Some thoughts on teaching little kids algebra

Teach Kids Algebra Project (TKA: 1st, 2nd, 3rd Grades)
I shall never forget these kids! 

The idea that little kids cannot learn algebra ideas is nonsense.

TKA is a response to reform math and Common Core.

Special Insert 10-30-17

"Equations are the lifeblood of mathematics, science, and technology," points out mathematician Ian Stewart (In Pursuit of the Unknown, 2012). It is the reason that students need to learn to write equations, rearrange them, deal with them and become skilled in solving them, all paper-pencil, not only for math class but also for chemistry and physics classes, etc. Also, knowing trig is important in physics courses. Right triangle trig used to be taught in good 7th-grade pre-algebra courses, but it is no longer the case. (Also, the 1970 Dolciani Algebra-1 textbook included a chapter on Geometry & Trigonometry. The trig problems were physics problems that are found in algebra-based physics courses. Other problems involved finding the vertical and horizontal components of a vector and the resultant of vectors. Students learned to solve trig equations that solved physics problems.) 

The use of graphing calculators and the shift away from solving equations using traditional algebra have produced weak algebra students over the years. It is one reason that early algebra is important in our schools.

End Insert 

TKA 3rd-Grade Students (2011)

Algebra is arithmetic plus variables. A variable is a symbol, such as x, y, or ☐, that represents an unknown number. For little kids, it is important to stress that a variable is a number. 

Students start with numerical equations (true/false) and the idea of equality, go to equations in one variable, proceed to equations in two variables (functions: input-output model), etc. Little kids solve equations using guess and check, memorized math facts, number and equality properties, procedures, and logic. This prepares them for algebraic methods for solving equations (e.g., the addition property of equations, etc.) by 3rd or 4th grade. Algebra ideas are not difficult if students know math facts (automaticity) and pay attention in class.  

● Variables, Equations, tables, graphs

Little kids, including typical 1st-grade students, can 

1. solve a range of linear equations using guess and check, number facts, and properties; 

2. write equations in one unknown to model a word problem (The idea is to translate words into symbols);  

3. build tables that show numerical relationships; and 

4. plot points and graphs of linear equations.

The idea of substitution is an important preparation for Algebra I. 

● Student Comments

1. "When you give us work it's fun because we don't know the answers." 

MV, 3rd Grade TKA Student

2. "I look to you to keep my brain working. I was able to understand what you were teaching. You have helped me with math I used to struggle with. The math that you taught me is amazing. You have taught me never give up, keep on trying. Thank you for all you have done for me."  MR, 3 Grade TKA Student

● Effort, persistence, and practice (EPP)

The idea that kids need innate ability to learn arithmetic and algebra is bunk. Kids need the skilled teaching of content and lots of practice to master arithmetic and algebra. Furthermore, they must work hard, i.e.,  effort, persistence, and practice count. Indeed, according to Daniel Willingham, a cognitive scientist, the “vast majority of K-12 students” can learn arithmetic and algebra; however, this should not imply that learning math is easy because learning math does not come naturally. Willingham points out, “It takes time, effort, and mastering increasingly complex skills and content.”  Math is hierarchical and should be taught so that one idea builds on another. It is logic. The logic begins with the transitive property of equality (Think Like A Balance) and true and false statements. Reasoning in math (new true statements are linked to other true statements) is different from reasoning in science (inferences are based on observations). In short, problem-solving is always domain-specific and requires domain knowledge.

● Equivalency (=)

Equivalency, which is a fundamental concept in mathematics, is seldom taught well. For example, 3 + 5, which is 8 {true}, is equivalent to 10 - 2, which is 8 {true}.

Therefore, 3 + 5 = 10 - 2 is a true statement. 

The logic behind the true statement is the transitive property of equality: two things equal to the same thing {8} are equal to each other. In Teach Kids Algebra, 1st, 2nd, and 3rd-grade students apply the "equivalency idea" (Think Like A Balance) and guess and check to find an unknown: 3 + 5 = x - 2. It is not possible to determine whether this statement is true or false without substituting a number. The box is a variable like x, and it can represent any number. But, to make a true statement, x is 10, not 8.

