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MHPS New 6th Grade MathLab

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Illustrative Mathematics provides resources and support for giving their students an enduring understanding of mathematics.

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At youcubed, we inspire, educate and empower teachers of mathematics, transforming the latest research on mathematics learning into accessible and practical forms.

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To get four squares in a row-vertically, horizontally, or diagonally.

How to Play

1. To begin the game, Player 1 moves a marker (green square) to a number in the factor list of numbers 1-9 along the bottom of the game screen.

2. Player 2 then moves the other marker (green square) to any number in the factor list (including the number marked by Player 1). The product of the two marked numbers is determined, and that product is colored red for Player 2.

3. Player 1 moves either marker to another number, and the new product is colored blue for Player 1.

4. Players take turns moving a marker, and each product is marked red or blue, depending on which player made the product. However, if a product is already colored, the player does not get a square for that turn.

5. Play continues until one player wins, or until all squares have been colored.

Credit

This Product Game Investigation was adapted with permission and guidance from Prime Time: Factors and Multiples, Connected Mathematics Project, G. Lappan, J. Fey, W Fitzgerald, S. Friel and E. Phillips, Dale Seymour Publications (1996), pp. 17‑25.

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To get four squares in a row-vertically, horizontally, or diagonally. To begin the game, Player 1 moves a marker (green square) to a number in the factor list of numbers 1-9 along the bottom of the game screen. Player 2 then moves the other marker (green square) to any number in the factor list (including the number marked by Player 1).

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Exercise your factoring ability against a human or the computer in this Calculation Nation game.

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Great hands on STEAM product!

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littleBits: Award-winning electronic building blocks for creating inventions large and small

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littleBits is an award-winning platform of easy-to-use electronic building blocks for creating inventions large and small.

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MindsetWorks | Growth Mindset | Growth Mindset Programs

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When people have a growth mindset they understand they can increase their intelligence through effective effort, deliberate practice, asking questions, and risking mistakes. A growth mindset causes people to persist in the face of setbacks, embrace challenges, and learn from criticism.

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The New York State Education Department discontinued support for the EngageNY.org website on July 7, 2022. The NYSED encourages educators to download any EngageNY content they wish to use in the future from our archive sites below.

All ELA and mathematics curriculum files will be available at the links below, and will remain free and licensed under the Creative Commons Attribution-NonCommercial-ShareAlike (CC BY-NC-SA) license.

Archived EngageNY Curriculum Files:

Grades 3-8 ELA and Mathematics Test Materials:
Questions regarding this change may be directed to the EngageNY Help Desk.

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The New York State Education Department discontinued support for the EngageNY.org website on July 7, 2022. The NYSED encourages educators to download any EngageNY content they wish to use in the future from our archive sites below.

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When schools started closing months ago, we heard two loud requests from teachers in our community. They wanted:

1. Written feedback for students.

2. Co-teacher access to student data.

Those sounded like unambiguously good ideas, whether schools were closed or not. Good pedagogy. Good technology. Good math. We made both.

Here is the new loudest request:

Self-checking activities. Especially card sorts.

hey @Desmos - is there a simple way for students to see their accuracy for a matching graph/eqn card sort? thank you!

Is there a way to make a @Desmos card sort self checking? #MTBoS #iteachmath #remotelearning

@Desmos to help with virtual learning, is there a way to make it that students cannot advance to the next slide until their cardsort is completed correctly?

Let's say you have students working on a card sort like this, matching graphs of web traffic pre- and post-coronavirus to the correct websites.

What kind of feedback would be most helpful for students here?

Feedback is supposed to change thinking. That's its job. Ideally it develops student thinking, but some feedback diminishes it. For example, Kluger and DeNisi (1996) found that one-third of feedback interventions decreased performance.

Butler (1986) found that grades were less effective feedback than comments at developing both student thinking and intrinsic motivation. When the feedback came in the form of grades and comments, the results were the same as if the teacher had returned grades alone. Grades tend to catch and keep student attention.

So we could give students a button that tells them they're right or wrong.

Resourceful teachers in our community have put together screens like this. Students press a button and see if their card sort is right or wrong.

My concerns:

1. If students find out that they're right, will they simply stop thinking about the card sort, even if they could benefit from more thinking?

2. If students find out that they're wrong, do they have enough information related to the task to help them do more than guess and check their way to their next answer?

For example, in this video, you can see a student move between a card sort and the self-check screen three times in 11 seconds. Is the student having three separate mathematical realizations during that interval . . . or just guessing and checking?

On another card sort, students click the "Check Work" button up to 10 times.

Instead we could tell students which card is the hardest for the class.

Our teacher dashboard will show teachers which card is hardest for students. I used the web traffic card sort last week when I taught Wendy Baty's eighth grade class online. After a few minutes of early work, I told the students that "Netflix" had been the hardest card for them to correctly group and then invited them to think about their sort again.

