# Fractions Day 5: Which are Equivalent?

Equivalence is everything when it comes to fractions. Kids who understand which fractions are the same size and how to create fractions that are the same size are the ones in math class that say, “It’s easy!”. It’s not easy; it makes sense to them. Since I am at home with my tiny human for A-WHILE, I decided we would spend a good grip of time on equivalence. The next 3 lessons (and many more after introducing the family of thirds in Lesson 8) will focus on understanding equivalence with fractions.

1. Play “Cover it Up!” twice. Please see Fractions Day 2 for how to play the game. My questions to Chris focused on when our amounts were the same, or how to make them the same. How much more would I need to tie you? Which fraction would you need to have the same amount as me? I still had him write the addition sentence once he covered up the whole.
2. I asked him to pull out one of his half pieces. How many eights would I need to cover up the half? (2). I then directed him to do the same with each fraction size. He had to cover up with only that size piece. We counted them out (One-eighth, two-eighths, three-eighths, 4-eighths…4/8 is the same as 1/2.) and wrote down each fraction amount that was equivalent to half. See image above for our work.
3. On the whiteboard, I wrote 3/4. How many eighths would it take to cover 3/4? How many eighths are the same as 3/4?
4. We continued looking at different fractions and finding the equivalent amount. The fraction sequence is below (See Equivalent Fractions Practice doc.), but looking at it now I would suggest less sixteenths and more fourths and eighths. Will need to do that tomorrow.
5. I let him create two of his own. This was not easy. He just kind of stared at me. (Soooo tough not to just tell him, but the struggle is good for him!) I rephrased: Choose a fraction and find another that is the same size. Though this limited him at first, it helped get the ball rolling. He chose 1=2/2 for his first one. Though not what I was looking for, it is true! He second was more interesting (Wish I had asked for 3!) He chose 2/1=4/2. I was shocked, because we haven’t even talked about anything over a whole. However, when concepts make sense to kids, they can naturally apply them to unique situations.

Hope you have a great day exploring new ideas and making fractions fun!

# Fractions Day 4: Which is Smaller?

I am a fan of repurposing (NEW WoRd!!!) problems. I figure, if you can read a book for different purposes, why not math problems? Plus, it makes planning that much easier for you! And right now we have enough on our plates!

1. Played “Cover it Up!” twice. Please see Fractions Day 2 post for how to play the game. The questions I used were focused on were:  Who has less? or Who is losing? How do you know? to front load for today’s lesson. NOTE: I still make him write the addition sentence at the end of each game to work on notation and such.
2. Using a whiteboard (or scratch paper), I drew the equal symbol (=). What does this mean? Today Chris said they were equal, or the same. We moved on.
3. Under that work, I drew the greater than symbol (>). What does this symbol mean? (The first number is bigger or greater than the second number.) Again, I had him choose two fraction pieces from the Fraction Kit to compare, with the first being bigger than the second. He actually did not choose the same ones as yesterday. We then wrote the number sentence that it represented.
4. I drew the greater than symbol (<). What does this symbol mean? (The first number is smaller or less than the second number.) I had him choose two fraction pieces from the Fraction Kit to compare, with the first being smaller than the second. He actually did not choose the same ones as yesterday. My little Sassy Sam just reversed the ones he had. We wrote the inequality and moved on.
5. I had Chris compare pairs of the unit fractions from the Fraction Kit and tell me which one was smaller and why.  They were the same ones I used yesterday. He didn’t notice!
6. We moved on to the fractions with different numerators (still only using the denominators from the Fraction Kit). I gave him (one at a time) pairs of fractions to compare. He could use the Fraction Kit pieces to determine which was smaller, circling it on the whiteboard. I asked him to convince me why one fraction was smaller than the other, and he verbally explained or showed me with his fraction pieces. See below for the sequence of pairs we explored (The red fractions are the smaller fractions.)I often asked, How many ________ would you need to make them equal?, just to start the seed of equivalent fractions (Day 5). I threw at him two unit fractions (1/2 and 1/7) to see if he could apply his understanding without always using the Fraction Kit pieces.

Happy Comparing!