● Algebra, the Higher Arithmetic

The algebra lessons I give to 125 1st, 2nd, and 3rd-grade students require them to use number facts, number procedures, and number properties or laws (arithmetic). In short, algebra is built on arithmetic. As Morris Kline states, “Algebra is the higher arithmetic.” If we want kids to learn algebra, then they must be good at arithmetic. In primary school, kids who have auto recall of number facts and experience with standard algorithms do better and understand more. 

● Mini-Lessons: Lecture And Feedback

Manipulatives and calculators are not used. Students do not color things, paste things, cut things out, or work in groups. I lecture (explain how things work with examples), write stuff on the board as I explain things (visual-auditory), and ask students questions as I go (interaction). Then, I hand out problems for students to try on their own (guided practice) and roam around the room talking to students, giving feedback, and providing individual help. The classroom teacher helps a lot too. All this is accomplished in a 30 minute period, which is often too short. Explicit instruction works best. 

FYI: I met one 3rd grade class twice a week for at least one hour each time. Typically, the session would last 15 to 20 minutes more than an hour We always ran out of time. (Ms. S., the classroom teacher, was generous with time.) The students work hard for an hour straight, usually longer, but they do not seem to mind. Time passes quickly. I was not only teaching fundamental algebra ideas, but I was also teaching persistence and effort. One 3rd grade student wrote to me, "I had a hard time with algebra, and because of you I got better with algebra. I enjoyed learning because you made my brain work." 

At first, the math is challenging, but I keep reassuring and encouraging students, often one-to-one, giving them support: "Try a different number. Do not give up! You can do this. Let me show you how. Try this. You can learn this--it's great stuff." And, they did. The success of the program hinges on talking to individual students and giving them important feedback and encouragement. Kids need adult support. 

● The Equation As Model: Translating words into symbols

When I present a word problem, let's say in 1st or 2nd grade, I model it with an equation. For example, Jill has some pencils. Bill gives her 5 more pencils. Now Jill has 12 pencils. How many pencils did Jill have before Bill gave her pencils? Jill has "some pencils" is represented by a variable I call x. Step-by-step, I piece together an equation: x + 5 = 12 on the board by asking questions: Do we add or subtract 5? What do we do with 12? The equation is the model. Kids do not need bar models to understand simple problems. They learn to write equations. They learn to translate words into symbols. It is important that students identify the unknown in a word problem. This starts in 1st grade. Identifying the unknown and solving problems from an algebraic perspective makes sense.  


For older kids (3rd grade), the equations become more complicated. Write an equation, then solve it. I am thinking of a number. Six less than triple a number is 15. What is the number? 
Equation: x + x + x - 6 = 15
Solution: x = 7  by inspection (7 + 7 + 7 - 6 = 15; 15 = 15). 


Note. Being able to translate words into symbols (equations) and being able to solve the equations is what algebra is all about.

Note. It is okay for students to struggle because math is not always about getting the right answer. It is about developing young minds and improving their effort, persistence, and reasoning in solving problems. Furthermore, math builds the brain. It makes kids smarter. In short, a cognitive struggle is a good thing.

● Myth

Willingham also observes that “our society has accepted the fact that math is not for most us;” however, he says that this “notion is a myth.” The idea “I am not good at math” is rooted in our culture. We need a radical change in our attitudes toward learning and schooling. 

● Cognitive Horsepower 

Attention is essential for learning. Attention is controlled by something called executive function. Gary Stix (How to Build a Better Learner, Scientific American, August 2011) says that executive function encompasses important cognitive attributes such as the ability to "be attentive, hold what you have just seen or heard in the mental scratch pad of working memory, and delay gratification. These skills (being attentive, holding stuff in working memory, and delaying gratification) often predict a child’s success in school. 

Children who have difficulty concentrating also have difficulty learning mathematics. Teaching little kids algebra is a blast, but learning algebra ideas requires mental fluency with arithmetic facts and procedures and sufficient concentration to hold stuff in working memory.  

