Multiplication, Proportional Relationships, and Slope Word Problems Updated

I posted previously about unpacking word problems involving multiplication or Slope. Here is an update. I have students draw a picture for the quantities identified. The multiplication situations become apparent when groups of objects are arise through the drawing, in contrast to an addition problem (right).

This also helps algebra students visualize the rate and therefore identify the slope for the line and helps middle school students with unit rate and constant of proportionality.

CCSS MATH Coherence Map – 3rd Grade Groups of Objects to HS Slope

A hidden treasure is the Common Core of State Standards Math Coherence Map. It is an interactive flow chart that shows connections between the various standards.

  • If you are teaching math, you can see the connections between what you are teaching, what was taught previously, and how you are preparing students for their future math education.
  • If you are a special education teacher, you can see the progression of prerequisite skills. If you write IEP objectives for grade level standards, you can address the prerequisite standards and progress made through these prerequisites can show progress towards mastery of the IEP objectives.

In this post, I show the progression from 1st grade standard on the = sign and 2nd grade standard of repeated addition all the way to interpreting slope in high school math.

After clicking “Get Started” you will narrow your focus to the grade, the cluster, and then the math domain.

The flow chart shows connections between a selected standard and others, including prerequisites. In this case 8.F.B.4 – 8th grade content that is a prerequisite for the high school math work. Click on the 8.F.B.4 standard and it pops up (below right).

In turn, the 8th grade standard is connected to a 7th grade prerequisite regarding ratios and proportions.

Notice that the 7th grade standard, 7.RP.A.2, appears to be a dead end (bottom left). I picked up the path by clicking on Grade seen at the top left of the screenshot and made my back to that standard and the connections to prerequisites appeared. (Same happened in 3rd grade shown further down in this post.)

The path continues from ratio and proportions in 7th grade to unit rate in 6th grade, multiplication word problems and multiplication in elementary school.

I want to emphasize that students are working on unit rate and slope problems in ELEMENTARY SCHOOL! 3.OA.A.1 below addresses groups of objects model for multiplication and 4.OA.A.2 addresses word problems involving multiplication.

I was recently working with a student entering middle school on multiplication word problems. To unpack the word problems and the concept of multiplication in context, I had her draw (photo below) groups and groups of objects to help her identify the unit rate (although we don’t use that term yet). This work will impact her math education through the high school math and even into college (slope has been a common gap for the college students in the math courses I taught at various colleges and universities).

This approach I used with the student could be used for high school students, especially those with special needs.

Math Intervention is Packing a Suitcase

In working with students with special needs on math programming and services, a common and important issue is that the student is behind and there is a tension between math intervention to fill gaps and addressing ongoing grade level content.

Unpacking the situation

There is no single grade level for math, as is the case for reading. Math progression is more like a web, not a line. For example, if a student can do 5th grade geometry but only 3rd grade level fractions, do we average out the grade level math to be 4th grade? (No.) Do we identify the student as working at a 3rd grade level? (No.) 5th grade level? (No.)

Like a suitcase, there is a capacity to the daily time a student has for school services. I often encounter situations in which the services recommended involve the student working on grade level content and catching up on the gaps during support time. If the student has only been learning 75% of the math content each year, he or she needs that support time to help learn the new content to get closer to 100%. There is too much being stuffed into the suitcase. Something has to give.

The focus of the services and programming often shifts away from post-secondary plans, with a focus on the short term. Like the situation facing the man in the image below, there are long term implications.

Recommendations

There are two recommendations I make in regards to addressing the gaps, without overstuffng the suitcase.

The IRIS Center is part of the Peabody College of Vanderbilt University.
  • Use triage to shift focus to the priority topics. For example, the parents of a student in 7th grade but working on math from lower grade levels wanted to pursue a math track that would allow the student to go to community college. I mapped out a long range plan (image below) that focuses on algebra as that is the type of math most likely encountered in a math requirement. Here is another plan which was to prepare a student to possibly work in a field related to cars.