I suspect that students gave the Netflix card some extra thought (e.g., "How should I think about the maximum y-value in these cards? Is Netflix more popular than YouTube or the other way around?") even if they had matched the card correctly. I suspect this revelation helped every student develop their thinking more than if we simply told them their sort was right or wrong.

We could also make it easier for students to see and comment on each other's card sorts.

In this video, you can see Julie Reulbach and Christopher Danielson talking about their different sorts. I paired them up specifically because I knew their card sorts were different.

Christopher's sort is wrong, and I suspect he benefited more from their conversation than he would from hearing a computer tell him he's wrong.

Julie's sort is right, and I suspect she benefited more from explaining and defending her sort than she would from hearing a computer tell her she's right.

I suspect that conversations like theirs will also benefit students well beyond this particular card sort, helping them understand that "correctness" is something that's determined and justified by people, not just answer keys, and that mathematical authority is endowed in students, not just in adults and computers.

Teachers could create reaction videos.

In this video, Johanna Langill doesn't respond to every student's idea individually. Instead, she looks for themes in student thinking, celebrates them, then connects and responds to those themes.

I suspect that students will learn more from Johanna's holistic analysis of student work than they would an individualized grade of "right" or "wrong."

Our values are in conflict.

We want to build tools and curriculum for classes that actually exist, not for the classes of our imaginations or dreams. That's why we field test our work relentlessly. It's why we constantly shrink the amount of bandwidth our activities and tools require. It's why we lead our field in accessibility.

We also want students to know that there are lots of interesting ways to be right in math class, and that wrong answers are useful for learning. That's why we ask students to estimate, argue, notice, and wonder. It's why we have built so many tools for facilitating conversations in math class. It's also why we don't generally give students immediate feedback that their answers are "right" or "wrong." That kind of feedback often ends productive conversations before they begin.

But the classes that exist right now are hostile to the kinds of interactions we'd all like students to have with their teachers, with their classmates, and with math. Students are separated from one another by distance and time. Resources like attention, time, and technology are stretched. Mathematical conversations that were common in September are now impossible in May.

Our values are in conflict. It isn't clear to me how we'll resolve that conflict. Perhaps we'll decide the best feedback we can offer students is a computer telling them they're right or wrong, but I wanted to explore the alternatives first.

2020 May 25. The conversation continues at the Computation Layer Discourse Forum.

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When schools started closing months ago, we heard two loud requests from teachers in our community. They wanted: Written feedback for students. Co-teacher access to student data. Those sounded like unambiguously good ideas, whether schools were closed or not. Good pedagogy. Good technology. Good math. We made both.

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Dan Meyer Date Added,Last Updated,Task Title,Desmos Activity,Lesson Plan,Act 1,Act 2,Act 3,Sequel,Standard 1,Standard 2,Standard 3,Standard 4,Suggested Question,License Background Information 3/10/2017, World Recor...

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I train mathematics educators who want their students to be better problem solvers.

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Louis Agassiz, the famous Swiss biologist, placed a fish specimen on the table in front of his post-graduate student.

"That's only a sunfish," the student said.

"I know that," Agassiz replied.

He continued, "Write a description of it. Find out what you can without damaging the specimen. When I think that you have done the work I will question you." 1

The Power of Observation

The student wrote for an nearly an hour, until he felt confident that he knew nearly all there was to know about this particular fish.

Much to the student's frustration, however, Agassiz did not return to see him that day. His teacher did not come the next day either. Nor for the entire week that followed. Eventually, the student realized Agassiz's game: the teacher wanted him to observe the fish more deeply.

After nearly one hundreds hours of study, the student began to notice finer details that had escaped his vision previously: how the scales of the fish were shaped and the patterns they made, the placement of the teeth, the shape of each individual tooth, and so on. When his teacher finally returned and the student explained all that he had learned, Agassiz replied, "That's not right." And walked out of the room. 2

Shocked and angry at first, the student eventually recommitted to the task with new vigor. He threw out all of his previous notes. He studied the fish for 10 hours per day for an entire week. When he met with Agassiz a final time, the student had produced work that "astonished." 3

The Art of Comparing Objects

After his investigation of the sunfish Agassiz's student wrote, "I had learned the art of comparing objects." How does this tooth compare to the one next to it? How does this scale compare to the one on the opposite side? How does the symmetry of the bottom half of the fish compare to the top half?

The art of comparing objects is a remarkably useful strategy in many areas of life. Take weightlifting, for example.

For the first five years that I lifted weights, I experienced mediocre results at best. I assumed that it was information that held me back. Like many people, I thought that once I found the right workout routine, then I would be set. I was under the assumption that I simply hadn't reached the next level yet because I hadn't come across the right information. What I didn't realize was my search for the perfect pre-made formula was preventing me from observing my actual results.

When I started to observe with greater care and focus, I realized that my body tended to respond better to higher volume rather than higher intensity. I noticed that my foundational strength in major movements like the squat and deadlift was lacking. I was able to use these observational discoveries to tailor my training to my needs and, subsequently, make much greater strides because of it. It was through comparing what I was doing with what was actually working for me that I made progress.

