Pre-Algebra Manipulatives and Apps

So, a few days ago I linked up with Teaching Trio for the Sunday Scoop and shared some example slides of pre-algebra problems from Common Core. The examples are from an interactive whiteboard application from Hands-On Equations. I accidentally deleted it - a long annoying story. I was going to repost it anyways and thought Tech Thursday would be the perfect opportunity! It relates to so many grade levels, whether you're teaching Common Core math or not.

I used Hands-On Equations with my 3rd-5th grade gifted and advanced students, but I also taught the basics to regular 3rd-5th grade classroom teachers from my two elementary schools. Your students do not have to be gifted to use it, and the program itself is developed with many levels, some of which I will illustrate below, so the exact same resources can be used from grades 2-8, and the students just work up to their own capacity.

The image above will give you an idea of how the materials needed, you will basically use the manipulatives to set up any problem involving a variable (missing number). The actual materials are very simple to create yourself using blocks, markers, or digital images, but the instructional progression of skills and problems for teachers and students come with the program. The examples below will show how the manipulatives can be used at various grade levels.

There are a few basic rules to using Hands-On Equations that come form the Properties of Operations. The first and most important one being, "Whatever you do to change the problem on one side, you must do to the other side." The second one is that you want to use inverse operations to simplify your equations! Subtraction should be used first, to get rid of extra values and simplify the equation, and then multiplication or division of values can be applied if necessary. Addition really doesn't enter in until the negative values are present; then students are adding values to both sides to make zeros, which are easier to work with.

The blocks are used to set up the equation or story problem. Red blocks represent positive integers. Blue markers represent positive variables. Green blocks represent negative numbers. White markers represent negative variables. The negatives become useful in the middle school standards. The red lines mean that values are being changed. If you use actual blocks, then you'd simply trade them for whatever you need next.

2.OA.1: 2 is subtracted from both sides. x = 5; 7 = 7.

The "Check" is part of the problem-solving routine. Kids have to redo the original problem with the new x value they've just found. It gets them in the habit of checking their work!

3.OA.4: Divide by 3 on both sides. x = 4; 12 = 12

 4.OA.3: Subtract 22 on each side. Divide by 3 on both sides. x = 12; 58 = 58.

For story problems and problems with large numbers, the pictorial method that allows students to draw out the blocks on paper, if they can't use real manipulatives at the time. Filled triangles represent positive variables; empty triangles would represent negative variables. Squares represent positive integers; circles would represent negative integers (see 7.NS.3 and 8.EE.2. below).

 5.OA.1: Add 15 to both sides. Divide by 3 on both sides. x = 11; 48 = 48.

6.EE.3: Subtract 6 on both sides. Divide by 3 on both sides. When 0 = 0, x can be any number!

                  7.NS.3: Add 7 to both sides. Divide by 12 on both sides. x = -12; -151 = -151.

             8.EE.2: Take the cubed root of both sides. Add 10 to both sides. x = 3; -343 = -343.

I have chosen a few examples based on the Common Core Standards for each grade level, but the complete program includes 26 different levels of increasing difficulty. It's a fantastic resource for differentiation or small group work with your higher students! Again, if buying a new program is not a possibility for you right now, you can definitely create your own materials and use it with your regular grade-level problems.

The materials you will need per student or pair are:

workman or model of a balance
2 dice with digits 0-5 (2 of each color)
2 dice with digits 5-10 (2 of each color)
8-10 colored markers (for each color)
(The actual colors are not important, as long as you have equal numbers of both.)

Hands-On Equations, the official program, can be found at http://www.borenson.com. Please visit the homepage and browse through some of the research and examples. I love this program so much, because as a visual-spatial learner, I always had trouble remembering the rules of equation transformation and equality. I was always taught Algebra as if it were just a big "to do" list of rules. The rules were hard for me to remember because I didn't have a concrete understanding of why we had to do it this way.  Consequently, I failed Algebra two different semesters, in 8th grade and in 10th grade. My mom hired a tutor and I started getting As again. My tutor didn't know about this program, but she did know how to explain concepts with concrete images instead of just rules.

I hope you found this helpful and will try using manipulatives for your pre-algebra lessons. If you're looking for more apps from Hands-On Equations, you can visit their Products page.

This month's Follower Freebie is a "Who Has" Game for practicing Pre-Algebra Expressions. Different sets allow differentiation for Grades 3-8! Sign up for my email Newsletter to get this exclusive offer! This offer will expire by Dec. 15th, 2014. Sign up today!