# Fractions Day 3: Which is Bigger?

Science and a Birthday gift of slime/putty jars kept us busy and at a very quick math lesson today. ‘Cuz you know…priorities!

1. Played “Cover it Up!” twice. Please see Fractions Day 2 post for how to play the game. The questions I used were focused on were:  Who had more? or Who is winning? How do you know? to front load for today’s lesson.
2. Using a whiteboard (or scratch paper), I drew the equal symbol (=). What does this mean? (Chris said it meant they were the same size, which I was fine with.) Show me which pieces would be equal. He showed the 1 whole and 2/2. I then wrote 1=2/2.
3. Under that work, I drew the greater than symbol (>). What does this symbol mean? (Chris said it was an alligator. More on that another blog. I am not in the mood for that one!) We discussed that it meant the first number you write must be bigger than the second number. The math vocab wasn’t very sophisticated, as I just want him understanding the idea. I had him choose two fraction pieces from the Fraction Kit to compare, with the first being bigger than the second. We then wrote the number sentence that it represented.
4. I had Chris compare pairs of the unit fractions from the Fraction Kit and tell me which one was greater and why.  See the unit fractions to the right for sequence of the pairs we explored.
5. Once he had the idea of comparing we moved on to fractions with different numerators (still only using the denominators from the Fraction Kit). I gave him (one at a time) pairs of fractions to compare. He could use the Fraction Kit pieces to determine which was greater, circling it on the whiteboard. I asked him to convince me why one fraction was larger than the other, and he verbally explained or showed me with his fraction pieces. See below for the sequence of pairs we explored (The circled ones are the greater fractions.).I often asked, How many more of ____ would you need to make them equal?, just to start the seed of equivalent fractions (Day 5). Then I threw a pair of equivalent fractions in our set to see what he would do. (He rolled his eyes and said they were equal. Duh, Mom!)

Overall, Chris did well with circling which fraction was bigger, so long as he could use the pieces to work through the pairs. This is appropriate, as he hasn’t learned any other strategies for comparing. We will move to which fraction is smaller tomorrow to continue comparing sizes of fractions and really understanding what the numerator and denominator mean with respect to the Fraction Kit pieces.

# Changing Times Call for Changing Posts!

Hi Everyone,

So I started the year wanting to read professionally and discuss within this format. THEN we got put on lock down. So that is stopping NOW (for the time being) and I am altering the nature of my posts.

Two different types of blogs will be coming through this page. The first will be personal vignettes of life as we are all stuck together. I have a Senior in high school who is digging sleeping in and working in his pj’s, but ticked off he is missing his state Art competition and uncertain how AP exams will roll out. I have a second-grader with dyslexia and I am on the fast track to learn how to help him learn to read and write (while being a math-geek). So these types of blogs will be real-time and messy, ugly, funny truths.

The second type will be the math I am doing with my second grader around fractions (for now, but if we stay out FOREVER I will have to move on…sigh…I love fractions…). I will post our daily lessons and try and include pics and videos when applicable. If I used a worksheet or blackline masters, I will attach them at the end of each lesson. These will be lessons you can share with other teachers and parents if you so desire.

Though these blogs are now completely meant for me and my well-being, I hope you find some humor and help as needed in this crazy hot mess of a time.

Take care of yourselves!

Jen

# I Get By With A LOTTA Help From My Friends

So NOT the post I thought I would be writing next, but this has been weighing heavy on my heart…

I am blessed to have worked with several educators for consecutive years. This is a true gift, for as a consultant I could be hired and fired at anytime. I treasure the relationships I have with so many educators, as they are in THERE doing awesome work with children, yet still wanting to learn. They fuel my soul, and make me want to give my very all PLUS more when I collaborate with them.

Last week I was working with Special Education teachers and Paraprofessionals. This is unique, as the District feels strongly that their Paraprofessionals should get the same trainings (as they often do the one-on-one work with the kiddos). The elementary team is near and dear to my heart, since my own child has Dyslexia and I tutor many kiddos that have learning disabilities.

During the morning break, one of the teachers came to me and suggested what I would like to call a “duh” moment. Should have already been something I knew, but didn’t. She suggested that, instead of naming our kids as “Autistic Kids” or “Dyslexic Kids”, might I restate it as “Kids with Autism” or “Kids with Dyslexia”?

