may be able to use it later.

7. Turn up another card. Follow the same directions as in Step 6. If you have been able to combine a card from the pyramid with the new card from the deck, you may also have freed up a card on the pyramid that can be combined with the card you previouslyplaced below the pyramid. Note: As you proceed through the deck and are forced to addmore cards to the top of this pile, you may use only the top card in this pile to combined with cards on the pyramid.

8. The game ends when you've turned over all the cards in your deck and you can't makeand remove any more 1Os, or you have picked up the entire pyramid.

Sweet 13You need:1 deck of cards, with face cards removed

Rules1. Shuffle the cards, then deal seven cards to each player.

2. Leave the rest of the cards face down. This is the draw pile.1

3. Player 1:Choose one of the cards in your hand, and place it face up next to the draw pile. Say the value of the card out loud.

4. Player 2: Place one of your cards next to the first one, and add its value to the first card. The total value must be thirteen or less. If you cannot play a card without going over thirteen, draw from the pile until you have a card that you can play. Say the total value out loud when you're done.

5. Play continues until one player brings the total to exactly thirteen. That player sets aside the trick (the cards totaling thirteen) and gets to lead with a new card.

6. The player to get rid of his or her cards first is the winner.

From Second-Grade Math. A Month-to-Month Guide by Nancy Litton.© 2003 Math Solutions Publications 205



Number Relationships with Numbers Up to 20:

STEP TWO: The children must then use these same relationships to work with numbers up to 20.

o Van de Walle, pages 54-58

• Ten and Some More (page 55)-Make sure that children see the "teen" numbersas "10 and some more".

• Count out a quantity of unifix cubes (between 13-19). Make a stack of ten. "Read" the number: 16 is 10 and 6 more." Repeat with other numbers. (Many 2nd grade students need this exercise.)

• Use a mat divided into 2 sections. Have the children count out 10 objects and place them on the left side. Have them count out 5 objects and place them on the right side. Together, count all the objects one-by-one. Chorus the combination: "Ten and five is fifteen." Turn the mat around: "Five and ten is fifteen." Repeat with other teen numbers.

• Extend More and Less Relationships (page 55)-Show 7 counters on theoverhead. Ask what is one more? one less? two more? two less? Add a filled 10- frame and repeat the questions. Repeat with other numbers. (You may want to leave the filled 10-frame on the overhead, but cover it when you as the questions about the single-digit number; uncover it to ask the same questions with the teen number.

o Developing Number Concepts:: Book 2 -pages 151-172

• • • • •• • • • •



Using 10-Frame Cards: Using Tens to Show Larger Numbers• Take the full 10-frames out of the stack of cards and place them on the table near you. Each of

these cards is a 10 card. • Now select 2-3 other cards from the set of 10-frame cards. Count a// of the dots on these cards

to find out how many dots there are in all.• Can you find one card that has that many dots? If you have to use more than one card to show

how many dots there are in all, make sure to use as many of the 10 cards as possible.

-+ For example, suppose you chose these 3 cards:

-+ When you count every dot, you get 13.-+ Now, use as many 1 O cards as possible to make 13 in a different way:

Use one 10 card and a card . that has 3 dots .

• Now try a different number. Put the 10 cards next to you to use again. Mix up the other cards.Draw 2-3 cards and count a// the dots to see how many there are in all. Use as many of your 1 O cards as you can to show how many dots there are in all.

Flashcard Activities with 10-Frame Cards:• Regular addition flashcards- Choose two cards at random and find the total. (If you do not

know the total, please count all of the dots to find out. Counting is very important when you are learning these facts.)

• Adding to the number 9- Place one card that has 9 dots face up on the table. Choose any other card and add it to the 9. (How can you use the empty space on the "9" card to help you find the answer?) Now choose another card and add it to the 9. Repeat. Do you see a pattern? Can you make a rule for adding a number to the number 9?

• Adding to the number 8- Place one card that has 8 dots face up on the table. Choose any other card and add it to the 8 (Use the 2 empty spaces on the "8" card to help you find the answer.) Continue with the other cards. Do you see a pattern or a rule for adding a number to 8? What is your rule for adding any number to the number 8?

