Lisa Waszak
CCLM^2 Project
Summer 2012
DRAFT DOCUMENT. This material was developed as part of the
Leadership for the Common Core in Mathematics (CCLM^2) project at the University of Wisconsin-Milwaukee.
Part 1: Grade: 2 Domain: Operations and Algebraic Thinking Cluster: Represent and solve problems involving addition and subtraction. Standard: 1. Use addition and subtraction within 100 to solve one-­‐ and two-­‐step word problems involving situations of adding to, taking from, putting together, taking apart, and comparing with unknowns in all positions. * While this standard includes 2-­‐step problems, for my analysis, I will be focusing on one-­‐step word problems. Part 2: (a) Explanation: As students progress from K-­‐2 they develop meanings for addition and subtraction through word problems. These story problems become increasingly more difficult and their representational and computational skills become more sophisticated. If children are to become fluent with addition and subtraction facts, they need regular and repeated practice with a variety of problem situations. The following is a chart showing which types of problem solving situations children are expected to work on for grades K-­‐2. You will then find a table that will give you examples of each type of problem with numbers to meet this second grade standard. For kindergarten and first grade, or for RTI, you may use the same format using smaller numbers. As enrichment for higher level learners, use the story problem format and numbers in the hundreds. Kindergarten Grade 1 Grade 2 Add To-­‐Result Unknown Take From-­‐ Result Unknown Put Together/Take Apart-­‐ Total Unknown Put Together/Take Apart-­‐ Both Addends Unknown Add To-­‐ Change Unknown Take From-­‐ Change Unknown Put Together/Take Apart-­‐ Addend Unknown Compare-­‐ Difference Unknown (more and fewer) Compare-­‐ Bigger Unknown (more) Compare-­‐ Smaller Unknown (fewer) Add To-­‐ Start Unknown Take From – Start Unknown Compare-­‐ Bigger Unknown (fewer) Compare-­‐ Smaller Unknown (more) Addition and Subtraction Problem Situations For Second Grade Result Unknown Change Unknown Start Unknown Add To 24 kids are on the field at recess. 32 more kids run onto the field. How many kids are on the field now? 24 + 32 = ? Take From 75 M&M’s were in the bowl. My mom ate 47 of them. How many M&M’s were left in the bowl? 75 – 47 = ? Total Unknown 41 people were sitting in the moving theater. Some more people came into the theater. Then there were 68 people in the theater. How many people came in to join the 41 people? 41 + ? = 68 Tia was building with 93 Legos. Her brother and took some of the Legos. Then Tia only had 86 Legos left. How many Legos did her brother take? 93 -­‐ ? = 86 Some dogs were playing at the dog park. 15 more dogs came to the park. Now there are 29 dogs. How many dogs were there at the beginning? ? + 15 = 29 My teacher put some pencils in the caddies. The kids in my class took 37 of the pencils to use to work in reading stations. Now there are 9 pencils left in the caddies. How many pencils did my teacher put in the caddies? ? – 37 = 9 Addend Unknown Both Addends Unknown Put Together/ Take Apart Sasha made a necklace with 22 red beads and 35 blue beads. How many beads are on her necklace? 22 + 35 = ? Marquis has a package of 40 mechanical pencils. 19 of them are green. How many pencils are not green? 19 + ? = 40 40 – 19 = ? Mrs. Smith had 58 star stickers on her homework chart. If the star stickers were gold or silver, how many of each color could she have? 58 = ? + ? Difference Unknown Bigger Unknown Smaller Unknown “How many more” version: Jessica is 60 inches tall. Her brother, Wesley, is 43 inches tall. How many more inches tall is Jessica than Wesley? “How many fewer” version: Jessica is 60 inches tall. Her brother, Wesley, is 43 inches tall. How many more inches tall is Jessica than Wesley? 