Word Problems with Unit Rates

Word problems are challenging. In middle school and high school, word problems for proportional relationships and linear functions are particularly challenging. This post shows an visualization approach to unpacking the unit rate component of such word problems.

Visualizing Unit Rate

Below is an excerpt from a handout used to introduce such word problems. It starts with the unit rate and unit quantity provided. This is more accessible for students and allows for a visual representation. This allows students to “see” the multiplication and make a connection to the word problem.

This is followed by situations in which the unit quantity is unknown. Students still draw a visual, but use the 3 dot symbol for etc. to indicate the unknown quantity.

Fading Visuals

The same problems are used but students now write the symbolic representation. First they write the expression with both factors provided, then with the unknown. The “?” emphasizes there is an unknown quantity before writing the variable.

The Handout

Here is a link to the handout. It is in WORD format to allow you to enter your own problems.

Intro to Google Jamboards

Visuals and manipulatives allow for a multi-sensory approach to presenting math topics. Google Jamboard makes implementation of both relatively easy and is effective.

Current Jamboard

Here is an image of Jamboard used to guide multiplication by a 2-digit factor.

Creation of the Jamboard

Below is a step by step visual guide on how I created a version of the Jamboard seen above. Here is a Facebook Reel version with music only. Here is a YouTube version with music only. Here is a PDF for the slides I show and a YouTube version of me talking about the sldes.

Introduction to Adding Integers

Operations on integers and integers in general is challenging for many students. Negative numbers are abstract. Whole numbers and fractions can be represented with images. The activity presented draws upon student prior knowledge of thumbs up and down in a vote to make negative more accessible.

The following images are from a Jamboard. Here are accompanying videos on FB Reels and Youtube showing how this works. There are 3 sets of images (or chunks) loosely following a CRA appoach and all referring to the same two situations.

  • Prior Knowlege drawing upon a classroom setting (concrete)
  • Transition using thumbs (representational)
  • Introduction of adding integers, using thumbs (more abstract but still supported by

Prior Knowledge

The activity starts with a couple of classroom votes using thumbs up and down.

This is followed by scaffolding to focus on how the voting works through a comparison of the quantity of thumbs up vs down.

Transition

This section introduces thumbs as counters for integers, which is a common instructional strategy (yellow for positive and red for negative). The scaffolding is the same.

thumbs from Educlips on TPT

Adding Integers

Finally, references to thumbs is replaced with the integers values. The thumbs tokens are maintained to allow for continued concrete representation.

Accessing Jamboard

To access the Jamboard you must make a copy.

Equal Parts of Fractions – Intro

The concept of fractions as some number of equal parts begins in 1st grade per the Common Core (image below). There are students who struggle with the idea of equal parts and this could undermine student work in subsequent topics. The activity cited in this post is designed to develop the concept of equal parts.

CCSS Coherence Map

Jamboard with Sharing Slides

The following images are from a Jamboard used as an introduction to equal parts activity (see photo at the end for access). The activity is chunked to incrementally present more of the ideas underlying equal parts. The use of the Jamboard can be viewed in a FB Reel and on YouTube.

First, the idea of equal is addressed by presenting a situation in which two students are sharing candy. Partitioning out pieces alludes to the set notation of fractions.

The idea of sharing equal amounts transition to sharing a single candy that can be broken into parts. The candy bar image is actually two images of parts. The a non equal sharing is used to unpack equal parts. This is continued for a circular shape and a triangular-ish shape.

Jamboard with Mathy Slides

There are additional slides to do more “mathy” work with equal parts. First, the students are asked to choose the shape that was cut into equal part (rectangle, circle, triangle). Then the students partition the shapes but with a dotted line as scaffolding.

Each shape can be connected to the food images from above. For example, the student may intuitively understand that a pizza is cut from the crust to the tip. I use pizza fractions to unpack the need for common denominators, which reinforces the significance of the concept of equal equal parts cited previously.

Handout

Here is an image of an accompanying worksheet. It draws upon the images from the Jamboard and follows the same sequence.

Accessing the Jamboard

The image below shows how to make a copy of the Jamboard in order to use it.

Intro to Slope of a Line

Slope is one of the most important topics in algebra but perhaps one of the most challenging topics. One issue is that students often think of the formula as opposed to the idea of slope. The images shown below are for a handout that unpacks the idea or concept of slope.

MathBootCamps

Handout

The images show sections of a handout. Slope is a measurement of a graphed line. It measures steepness and also indicates direction. The handout starts by drawing on prior knowledge of measurements for newborns.

Prior knowledge of steepness and up and downhill are invoked.

How to measure steepness is introduced through a focus on stairwells.

The measurement of the length and height of steps leads into the rise and run of a hill. I use “right” and “up” or “down” which focus on direction. The terms “rise” and “run” can be addressed subsequent to the introduction. This part of the handout is scaffolded to reduce task demand and focus on the concept.