In Teach Kids Algebra lessons, for example, students deal with a lot of new stuff all at once in a short time. This s t r e t c h e s working memory and pushes students to concentrate. Knowledge of key math facts and procedures in long-term memory (automaticity) helps a lot because of working memory space, although somewhat "plastic," has limitations. In problem-solving, too often students figure out simple facts that should have been memorized (e.g., 3 + 8). This "figuring" wastes time and working memory space and distracts from solving the problem. Facts should be memorized and stored in long-term memory for a child to excel in mathematics. The focus in working memory should be on solving the word problem, not on figuring out simple facts.   

A child's working memory structure, including the mental scratchpad, is in place by age 6. It has less capacity than an adult's working memory, but it improves somewhat as children grow. Working memory, however, has limitations. It can hold only so much stuff at a given time before becoming overloaded. In contrast, long-term memory does not have this limitation. In mathematical problem solving, it is important that key math facts and procedures are in long-term memory (background knowledge) so that working memory can hold all the essential information from the problem to devise a plan for solving. Also, negative thoughts about math (e.g., math anxiety) can often crowd working memory, which is another concern. (Information from From Stix, Willingham, Beilock)

According to Sian Beilock (Choke), "Working memory is your cognitive horsepower. It involves the ability to hold information in mind (and protect that information from disappearing) while doing something else at the same time." Beilock says that "working memory is one of the major building blocks of IQ." Working memory can be developed, so it is important to practice, stretch, and exercise working memory with challenges to improve cognitive muscle. Beilock points out that "practice shapes your brain." To learn math well, for example, requires both practice and challenges. Also, a student will not learn much math if he is easily distracted or has difficulty with attention in class.

Daniel T. Willingham, a cognitive scientist, states that students learn what they are thinking about, which takes sharp attention. The ability to control attention and hold information are skills that can be trained. Bronson & Merryman (Nurture Shock), suggest that "being able to concentrate [cognitive control] is a skill that might be just as valuable as math ability, or reading ability or even raw intelligence." 

Of concern is that a child's attention span or ability to concentrate has been declining over the past 20 years. The Net-Generation tends to bounce from task to task. Click, Click, Click! They often have difficulty focusing on one task and doing it well. They do not reflect or think through things. Nicholas Carr (The Shallows) says the Net causes brain changes. The biggest change is that students have difficulty concentrating. He observes, "Tests of memorization, vocabulary, general knowledge and even basic arithmetic have shown little or no improvement."  The Net does not make you smarter; school makes you smarter. The Net, says Carr, should not be a "replacement for memory." Remembering is a fundamental cognitive skill that is needed for problem-solving. Math facts and procedures must be retrievable from long-term memory so they can be used in working memory to solve problems.

Aimee Cunningham (Kids' Self-Control Is Crucial for Their Future Success, Scientific American, July 25, 2011) points out that a child’s “self-control is crucial for their success.” She writes [Long Quote], “Self-control—the ability to regulate our attention, emotions, and behaviors—emerges in childhood and grows throughout life, but the skill varies widely among individuals. Past studies have reported that self-control is partially inherited and partially learned and that those with less self-control are more likely to be unemployed, engage in unhealthy behaviors such as overeating, and live a shorter life.” 

● Common Brain Myths

Teachers and parents should be aware of common brain myths. Gary Stix (Scientific American, August 2011) lists five myths from Mind, Brain, and Education Science (Takuhama-Espinosa, 2010). Here are two of the myths. 

1. “Left-brain” and “right-brain” people differ. No. “Brain-imaging studies show no evidence of the right hemisphere as the locus of creativity. And the brain recruits both left and right sides for both reading and math.” 

2. Each child has a particular learning style. No. There is little evidence to support this claim. “For this and other myths, public perceptions appear to have outstripped the science.” Many parents tell me their child is a visual learner or a kinesthetic learner (not an auditory learner). These popular perceptions held by many parents and educators are not backed by evidence. 

In education, "there is an enormous supply of totally untested, untried, and not very scientific methods.”   