• What do you think the pattern or rule will be if you add a number to the number 7? Now try adding a number to 7 to see if your rule works.

Use multiple informal representati

ons to convey matthematical

2nd Nine Weeks: Addition Strategies: Week 3


Other Number Relationships to Work On:

Doubles:3 + 3 = 6 {bug double: 3 legs on each side)4 + 4 = 8 (spider double: 4 legs on each side)5 + 5 = 10 (hand double: 5 fingers on each hand)6 + 6 = 12 (egg carton double: 6 eggs on each side) 7 + 7 = 14 (two weeks on a calendar)8 + 8 = 16 (crayon box: 2 rows of 8)9+ 9 = 18 ( 18-wheeler: 9 tires on each side)

MONSTER DOUBLES: from Kathy Litton (pages 57-58 in Second Grade Math)• Read the book My Monster Mama Loves Me by Laura Leuck to the students• Have each student draw a monster or use folded paper to cut out a monster shape:

o Fold a paper in half and cut an interesting shape. When opened, decide what creature or monster can be made out of the shape.

o Have the children make doubles in one of the following ways:• Use pipe cleaners or paper strips to glue arms or legs to the creature-making

doubles (3 on each side or 8 on each side)• Give the monster "double" eyes-perhaps 4 on each side of his face• Make doubles out of the ears-perhaps 3 ears on each side• Other possibilities-whiskers, noses, mouths, warts, etc.

A recording sheet for small pictures for doubles from Kathy Litton is attached.

Near Doubles:2 + 2 = 4, so 2 + 3 = 53 + 3 = 6, so 3 + 4 = 74 + 4 = 8, so 4 + 5 = 95+ 5 = 10, so 5 + 6 = 11

6 + 6 = 12, so 6 + 7 = 137 + 7 = 14, so 7 + 8 = 158 + 8 = 16, so 8 + 9 = 179 + 9 = 18, so 9 + 10 = 19

NOTE: A set of facts that children need to generate strategies for is attached (from Kathy Litton).


Adding DoublesThink of ways to represent all the doubles; from 1 + 1 = 2 to 1O + 10 = 20. For example, during the World Series there are 18 players for each game. Nine of the players are Yankees, 9 are Padres. Brainstorm ideas with your family. Use numbers, words, and pictures. Practice your doubles when you have finished. Knowing your doubles can give you strategies for adding other number combinations.

210 From Second-Grade Math: A Month-to-Month Guide by Nancy Litton.© 2003 Math Solutions Pub! ications



Number Relationships with Numbers Larger Than 20:STEP THREE: The children apply the relationships to numbers larger than 20 using single concrete objects.

NOTE: The activities children participate in should relate to the work that the children have done making groups of tens and leftovers using single objects-unifix cubes, beans and cups, color tiles, blocks, paper clips, etc.

a Connect the work to indicator to 2-2.1 (estimating quantities and length) Have children first estimate the length of an object or the number of tiles it takes to cover shapes

o Developing Number Concepts: Book 3, pages 78-95-Look at the level 3 activities that are named "How many more?" in these activities children find the difference in their estimates and the actual numbers.

o "Area with Color Tiles" - a covering-shapes activity from Marilyn Burns is attached. Children estimate which shapes holds more and then fill the shapes to find out. (Note: The emphasis is on base-10 counting. Even if you use the word "area", the children are not accountable forthe concept of area-only for base-10 concepts.)

• Children can find the difference in their predictions and the actual count using these shapes. Ask: "How many more tiles would you have to add (or subtract) from your prediction to get the actual count?" Then say: "Now that you know how many tiles it takes to cover this shape, can you use that number to help you predict how many tiles it will take to cover the next tile? Do you think that it will take more or less than_?"

