Investigations Blog

Q&A: The Definition of a Trapezoid

Question: Why did you decide to use the exclusive definition of a trapezoid? As the question suggests, there is more than one definition of a trapezoid. Mathematicians define trapezoids in one of two ways: Using the inclusive definition, all parallelograms (which include rectangles, squares, and rhombuses) are trapezoids. Using the exclusive definition, they are not. In determining which definition to use, we thought about a couple of things: Most elementary textbooks use the exclusive...

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And Then, She Waited

Have you ever been teaching (or leading professional development) and asked a really good question only to be met with silence? We all have teacher moves in our back pocket for situations like this—maybe do a turn and talk, ask the student if they’d like to call on someone to help them, or ask a different question. I recently observed a third grade lesson when I saw a teacher face this exact situation. The lesson (Unit 5, Session 3.4) focused on strategies for solving division problems. The...

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What Does It Mean to be a Math Person?

“I’m just not a math person.” I don’t know how many times I heard this sentiment over the course of a three-day workshop kicking off a new project focused on professional development for paraeducators. But, it surprised me that it was Tonya saying, “I’m good at it, but I’m just not a math person.” In the three days we spent looking at student work, solving problems together, discussing strategies, and playing games, Tonya was comfortable, active, and engaged. She shared her strategies for...

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From “Defective” Fractions to Infinite Equivalents

On a recent site visit, I was observing in a fourth grade classroom. The teacher started the lesson (Unit 6, Session 2.1) by writing “3/2” on the board and asking students to name the fraction. Most said “three halves” although one or two said “two thirds.” The teacher then displayed two blank 4 x 6 rectangles. She established that one rectangle was the whole, and asked students to use their copy of the rectangles to draw a representation that showed 3/2. The math coach called me...

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“That Seems Way Too Big”

On a recent visit to a small district in the Midwest, I got the chance to visit a third grade class that was working on division (3U5, Session 3.4). When I joined Nicole, she was in the middle of working on the following problem: Gil loves toy cars. He saved enough money to buy 32 toy cars. How many 4-packs of toy cars did he get? (SAB p. 333) Below the problem, Nicole had written I asked her to tell me about her thinking. She said she wanted to start with something she knew, which I agreed...

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A Division Solution: Amazing or Perplexing?

At a recent professional development session on multiplication and division, my colleague and I asked participants to examine some student solutions to the problem: 1564 ÷ 36. Before looking at the student work below, think about how you would solve the problem. The following solution from a fifth grade student seemed to either amaze participants or they had a hard time making sense of it. I’ve been thinking a lot about their reactions. Take a moment to look at this work. What do you notice?...

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Counting is Serious Business

“One, two, three, four, five…” I was interested in how quickly Owen and Ravi figured out a way to count the set of 40 yellow hexagons in their Inventory Bag. The boys took turns saying a number as they placed the hexagons in a line which started to snake across the rug in the meeting area of their kindergarten classroom. “Twenty-eight, twenty-niiiinnnne…” Ravi pauses unsure about what number comes next. Owen whispers “30” and Ravi says “30” as he places the thirtieth block in the line....

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A Cross-Grade Q&A: Quizzes in Investigations 3

Question: The Quizzes in Investigations 3 are new to us. We are used to assessing the benchmarks with the Meeting/Partially Meeting/Not Meeting system outlined in the Assessment Teacher Notes. Can you help us get a better sense of how to use the Quizzes as they relate to the Unit’s benchmarks? Quizzes are included in grades 1-5 of Investigations 3, to give students experience with next-generation test formats, such as: multiple choice fill-in-the-blank questions with more than one right answer...

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A Grade 3 Q&A: Assessing the Multiplication Facts

Question: Why do the assessments of the multiplication facts in Grade 3 include a time limit? In Investigations, the overwhelming majority of students’ work with the facts is focused on making meaning of the operation of multiplication, building connections between problems and images that represent them (e.g. problems about things that come groups, arrays), and using what they know to solve what they don’t (e.g. how can knowing that 3x4=12 help with 6x4?). This work happens in Ten-Minute Math...

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Asked and Answered: Why Ask the Same Question When You’ve Already Gotten a Perfectly Good Answer?

I was watching one of those legal shows on TV the other night. The prosecutor was asking the defendant a version of the same question for the third time. The defendant’s lawyer, getting annoyed, objected: “Asked and answered!” I’ve heard this phrase a hundred times in the (made-up TV) legal context, but this was the first time it struck me how pervasive this idea was in my own mathematics education, and how powerful it still is: If a student has given a perfectly good answer to a math...

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