“The Disability isn’t what defines them.”

Damn.

Absolutely I should rethink my language. Absolutely I need to rethink how I refer to a certain group of students. ABSOLUTELY it does NOT define them!

I love that this amazing person came to me and called me out (privately). I love that I have friends that will tell me when I need to rethink and consider different options.

Consider your cadre of teacher friends. Do you feel like they will call you out when needed? Support you, yet let you know when you can do better? BE BETTER? If not, get a new group of teacher friends!

Thanks, Rachel. You make me want to be a better human.

# The Purpose of Hand Raising

Here we goooooo!!!! I have just started the book, Hacking Questions: 11 Answers That Create a Culture of Inquiry in Your Classroom, by Connie Hamilton. Below is my first    ah-ha moment.

Raise your hand if you have ever asked a question to the class and…

• The one hand (that always goes up) speeds through the air like Hermoine Granger’s in potions class…
• Several hands go up and now you must choose…
• No one’s hand goes up

Yeah, me too. And if you say you have never done this…

We have all done this. And, before reading the first chapter, I would still be doing this. It is what we did in school; it is how we have seen others questions students; it is how it has always been done. But just because that is our norm, should it be the norm for questioning students???

Let’s go back to the three hand-raising options. Here is what I have been thinking about and how this might roll in my class:

• The one hand (that always goes up) speeds through the air like Hermoine Granger’s in potions class… So now everyone else can breathe a sigh of relief as she answers. Or if I don’t let her answer, she gets all impatient and sighs vehemently.
• Several hands go up and now you must choose… I spend 4-5 precious minutes going through multiple responses and get some answers I wasn’t ready for and now I have to think on the fly about a positive way to respond.
• No one’s hand goes up…I answer for the class and basically take the learning away from them.

Here is what hit me: “Disengagement is the enemy of learning. We unintentionally create the conditions for disengagement when we allow students to keep their hands down”    (p. 21).

Say whaaaatttt??? THIS was not my intent when I ‘cold-called‘ (the term in the book for hand-raising) on students. These are my questions I want to reflect on for this week as I move forward with this slap-me-in-the-face realization. I invite you to think about them as well:

• What is my purpose for using ‘cold-calling’ in class?
• What is the purpose of questioning in my class? What do I want to get out of my questions?
• How am I asking students these questions? Are there ways in which I can ask questions and get more feedback and engagement?

I am hoping to get into some elementary math classes this week and reflect on these as I teach. I encourage you to comment below or email me your reflections. Let’s learn together! jen.moffettllc@gmail.com

# Lesson Study: Lines of Best Fit

I had the privilege of working with Integrated 1 teachers last week on statistics. We voiced our concern that students are often given the line of best fit for a data set rather than exploring how to create it themselves. We also wondered why their stats chapter was not right after exploring equations in slope-intercept form, as that how we wanted students to write the equation for the line they created.

Our goals for creating the lesson were as follows: 1) Students would review writing a linear equation in slope-intercept form from a graph, 2) Students would create lines of best fit for data plots on a graph, and 3) Students would write the equation for the line of best fit in slope-intercept form.

After searching several open resource sites, we settled on using some graphs from an earlier exploration to review writing equations of lines (please see the attached worksheet). We felt that these graphs would be easier for students to be able to read, and allowed students to determine the slope either by counting or by using their slope formula when given two points. We didn’t want the lesson to get stuck in this review section, so the cleaner the graphs were, the better. We strategically chose two with positive slopes, two with negative slopes, some with fraction values and one with a y-intercept of 0 so that we could frontload the necessary skills before moving to the new concept of creating lines of best fit.

The second component of the lesson was the Desmos activity, Fit Fights (see link below). We wanted to use the activity to build understanding for how lines of best fit are created. We used the following questions to drive the discussions:

• What do you notice?
• What do you think the gray lines represent?
• Do the data points make a line if we connect them? Is this a linear function? Why or why not?
• Do the data points gravitate towards making a line? Would it be a positive or negative (or zero or undefined for the graphs they created) slope?
• If we drew a line to ‘best fit’ our data points, where might it go?
• How can you tell if your line is a good fit for the data points?