• They can then order the shapes from smallest to largest and/or compare shapes with<, >, =

2nd Nine Weeks: Addition Strategies: Week


o Combining shapes or combining lengths-- Developing Number Concepts: Book 3, pages 124-130 (Similar to the first activity above, but the children will add to find the total amounts)

o The children cover 2 shapes with concrete objects or measure 2 lengths by lining up concrete objects next to each one, find the total for each, and then combine the two amounts to find the answer.

o Race to 100 I Race to 0 (Developing Number Concepts: Book 3 (page 98) This version asks that children use Unifix cubes and Place-Value Boards (or mats) to play the games. Children need to understand that only "complete" 1Os can be placed on or remain in the "TEN" section of the board.

a Pages 99-115 of Developing Number Concepts: Book 3 Developing Number Concepts: Book 3- "Addition and Subtraction of Two-Digit Numbers" describes many strategies that children can use to add and subtract without using the algorithms.

o Have children estimate sums (and differences) by looking at "numbers" made from Unifix cubes before using the manipulatives to find the actual answers. Refer to Developing Number Concepts: Book 3- pages 118-122

o Child A makes a number by grouping Unifix cubes into 1Os and 1s. He places his "number" on a mat ( 1Os on the left and 1s on the right).

o Child B makes also makes a number out of Unifix cubes and places his arrangement below the mat (horizontally).

o Both children use their estimation skills and mental math skills to find the answer before they move the Unifix cubes.

o Then each child takes a turn using the Unifix cubes to explain to the other how he got his answer.

o Story Problems ideas can be found on page 131 of Developing Number Concepts: Book 3.

2nd Nine Weeks: Addition Strategies: Standard: 2.NBT.5 Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction.

Number Relationships with Numbers Larger Than 20:

"MAKING TENS" TO ADD LARGER NUMBERS:(Making tens works best for the numbers 7-9. Some children successfully use this strategy for numbers 6-9. After practicing the strategy with numbers 7-9, allow children to experiment with using the same strategy with the number 6.)

Model "making a ten" as a strategy that helps you mentally add numbers.• For instance, place 9 beans on one of the ten-frames (on the place-value mat).• Tell the class that you need to add 6 more. "I think that I know what the answer will be

because I know that if I use one of the 6 beans to fill the ten-frame, I will have 5 beans left. My answer should be one full ten frame and 5 more- 15."

• Add the 6 beans to "show" that your thinking was correct.• Demonstrate/model with other combinations, such as 8 + 6. "I know that it will take 2 beans

to fill the ten-frame, so I will have 4 beans left. I think that the answer will be one full ten frame and 4 more- 14."

• Have a child explain how making a ten can help him/her add 7 + 5.• Use other combinations for the children to practice.

o Have them explain how they use the "making ten" strategy to their partner.

AFTER children are successful with the "making ten" strategy to add single-digit numbers, move to the next step. (Begin with numbers that end with 7, 8, 9. Use the place value mat with double 10 frames.)

MODEL:• Tell the students that you are going to try the "making ten" strategy to see if it works with

larger numbers.• Use cups of beans and loose beans to make a number on the overhead, such as 38.• "I am going to add 6 beans to this number. (Show the children 6 beans in your hand.)• "If I can use the 'making ten' strategy, I know that it will take 2 of my beans to fill this ten

frame. I will then have 4 beans left."• Demonstrate filling the first ten-frame and placing the other four beans on the other ten

frame.• "My place-value chart now has 3 tens on the left and 14 ones (beans) on the right. So I

need to fill another cup with ten beans and move it to the tens side of the mat."• "My new number is 4 tens and 4 ones- 44. Is this correct?"• Add on to 38, using 6 fingers. "I have 38." As you touch each finger, count "39, 40, 41, 42,

43, 44. Yes, I see that 38 + 6 = 44."

GUIDED PRACTICE: • "Let's try one together.·· Have children put 2 tens and 9 single beans on the place-

value mats.• Ask: "How will we use 'making ten' to add a number to 29?"• Have children explain. Clarify their thinking: "If we add one more bean to this mat, it will

be full. It will make another ten. The rest of the beans will go on the other mat."• Give the children a single-digit number to add to 29.• Have them explain the process they used to get the answer.• Repeat with other numbers-always having children explain to partners, or by ca Iling on

one or two children to explain to the whole group.

AFTER children are able to successfully add to a larger number by "making ten" with beans &cups and with tens-frames:

• Show children how to check their answers using a hundreds chart.

2nd Nine Weeks: Addition Strategies: Note to Families

Single-Digit AdditionMake sure you know your doubles. lf you don't, practice them every day,

1+1 2+2 3+3 4+4 5+5 6+6 7+7 8+8 9+9 10+10

In class we looked at the chart for addition facts from 0 + O through 1o + 1o. There were 121 facts! That's a lot.

