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My question pertains to this excerpt from 3.OA.8: “Represent these [two-step word] problems using equations with a letter standing for the unknown quantity.” It is unclear whether third graders should be expected to represent two-step word problems with a single number model, or if multiple equations are acceptable. The progressions documents don’t clearly address this either, saying only, “More difficult problems may require two steps of representation and solution rather than one.” Is the spirit of the standard that third graders should be able to represent a multi-step problem with a single number model (e.g., 7 x 3 – 2 = S) as well as with multiple models (e.g., 7 x 3 = 21; 21 – 2 = S)? I appreciate your thoughts on this.
I would really appreciate any feedback you could offer on this topic. I am unsure whether I should expect my third graders to be able to write a single number model to represent a two-step number story, or if two equations are acceptable. My colleagues and I interpret the expectation of this standard differently. Thank you so much.
I don’t see anything that constrains the number of equations used to solve a problem in 3.OA.8 to one. Note that 2.OA.1 uses a very similar formulation but goes on to elaborate:
Represent and solve problems involving addition and subtraction.
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, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.
As you note, page 28 of the OA Progression says:
As with two-step problems at Grade 2,2.OA.1, 2.MD.5 which involve only addition and subtraction, the Grade 3 two-step word problems vary greatly in difficulty and ease of representation. More difficult problems may require two steps of representation and solution rather than one.
“Two steps of representation and solution” sounds to me as it includes solutions that involve two (or more) equations or tape diagrams, as in the example in the margin.
It might be that part of the concern is whether students should be able to interpret things like 3 × 10 + 5. That’s discussed here: http://commoncoretools.me/forums/topic/expanded-notation-and-order-of-operations/
Thanks for the feedback. I’m less concerned about equations being “constrained” to one, but more whether kids are expected to be able to read a number story and write one number sentence to represent it. At this point in third grade, it seems it would be much more natural for kids to use two or more number sentences. For example, in the sample problem I provided above, take this number story:
Julie had 7 bags of apples. Each bag contained 3 apples. She gave one apple to Amanda. How many apples does she have left?
Most third graders I know would write two equations: 7 x 3 = 21, and then 21 – 2 = S. It is a more advanced process to see this equation as a single number model: 7 x 3 – 2 = S. It gets even more complicated if we expect them to interpret a number story that involves application of the order of operations (which, of course, they are expected to apply correctly), or where the unknown amount is the start or the change.
Is the intent of the standard that kids must be able to represent a multi-step number story with one equation?I’ve been watching teachingchannel.org (and Phil Daro videos on vimeo.com) these days and I envision this excellent number story playing out like this: Kids work in pairs or small groups to figure out the problem. Some kids would draw pictures of 7 bags, each with 3 apples and figure it out that way. More advanced students would, as you say, write two equations. Even more advanced students write one equation. At each level, the teacher serves as coach, drawing the best out of the students as they think and talk with each other.
Next, a whole class discussion begins with the picture-students who explain their thinking. The discussion then morphs to students who used two equations and capped with the single equation. Lastly, the students are given time to think about how subtracting before multiplying would mess up the answer.
3.0A.8 says, “Represent these problems using equations with a letter standing for the unknown quantity.” In your task, the variable is already isolated. At some point we need the students to be able to solve this: Julie has some bags of apples, each with three apples. If she has 21 apples total, how many bags does she have? 21 = 3b. The next level of complexity would be something like, “Julie empties the bags onto the table and Amanda takes one of them away. Julie sees there are 20 apples left. How many bags did she have?” 20 = 3b -1
This might seem “over the top” with a class of third graders, but working in groups to solve problems like this tends to pull their thinking skills impressively upward! What we don’t want is for students to see there are two numbers in the problem and automatically use their favorite operation to “get the answer.”Oh, I should have also mentioned Illustrative Mathematics has a couple other great tasks you might like to see. Illustrative Mathematics is linked to this site, but here’s the particular tasks:
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