Directions: Using the Integers 0 to 9, each only once, how many different ways can you fill in the blanks to make the statement true?

# equivalent equations

# Open Middle Problem- Open Number Sentence

Directions: Using the Integers 0 to 9, each only once, how many different ways can you fill in the blanks to make the statement true?

# 2 + __ = __ + __

# 40 Yard Dash -3Act Math

**ACT 1:**

This video was first brought to my attention from Dan Meyer’s My Opening Keynote for CUE 2014. Turn the volume on mute when you show this to your students. I also do not show the individual run, I start the video 15 seconds in. I show the clips of Jacoby Ford and Terrence Cody ending at a minute in.

*What questions do you have when you watch this video?*

Ask students to write down their questions, I normally ask students to find at least 3. When I observe that most students have questions written, I ask them to share those questions with their neighbor. I then throw up a Microsoft Word document and start typing down questions students supply. Students from my classroom came up with all sorts of different questions, some we can easily answer and others that we can’t. I am looking for a key question or questions to start this lesson. If students do not ask one of these questions, I tell them that I hope I can answer most of the questions provided, but that I need them to consider one of these questions first.

*How fast are they running?**How much of a lead does Rich get on run 2? run 3?**How much does Rich lose by each time?**How big of a lead does Rich need to tie? to win?*

Any of these type of questions will lead students down the inquiry I hope to explore with them.

**ACT 2:**

Once again, this video can create a few different paths of exploration. We can explore:

- The rate of the runners
- Graphs of the runners
- Equivalent equations

These are all excellent topics and students generate a lot of classroom discourse discussing each one.

- Rates: This is one that creates a lot of arguments about precision. Students normally start trying to time Rich by using the clock on the wall, or their wrist watch. Some will break out the stopwatch feature. I have other students use the watch feature on their phones. Timing issues like accuracy starting or stopping the time, cause quit a disturbance with the students.
- Graphs: I love this part. I normally show a clip twice and have students graph the race. Independent and Dependent variables, scale factor on axis, and the solution of two lines are great topics to discuss. Students really enjoy graphing the races and are really good at evaluating work and refining the process and answer.
- Equations: This normally involves at least one of the first two processes and builds upon that. Finding the rate of each runner (their slope) and setting their expressions equal to each other leads to when Rich is overtaken. Having these expressions will also allow us to find the exact time Rich is passed and how much of a head start he would need to either tie or finish first. I can’t think of a better introduction into solving systems of equations.

**ACT 3:**

I normally show the opening video to answer how fast Rich runs, and turn up the volume to allow students to know how much of a head start he is given in the other races.

**Extensions:**

Show them the 3 man race (1:10 into the video) and let them loose, it’s fun to watch.

# Using Pictures to Solve Compound Fractions

Today I find that Bridging the Gap stole my thunder (sort of) by posting on a method I have been using in class recently. We have been looking at compound fractions when writing equivalent forms of equations. Changing a standard form equation such as 2x + 1/2y = 4 to slope-intercept has really thrown a curve to my students, and it’s all about one concept- fractions. My students admit they struggle with fractions and then to create a compound fraction? Most just put the pencil down, worry about other things and try to nap in class. I have tackled the problem by using pictures of fractions to help students visualize what is going on.

I first start with an improper fraction example:

We work on dialogue on what 4/2 means, and students came up with the wording you now see on the board. They then draw 4 circles and split them into groups of 2.

We then look at an example when there is a fraction in the denominator:

Students once again write their wording to the fraction and attempt to complete it as instructed. I have had students struggle on how to make groups of 1/2, but when you ask them to draw 1/2 they easily give you a semi-circle. They they have that *aha!* moment and quickly draw lines bisecting the four wholes. They created eight 1/2 pictures.

Students really grasp the first two examples quickly, and we then throw them the curve, a fraction in the numerator:

Many students automatically say they can’t split a half of a circle into a group of four, so I will go back and change the fraction to a whole number, 4. they quickly draw four lines between the wholes and say that this problem is nothing like the one I asked them. I have them explain their process to find 4/4, and then write it as 1/4. Some students will see that you can cut the circle into four pieces, but others will still struggle- and this is where as an instructor I have to realize that inadvertently I gave them a specific context in which to look at creating groups. I gave them a circle, so I ask the students if we need to start with a circle, and many say no. They draw a rectangle on the board and are able to make it into a group of four:

Once they see this, they quickly come to the idea that they can cut the half into four pieces with either a rectangle or circular representation. They identify the new pieces as eighths.

We do a few other problems where students practice with both cases, and then I ask students to find a pattern. Although they do not tell me to multiply by the reciprocal, students understand what numbers to combine and why they are doing so.

The next step, of course is to extend this concept beyond the unit fraction…