When we looked at the chart closely, we discovered that there were many facts that everyone already knew, so we crossed those out.

The facts listed below are the ones that you may still need to practice. The important thing is to have strategies that make sense to you so you can quickly find the answers when you need them.

4+3 5+3 6+3 7+3 8+3 9+35+4 6+4 7+4 8+4 9+4

6+5 .7+ 5 8+5 9+57+6 8+6 9+6

8+7 9+79+8

Each day, choose three or more of these addition facts. Write your strategies for getting the answer to each fact on a separate piece of paper.

Note to parents: The purpose of this assignment is to encourage children to use their number sense (especially the ability to decompose numbers and knowledge about how numbers relate to one another) to develop strategies for efficiently doing single digit addition. Different children may come up with different strate gies to solve the same problem-the important thing is for, each child to think about what works for him or her.

Memorization is less important than having a quick and accurate way of coming up with the answer. For instance, when I want to answer 8 + 9, I say, "8 +a= 16 so 8 + 9 = 17."I can do this in the blink of an eye. In class we're talking about the kinds of strategies that make us mathematically powerful. Your child should have many ideas to draw on. You might want to share some of your own strategies with your child. Just remember that your child may go about solving the problems differently and that's fine. We must construct understanding and met holds that work for each one of us.

From Second-Grade'" Math: A Month-to-Month Guide by Nancy Litton.© 2003 Math Solutions Publications 209

2nd Nine Weeks: Addition Strategies: Practice Using All 4 Relationships Together with Beans (Bi-Colored Counters) and Cups Model

a. Positioning place-value mat so that ten-frames are on the right side. You may want to draw asmiley face in the corner of the mat so children will know that they have it turned the correct way. (Note: When you pair children to work together, place them side-by-side so they both will see the tens on the left side of the mat and the ones on the right side of the mat.)

b. Alternating colors of beans (bi-colored counters) to keep up with the numbers as they are added to the mat

c. Filling one ten-frame at a timed. Completely filling one frame before starting another - stress the part-part-whole

relationship; I have 7 counters on the mat. I see 3 empty spaces, so I need 3 more counters before I will have a "1O."

e. Placing 10 beans from a full frame into a cup Cups can only be used if there are 1 O beans in them!

f. Moving full cups of beans to the left side of the place-value mat.g. Counting the total number of beans each time a new number is madeh. Referring to the "Base Ten Number"- "I now have 2 cups of beans and 4 other beans on the

mat. My base ten number is 2 tens and 4 ones."i. As the children become more comfortable with the processes involved in using the place

value mat, model more math terminology: standard form, the value of the tens (6 tens has a value of 60), and the expanded form.

j. As lessons progress over time, model how you apply each of the addition strategies as you fill you ten-frame (making tens, part-part-whole relationships, doubles/near doubles, etc.)

Guided practice a. First with teacher calling out the number of beans to add to the ten-frames

RECOMMENDATION: Use a story problem with each addition of beans. b. Later a die can be used to find number of beans to add to ten-frames.c. Ask a child to tell how many beans (apples, toys, pieces of candy, etc.) are on the mat after

each roll. "How many cups of ten do you have? How many counters are still on the ten frames?" "What is your base-ten number?" "What is another way to tell how rn:::iny beans this is?"Note: Spend extra time working on the teen numbers.

d. As the children become more comfortable with the processes involved in using the place value mat, add more math terminology: standard form, the value of the tens (6 tens has a value of 60), and the expanded form.

Estimating and Rounding to the Nearest Ten (Indicators 2-2.1 and 2-2.9)oUse the small 10-frame cards. Have students work in pairs.oHave each child make a 2-digit number using the least number of cards possible. The children place

their cards side-by-side and estimate which ten the new number is closer to.o The children should be allowed to move the cards around as they generate and discuss

strategies.oEmphasize that they explain their strategies:

• For seeing how many "full" frames there will be.• For figuring out whether the "dots that are left over" will make a 10-frame that is

closer to being empty or closer to being full.

