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 deflines of best fit 1initely want to use this lesson again, and hope you find it helpful as well!

lines of best fit 2

 

 

 

 

 

 

Desmos Fit Fights: https://teacher.desmos.com/activitybuilder/custom/5c5bc938531af975828d9e92

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

 

Lesson Study: Lines of Best Fit

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!

Thinking Rationally With Your Tween

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?IMG_8202

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.)IMG_8205

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?IMG_8208

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: IMG_8207

  1. Know your goal!
  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

 

 

Creating Structure for Context in Math

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.

Belief chart

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!

For the beliefs survey: 2017_06_19_Holstrom_Grady_2BeliefsSurvey

 

How Do Our Beliefs in Math Affect Our Students?

Equations and “Flow Charts”

A group of seventh grade teachers and I were trying to figure out how to move from concrete representations of solving equations (some used chips/cups and some used tape diagrams) to the more symbolic procedural (traditional) representation. While students were able to model the “moves” with the concrete, some still struggled to move from that to solving on paper.

I had recalled a method a dear colleague, Bruce Grip, had shown me years ago using a flow chart. We decided to try it out ourselves.

Starting with expressions, we discussed what the “moves” are when simplifying. Order of operations made a showing, and we moved through the flow chart. We decided this was a valuable use of time, as it built understanding of the structure of numeric expressions and fluency with integers (Which, let’s be honest; they need LOTS of practice with!). IMG_1854   IMG_1856

IMG_1857

From there, we decided to bust out a single step equation. We started the same way we did with expressions, using x as our starting value. “What moves am I making to x in this equation?” We then built our flow chart. HOWEVER, rather than simplifying (as in the expressions), we know the value we want to get. So the flow chart looks like this:

IMG_1858

To solve for the value of x, we need to work backIMG_1859wards through our flow chart. If I had added 2 to a value to get -5, then I need to subtract that 2 to figure out what I started with. We could then parallel the flow chart with the more traditional algorithm for the students.

Below are several of our examples, limited to the structures seventh grade explores for CCSS.

We also explored some “messier” problems, as shown here.IMG_1863 It illustrates the fluency with the distributive property piece of “When do I need to distribute and when is it efficient to divide out the factor first?”. We liked that the students could show both ways and determine which route to take.

 

 

Our big commitments to this flow chart method:

  1. Start with the concrete/visual. This is not a substitute for chips/cups nor the tape diagram. This is the next step for students who need it.
  2. Next year, use the flow chart when exploring simplifying expressions so we can build on that understanding for solving equations.
  3. Use friendly numbers (NUMBER CHOICES MATTER!!!) first to build understanding.
  4. Bring in some messier problems to seal the deal and discuss different moves they can make based on the given numbers in the equation.
Equations and “Flow Charts”

Number Choices Matter

I had the privilege to work with fourth and fifth grade teachers this week. We explored multiplicative comparison problems in fourth and division with fifth (more on those in another blog). What I came away with is this: NUMBER CHOICES MATTER.

We don’t get much choice on the concepts we teach, nor often on the program we have to use. But we do get to choose what numbers we use with children. This could make all the difference for kiddos. If we choose our numbers wisely, we can build understanding through the patterns they see, the differences that appear, and talk about why those differences happened.

For example, consider the following set of problems:division setWhat is the same about each? If I am thinking about this as a partitive model and using base 10 blocks, I am sorting the amount I am given into 3 equal groups each time.

What is different about each? The amount I am giving to each of the 3 groups.

In the first example, 36 can be created by using 3-tens and 6-ones, and each can be fairly shared without any problems. Each would get a ten and two ones, or 12.

In the second example, I can still give out a ten to each of the 3 groups, but now I have to figure out what to do with the leftover ten and eight ones. This one builds off the first example, but pushes students to think about exchanging (regrouping).

The final example builds off the 48, but leaves 2 left that students have to consider. This allows to have a conversation about remainders.

These three build understanding of division, regrouping and remainders through strategically chosen problems to build from one to the next. Students have something to grasp on to when negotiating meaning with this tough tough subject.