The third component of the lesson was to relate the lines of best fit students created to writing equations of linear functions in slope-intercept form. We took snap shots of the lines of best fit from the Desmos activity, strategically choosing lines that had at least two whole number coordinates. We purposefully related the warm up to the lines of best fit, using similar language and questioning.

### Our Take-Aways

• Our students are successful in determining the slope by counting rise/run, but need to continue to work on using the slope formula when given two points.
• The Desmos activity only had negative slope lines of best fit. Consider having students create data points that could have lines of best fit that have other correlations so they don’t overgeneralize and think it will always be a negative correlation.
• Students struggle when the axis are scaled differently. Need to build in some warm-ups that continue to explore differently scaled axes.
• We need to be clear in our questioning. Using questions such as, “Are we good? Do you get it?, etc.” does not give us formative information to move forward.

Below are some of the students’ work and what they felt they had learned. Overall, students were more engaged with this lesson than they have been in the past when learning this concept. We definitely want to use this lesson again, and hope you find it helpful as well!

Lesson Study Worksheet (Shout out to Fontana High School!): Stats Lesson Study Worksheet

# Thinking Rationally With Your Tween

Students typically start exploring positive and negative numbers towards the end of elementary or early middle/junior high school. And it can be a bit weird. It is literally the opposite of what they had been taught for the last ten years in a number of ways (Puns intended.). Here are some ways you can support your child in their negative number journey to make it a positive experience (Dang, I am on FIRE!).

1. Bring in finances. “In the Red” and “Black Friday” are references to business. When a company in “in the red”, they are in debt. They owe money. Traditionally “Black Friday” (The Friday after Thanksgiving) was the first day of the fiscal year companies got out of the red and posted a growth. Credit cards are another great place to explore debt and credit. A debt would be what you owe and would be represented as a negative number. A credit would be a positive number. Show them your mortgage and credit card statements and discuss terms such as “deposit, credit, debt, with drawl, etc.”. The stock market is a great place to discuss negative and positive fractions and decimals. Pull up the daily NYSE and discuss which companies have an increase (or positive) and which have a decrease (or negative) change that day.
2. Football Season!!! This is a fantastic place to bring in integers (positive whole numbers and their opposites). If I gain six yards on a drive, how could I represent that change? (+6 or 6). What if the QB gets sacked? How many yards did they lose? (ex: -7). How far do they now have to go to get a TD? If your child is interested in football, use it to your advantage!
3. Playing Cards. Below are a couple of games you can adapt to include negative numbers. I prefer to omit the face cards and only use numbered cards, but you can make Aces = 1 (and -1) and the face cards values after 10 (and -10).

War! Black Cards are positive values; Red Cards are negative values

1. Shuffle and divide the cards evenly among players. Keep your cards in a pile face down. Everyone flips over their first card. Player with the greatest value wins all the cards for that round. Tie? Flip another card and whoever has the greatest value that round wins all the cards from both rounds. Whoever has all of the cards at the end (or the most cards when you get bored) is the winner.

Example: I flip over a red 9 (-9) and you flip over a black 2 (2 or +2). A gain of 2 is greater than a loss of 9 so you win the cards.

You can also play that the winner is the one with the smallest value.

1. For students who are learning to add integers. Shuffle and divide the cards evenly among players. Keep your cards in a pile face down.

Everyone flips over TWO cards and finds the sum (add them). Whoever has the greatest (or least) sum wins the cards for that round.

Example: I have a black 2 and a red 4. 2 + (-4) = -2.

You have a red 4 and a red 2. -4 + (-2)=-6.

Since -2  is greater than -6 (a loss of 2 is better than a loss of 6), I would win.

1. Go Fish! Black Cards are positive values; Red Cards are negative values

Shuffle and hand each player 7 cards. The rest are in a pile in the middle face down.  The objective is to be the first one out of cards.

How do you get rid of cards? By making matches of cards that have a value of 0.

Example: Jen has a 5 black (5 or +5). She says to Chris, “Do you have a negative 5 (5 red)?” Chris does, and hands the 5 red (or -5) to Jen. Jen takes the positive 5 and the negative 5 and lays the pair in front of her.