2nd Nine Weeks: Addition Strategies:

Number Relationships with Larger Numbers - Using Base 10 Pieces

STEP FOUR: The children apply the relationships to numbers larger than 20 using base-10 pieces. oPart-Part-Whole relationships with the larger red/green/yellow base-10 pieces.

o Begin with the question, "How far am I from 100?"• Ask children to build on the red mat.• Give them a number to build. Begin with the "nice" numbers (multiples of 10, of 25).• Ask the children to start at the left hand side of the red mat and add yellow longs

and ones to make the number that you gave them.• Ask them to use the green pieces to build the other part of 100.• Repeat with many nice numbers before attempting to find the missing part using

other numbers.

o How far am I from ? (Use numbers smaller than 100)• Give children a number to build, such as 40. Ask them to build 40 using yellow.• Tell them that you are going to look at the parts of 40. Ask them to build 25 in

green on top of the yellow 40. "Now part of the 40 is yellow and part is green. The green part is 25; what is the yellow part now?" (15)

• Continue working with 40. Give the children other numbers to build on top of the yellow pieces so they can see both parts of 40.

o How far am I from 200?• Have children place 2 red mats in front of them. Give them a number to build in

yellow. Ask them what part of 200 is still red.• Repeat with different numbers.

STEP FIVE: Target Numbers: (adding on)o How far am I from_? (start with "nice" numbers)

• Give the children a number to build.• Ask them what they would have to add to what they already have to change the

number too How far am I from 500?

• Give the children a number to build using the base-10 pieces, such as 250.• Ask them what you would have to "add on" to the number that they have in order to

have 500?• Have children add more base-10 pieces so they will have 500.

Use multiple informal representations to convey mathematical

BLM 15-Place-value mat (with ten-frames)


Missing Addends

Support Document:• Students must be very flexible with addition and subtraction up to 20 and be able to show inverse

relationships• Sequence to follow:

o Move from concrete to pictorialo Next connect concrete/pictorial models to writing numerical sentences o Then find missing addends and subtrahends in combinations to 20

• Use number balance:o Stress "equivalence" (is the same as) for the equal signo Connect work with number balance to working with equations (eg: 4 + = 7

Question: What do I have to add to this side of the balance to make the two sides equal, orthe same? Question: What do you need to add to 4 to make it equal

7?) o Gram weights - (eg: two stacks-One stack of 6 and another stack of 2-

Question: What do I need to add to this stack of 2 to make it as tall as the stack of 6?)

Sample LEQs:• How do I explain what the equal sign means?

(As you model, stress "is the same as" and "is worth the same" when using the balance and the stacking gram weights.)

• How can I use gram weights to show what number is missing in this equation?• How can I use a balance to find what number is missing in the equation?• How can I use part-part-whole relationships to find missing numbers?•What strategies can I use to solve part-part-whole problems?•How can I use what I know about addition and subtraction to find missing numbers?

Materials:Developing Number Concepts: Book 2, pages 145-149


Missing Addends

Materials: Developing Number Concepts: Book 2, pages 145-149

Sample Activities:• Model part-part-whole concepts using Unifix Cubes: Build a stack using red and blue cubes.

oAddition: Hold up the stack of cubes and say the number sentence that goes with the stack. (8 blue+ 9 red = 17 cubes.) Flip it upside down and say the related fact. (9 red + 8 blue = 17 cubes)

oSubtraction: I have 17. The blue part is 8. What is the red part? : I have 17. The red part is 9. Whatis the blue part?

oMissing part: I have 17 cubes. 8 blue cubes are in my hand. How many red cubes are in the bag? a Have the children repeat similar activities with many different combinations.

"EQUAL"-• Use the words "the same as" and "is worth the same" often to help children expand their

understanding of the equal sign.• Build the concept of "equal" by using it in many different contexts, such as with the pan balance

and/or by stacking gram weights-making the stacks the same height, but with different combinations/colors of weights. For example, ask "How many different ways can you make a stack that is worth 20 grams?" (A sample recording sheet is attached.)

