# Investigations Blog

## Robot Fingers and Multiplicative Structure

In a blog post over a year ago, I wrote about the importance of using story contexts to support students in developing mental images of the operations. In that post, I concluded: “Along with pictures, drawings, diagrams, equations, and physical models, story contexts are critical parts of a student’s repertoire of representations. In fact, it is often the case that three kinds of representations are in play: a numerical representation, a picture or diagram, and a story. Moving among these to...

read more## Multiplication in 5th Grade: What Are Some Issues?

Last year, my colleague Keith and I worked a few times with a group of 5th grade teachers. One of the questions they asked us to help them think about related to this 5th grade benchmark: “Fluently solve multidigit multiplication problems using a variety of strategies including the U.S. standard algorithm.” They told us that they had students who could multiply 2-digits by 2-digits successfully but struggled with 3-digit by 2-digit multiplication problems. They wanted to discuss the...

read more## 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...

read more## “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...

read more## 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?...

read more## 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?Answer: 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...

read more## What Does It Mean To Be Smart?

“Wow, you’re so smart.” These words drew my attention to a pair of 5th grade girls in a class I was visiting, who I’ll call Cassie and Sophia. They were mid-way through a turn and talk, each sharing her strategy for solving 84 x 59. I casually moved closer, curious about what prompted the comment and trying hard to see each girl’s strategy, recorded in their math journals. Upon hearing Cassie’s comment, Sophia responded in an inviting tone, “No, no. Explain to me what you did.” She...

read more## A Grade 3 Q&A: Why start with multiplication and division?

Question: Why does 3rd grade start with a multiplication/division unit and not addition/subtraction?Answer: We often hear from people who wonder why Grade 3 starts with a multiplication and division unit—just like it did in the 1st edition!—rather than an addition and subtraction unit. As we decided on the sequence of the units in any grade, we considered many different things. The most important was the development of mathematical content within and across grades.In Investigations 3,...

read more## “What Was Thomas Thinking?”

I often think about a lesson, captured on video some years ago. Liz, a fifth-grade teacher, gave her students two-digit multiplication problems—12 × 29 and 36 × 17— and asked them to come up with strategies other than the conventional algorithm to perform the calculation. For 12 × 29, Jemea thought about twelve 30s and then subtracted 1 for each of 12 groups. 360 – 12 = 348. For 36 × 17, Duane thought of 36 bowls, each holding 17 cotton balls. Ten bowls hold 170 cotton balls, and there are...

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