(To Be Revised)

LT, Guest Teacher

LT, Founder of  ThinkAlgebra

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Updates: 6-19-11, 6-20-11, 6-21-11, 6-30-11, 7-1-11, 7-2-11, 7-9-11, 7-20-11, 7-23-11, 7-24-11, 7-25-11, 7-26-11, 7-31-11, 8-1-11, 9-21-11, Minor grammar corrections made on 4-29-17
Photos by 3rd grade classroom teacher: CSmith 

Most kids do not understand science

NAEP recently released science results. Most kids do not understand science. Only 34% of 4th graders, 30% or 8th graders, and 21% of 12th graders are proficient or above in science. Science education is just as bad as math education. Dr. Mark A. McDaniel says that before engaging students in inquiry-based problem-solving in science or mathematics, they should have a sound knowledge base (background knowledge). In short, background knowledge of fundamentals in long-term memory is key to problem-solving in math, science, and other academic disciplines. Furthermore, background knowledge should be domain-specific. (Note. The idea that math or science always has to be fun or a game to play is nonsense. There is nothing wrong with rigor or struggle. To learn math or science well takes plenty of effort and hard work.) 

Unfortunately, elementary teachers are told to focus on math and reading (NCLB) at the expense of science and other subjects. Even so, achievement in math and reading falls short of NAEP proficiency levels for most students. Furthermore, very little science is taught in elementary school; K-8 science textbooks tend to avoid math. In contrast, Science--A Process Approach (1967), from the Sputnik era, focused four of the six processes taught in first grade on mathematics: Using Numbers, Using Time/Space Relationships, Using Communicating (graphing relationships), and Measuring. The math was ahead of the normal 1st-grade math curriculum. 

To improve science achievement at the high school level, students should take more than biology. They should take both chemistry and physics and avoid integrated courses. The coursework taken in high school makes a big difference in science learning. Chemistry and physics also require a good background in algebra and solving word problems. (FYI: No calculators are permitted on AP Chem or AP Physics exams.) 

Systemic Approach, Not Bits & Pieces
The lack of adequate math and science achievement in the United States is partly a result of system failure. One cannot fix one part, such as standards, in isolation from other parts, yet this fragmented approach seems routine in education. Better standards will not fix a system failure without concurrently addressing teacher education and schools of education, instruction and curriculum, textbooks and assessments, policies and teacher support, learning attitudes, and so on, as all are interconnected. In short, a complete system overhaul is needed, not bits and pieces now and then. 

Return to ThinkAlgebra

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Equation Notes

An Equation Is Like a Balance 

Teach students to "think like a balance" (left side = right side). An equation is a puzzle. The equal sign does not mean "calculate the answer." For example, in the equation 12 + 5 = ❏ + 8, box is not 17. First grade students should start with simple equations (e.g., 4 + ❏ = 7) and advance to harder equations: (❏ + ❏ + ❏) - 9 = 24), using the algebra rule for substitution. The equations should relate to the arithmetic facts being learned. With practice, primary students can solve equations in one unknown (❏ is a variable like x in algebra). 

Examples:
4 + 3 + ❏ = 20
4 + 5 + 12 + ❏ = 33

❏ + ❏ + ❏ - 9 = 24

(❏ + ❏) - 2 = ❏ + 5

Students should learn number properties and memorize number facts for automaticity. Number properties, such as the equality axioms, control how numbers work and are essential for understanding arithmetic. Young children should learn how numbers behave.

In arithmetic instruction, a competent teacher can integrate basic algebra ideas (e.g., variables, equations, equality, algebra rule for [variable] substitution, etc.) along with reasoning and “guess and check.” These ideas can be taught as first-grade students learn arithmetic facts and procedures. Furthermore, an equation approach is a powerful opportunity for students to practice arithmetic and do “number property” reasoning. The equations for 2nd and 3rd-grade students move up a notch in difficulty. For example, third-grade equations should include fraction ideas, multiplication/division, and multiple operations. Examples of equations in one variable at grade 3 level:

    ❏ + ❏ = 5
    ❏ x ❏ + 12 = 21 
    12 x 4 x ❏ = 12 
    ❏ ÷ 4 + 12 = 19. 