Finally, the students are provided a line and apply the ratio procedure. Note: the slopes are not the same as in the prior section.

Jamboard

An accompanying Jamboard will eventually be shared in this space.

Intro to Ratios using Jamboard

Below are images from a Google Jamboard for a hands on introduction to ratios. (See image at the bottom for how to make a copy in order to use it.) The images are from Clever Cat Creations and provide a visual representation. The moveable items engage the students kinesthetically. It also helps unpack the concept of ratio as a comparison of two quantities as the students count out the quantities and represent them as numbers in a ratio. The scaffolding guides the process.

First, students move the terms to make a connection between the statement and the ratio.

Then the objects are counted and moved.

Then the ratio is written.

The quantities can be flipped to show an alternative ratio.

There is a blank to create your own and another with shapes.

You have to make a copy in order to move the pieces.

Modified Multiplication Table – Area Model Included

The idea is that the student will have to count squares and eventually skip count by how many rows. By doing so the student is more engaged (or less passive) in determining the product byy engaging the visual representation. I am interested in feedback and will revise if this could be useful.

Here is a link to the document.

Intro to Multiplication – A Sequence of Lessons

Below are photos from multiple lessons to introduce multiplication. They are combined into a single document. I use a task analysis approach to first develop conceptual understanding of multiplication as repeated addition. This is followed by skip counting and then using skip counting to multiply. The lessons are not necessary completed in a single day.

Lesson 1 focus is to unpack repeated addition vs simple addition to build on prior knowledge.

Lesson 2 focus is to unpack arrays by identifying rows and columns which are the factors in a multiplication problem. It builds on the previous lesson with repeated addition of groups that are then converted into arrays of items and then into arrays of circles and squares.

Lesson 3 transitions from repeated addition to skip counting (with a future focus of multiplication by skip counting vs fact memory).

Lesson 4 combines skip counting and the rows and columns of arrays into a multiplication sentence.

Lesson 5 uses skip counting to multiply, first with arrays and groups, then as multiplication problems. Here is the link to a post about the Grumpy Cat Jamboard cited in the document.

The nature of the task analysis approach is a sequence of topics building towards the objective of multiplying single digit numbers. Mastery of each of the steps or lessons can be recorded as progress towards mastery of the overall objective. Below is an excerpt from a Google Sheet that is used to record such progress. This can be shared with the team, including parents.

Unpacking Graphs Using a Rollercoaster

I introduce key characteristics with parabolas and use the analogy of a rollercoaster. Riding once (and never a again) the Superman rollercoaster at Six Flags New England got me thinking about this. At one point the roller coaster hits ground level (a zero) and then goes underground (negative y values).

Handout

Here is a handout I use for the introduction. Here are images showing how I use the handouts. The table helps students visualize the x-values and y-values when looking at the graph. The rollercoaster provides extra context for the various characteristics, e.g., increasing means the rollercoaster is going up. Note the scaffolding by adding context clues for each characteristics.

I start with max height of the rollercoaster. I highlight the actual graph first, then the y-values in the table. Then point out we are looking for the x-values for what we highlighted.

The issue of highlighting the vertex for the increasing and decreasing values would be addressed when writing the interval. The idea of the rollercoaster at the tip top provides context to develop the concept.

Similarly, I start with the zeroes. Again, highlight the graph, then the y-values, then the x-values.

The issue of highlighting the zeroes for the positive and negative values would be addressed when writing the interval. The idea of the rollercoaster at the ground level provides context to develop the concept.

I has been effective to have students highlight the parts of the respective axis when discussing the domain and range (not discussed yet). A common challenge is understanding that the x-values continue to the right or left when it appears they simply go down. To address this, I use a very wide parabola to show more lateral movement.

Jamboard

Here is a Jamboard. You have a make a copy to access it (see photo at bottom). Here is a FB Reel and a YouTube video showing how it works.

Intro to Solving Equations – Seesaws and Oranges

Here is an introduction to solving equations using a Jamboard (see photo at very bottom for how to make a copy). A seesaw is used to unpack the concept of an equation as two sides that are equivalent. The box is used to unpack the concept of a variable representing an unknown number (or oranges in this context). The form of a solution for an equation is established, with students revising to create other solutions. Here is a YouTube video and a FB Reel showing how it works.

Students are then provided a couple slides to match seesaw representations with actual equations. The matching provides a scaffold to support the connection between representations.

Then the seesaw and box representation is used to unpack the steps. Students are provided the steps as written directions on how to model the given solving steps with the seesaw.

The students are then provided the equation and seesaw representation, along with the solving steps provided as moveable pieces. Students slide pieces and move seesaw objects to make the connection between the two. Here is a link to a post with an updated version of this handout.

Finally, students are given the equation and tasked with completing all of the steps including the initial set up. This slide can be copied with new equations entered for additional practice (including having the variable on the right or writing the number before the variable (e.g., 5+m=8).

Make a copy of the Jamboard so you can edit it.