K and 1 st grade  What math abilities do your students have when they enter kindergarten?  What math abilities do you hope they will have when they leave kindergarten? When they leave1 st grade?  Take a look at the diagnostic assessment to consider the range of abilities at each grade.

Our Purpose and Agenda  Important outcomes for Kindergarten Number relationships including part/whole Adding and subtracting situations Facility with number combinations to 5  Important outcomes for 1 st grade Additional adding and subtracting situations Strategies for adding and subtracting Fluency with single-digit numbers

K-1 Number Concepts  More, less, equal  Counting objects, “counting on”  Number relationships  Addition and subtraction problem types  Place value (two digits) Elementary Math ResourcesElementary Math Resources – Van de Walle and Lovin

More, less, equal  Is this the same as “Which of these numbers is larger: 3 or 8?”  This is where you are taking the children: To knowledge of 3-ness or 8-ness.

Counting objects, “counting on” Counting on from any number. Counting past the decade numbers. Teaching Channel video Let’s Count:Let’s Count: Learning Numbers in Meaningful Ways Activities 5, 6 & 7

How many more (less)? Developing Number ConceptsDeveloping Number Concepts by Kathy Richardson See handout

Number relationships Numbers are related to each other through a variety of number relationships. The number 7, for example, is  more than 4, less than 9,  composed of 3 and 4 as well as 2 and 5,  is three away from 10,  and can be quickly recognized in several patterned arrangements of dots.  Number relationships for 7 further extend to an understanding of 17, 57 and 370. Starting at 17, how many more are needed to make 20?

Simon Says  Simon says, Show six fingers.  Janice, tell us about the way you showed six fingers.  Peter, yours is different. Tell us about yours.  Does anyone have a different way to show six fingers?  Simon says, Show nine fingers.

Number relationships Activities 11, 12 & 13

Spatial relationships Activities 8, 9 & 10

5-frame and 10-frame cards Number Talks: Ten Frames and Dot Cards

Number relationships 1-10 Graphics are from Van de Walle & Lovin, Teaching Student-Centered Mathematics: Grades K-3 Activities 17 – 22 Teaching Channel: Quick ImagesQuick Images Number Talks: Rekenreks

Number relationships 1-10 Activities 14, 15 & 16

Anchors to 5 Show the child 5 counters, then put them under your hand. Pull out several of the counters, and ask: How many are hiding?  Notice how the child solves the problem. If not visible, ask them to explain how they got an answer.  Repeat with different numbers of counters.  Have the child write the two numbers.  You can do this with any number, but 5 is a useful “anchor” number.

Anchors to 10  The frame for seven provides a visual model of seven as two more than five and three less than ten.  Working with this model supports the eventual connection to = 7 and 10 – 3 = 7.  Objects, pictures, symbols  Packet on wiki Packet on wiki  Tasks in Envision Teaching Channel: Popsicle Stick Math: Making 10Popsicle Stick Math: Making 10

Number Talks …to develop basic number concepts What do you like about this approach?

The teacher’s role “By changing my question from ‘What answer did you get?’ to ‘How did you solve this problem?’ I was able to understand how they were making sense of mathematics.”

Number Talks book  K.OA.3 Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = and 5 = 4 + 1). As each number talk is shown, ask students, “How many dots do you see? How do you see them?”

 K.OA.4 For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.

Early Number Concepts  What do children need to understand about numbers?

Adding and Subtracting  The end goal for K and 1 st grade:  Rachel has 6 beads. How many more beads does she need to collect to have 13 beads altogether?

The path to fluency Counting & number relationships are the basis for Problem solving which leads to Fluency with whole numbers

What operation is this?  Steven had 4 toy cars. He wanted 9. How many more toy cars would Steven need to have 9 altogether?  How might a K or 1 st grade student solve this?  Is this an addition or subtraction problem? Or something else?  4 + ___ = 9

Modeling the Action  Liz had 8 cookies. She ate 3 of them. How many cookies does Liz have left?  Liz has 3 marbles. How many more marbles does she need to buy to have 8 marbles?  Liz has 3 fish. Tom has 8 fish. How many more fish does Tom have than Liz?

Rachel’s Problems  Try each of the problems. Think about how students might model the action in the problem.  Discuss your solutions with a partner.  As you watch the video, think about which problems seem harder for Rachel. CGI Interviews – Direct Modeling

CCSS Kindergarten Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from.  K.OA.1 Represent addition and subtraction with objects, fingers, mental images, drawings (drawings need not show details, but should show the mathematics in the problem), sounds (e.g., claps), acting out situations, verbal explanations, expressions, or equations.  K.OA.2 Solve addition and subtraction word problems, and add and subtract within 10, e.g., by using objects or drawings to represent the problem.

CCSS 1 st Grade  1.OA.1 Use addition and subtraction within 20 to solve word problems involving situations of adding to, taking from, putting together, taking apart, and comparing, with unknowns in all positions, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem.

Basic assumptions about children’s learning of mathematics  Very young children know how to solve math problems.  Children develop mathematical understanding and acquire fluency with whole number computation by solving a variety of problems in any way that they choose.  Children learn more advanced computational and problem solving strategies by watching their classmates solve problems.