Examples of challenging equations at 3rd or 4th grade level:
(❏ x ❏) - (5 x ❏) + 6 = 0     (❏ x ❏) - (13 x ❏) + 22 = 0.


Commentary

These are notes on equations from ThinkAlgebra. They are like “cheat sheets” for parents and teachers. They will help you understand the background knowledge students should have, including auto fact recall. Students learn true/false statements, variables, equations, axioms (e.g., equality), conventions, and the algebra rule for variable substitution. Furthermore, students use reasoning and apply a “guess and check” strategy to find unknowns.


I cannot overemphasize the primacy of background knowledge in long-term memory, both factual and procedural. The key to learning is practice to mastery. Furthermore, understanding and problem solving are rooted solidly in background knowledge. Practice builds factual and procedural knowledge in long-term memory.

FYI: I taught these ideas to my 1^st graders, but the equations were easier. My first-grade students memorized the “Make 10” and “Doubles” in the first couple of months of school. This is not the usual order for learning math facts. “Make 10” and expanding numbers by place value (e.g., 13 = t + 3 or 10 + 3) are background knowledge for understanding place value. 

Two Notes

1. No calculators should be used in elementary school. 

2. Students in 3^rd or 4^th grade do not use algebraic methods for solving these equations. They used guess and check, math facts, procedures, conventions (order of operations), axioms (equality properties), the rule for substitution, etc.

Note: In 2016, I taught 4th-grade students to use inverse operations to solve one-step equations such as x + 5 = 12.  Students also solved two-step equations such as 3x + 5 = 32 using inverses. Using inverses is an algebraic technique.

The commentary below is for specific equation types that can be introduced at this level for prepared students; i.e., students who have sufficient background knowledge. There are prerequisite skills. Teachers with sufficient knowledge in mathematics can figure out what these skills are and teach them in a coherent manner.  

“Think like a balance” illustrates the transitive property of equality, which is the reason  “2 + 3 = 4 + 1” is a true statement (left side = right side or 5 = 5). The transitive property of equality, roughly stated, implies that if two things (2 + 3 and 4 + 1) are equal to the same thing (5), then they are equal to each other. Thus, 2 + 3 = 4 + 1. This is a “number property” approach to arithmetic. In short, both “2 + 3” and “4 + 1” name the same point on a number line, which is 5. The equivalency idea is fundamental in 1st-grade arithmetic, but it is seldom taught this way. 


The transitive property of equality is the reason that 3 + 7 = 8 + 2 is a true statement. Both 3 + 7 and 8 + 2 names the same point on the number line (10) and, therefore, are equal to each other: 10 = 10. 
  

Equivalency is also fundamental in fractions, ⅛ = .125, or  ¾ = 3 x ¼ = ¼ + ¼ + ¼, etc. Substitution of equivalent forms is a fundamental idea in arithmetic, algebra, trig, and calculus. 

Grade 3 Level Equations

NOTES/Draft 1 To Be Revised

1.  ❏ + ❏= 5

2. ❏ x ❏ + 12 = 21 

3. 12 x 4 x ❏ = 12 

4. ❏ ÷ 4 + 12 = 19. 

5. (❏ x ❏) - (5 x ❏) + 6 = 0 

   

The equation approach for practicing arithmetic and learning algebra ideas is challenging for students. Kids need a “cognitive” stretch to improve math ability. To match or surpass kids in other nations, young students should learn substantially more mathematics than is currently taught. 

An equation is like a balance (left side = right side), which is the equality concept. Students should learn to think like a balance. While these equations are for prepared 3rd-grade students, they can also be used in 4^th grade. I taught some of these ideas, such as the algebra rule for a substitution or the idea of thinking like a balance (left = right), to my 1st-grade class.  

Double Box: ❏ + ❏          (x + x or 2x)

Triple Box: ❏ + ❏ + ❏      (x + x + x or 3x)

The Box is a variable like x in algebra. A double box or triple box means the same variable (box). Thus, the same number must go in all boxes in an equation or expression. This is the algebra rule for substitution. 