• Balance: Place weights on both sides of the balance, but have one side heavier than the other. Ask: "What do I need to add to this side to make it the same as the other side?" Emphasize that the same weights do not have to be used-different combinations of weights can make both sides the

• Counters on 10-frames or double 10-frames-for example, have the children place 8 counters on the frame. Ask, "How many more will you need to add so that you will have 15?""How many more will you need to add so that you will have 17?" Continue with other numbers that would need to be added to 8.

o Move to pictorial by giving the children a paper that has several double 10-frames on it. Have them make several combinations that use 8 as one of the addends using two different colors. They can write the addition fact below each picture. (See sample attached.) Math "ten frames_addition_missing addends

What are the missing addends? Color in the missing numbers and finish the number sentence.

2nd Nine Weeks: Addition Word Problems: Week 5Standard: 2.NBT.9 Explain why addition and subtraction strategies work, using place value and properties of operations (Explanations may be supported by drawings or objects.)

• First concrete objects - then pictures of arrays and equal groupings• Sequence of presentation-

o Have students:• Act out stories (use stories about the children)• Show story problems as equal groupings of concrete objects• Arrange objects from equal groupings to make arrays

Sample LEQs:• How do we "act out" a story problem?• How can I use objects to show what a story problem means? How do I explain how I used

these objects?• How do I change equal groups of objects into an array?•How can I change an array into equal groups?• How can I show equal groupings in my pictures? How do these equal groupings help me find my

answer?• How can I draw and label arrays?•What story problem can I create that "fits" an array?•How can I use words (sentences) and drawings to explain how arrays show multiplication? Extension I Challenge LEQs: • How many different arrays can I make that use 12 objects? What story can I write to match each

array? (Change "12" to different numbers as needed to meet the ability levels of the students.)

Materials:Book Three (3): Developing Number Concepts, pages 134-179

Information on Arrays:•Arrays must be in the shape of a rectangle.•There must be an equal number of objects in each row of an array.•Arrays are named using the length of the sides (dimensions of the rectangle).

o Names are written as 2 x 3 or 4 x 8.o Names are usually read as 2 "by" 3 or 4 "by" 8. (With second graders, it might be a

good idea to read the arrays both ways: 2 times 3 and 2 by 3.)• It is easier for children to "see·· arrays if they are made with color tiles. Even though arrays can be

made with beans, chairs, stickers, etc., use color tiles as much as possible in the beginning.•When doing work with arrays, follow the rule: concrete, concrete with pictorial, pictorial alone

o Concrete-build arrays with color tileso Pictorial-record arrays on grid paper• At the beginning, use 1-inch grid paper and 1-inch color tiles.• Once children are more comfortable with arrays, you can switch to 3/4-inch or

One-inch grid paper. This will allow children to record larger arrays.

Unit 2 Week 5: Problem Solving, GroupingUnit 3 Weeks 9 & 10 Repeated Addition

Equal Grouping (Multiplication)Operations and Algebraic ThinkingStandard: 2.OA.4 Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends.

Sample Activities Using Suggested Sequence:1. Have students act out story problems-Think of many different situations in which you can

make equal groupings of children in the classroom. Have the children act out each situation. Then change the numbers and have them rearrange themselves:

o There are 6 board games. 2 children play each game. (4 board games, 3 play each game)

o We have 3 tables. 4 children sit at each table.o Basketball game: 2 teams, 5 players on each team

2. Show story problems as equal groupings of concrete objects-Begin by using actual objects-books, chairs, pencils, etc. Have children make each arrangement. (See page 150 in Developing Number Concepts: Book 3)

o 4 stacks of books, 3 books in each stack (then 5 stacks of books, 2 books in each stack,etc.)

o 2 rows of chairs, 6 chairs in each row (then 3 rows of chairs, 4 chairs in each row, etc.)

3. Use manipulatives to "represent" people or objects-Place objects in equal groups.(Developing Number Concepts: Book 3-page 151 has sample problems to use)

4. Make equal groupings-then rearrange objects to make arrays (pages 152-162 in Developing Number Concepts: Book 3 has many different suggestions and sample stories to use)

5. Independent activities (see pages 163-179 in Developing Number Concepts: Book 3 for ideas)

Equal Groups:oCircles and Stars (Nancy Litton) (attached)oStudents roll a die once to find out how many circles to draw and again to find out how many stars to

draw in each circle.