43 + ? = 60 Version with “more” James scored 18 more points than Chloe on the video game. Chloe has 51 points. How many points does James have? Version with “fewer” Chloe has 18 fewer points than James on the video game. Chloe has 51 points. How many points does James have? 18 +51 = ? Version with “more” Cameron read for 13 more minutes during independent reading time than Lilly. Cameron read for 33 minutes. How many minutes did Lucille read? Version with “fewer” Lilly read for 13 fewer minutes than Cameron during independent reading time. Cameron read for 33 minutes. How many minutes did Lilly read? 33 – 13 = ? Compare 60 – 43 = ? 51 + 18 = ? ? + 13 = 33 * Pay close attention to where the unknown (?) is in each equation. Be sure that your unknown is in different positions. For example: 12 + 54 = ? ? = 12 + 54 66 – 12 = ? 66 + ? = 12 12 + ? = 66 54 = 66 -­‐ ? ? = 66 – 54 54 + ? = 66 (b) Examples: Children should be given daily mathematical practice working through problem situations. There needs to be purposeful manipulation chosen for specific problems in order for children to move along the problem-­‐solving continuum. Teachers need to: * provide manipulatives for children to use while problem solving. (counters, cubes, base ten blocks, number lines, double ten frames, tape diagrams) * encourage use of drawings and symbolic representations. * post different strategies for specific problem situations. * facilitate daily discussions around problem solving strategies. * help children become Level 3 problem solvers. Below is the progression of levels of problem solving. Children my go back to a lower level as they encounter unfamiliar or more challenging situations. Levels 6+ 8 = 14 14 – 6 = 8 Level 1: Count all Level 2: Count on Count all 1 2 3 4 5 6 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Count on 6 7 8 9 10 11 12 13 14 Take away 1 2 3 4 5 6 1 2 3 4 5 6 7 8 9 10 11 12 13 14 To solve, count on 6 + ? = 14 6 7 8 9 10 11 12 13 14 6 to 14 is 8, so 14-­‐ 6 = 8 Level 3: Recompose Make a ten: One addend breaks apart to make 10 with the other addend Recompose: Make a Ten 14 – 8: I make a ten for 6 + ? = 14 10 + 4 6 + 4 + 4 8 Make a ten (from 5’s within each addend) 6 + 8 = 14 10 + 4 Doubles + n 6 + 8 = 6 + 6 + 2 = 12 + 2 = 14 * By the end of second grade, children should be Level 3 problem solvers. Part 3. (a) School Mathematics Textbook Program : Scott Foresman * In examining the Scope and Sequence for this math series, the following chart shows the progression of Problem Solving Strategies from first through third grade. grade introduce develop maintain and apply first * act it out * try, check, revise * look for patterns * make list/table * logical reasoning * use objects * write number sentence second * same as first grade * same as first grade third * logical reasoning * try, check, revise * work backwards * solve a simpler problem * write number sentence * choose strategy While these are the strategies introduced, the children are seeing problem solving in a variety of areas within the curriculum. For each lesson, in each grade level, there is a: -­‐ “Problem of the Day” -­‐ problem solving application page -­‐ single problem solving situation at the end of the student workbook page -­‐ “Discover Math in Your World” pages -­‐ “Reading for Math Success” pages -­‐ Problem Solving Masters These questions may consist of a story problem, analyzing a graph/table or other problem solving situations. Analysis of the addition and subtraction situations in Scott Foresman, grades 1-­‐3 Reference : Table: Addition and Subtraction Problem Situations First grade: Majority of problems are : -­‐Add to, Result unknown -­‐Take from, Result unknown -­‐Put together/Take apart, Total unknown * Through the “Problem of the Day” students have several opportunities to solve Compare/Difference Unknown situations. * Students are introduced to one strategy per lesson in Chapter 2: pictorial models, drawing, doubles, count on, count back, tens frames, number line. After Chapter 2, problem solving situations are found in the “Problem of the Day” and in the last problem of student workbook pages. Second Grade: * In Chapter 1, problem situations include: -­‐ Add to, Result unknown Example: Lesson 1.7-­‐ Trevor has 7 erasers. He buys 2 more. How many erasers does he have? -­‐ Take from, Result unknown Example: Lesson 1.6 – There were 10 butterflies. Then 2 butterflies flew away. How many butterflies are left? -­‐ Put Together/Take Apart, Total