Suppose the equation is ☐ + ☐ = 8. If we put a 3 in each box, then we would get 3 + 3 = 8, which is a false statement because the left side is 6 and the right side is 8 (6 ≠ 8). Think like a balance. If we put a 4 in each box, then we would get 4 + 4 = 8, a true statement (8 = 8). It is true because the left side (4 + 4) is equivalent in value to the right side (8). In guess and check, students find a number for box that makes a true statement. 

Let’s say the student places 5 in the first box and 3 in the second box, doesn’t that make 8? Well, indeed 5 + 3 = 8; however, the substitution for box is not allowed because box cannot be both 5 and 3 at the same time. In the equation ❏ + ❏ = 8, if you place a 5 in the first box, then you must place a 5 in the second box. If the equation were box + triangle = 8 or  ❏ + ∆ = 8, then box and triangle are two different variables, and each could be a different value or even the same value. Thus, one valid solution for the equation ❏ + ∆ = 8 is 5 + 3 = 8. In short, box is 5 and the triangle is 3. Another solution can be box = 4, triangle = 4, etc. Instead of ❏ + ∆ = 8, we write x + y = 8 in algebra. There are several whole number solutions to this equation, including 0 + 8 = 8, and an infinite number of “real number” solutions, such as 2.655 + 5.345 = 8, etc. 

Okay. In the first equation, ❏ + ❏ = 5, third-grade students might try whole numbers (guess and check). For example if a student tries 3, she gets 3 + 3 = 5, which is a false statement (6 ≠ 5). If she tries 2, she gets 2 + 2 = 5, which is a false statement (4 ≠ 5). Thus, the student should reason that the number that works must be between 2 and 3. (This idea can be taught in 2^nd grade, even 1st, if students have had experience with fractions and mixed numbers.) The answer is 2½ because 2 ½ + 2 ½ = 5. Another way to think about the double box is that the box must be half of 5. In short, split 5 into two equal parts. Students in 2^nd and 3^rd grade should have experience with fundamental fractions. It is called background knowledge, especially adding like fractions, the idea that ¾ is 3 x ¼ or ¼ + ¼ + ¼, and simple multiplication of fractions, including reciprocals. What is half of 5? (Sometimes, a number line with halves is helpful. Points between whole numbers are fractions. Thus, fractions, like whole numbers, are simply points on the number line. Thus, fractions are a logical extension of whole numbers.)

In the second equation, ❏ x ❏ + 12 = 21, students start with guess and check. The equation is a combination of multiplication and addition. Remind students that, by convention, multiplication is done before addition. This is part of the order of operations. If a student guesses 4, then 4 x 4 + 12 = 21, is a false statement. The left side (16 + 12) is 28, the right side is 21 (28 ≠ 21). So, 4 is too high. Try box = 3. Thus, 3 x 3 + 12 = 21, which is a true statement. The left side (9 + 12) calculates to 21 and the right side is 21 (21 = 21).

FYI: There is a second solution, which is -3. But the solution is ignored until students study negative numbers.  

In the third equation, 12 x 4 x ❏ = 12, the idea of reciprocal is important: n x 1/n = 1. Two numbers are reciprocals of each other if and only if their product is 1. (Again, this takes background knowledge. Students should practice multiplication of fractions that involve reciprocals. In short, we need to beef up content in 3^rd grade.) The student should reason that 4 x ❏ must equal 1 because 12 x 1 = 12, which is the Multiplication Property of One. Thus, 4 x ¼ = 1. Again, it is important for students to work with fractions and their reciprocals in 3^rd grade. FYI: The idea of “dividing by 2 is the same as multiplying by ½” is important in the division of fractions. 