Where is the unknown? (action) 1. Lucy has 8 fish. She wants to buy 5 more fish. How many fish would Lucy have then? 3. Janelle has 7 trolls in her collection. How many more does she have to buy to have 11 trolls? 2 TJ had 13 chocolate chip cookies. At lunch she ate 5 of those cookies. How many cookies did TJ have left? 4. Max had some money. He spent $9 on a video game. Now he has $7 left. How much money did Max have to start with? Taking fromAdding to

Result UnknownChange UnknownStart Unknown 1. Lucy has 8 fish. She wants to buy 5 more fish. How many fish would Lucy have then? 3. Janelle has 7 trolls in her collection. How many more does she have to buy to have 11 trolls? 2 TJ had 13 chocolate chip cookies. At lunch she ate 5 of those cookies. How many cookies did TJ have left? 4. Max had some money. He spent $9 on a video game. Now he has $7 left. How much money did Max have to start with? Taking fromAdding to

Problem Types - Action Result Change Start Unknown Unknown Unknown Join5 + 2 =  5 +  = 7  + 2 = 7 Separate8 – 3 =  8 –  = 5  – 3 = 5 Notice how these equations are shorthand for the action in the problem. Students should write equations to represent the problems in order to eventually “decontextualize” the mathematics.

CCSS – 1 st grade “unknowns”  1.OA.8 Determine the unknown whole number in an addition or subtraction equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 + ? = 11 5 = ＿ – 3 (Very hard problem – trial and error) = ＿. Can you see action in each of these problems? Concrete-Representational-Abstract (C-R-A) Start with modeling the problem using objects. Then pictures. Only then, the equation. I have 5 buttons in my hand (show them, then hide some and show what remains). How many did I hide?

No-action problems – part/whole Understand addition as putting together and adding to, and understand subtraction as taking apart and taking from. (K and 1 st )  6 boys and 4 girls were playing soccer. How many children were playing soccer? (whole unknown)  10 children were playing soccer. 6 were boys and the rest were girls. How many girls were playing soccer? (part unknown)

No-action problems - comparing  Mark has 3 mice. Joy has 7 mice. Joy has how many more mice than Mark? (comparing)

No-action problems  Equalizing: Abby has read 6 books so far this summer. Mollie has read 4. In order to have read as many books as Abby, how many more does Mollie need to read? (number sentence?)  Missing part: There are 14 hats in the closet. 6 are red and the rest are green. How many green hats are in the closet? (number sentence?)  Comparative subtraction: Jessica found 6 rocks on the trail and Tia found 7. How many more rocks did Tia find than Jessica? (number sentence?)

Are some more difficult?  There are 14 hats in the closet. 6 are red and the rest are green. How many green hats are in the closet?  14 birds were in a tree. 6 flew away. How many birds were left?  The structure of a problem determines how difficult it is for children to solve and determines their initial solution strategies.

Abstract problems – no context

Solution Strategies  There were 7 apples on the tree. A farmer came along and picked 5 of the apples. How many apples are still on the tree? How many different ways can this be figured out?  4 ladybugs were crawling in the grass. 3 more came to join them. How many ladybugs were there then?

Solution Strategies  Direct modeling of the action in the problem  Counting strategies  Derived facts  Fluency 4 ladybugs were crawling in the grass. 3 more came to join them. How many ladybugs were there then? CGI Interviews - Strategies

 Children learn more advanced computational and problem solving strategies by watching their classmates solve problems.

CCSS Kindergarten Strategies  K.OA.3 Decompose numbers less than or equal to 10 into pairs in more than one way, e.g., by using objects or drawings, and record each decomposition by a drawing or equation (e.g., 5 = and 5 = 4 + 1).  K.OA.4 For any number from 1 to 9, find the number that makes 10 when added to the given number, e.g., by using objects or drawings, and record the answer with a drawing or equation.

CCSS 1st Grade Strategies  1.OA.3 Apply properties of operations as strategies to add and subtract (e.g. 5+3 = 3+5)  1.OA.4 Understand subtraction as an unknown-addend problem. ( can be “what do I add to 8 to get 12?)  1.OA.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten; decomposing a number leading to a ten; using the relationship between addition and subtraction; and creating equivalent but easier or known sums, e.g. (7+6 = “doubles plus one”).

Solution Strategies  Try “How Would Children Solve These Problems?” using each of the types of strategies.  Then try “Finding a Problem for a Strategy” – the Jeopardy Game of CGI.

Try this at least twice every week  Present a problem to the whole class, let them work on it individually, then have several students present their approaches.  Keep track of their solutions  Use problems with numbers that are appropriate for your students.  A good resource: CGI ProblemsCGI Problems  The teacher’s role is to guide student’s learning by knowing each child’s cognition.

Fluency with “math facts”  The use of manipulatives, counting and derived-fact strategies eventually grows into knowledge of most math facts.  Explicit instruction on strategies can be helpful for building math facts that haven’t come naturally through problem solving, but that isn’t necessary until 2 nd grade.  K.OA.5 Fluently add and subtract within 5.  1.OA.6 Add and subtract within 20, demonstrating fluency for addition and subtraction within 10.

Number Talks and Math Centers  Number Talks and partner games don’t take the place of problem-solving, but they do supplement it and provide more opportunities for learning and practicing strategies.  8 birds are in a tree. 6 more birds fly up to the tree. Now how many birds are in the tree? Number Talks: 8+6 Number Talks: Intro

Common Core Checklist  Read through the Common Core for both grades to see how what we’ve addressed fits with the core mathematics curriculum. Mark anything you have questions about or want to discuss further.  Then create a checklist to help you monitor students’ progress on important math understandings and skills.

Your Curriculum As you teach from enVision, keep in mind these key goals for Kindergarten and 1 st grade mathematics.