Have you ever thought about how similar
rulers and number lines are?
The CRA Process for Learning About
Word Problems with Length Measurements
One of this month's MathSnack focus Standards for 2nd grade is 2.MD.B.5 and, boy, is it packed with all sorts of interesting things for our 2nd Graders to learn about:
- addition with sums to 20 at this time (we'll work within 100 in later Modules)
- subtraction with minuends (the starting value) that are 20 or less at this time
- word problems (be SURE to review the Close Reading Process from an earlier post!)
- measuring lengths within a single unit (just inches or just miles or just cm)
- the use of rulers as a measuring tool
- rulers performing as a number line
- using variables (letters) to replace an unknown value in in problem
Since we are measuring stuff anyway, let's go ahead and throw in some activity ideas for integrating Standard 2.MD.D.9, too. For this Standard, we will be teaching students how to organize the data they are collecting about the measurements they take. Below is an image of what a line plot might look like for this Standard if you were to measure the length of your shoe in inches.
Phase I: CONCRETE
Let's begin by giving our students a ruler and referencing the number line that hangs at the front of our classroom or the number lines that may be stuck to your students' desks.
- Begin by asking students to look carefully at the two different tools (the number line and the ruler). Tell them that you want them to find ways that these two tools are DIFFERENT.
- Give plenty of explore time. Encourage conversation among groups.
- Ask students what did they discover about how the two things were different.
- Try to help students generalize their discoveries -- for example, if one of the differences was that their number line had numbers written in blue and the ruler had numbers written in black, ask them if they believe that ALL number lines have numbers written in blue. Help them to see that this is a difference between these specific tools, but not between rulers and number lines, in general.
- After you have discussed several differences, ask student to look at the two tools again. This time, ask them to find things that are the SAME between the two tools.
- both have numbers
- both begin at zero (even if the zero is not seen on a ruler, point out where it would be written - this is an important understanding when using a ruler for measurement)
- both have equally spaced numbers within the tool
- the numbers go in sequence 1, 2, 3, 4,... in a predictable pattern
- the numbers have a larger value as they progress to the right of both tools
- both could be used to help you perform addition or subtraction
Next, we will provide significant time for students to explore measuring. Resist the urge to over-teach this at the beginning. Briefly (very briefly) demonstrate measuring with a ruler then give each student a ruler and each group a bucket of "stuff.
Ask students to work with a partner. The first child measures the item and announces how long it is while his partner observes. Then the first child hands the item to the second child. The second child measures while the first child now observes and the two compare their measurements. It may not be perfect this time around – remember, they are exploring how to manipulate the rulers. As students measure, move from group to group, quickly redirecting as needed looking for these common errors:
- Not beginning with the edge of the item at zero
- Measuring from the wrong end of the ruler – to correct this, have them find the zero side and encourage them to begin at zero when measuring
- As the question arises for individual students, teach them how to approximate when an item falls between two whole number values
- For students who are struggling to manage the tools, it is generally easier to lay the item flat on the desk and then slide the ruler to align it properly rather than holding the item and/or ruler up in the air
- Some students may need help understanding that the number where the item ends on the ruler indicates the length of the item
- When partners disagree on the measurement, encourage them both to remeasure very carefully just like a mathematician would do
Phase II: REPRESENTATIONAL
Once students seem to grasp that rulers and number lines have many features that allow them to be used in a similar fashion AND they are able to measure items in a way that is reasonably accurate, we are ready to move to the Representational phase of this activity.
Ask students to use their Close Reading strategies as they read various story problems like the ones below. Model how they can draw a representation of the story using the number line to help them determine the solution.
Phase III: ABSTRACT
Problem structures are an important part of learning about mathematics equations. Be sure to look at the back pages of your Instructional Organizer or the back pages of the Common Core Companion book for a wide variety of examples of Problem Structures. Our focus standard calls for us to replace unknown values with a variable (yep, that's algebra in 2nd grade!) Take a look at the first word problem from the student page above – what type of problem structures could go with this story?
Let's take a quick look at how we can integrate Standard 2.MD.D.9 with what we have been doing. Imagine that your students have been measuring items and collecting the data for items such as their shoe lengths. Now that we've collected all of this data, we will want to organize it in a way that makes it easy to interpret. Remember, math is about patterns and relationships of numbers. When we organize the data, it helps us to see if there are any patterns or relationships among the data. Teach students how to both INTERPRET data on a line plot and how to CREATE a line plot like the one pictured below.
Then have students practice creating their own math-based questions to ask and answer based on the data presented on the graph.
- Do you see the equation 18–11 = n ?
- But can you also recognize why 11 + n = 18 could also be used to solve this problem?
Let's take a quick look at how we can integrate Standard 2.MD.D.9 with what we have been doing. Imagine that your students have been measuring items and collecting the data for items such as their shoe lengths. Now that we've collected all of this data, we will want to organize it in a way that makes it easy to interpret. Remember, math is about patterns and relationships of numbers. When we organize the data, it helps us to see if there are any patterns or relationships among the data. Teach students how to both INTERPRET data on a line plot and how to CREATE a line plot like the one pictured below.
Then have students practice creating their own math-based questions to ask and answer based on the data presented on the graph.
- Which length of shoe is the most common in our classroom? (8 inches)
- Which length is least common? (6 inches with 0 students having that length)
- Do more students have a 7-inch shoe or do more have an 8-inch shoe? (8-inch shoe)
- How many more classmates have an 8 inch shoe than a 7-inch shoe? (1)
- If you line up the 4 shortest shoes in our classroom, how long will the line of shoes be? (4+4+5+7=20 inches)
- What is the difference between the longest length and the shortest length? (4 inches)
- How many more students have a 7-inch shoe than a 4-inch shoe? (8-2=6 students)