In the fourth equation, ❏ ÷ 4 + 12 = 19, there are two operations, division, and addition. Children should be instructed that division, like multiplication, is done before addition (order of operations). Students should have an excellent grasp of their multiplication tables by Christmas in 3^rd grade. As in the other equations, students use guess and check and reasoning. Try box = 8. Thus, 8 ÷ 2 is 4, and 4 + 12 = 16, not 19, so 8 doesn’t work (16 ≠ 19). Try box = 12. Thus, 12 ÷ 4 is 3, and 3 + 12 are 15, not 19 (15 ≠ 19). Getting closer. Try box = 16. Thus, 16 ÷ 4 is 4, and 4 + 12 = 16. It is closer to 19, but the statement is false (16 ≠ 19). Try box = 28. Thus, 28 ÷ 4 is 7, and 7 + 12 = 19. Finally. 28 ÷ 4 + 12 = 19. True! The left side of the equation is 19, and the right side is 19 (19 = 19). Always think about balance: Left Side = Right Side 

Note Well. Instead of guess and check, another way to figure out the “number for the box” is to internalize (reason) that ❏ ÷ 4 must be 7 because 7 + 12 = 19. Thus, box is 28 because 28 ÷ 4 = 7. Students must know math facts. The more experience the student has, the better the student will reason in arithmetic. Showing students how this works is important. Students will need step-by-step practice with easier equations that lead to this equation. 

In the fifth equation (4^th grade), (❏ x ❏  - (5 x ❏)  + 6 = 0, start with guess and check as usual. This is a challenging equation for 3^rd, 4^th, or 5^th grade students. I used it in my 4^th grade class. Try box = 7. We get (7 x 7) - (5 x 7) + 6 = 0. The arithmetic is simple enough: 49 - 35 + 6 = 0, but 20 ≠ 0), so box = 7 doesn’t work. Too high. (Be sure to show how the arithmetic works. Also, remind students to do parentheses first, then subtract.) Try a lower number for box. Try Box = 4. We get (4 x 4) - (5 x 4) + 6 = 0, which is 16 - 20 + 6. How does a 4^th grade student handle this situation without getting into negative numbers? Simple. Add 16 and 6, which is 22, then subtract the 20. Thus, 22 - 20 = 2. But, 2 ≠ 0, thus box = 4 does not work. Continue guess and check until both solutions are found to (❏ x ❏) - (5 x ❏) + 6 = 0. They are not hard.

FYI: We can add 16 + 6 first because 16 - 20 + 6 is actually 16 + -20 + 6, which can also be rewritten as 16 + 6 - 20 (commutative property and definition of subtraction in algebra: 

a - b = a + -b). This explanation is not necessary at the 3^rd grade level. 

Practice equations like 12 - 14 + 6 = ❏, first. 

Before this, students should know what I call the “add in any order” property of addition, which I taught early in first grade. For example, to find the sum of  3 + 12 + 7, students recognize a “make 10” combination (3 + 7), add them first, and then add 12. Thus, 10 + 12 = 22, using mental arithmetic. Students should work extensively on “10 more than a number.” 

  

FYI: This equation is quadratic because box is squared. In algebra, the equation is written x² - 5x + 6 = 0 or y = x² - 5x + 6. By setting y equal to zero, the x values that work are the x-intercepts. Note the graph shape of the quadratic equation is a parabola. The x-intercepts are at x = 2 or x = 3. These are called the roots of the equation. 

I tell students there are two secrets. If you figure out the secrets, then you can look at the equation and figure out the two roots in 10 to 15 seconds. In the set of real numbers, quadratic equations can have zero solutions (no solution in real numbers), one solution (called a double root), or two distinct solutions.  

(❏ x ❏) - (2 x ❏) + 1 = 0

This equation has one solution (called a double root). The parabola intersects (bounces off) the x-axis once. 

I wrote quadratic equations in this form so that students can figure out two secrets. The idea is from the Madison Project. The equations are constructed so that the roots are integers. The idea of solving quadratic equations in this form in elementary school comes from the Madison Project, 1957. However, at this stage, the emphasis is finding the two secrets, applying the secrets, and practicing arithmetic.

There is a logical sequence of equations, step-by-step, that leads to each one of these five equations. Also, students should practice a lot. For additional information about equations, e-mail LT at ThinkAlgebra@cox.net.

LT

Visit my website: ThinkAlgebra

11-17-2010 Draft 1

Additional information added on 12-31-2010, 4-30-17