This is not an exhaustive list so I will be adding other fun ways to integrate math into your home conversations. Let’s make math a positive experience for our kiddos!

# Creating Structure for Context in Math

I was honored to facilitate lesson study with IM1 teachers today. Their students are struggling (due to high EL/SPED population) with solving word problems. I dug deeper, and we decided the struggle is really the first step: creating equations from situations.

We decided our goal as educators this year is to work on teacher clarity: making our lessons streamlined and very goal-oriented. If we know our goals for the lesson, then every move we make (every breath we take…) is for the goal. So how do we clarify translating context to equations?

We started from the end: the benchmark. We took a problem the students struggled with, and tweaked it several times, each time only altering only one component. Students had to work from the original version (which we used simple numbers to keep it accessible) for each new “version”. They discussed what changed from situation to situation and how that affected the prior equation.

Version 1: Troy works for an ice cream cart vendor. He receives \$10 for taking the cart out for a shift, plus a commission of \$2.00 for each item he sells. Troy worked a shift Saturday and earned \$60.  How many items did he sell?

Version 2: Troy works for an ice cream cart vendor. He receives \$15 for taking the cart out for a shift, plus a commission of \$2.00 for each item he sells. Troy worked a shift Saturday and earned \$60. How many items did he sell?

Version 3: Troy works for an ice cream cart vendor. He receives \$15 for taking the cart out for a shift, plus a commission of \$1.25 for each item he sells. Troy worked a shift Saturday and earned \$60. How many items did he sell?

Version 4: Troy works for an ice cream cart vendor. He receives \$25 for taking the cart out for a shift, plus a commission of \$0.10 for each item he sells. Troy worked a shift Saturday and earned \$52.90. How many items did he sell? (Problem from the benchmark.)

We used 3 scenarios. In each, we kept our questions as consistent as possible (again, clarity):

• Which part is varying (changing)? How do you know?
• Which quantity would be the coefficient? How do you know?
• Which quantity would be the constant? How do you know?
• (From version to version) What has stayed the same? What changed? How does the changed quantity affect our equation? Why?

Students were engaged, writing on their tables and willing to discuss with each other. They had many moments of “ohhhhhh” and “oops!” and learned quite a bit about the components of 2-step equations. They definitely need more time, and the teachers have committed to continuing the work as warm-ups or on modified days.

Oh! And did I mention this was a co-taught Special Ed class, with many English Learners?! Amazing!

So our major takeaways were:

2. Keep your goal in mind when creating the tasks/lesson and questions for clarity and focus.
3. Breaking the situations into translating and solving (working on a single component) allows students to focus and interpret.

Below is our ppt. Hope it is useful! Happy Math-ing!

Linear Equations in Context LS 8.27.19

# How Do Our Beliefs in Math Affect Our Students?

I was honored to work with amazing teachers this week. We took a survey from NCTM (National Council Teachers of Mathematics) on our beliefs regarding student learning and our instructional practices in mathematics. This, in itself, led to amazing discussions about what we truly believe math IS and how we interpret that into instructional decisions within our classrooms.

But then we took it further. We got into groups and discussed not so much whether we agreed or disagreed, but whether it was a productive or unproductive belief in respect to student access and learning.  Here are two to consider:

There were fantastic discussions about these particular ones, especially for educators of EL and SPED. We also considered how parents might respond to these. Powerful conversations around access, flexibility in thinking, understanding conceptual and procedural mathematical ideas, and yes, fluency.

Here was the point. Our beliefs, whether productive or unproductive, affect our attitudes towards mathematics and the children we are blessed to teach. Those attitudes affect the actions we take. Who gets to answer which questions? Who gets the “tough” tasks and who has to keep doing drill and kill worksheets? Who gets to explore puzzles and who has to retake tests or do homework (because their home life doesn’t lend itself to being able to do it at home)? And those actions MATTER. They affect the results you will get from your students.

So as you gear up for this school year, consider taking the beliefs survey yourself. Even better, have your team take it and REALLY dive in to what beliefs are productive an unproductive. The more we reflect, the more we can grow and be effective at what we truly want; to teach students to love, learn and understand mathematics. Have a great year!