# data[].attributes.body
When students were prompted to reach $2 using different combinations, I then asked, "How did you figure out a way of reaching $2?" I was surprised at the variety of answers, such as a student who reasoned that a quarter was one of the larger coins, so they thought how many quarters they could use and then make another combination of mostly quarters and then used dimes and nickels for the rest. Many students said they divided $2 by the value of a coin. Some students said they added coins until they reached $2 and one student said she thought about how to make $1 and then doubled it.
Another question I asked was "How can you show the data from a graph in a table?" and students responded by using the points, using the points on the line, using ordered pairs, using x and y values, putting one variable in the first column and another variable in the other, and using the same equation that (to find solutions) that was used to make the graph.
A third question I asked was, "How can you tell from a graph when a point is a solution to two equations?" Responses I received were "when two trend lines cross each other", when the lines meet at a certain point, when the lines intersect, when the lines "match up", when the lines cross paths, when the lines go through the same point, and when the lines have a common value. These responses reiterated that even as I look for a certain response, there are several different ways to explain or word that response.

# data[].attributes.body
"I have $2 in my pocket. What might be in my pocket? What else might be in my pocket?" How did you figure out a way to reach $2? This was designed to elicit thinking (share information/explanation of their representations). Students came up with many different ways to reach $2 and then took turns sharing how they got to $2.  i.e. "I figured out a way of making 2 dollars by remembering how much each currency is then adding it up into $2." "I figure it out because 1+1=2 and each dollar has 4 quarters in $1 so 4+4=8."  pennies 200-100 pennies = $1 so times 2 8quarters- 4 quarters =$1 so I just times 2."

For connections to representations, we asked students how they can show the data from a system in a a data table (make sense of different ways to connect representations). S: "By putting the time in minutes on the left and number of completed signs on the right. Then putting the information you got from the graph into the table."  Some students make a data table and filled it out.  S- "You can show the data in the table by seeing how much they each did around the same time or different times or when they did the same amount of signs." S-" You write in some of the minutes and signs that they have done."

Essential Understanding:

1. *How can you tell from a graph when a point is a solution to two equations?

1. *How can you tell from a table when there is a common solution for two equations?

Not all students got to these questions due to time. Here are snippets from those who did
S: In a graph, it can be when two lines intersect at the same point
In a table when two numbers are the same during the same time or whatever the factor is.
S: 1. both lines should lead up to that singular point meaning they have the same value
2. if both sides of the table are equal
S: -it's the point where the 2 lines meet

# data[].attributes.body
students were given information about functional relationships in tables and graphs and asked to think about how quickly the output of a function changes and interpret what it means in the context of the situation.

I asked students, " Which representation, a table or a graph, can give us a better sense  of the general trend of a functional relationship over a certain interval?"
Students said a graph shows it better.
I asked Why?
Because in the graph you can see the slope and if it is increasing or decreasinG.
I asked, "when we looked at the temperature change over certain time intervals what else did you notice about the rate of change from the graph?
Students said, "When the temperature was getting colder faster, the slope was steeper

# data[].attributes.body
Lesson 7 >Using graphs to find average rate of change
throughout the lesson students were given information about functional relationships in graphs

# data[].attributes.headline
Unit 4 Lesson 6: Increasing and Decreasing

# data[].attributes.subject
Unit 4 Lesson 6: Increasing and Decreasing

# data[].attributes.body
During the lesson the students had correctly established that percent means out of one hundred. They were also familiar with tape diagrams from the previous lesson and were able to create tape diagrams of  fractions that were more or less in given situations.

The learning targets are; a.  I can draw a tape diagram that represents a percent increase or decrease. b. When I know a starting amount and the percent increase or decrease, I can find the new amount.

Students were placed in groups of 3. They were given the problem below.

A cereal box says that now it contains 20% more. Originally, it came with 18.5 ounces of cereal. How much cereal does the box come with now?

This was one of the questions that was asked in the lesson; What percentage of the original amount of cereal is the new amount of cereal?

After students tried to solve the problem they were are asked:

What strategies did you use to arrive at your answer?

