Activities
The Virginia Standards of Learning state that the problem-solving process should be integrated early and continuously. So, I plan to work through the Common Core taxonomy of problem types right from the start of the school year. I plan to spend about 15 minutes of the daily math period on problem solving, introducing the 23 types of problems.
I will slowly introduce the types (i.e. Change Plus) one subcategory (i.e. result unknown, change unknown, and start unknown) at a time. I anticipate using the I do, we do, you do approach for each subcategory. So, the teacher will model, students will do one with me, and then students will do one alone. After a type has been fully introduced, I will allow time for a mixed review and students will engage in a Math Talk so students can share their solutions and misconceptions.
After students have worked with all 23 types, the focus will shift to multi-step problems, and will continue throughout the year. Below is the schedule I plan to follow beginning the second week of school.
Day 1 – Change Plus +/- result unknown
Day 2 – Change Plus +/- change unknown
Day 3 – Change Plus +/- start unknown
Day 4 – Mixed Change +/- plus problems and Math Talk
Day 5 – Change Minus +/- result unknown
Day 6 – Change Minus +/- change unknown
Day 7 – Change Minus +/- start unknown
Day 8 – Mixed Change Minus +/- problems and Math Talk
Day 9 – Mixed Change Plus +/- and Change Minus +/- problems
Day 10 – Part-Part-Whole +/- whole unknown
Day 11 – Part-Part-Whole +/- part unknown
Day 12 – Mixed Part-Part-Whole +/- practice and Math Talk
Day 13 – Mixed Change Plus, Change Minus, & Part-Part-Whole +/- practice and Math Talk
Day 14 – Compare Difference Unknown +/- more
Day 15 – Compare Difference Unknown +/- less
Day 16 – Compare Bigger Unknown +/- more
Day 17 – Compare Bigger Unknown +/- less
Day 18 – Compare Smaller Unknown +/- more
Day 19 – Compare Smaller Unknown +/- less
Day 20 – Mixed Compare Practice +/- and Math Talk
Day 21 – Mixed Change +/-, Part-Part-Whole +/- , Compare Practice +/- and Math Talk
Day 22 – Equal Groups ×/÷ – Product Unknown
Day 23 – Equal Groups ×/÷ – Group Size Unknown
Day 24 – Equal Groups ×/÷ – Number of Groups Unknown
Day 25 – Mixed Equal Groups ×/÷ Practice and Math Talk
Day 26 – Mixed Review Addition/Subtraction Change, Part-Part-Whole, Compare & Multiplication/Division Equal Groups ×/÷ practice and Math Talk
Day 27 – Area/Arrays ×/÷ Product Unknown
Day 28 – Areas/Arrays ×/÷ Group Size Unknown
Day 29 – Areas/Arrays ×/÷ Number of Groups Unknown
Day 30 - Areas/Arrays ×/÷ Mixed practice and Math Talk
Day 31 - Mixed Review Addition/Subtraction Change, Part-Part-Whole, Compare & Multiplication/Division Equal Groups and Areas/Arrays practice and Math Talk
Day 32 – Compare ×/÷ Unknown Product
Day 33 – Compare ×/÷ Smaller Quantity Unknown
Day 34 – Compare ×/÷ Comparison Factor Unknown
Day 35 - Mixed Compare ×/÷ practice and Math Talk
Day 36 – Mixed Review Addition/Subtraction Change, Part-Part-Whole, Compare & Multiplication/Division Equal Groups, Areas/Arrays, Compare practice and Math Talk
Day 37 - Mixed Review Addition/Subtraction Change, Part-Part-Whole, Compare & Multiplication/Division Equal Groups, Areas/Arrays, Compare practice and Math Talk
Day 38 - Multi-step Problems
Day 39 - Multi-step Problems
Day 40 – Multi-step Problems and Math Talk
Day 41 – Multi-step Problems and Math Talk
Day 42 - Multi-step Problems and Math Talk
Day 43 - Multi-step Problems and Math Talk
Day 44 - Multi-step Problems (ongoing throughout the year) and Math Talk
Day 45 – Problem Solving Assessment
I am selecting four activities from the above schedule to highlight and flesh out in more detail. I have selected Day 1, Day 4, Day 27, and Day 40 purposefully. Day 1 will introduce how I plan to go through each type of problem. Day 4 provides an example of the end of a type of problem review and lays out the format for the Math Talk. Day 27 is the first day of area/array problems and since this type of problem does not lend itself to bar models, it is important to lay out an approach. Finally, I chose Day 40 to share a Math Talk from the multi-step section. The problems will initially come from the collection of problems, and I will categorize and make up more as needed. I also will likely utilize some problem sets from curriculum units developed in the Yale National Initiative ® seminars with Roger Howe in 2015, Problem Solving with the Common Core, and 2017, From Arithmetic to Algebra: Variables, Word Problems, Fractions, and Rules.
Day 1 – I plan to kick off the unit by letting my students know that we will focus on problem solving this year, and will spend about 15 minutes each day working on problem solving strategies. We will also learn to solve many word problems using Singapore bar models because Singapore is ranked relatively high in math in the world. I will show them where Singapore is on a map.
Then I will introduce a Change Plus problem with a result unknown. I will not categorize the problem type, but I will provide three examples of a change plus result unknown problem. The first one I will model how to read the problem carefully, how to pull the facts out, how to write the question, how to draw a bar model, and how to go back to the question to write the answer in a statement. Then we will reflect as a class if the answer seems reasonable and makes sense.
Next, we will do another Change Plus problem with the result unknown. This time, the students will help me. I will ask them what the facts are. As they share a fact, I will record it and project it for the class to see. Once the facts are pulled out, a student will tell me the question to write down. Next the students will guide me to draw a bar model, solve the problem, and write the answer as a statement. I will ask students to share their thoughts on the reasonableness of the answer.
With the third problem, again a Change Plus result unknown, students will solve it independently following the steps that were modeled with the previous two problems.
Day 4 – For the mixed review, I plan to use one of each type of Change Plus problems. Students will solve three problems on a handout (using the steps learned in class.) Once students have completed the problems, we will share solutions, and students will have the opportunity to share different solutions or bar models. The Math Talk will begin organically as students share solutions, but I will work to guide students to understand the importance of providing a safe environment for sharing both correct and incorrect solutions. I also will stress the importance of learning from mistakes, yours and others, and how we can all benefit from talking about and analyzing misconceptions.
Day 27 – This is the first day area/array problems are addressed. Since these problems do not lend themselves to a bar model, I plan to point this out to students so they understand that a bar model is a linear representation and an array or area problem is two-dimensional. I intend to have students follow the same procedure as explained above, but instead of a bar model, students will draw a picture of the array or an area model using a rectangle.
Day 40 – Now that all of the problem types have been taught, students will now tackle two-step problems. Students should know that these are two-step problems, but that each step is similar to problems they have already solved. I anticipate these problems will lend themselves to a more robust class discussion as students have various solutions, particularly regarding the order in which the two steps are solved. The drawings may include two distinct bar models as shown in the multi-step problems previously described.
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