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# Introducing Problem 1.1

Presented by Cynthia Callard & Jennifer Kruger

Cynthia and Jennifer discuss a variety of purposes and possibilities for the use of Problem 1.1 and how to get the most “bang for your buck,” including how Problem 1.1 can be used as a formative assessment tool.

## Problem 1.1 from Grade 6 CMP Units

### Prime Time

Students play the Factor Game and develop strategies for winning the game. They use their knowledge of finding factors and proper factors to develop these strategies.

### Comparing Bits and Pieces

Students analyze a variety of comparison statements about fundraising goals. They are given seven different comparison statements and have to decide whether each claim is true or false, and then explain their reasoning. Students are, also, asked to write their own comparison statements about the given fundraising goals.

### Let’s Be Rational

Students play the Getting Close game that requires the use of estimation by using benchmark numbers. After playing the game, students are given fractions and decimals and have to answer a variety of questions around estimating sums with the given fractions and decimals.

### Covering and Surrounding

Students design bumper car rides with given area and/or perimeter constraints. After they make designs, they, then, analyze four designs and answer a series of questions based on area and perimeter of the four given designs. The problem ends with students exploring the area and perimeter of rectangles, and developing formulas for area and perimeter of a rectangle.

### Decimal Ops

Students identify operations needed to solve given word problems. They explain how they decided upon the chosen operations, and then they write number sentences and solve the problems.

### Variables and Patterns

Students conduct an experiment to collect data on how many jumping jacks a classmate can complete in two minutes. After collecting data, they make a data table and graph of the data, and answer questions that analyze the data. At the end of the problem, students are given constant rates and asked to complete rate table and coordinate graphs based on the given constant rates.

Students analyze name-length data for two different classes of students, one from the United States and one from China. Students create frequency tables and line plots for the data. Students, then, focus on describing typical name lengths, describing distributions, and comparing distributions.

## Problem 1.1 from Grade 7 CMP3 Units

### Shapes and Designs

Students are given a set of 22 polygons. Students sort the polygons into categories that make sense to them. Then, they look at triangles and quadrilaterals and sort them into subgroups based on their properties. The problem ends with students analyzing a set of polygons that were placed into a group by another student.

### Accentuate the Negative

Students analyze the game Math Fever (which is like Jeopardy). They work with different scores and sequences of plays from different teams. Students have to find total scores and differences of scores and write number sentences to represent a series of problems.

### Stretching and Shrinking

Students use rubber-band stretchers to enlarge a figure. Then, they compare the original figure to the image created by the rubber-band stretcher, specifically looking at general shape, lengths of line segments, areas, perimeters and angle measurements. They use their findings to develop conjectures about what they think it means to be similar in mathematical terms.

### Comparing and Scaling

Students are given four different claims (2 ratio statements, one difference statement, and one percent statement) based on survey results about cola preferences between Bolda Cola and Cola Nola. Students have to describe what each type of statement means, if they could mean the same thing, what the best statement is, and create additional statements that could represent the survey data.

Students determine their walking rates in meters per second as a ratio of distance to time. They answer questions about time and distance using their constant walking rates and write an equation that models the distance walked over time at their constant walking rates.

### What Do You Expect?

Students collect data of coin tosses to determine the probability of tossing a heads. Students collect data from a relatively small number of trials and then pool their data with classmates to collect data from a larger number of trials. They, then, compare the similarities and differences of their personal data to that of the entire class.

### Filling and Wrapping

Students analyze a variety of rectangular prisms by looking at their length, width, height, diagonals, surface area and volume. They are asked to calculate surface area and volume of the boxes when given the dimensions of the boxes.

### Samples and Populations

Students analyze math test scores from sample data belonging to two students. They use measures of center and variability to compare performances, calculating the means, medians, ranges, and MADs. Students examine how adding data values changes the summary statistics. They also judge performance based on consistency and see how different measures might result in different conclusions.

## Problem 1.1 from Grade 8 CMP3 Units

### Thinking With Mathematical Models

Students conduct an experiment to explore a linear relationship as they test how bridge thickness is related to strength. They display their data in a table and a graph, look for relationships, and use the relationships to make predictions.

### Looking for Pythagoras

Students are asked to give coordinates of landmarks on a map that is shown on a coordinate grid. Then, they find distances between points. One type of distance is distance along grid lines (which is represented by driving distances along city streets) and the other is straight-line distances (which is represented by flying distances).

### Growing, Growing, Growing

Students investigate the growth in the number of ballots created by repeatedly cutting a piece of paper in half. They make a table and examine it to find a pattern. Then, they use the data to create a graph and write an equation to represent the pattern they discover.

### Butterflies, Pinwheels, and Wallpaper

Students analyze a line reflection. Then, they use the information they discover to place a line of symmetry between a new shape and its image. At the end of the problem, they are given a shape and a line of symmetry and they have to use what they’ve learned to describe how to find an image of the original shape under a line reflection.

### Say It With Symbols

Students write expressions and equations to represent the number of border tiles needed to surround a square pool of side length, s. They create tables and graphs of their equations to help justify the equivalence of different expressions.

### It’s in the System

Students work with a description of a linear equation that has two variables. The context is profit earned by selling t-shirts and caps. Students determine profit for given amounts of shirts and caps. Then, they search for specific numbers of shirts and caps that can yield a given profit. They plot ordered pairs found for the given profit on a coordinate grid and then analyze the results. Students are then asked to transfer their findings from the profit context to equations that are out of context.