There were multiple responses: A couple students said that they multiplied 1.20 by 18.5 to get their answer. I then asked where did the 1.20 come from? Some students said that it came from the 120% increase ( 120% as a decimal)

Students said that they found 20% of 18.5 then add what they got to 18.5

Two students said that they divided 100 by 20  to get 5 then 18.5 by 5 then add the result to 18.5.

One student said that the answer had to be greater than 18.5 as the percent was greater than 100

Student were then guided into creating tape diagrams to illustrative their answers.

# data[].attributes.headline
Alg 1, Unit 4, 8.2 (Flag Raising Part 1)

# data[].attributes.subject
Alg 1, Unit 4, 8.2 (Flag Raising Part 1)

# data[].attributes.body
The basic premise is that students had to interpret different graphs and decide which was most and least realistic in the context of a flag-raising ceremony.

Students were working on groups for this, but these were some of the questions I asked and responses I heard as I visited different groups.

Why do you say graph A is most realistic?

A student replied that it was going up at a constant rate. Another student replied that it means it COULDN'T be most realistic because no one can raise a flag perfectly like that.

Why do you say graph E is most realistic?

Many students mentioned that if you're raising a flag in real life you're going to pause or slow down as you grab to pull the rope down more. Some students mentioned that most of the others started from a height of 0, and a flag would never touch the ground.

Which graph do you think is least realistic?

Student responses varied, many said D or B.

Why do you think that?

Students who chose D said that it would mean someone was frantically jerking the flag up and down. Students who said B said it would mean the flag just appeared at that height without being raised.

Remember that we don't know what the scale on the x-axis is. Could there be a scale that would make graph D more realistic?

Thoughtful faces. A couple of students figured out that if the scale were days, then the graph would be showing the flag being raised and lowered once each day, which could be more realistic. Thinking about the scale in this way also got some students thinking that maybe graph A could be more realistic than they thought if the scale was very small. They pointed out that maybe if the scale was in milliseconds it would be possible to have a moment where there was a constant rate of change.

What's the story graph C is telling us?

Students said it would mean it got raised slowly, and then someone paused, and then it got raised quickly. One student suggested maybe someone who wasn't so strong started raising the flag, and then he got tired and so someone who was "really buff" took over.

What's the story graph F is telling us?

Students said it got raised, and then lowered halfway. Some students suggested that maybe partway through the day someone important died and so the flag was brought to half staff.

Can this (a different graph with a vertical line) represent the height and time of the flag?

Students agreed right away that it could not, but had different levels of understanding about why. Some said it would move "too quickly" or that it "went instantly to 6" or that it didn't make sense that it was starting at a time of 3. But they built off of each other's responses to get to the understanding that it wasn't just showing an instant output of 6, it was showing that the flag was in infinite locations at once and that would not make sense or be a function.

Sorry I wrote too much. This lesson was a lot of fun, and I was enjoying remembering it.

# data[].attributes.body
The learning target for this lesson is " I understand and can solve problems about commission, interest, markups and discount." In the previous lessons, students worked on percent increase and decrease using tables, double number lines, tape diagrams and equations.

Throughout the lesson, students practiced applying their knowledge of percentages to find commission, retail price, mark ups and markdowns. At the end I asked three questions to conclude the lesson:

1.) What are some situations in life in which people encounter percentages?

-Most students said car dealerships, stores, taxes, tipping at a restaurant, phone battery, grades, etc.

2.) When an item is marked down 10%, why does it make sense to multiply by 0.9?

-Some students said that if we multiply by a number less than 1, the price will go down, which shows that there is a markdown but then got stuck when I asked where the 0.9 came from. I then said, "Instead of finding 10% and then subtracting from the whole, we can take our 100% (or our whole) and subtract 10% (the discount) and what are we left with? 90%." Then students were able to see that 90% of a number is equivalent to a 10% discount.

3.) When an item is marked up 25%, why does it make sense to multiply by 1.25.

-Most students were able to say that 1.25 is greater than 1 so that means there is a markup. 1 represents the whole (100%) and .25 represents the 25% increase.

I was surprised with how well students did with this lesson! We talked about real life situations and had great conversation!

