supporting the ccss operations and path · kindergarten activities for partner expressions unit 2:...
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Supporting the CCSS Operations and Algebraic Thinking Learning Path
Professor Karen C. FusonNorthwestern University
Learning paths within and across grades for•situations (problem types) that give meanings for operations•single‐digit computation (+‐ and x÷)
Students represent using drawings/diagrams and/or equations, then solve.
Students understand and apply properties of operations and the relationship between addition/subtraction and multiplication/division).
OA: Operations and Algebraic Thinking
What is new in OA?
a) Solve problems with all 3 unknowns. Each situation can have 3 unknowns.This creates a learning path of difficulty fromKindergarten to Grade 1 to Grade 2.
b) Show the situation with a math drawing or diagram.
Represent the SituationOA: Operations and Algebraic Thinking
Grade 1 and Grade 2 subtypes involve algebraic thinking:
Represent the situation with a drawing, diagram, and/or an equation.
Then decide how to solve for the answer.
Grade 2 Labeled Math Drawings for a
Start Unknown Problem
The key to solving story problems is understanding the situation. Students’ equations often show the situation rather than the solution. Students drawings should be labeled to show which numbers or objects show which parts of the story situation.
Yolanda has a box of golf balls. Eddie took 7 of them. Now Yolanda has 5 left. How many golf balls did Yolanda have in the beginning?
Grades 3 and 4: Represent a Start Unknown Situation
Grades 3 and 4: Represent a Start Unknown Situation
The Math Practices in action
A teacher asks every day: Did I do math sense‐making about math structure using math drawings to support math explaining?
Can I do some part of this better tomorrow?
Learning Paths for Single‐Digit Calculation
OA specifies learning paths for single‐digit calculation.• Addition/Subtraction• Multiplication/Division
Each operation has levels of increasing abbreviation, abstraction, and internalization.
We will now focus briefly on addition and subtraction.
Concepts for Each Level
Level 1: Each new addend and the total are separate.Level 2: Both addends are embedded within the total.Level 3: Addends are recomposed to make new
addends (e.g. 8+6 becomes 10 + 4)
Make‐a‐ten prerequisitesa. Partner of the larger addend to 10 (K.OA.4)b. All partners of the smaller addend to find how
much over ten (K.OA.3)c. 10 + n for n = 1 to 9 (K.NBT.1)
Kindergarten Activities for Partner Expressions
Unit 2: Use 5‐groups to show quantities, addition expressions, and total for numbers 6 to 10
Use Unit 1 Steps A, B, C with a group of 5 and some units:one unit of 5 red or blue squares, each with a dot on one side or
one unit of 5 pennies drawn in squares on a strip.
Children put tiles for the total to the leftand for an addition expression for partners (addends) to the right.
Kindergarten Partners and Expressions
Unit 3: Partners of 2,3,4,5 and 6 with tiles, break-apart stick, total, and addition expression
A) Make a number with a numeral tile and that many things.B) Elicit partners of that number.C) Use a break-apart stick to show the partners.D) Use number tiles and the + tile to show an addition expression for the partners and say
the partners: Six is four plus two.E) Repeat for different partners of the number.F) Repeat all steps with a different number.
A-C) D)
Percentage Correct on Partner (Addend) Tasks for Kindergarten Children
Percentage Correct on Partner (Addend) Tasks for Kindergarten Children
Kindergarten Number Patterns in Order
K Count from 11 to 20 Quick Practice Unit 2
Number Pattern Poster
K Unit 3: See Teen Numbers as Ten Ones and More Ones
The Ten Bug helps children see a group of ten things or two groups of five (that make ten).
The 10-Penny Strip
and 2 loose pennies
Number Tiles show the ten
and ones.
K Make Teen Numbers with Real Objects
Math Talk Questions:
• Why did we make a group of ten in each teen number?
• Does it matter what objects we use to show a teen number?
• Does it matter how we arrange the objects?
Grade 1Related Partner Supports
Grade 1Partner Switches
Grade 1 Make‐a‐Ten Adding
9 + 5
Use the Stair Steps
Use Math Drawings
Grade 1 Fluency to 10
Grade 1 Strategies Within 20 and Grade 2 Fluency Within 20
Addition
Make-a-Ten Cards Unknown Addend
Subtraction
Smarter Balance and PARCCDr. Fuson has consulted with both PARCC and Smarter Balance about item distribution and complexity. Details of this are confidential, of course.
These tests will test the CCSS‐M standards, so it is crucial to use a program that implements the standards thoroughly and well and helps teachers to do this.
Start sooner rather than later with this process.
It is impossible to predict exactly the shape or complexity of the final tests. So far the complexity of the items is being reduced to sensible levels.
Learning Paths for Single‐Digit Calculation
OA specifies learning paths for single‐digit calculation.• Addition/Subtraction• Multiplication/Division
Each operation has levels of increasing abbreviation, abstraction, and internalization.
The learning path for multiplication/division begins and ends in Grade 3. It is crucial and takes much of the time in Grade 3.
What to Emphasize and Where to Intervene as Needed
Grades K to 2 are more ambitious than some/many earlier state standards:•K: The ten in teen numbers•G1: + within 100 with composing a new ten; ok if many children still use math drawings; no subtraction without decomposing a ten•G2: a) +‐ total ≤100 with composing and decomposing a ten; use math drawings initially, but fluency requires no math drawings• b) +‐ totals 101 to 1,000 with math drawings; vital get mastery by most so that G3 can focus on x÷; intervene with as many as possible to get G2 mastery
Grades 3 to 6 are less ambitious than some/many earlier state standards:•G3: Fluency for G2 goals +‐ totals 101 to 1,000, so no math drawings; no new problem sizes so can focus on x÷ [intervene for x÷ all year]•G4 and G5: a) x only up to 1‐digit x 2‐, 3‐, 4‐digits and 2‐digits x 2‐digits; so not really need mastery of 1‐row methods for multiplication [have time for fractions]•b) division has only the related unknown factor problems; 1‐digit divisors G4 and 2‐digit divisors G5; fluency G6