unknown Example: Lesson 1.2-­‐ There are 3 blue chairs. There are 2 red chairs. How many chairs are there in all? * Chapter 6 has several lessons on Compare, Difference Unknown situations. Example: Lesson 6.5 – Maria fed the fish 65 flakes. Paul fed the fish 50 flakes. How many more flakes did Maria feed the fish than Paul? * Note, as the standard states, the numbers being used are within 100. * Periodically, there are lessons or individual problems introducing children to: -­‐ Add to, Change unknown Example: Lesson 2.5 – Harriet had 7 trees in her yard. She planted some more. Now she has 16 trees. How many trees did Harriet plant? -­‐ Put Together/Take Apart, Addend unknown Example: Lesson 8.17 – 22 children in class voted for their favorite butterflies. If 12 children chose painted ladies, how many children chose other kinds of butterflies? -­‐ Unknowns in different positions Examples: Lesson 1.11 – There are 12 books in all. If 8 books are not in the bag, how many are in the bag? The book shows: 8 + ? = 12 Lesson 2.12-­‐ A father bird made 20 hunting trips in 2 hours. He made 10 trips the first hour. How many trips did he make in the second hour? The book shows 10 + ? = 20 * Like in first grade, students are given one problem addition/subtraction situation type per lesson. They are also given one preferred strategy with which to solve these problems. In many lessons, the accompanying “enrichment” book has examples of a variety of story problem situations. Children use the same strategies to solve problems as in first grade. In first grade numbers are within 20 for problem solving and in second grade the numbers are within 100. Third Grade: addition/subtraction problem solving situations : -­‐ Problem Solving Application page -­‐ Daily Spiral Review page -­‐ “Problem of the Day” *You will find similar problem types/strategies using larger numbers to one thousand. Example: 2.11-­‐ If a suited-­‐up firefighter weighs 220 pounds and his gear weighs 40 pounds, how much does a firefighter weigh without his gear? 1.8 -­‐ Darren had 500 baseball cards and 203 football cards. How many cards did Darren have in all? * In Chapter 2, “n” is introduced for the unknown. Example: 2.4-­‐ You can spend $15. The socks cost $6. How much money will you have left? $15 -­‐ $6 = n * Introduction of multi-­‐step addition and subtraction problems. Example: 2.12-­‐ Certain packages that weigh up to one pound can be mailed for $3.80. Wanda sent three of these packages and paid the postage with a ten-­‐dollar bill and a five-­‐dollar bill. Estimate to decide if she had enough money left for a $3 lunch. * Connects multiplication story problem situations to repeated addition. Example: 5.1 – There are 4 groups of 3 paintbrushes. How many paintbrushes are there altogether? 3 + 3 + 3 + 3 = 12 or 4 x 3 = 12 (b) Conclusions and Suggestions: While Scott Foresman has a variety of story problems throughout the second grade series, there are several things that a teacher can do to help the students meet the standard 2OA.1. * Instead of focusing on one strategy per lesson, the children need time to explore a variety of strategies, daily. Teachers should then post these strategies so students are able to choose the most efficient strategy for different story types. * An important part of this standard is that the children are able to work with the unknown in different positions. (See *note on page 2 of this document.) Since Scott Foresman does not have many lessons where this is the case, teachers will need to supplement problem situations in which the unknown is in different positions in the number representation. Also, introduce a symbol to represent the unknown in a problem as shown in the examples on page 2 of this document. (question mark, box, letter…etc) * In order to meet this standard, teachers will need to supplement certain problem situations that are found infrequently, or not at all, in the Scott Forseman second grade series. *Add To/Start Unknown * Taken From/ Change and Start Unknown * Compare/ Smaller and Bigger Unknown * Put Together/Take Apart-­‐Both Addends Unknown *See Table on page 2 of this document for examples of these problem types.