Other activities using arrays:• Make as many "2-bys" as you can. Name the arrays that you made using the dimensions.• Make as many "3-bys" as you can. Name them.• Continue with "4-bys" and 5-bys".•Count out 8 tiles. How many arrays can you make that have 8 tiles in them. Each time you find a

different array, draw it on grid paper and write its name.• How many different arrays can you make with 12 tiles? Draw and name each one that you find.•Challenge: from Marilyn Burns "Exploring Rectangles (Arrays)". Some of your students may be ready

to try some of these activities:o What squares did you make when you made 2-bys, 3-bys, 4-bys, and 5-bys?o What other squares can you make? Draw them and name them. Tell how many tiles are

in each square.o What is the smallest number that has two different arrays? (2x3 and 3x2 are considered

the same array, since you can rotate a 2x3 and make it a 3x2.) o What is the smallest number that has three different arrays?o What is the smallest number that has four different arrays? o What numbers have only one array?

164

Second Grade Math

FlG!JRE 7-3 >Things that come ingroups.

2s eyes ears

shoeswheels on a bicycle

3s Wheels o n a tricyclecorners on a triangle

4swheels on a car

corners on a square

5spoints on some starsfingers on one hand

toes on one foot

6slegs on an insect

points on a star of David

Circles and StarsThis activity provides a strong visual interpretation of multiplication as repeated addition. It is a fine way to introduce the language and symbols of multiplication, and is described more fully in the book Teaching Arith metic: Lessons for Introducing Multiplication, Grade 3 by Marilyn Burns (2001). The following lesson is a pared-down version of the original and is suitable for second graders.

Materials 1 die for every 2 studentso 81/2 -by-11-inch paper for every student

Instructions

1. Teach the game by playing it with the class as your opponent. You'll need one die and the chalkboard. Have a piece of 81/2-by-11-inch paper on hand to show the children how to fold their paper into quarters to make a place to record four rounds of the game. Divide the chalk board into two sections-one for you to draw your circles and stars on, and one to record the class's work.

2. Explain that you’ll roll the dice one to determine how many circles to write on your side of the board. If you roll a three, draw three large circles on your side of the board. Roll again ar1d let the children know that this second roll tells you how many stars to put in each circle. If you roll a four on your second roll, your side of the board would look like this:

April: Taking Stock

3. Say, "That's why we call the game Circles and Stars. The first roll tells how many circles to make, the second roll tells how many stars to put in each circle. Now have to figure out how many stars have altogether. Who has an idea what three groups of four is altogether?" Take suggestions from your students and then write your total for this round.

4. Now ask a student to roll once to determine the number of circles to draw on the other side of the chalkboard. Then ask someone else to roll again to determine the number of stars. Have volunteers sug gest ways to figure out the total number of stars that the class has accumulated.

5. If you feel that the children understand the basics of the game, demon strate folding an st-by-11-inch piece of paper into quarters, and explain that partners will now play the game with each other, play ing four rounds. For each round, players should figure out their total number of stars. When all four rounds are over, each player should determine the grand total number of stars he or she has accumulated. A finished paper would look like Figure 7-4.

6. After the children have all played four rounds of the game (perhaps on Day 2), have a class discussion about their results. You might ask questions such as:

"What was the greatest number of stars you got in one round?" "Did anyone get more than thirty-six?""How did you get thirty-six? So, when you get six groups of six,your score is thirty-six. Is there another way to get thirty-six?" "What was the fewest number of stars?""Did anyone get twelve stars? If so, what did your paper look like?Did anyone get twelve stars in a different way?""Did anyone get seven.. stars in one r o u n d ? W h y n o t ?

As the children respond to your questions, use the language "two groups of four give you eight stars" or "two sets of four is eight altogether."

I'

166

FIGURE 7-4

Joelle had thirty-four stars after rounds.

Second Grade Math

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7. On the board draw three circles with four stars in each one and label it in three ways:

3 groups of 4 is 12o 3 sets of 4 is 12

3 x 4 = 12

Explain that the x is the mathematical symbol that means groups of or sets of Say, "Now I'd like you to use one or more of these ways to label each of your rounds for the game. Use the language that makes the most sense to you. When you finish, turn your paper over and play four more rounds with your partner. Again, label each round -vith at least one of these ways for describing your circles and stars and totals.