Do the Work For Yourself

"I never pay attention to anything by 'experts.' I calculate everything myself."
-Richard Feynman

When Richard Feynman, the brilliant physicist, was working on a new theory of beta decay he noticed something surprising. For years, experts had been saying that beta decay occurred in a particular way, but when Feynman actually ran the experiments he kept getting a different result.

Eventually, Feynman investigated the original data that all of the expert's were basing their theory on and discovered that the study was flawed. For years, nobody had bothered to read or repeat the original study! All of the experts just kept quoting one another and used their mutual opinions as justification for the theory. Then Feynman came along and turned everything upside simply because he did the calculations himself. 4

Look, And See for Yourself

"Take the facts into your own hands; look, and see for yourself!"
-Louis Agassiz

Pick any industry of life and you'll find that very few people actually do the work.

Rather than read the original study, most people cite the headline from a secondary source. Rather than spend 100 hours observing every detail of a fish, most biology students would look up the description of the fish online. When most people say, "I read an article on climate change," what they really mean is, "I read the title of an article on climate change."

This is exactly why doing the boring work more consistently is actually a competitive advantage. Ignore the expert advice and pay attention to what gets results for you.

Look, and see for yourself.

Read Next

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Famous Biologist Louis Agassiz on Learning by Observation

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Louis Agassiz, the famous Swiss biologist, placed a fish specimen on the table in front of his post-graduate student. "That's only a sunfish," the student said. "I know that," Agassiz replied. He continued, "Write a description of it. Find out what you can without damaging the specimen.

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Figure This Math Challenges for Families - Challenge!

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Home · Getting Started · Answer · Try These · Think About This · Did You Know? · Resources Try Another Challenge · Challenge Index · Math Index · Printing the Challenges · En Español Family Corner · Teacher Corner · About Figure This! · Purchase the CD

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6 year old solves the sum of counting numbers from 1 to 25

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6 year old solves the sum of counting numbers from 1 to 25

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Intended Audience: Math teachers, prospective teachers, and parents (public, private, homeschool).

In this video, 6-year-old Autumn shows how to find the sum

1+2+3+4+...+24+25.

This problem actually has a rich history involving one of the greatest mathematicians who ever lived: Carl Friedrich Gauss. Please share Gauss' fascinating story with your children or students. The story goes like this: When Gauss was seven, his elementary instructor gave what the teacher thought was an extra-pointless exercise of adding up the first (say) 25 numbers just to keep his students busy for an hour. Almost immediately, young Gauss threw down his slate and declared, "There it lies." When the hour was up, the teacher inspected Gauss' answer and found it to be correct. An interesting article investigating the full story can be found here.

Autumn's solution to this question was probably very similar to how Gauss solved it. In the video, I mention that Autumn ran up stairs and came down 10 minutes later with the solution. Here is her actual work (click Autumns_Work for a .pdf version):

In this picture you can see her thinking: Pairing up 1+25, 2+24, 3+23, etc. and keeping track of which terms she summed on the line below (you can also see the left over 13 at the end). I think what happened next is that she recognized (in her mind) that

because she writes "same as 13 x 25." Regardless, you can see her calculating the

just as she did in the video, and you can see me checking her work underneath (and actually showing her another mental math technique-look!).

For Teachers who teach Eureka Math: This sum is called a finite series because it is a sum of a finite sequence of numbers. We explore this particular series in many different places in the high school curriculum, but a particularly interesting discussion around this series occurs in Lesson 8 of Module 3 of Grade 9 (Algebra I). In that discussion we show how the sum can be visualized as "triangles:"

-Grade 9, Module 3, Lesson 8 of Eureka Math/EngageNY

This visualization quickly leads (through pictures!) to the general formula of the sum of the first n positive integers:

Plugging 25 in for n shows that , which is what Autumn derived. If you are teaching Lesson 8 (or any of the other lessons where this series shows up), think about sharing this video of Autumn with your class as a way to stimulate a discussion with your students.

As always, comments are welcomed! In particular, we are trying out different places to do math. The math in this video was done on the hood of a Porsche 914. We have friends with all kinds of cool sports cars who may be willing to let us borrow them for an afternoon, so look for more "Math on the hood of sports cars" soon (especially if we get positive feedback to do more videos like this).

CHANNEL: Growing up with Eureka
© 2015 Autumn Baldridge and Scott Baldridge
Partially supported by NSF CAREER grant DMS-0748636

About Scott Baldridge

Distinguished Professor of Mathematics, LSU. Geometric topologist: gauge theory, exotic 4-manifolds, knot theory. Author: Elementary Mathematics for Teachers.

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Intended Audience: Math teachers, prospective teachers, and parents (public, private, homeschool). In this video, 6-year-old Autumn shows how to find the sum 1+2+3+4+...+24+25. This problem actually has a rich history involving one of the greatest mathematicians who ever lived: Carl Friedrich Gauss. Please share Gauss' fascinating story with your children or students.