So where can you build understanding through your number choices? I challenge you to think about what you want your students to learn next week and how your number choices can contribute to students understanding those goals!!!!

 

 

Number Choices Matter

Road Trippin’: Math Games for the Car

Special thanks for requesting this, Jen Duley!

Many of us are taking summer road trips with our tiny humans. Here are some ideas for keeping math in their brains and ditching the ‘summer slump’.

Counting Circle (sort-of)

This is something I have blogged about before. Kids must practice rote counting. Count up and down. Each person in the car takes a turn, counting by what the designated amount is. Below are a few ideas that I hope your kids will enjoy (And save your sanity!).

  1. Count by 1’s, first starting at 1, then building to start at a different value. Count up and down!
  2. Count by 2’s, 3’s or 5’s. Again, start with the value and practice skip counting, then start with different values. (Don’t forget to go backwards too!) One of the best things I ever did was make my clock in the car off by 5 min (too fast). My oldest had to figure out the time every time he got in the car. He had to regroup in his head almost every time!
  3. Count by 10’s, first starting at 10 to 100, then back down. Also start with different multiples of 10’s, different values other than tens, etc. Example: Start at 12 and count up by 10’s. Or start with 87 and count down by 10’s. (Super important for regrouping and subtraction!)
  4. Older kiddos: count by fractions. Start at 0 and add 1/2 each time. Start at 3 and count back by 1/3 each time (Gearing up for mixed number subtraction). Start at 1 2/3 and count up by 1/6.
  5. Older kiddos: count by integers. Start at 0 and add -2. You get the idea.

Guess My Number

Again, one that I have previously discussed, but super important.

  • I am thinking of a number between 11 and 13. What is my number?
  • I am thinking of a number that is less than 40 but greater than 35. It is odd. What is my number?
  • I am thinking of a number that is less than 100 and a multiple of 5. Now let them start asking yes/no questions to narrow it down.
  • I am thinking of a number between 11 and 12. Start them on fractions!!!!

Count the Cars

Choose a color, type, make, or model. Kiddos count all of that category of vehicles. Each child can count a different type (EX: Ev counts all the blue vehicles and Chris counts all the red ones) and whoever gets the most when you park wins!

Find the Number

Print out a 100 chart. Put it in a sheet protector and clip with a clipboard. With a dry erase pen (I attach the pen with yarn to the clipboard.) he/she crosses out every number he/she sees. Look at license plates, signs, billboards, etc. See who can cross out the most in a trip! You can also print out a partially filled in chart and they have to fill in the missing numbers before playing. For PDF 100 Charts: https://www.homeschoolmath.net/worksheets/number-charts.php

Three in a Row

Two options: print out a Three in a Row page for each child, put in sheet protector, and clip to clip board. Or print the blank, and allow them to fill in values 1-10 (they will have 1 missing since there are only 9 spots for 10 numbers).

Call out either addition or subtraction problems. You can just do number problems or put them as word problems. Example: Chris has 3 Stormtroopers. If he loses one in his car seat, how many does he have left? 3-1=2, so they cross off the 2 on their game board (if they have a 2). Once they get 3 in a row, they win! Link below for PDFs created for you.

3 in a Row

Target Value

Print out the Target Value sheet and put in the sheet protector. Clip to clipboard.

Give your child a target value (EX: 10). They write 10 in their Target Value box. They create as many addition problems that add up to 10. I honestly would rather just have the target box and let them make all kinds of equations (such as 1 + 4 + 5 or 20-10), but below is an example you are free to use.

Target Value

Shape Spotting

Great idea from my amazing friend and colleague, Kelli Wasserman! I found geometry cheat sheets if they need it. https://www.math-salamanders.com/geometry-cheat-sheet.html   Just put it in the handy-dandy sheet protector! Have your kids see who can spot each shape (square, rectangle, circle, etc) and they can cross it out on the sheet with a dry erase pen. Or, give them a shape to spot, and see who can spot the most. Love it!

 

Road Trippin’: Math Games for the Car