8. On future days, encourage students to keep playing Circles and Starsat school and at home.

lji

What Makes a Good Shape for a Quilt?Materials

• 4 6-by-6-inch construction-paper squares for the class demonstration• color tiles for your students, 24 for each pair

Extended Common Core: p.12 Flip book Equal Grouping (Division)

2nd Nine Weeks: Addition Word Problems: Week 5Unit 4 Weeks 7 & 8 Repeated Subtraction

Standard: 2.NBT.9 Explain why addition and subtraction strategies work, using place value and properties of operations (Explanations may be supported by drawings or objects.)

Sample LEQs:•When is subtracting numbers the same as dividing numbers?• How does subtraction of equal groups show division?• How many times can I take _ objects from a group of objects?•How can I use manipulatives to show subtraction of equal groups? How do I record each time that I

subtract objects?• How many piles of§ can I make out of this group of 30 objects? How do I make an array out of these

equal groups? (Change the numbers as new problems are presented.)• How do I count the objects in an array? When do I use the word "vertical" when I count objects in an

array? When do I use the word "horizontal" to count objects in an array? How can I use the results of my counting to explain division?

• How can I use the words vertical and horizontal to give directions to my partner?(Have one child give directions and the other child build the array. Both children write the number sentence that matches the construction. Then have the children switch roles.)

• How can I use manipulatives to show a story problem?• How can I use manipulatives and drawings to show a story problem?• What is a story problem that matches this array?

Materials:Book Three: Developing Number Concepts, pages 182-210

Extended Common Core: p.12 Flip book Equal Grouping (Division)

2nd Nine Weeks: Addition Word Problems: Week 5Unit 4 Weeks 7 & 8 Repeated Subtraction

Sample Activities Using the Suggested Sequence:1. Repeated Subtraction-Begin with the children. (Have the children act out each situation. Model

the subtraction on the board each time part of the children leaves the group. After modeling several problems, have the children record each step of the subtraction on their whiteboards.)

a. 15 children are standing. 5 children sit at this table. 5 children sit at the next table. 5 children sit at the last table.

b. 12 children are getting ready for recess. 3 children get their coats. 3 more children get their coats ....

c. 10 children are going to play this game. 5 children are on team A. 5 children are on team B.

2. Repeated Subtraction-Use actual objects-books, pieces of paper, candy, pencils. (See page 192 in Developing Number Concepts: Book 3)

a. I have 15 pieces of candy. 5 children sit at this table. How many pieces of candy will eachchild get?

b. I have 20 books to return to the library. 4 children will carry the books for me. How many books will each child carry?

3. Repeated Subtraction-Use manipulatives to represent objects.a. (See page 193 in Developing Number Concepts: Book 3)

4. Translate equal groups to make arrays. Give the children Unifix cubes (or color tiles) and plastic cups to problems similar to these (see page 199 in Developing Number Concepts: Book 3):

a. Use 4 cups & 12 cubes. Place the same number of cubes in each cup. How many cubes will be in each cup?

b. Now make a row out of the cubes in each cup. How many rows can you make? (or make acolumn out of the cubes in each cup. How many columns can you make?

5. Independent/Partner Activities-See pages 202-210 in Developing Number Concepts: Book 3)

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Grade 2, Multi-Use 37

19 Apples-Extended Number Pattern

You WHI N-e d > Chapter 3, Patterns; Chapter 6, Multiplication and Division<» an apple and a sharp knife,, a 4 x 4 sheet of white paper, crayons, scissors for each child• a large sheet of bulletin board paper for display and glue --

Your lesson If you cut an apple horizontally and look at the inside of one of the parts, what type of design will you see? How many points will it have? What type of seashell will it look like?

Answer these questions by cutting an apple as shown.

Have the children recreate the 5-point design by coloring a circular shape cut from paper. Glue some of these shapes on the bulletin board paper to create the groups shown below and identify the multiples of 5 points seen in each group.

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I apple. VJiH15 poill S

I xs -==S1·5-=5

2 aep\ s wit\.,5 po1nh eoc.n

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Ask the children to visualize the 10th group in this display. Ha\·e volunteers build this group without building those in between and discuss reasons for their answer. Here is one picture of the 10th group.

28

CONTACT LESSONS

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