# data[].attributes.headline
6.3.7 Equivalent Ratios Have the Same Unit Rates

# data[].attributes.subject
6.3.7 Equivalent Ratios Have the Same Unit Rates

# data[].attributes.body
We spent a lot of time working with ratios in the previous unit. In this lesson the students were working with prices of items to show them that equivalent ratios have the same unit rate.  We had discussed unit rate previously and practiced finding the unit rate of items to identify which was the "better buy", but had not made the connection between unit rate and equivalent ratios.

When the students were given a table I allowed them a few minutes to think independently for the following questions:

1) Two burritos cost $14. Complete the table to show the cost for 4, 5, and 10 burritos at that rate. Next, find the cost for a single burrito in each case.

2) What do you notice about the values in this table?

Some students filled in the table while others sat silently.

I then broke them up into break out rooms to discuss with a partner their observations about the values in the table.

Majority of the students were able to complete the beginning of the table correctly but had trouble applying the steps to use a variable.  Since we had not discussed expressions and equations yet, this was very abstract for students and they were very confused.  They wanted to put a number in for the letter and continue with what they were doing above.

We discussed as a class what the letter represented. I then asked them what they did for each one of the items above, instead of putting a number in for the value that changed I showed them how we could put the variable in. I felt like they spent most of their time trying to figure out this question.  I took note of this for when we move into the expression and equation unit.

When I put them into break out rooms some classes did an excellent job, but I still have the challenge of encouraging verbal communication.  The same students consistently participate and share their ideas while others do not add to the conversation even when they are called on or prompted.

# data[].attributes.body
We are studying various strategies for how to solve linear equations with one variable (on both sides). Students were shown several equations, asked, "Which would you rather solve?" and to select three equations that appeared the most difficult and three that appeared to be the least difficult to solve. Most students in 5 out of 6 classes selected equations A, C, and F and I followed up by asking what those equations all have in common and students recognized that they all had fractions. Some students stated that the fractions were the reason those appeared most difficult. What I noticed about the class that did NOT choose those equations is that each student did not appear to have a reason for why they selected certain problems. For example, students chose a problem with a fraction, a problem in which you needed to distribute or factor on both sides, and a problem that had parentheses on just one side of the equation. That information told me that these students were really struggling with solving equations in general, did not know what to do first, and could not differentiate among the problems.

After discussing various strategies, (such as making predictions, multiplying an equation by the LCD to eliminate the fractions, factoring as a means to eliminate parentheses, etc.) students were asked,

Were there any equations that were more difficult to solve than you expected? Which ones and why?

Were there any that were less difficult to solve than you expected? Which ones and why?

the majority of student responses referred to the problems with fractions such as, "[Problem] C seems less difficult because you just had to [multiply] then you did it like a normal problem." Although some students stated that the equations still seemed just as difficult because there were so many steps, I was pleased that the majority took the viewpoint like the one quoted above.

# data[].attributes.body
We are in Unit 4 on Solving Equations and Systems. The learning target is "I can solve linear equations in one variable." The warm up has students finding the value of x and the perimeter of a triangle and square with expressions as side lengths. Some of the equations being asked are: "What expression represents the perimeter of the triangle? What was your strategy in making an equation? What does x mean in the situation? Looking at the figures, are there any values that x could not be? Explain your reasoning." The goal of this lesson is to get students really thinking about what values of x could make an equation true. The purpose of the activity is to shift the focus from solving an equation to thinking about what it means for a number to be a solution to an equation. Can x be positive, negative, or zero?

# data[].attributes.body
We had just started our Unit 4 lesson that focuses on proportional relationships and percentages. The learning objective for this lesson was "I can choose and create diagrams to help me solve problems about percentages." Knowing that we had not covered percentages a lot in 6th grade the previous academic year, I decided to add two slides that asked the following questions.
1) What is a percentage?
2) When do we use percentages?
The purpose of the first question was for students to not only think about what a percentage was but also for me, as the teacher to see how much knowledge students had on percentages. The second question was used for students to connect percentages to the real world. I noticed that one students noticed how much they used percentages in their real life (ex: their phone batter), they were able to explain what a percentage was. We went back to the first question where students were able to use their real like example to better explain what a percentage is and were able to have deeper conversations on what a percentage was.