27.1 Early Algebra: Patterns, Equality, and Unknowns

27.1 Early Algebra: Patterns, Equality, and Unknowns

Quick answer: PECT 0012.7 (Module 3 — Mathematics, Science, and Health, 100% of Module 3 together with 0013–0014) groups algebra, geometry, and measurement. This section is the algebra portion: repeating and growing patterns, equality as balance (not "the answer comes after ="), unknowns/boxes, properties as algebraic thinking, and function tables at Grade 3–4 grain. Contrast 3 + ? = 10 with 3 + 7 = ?. Pearson 0012.5 already covered prenumeracy pattern play. Here the exam scores relational thinking.

Pennsylvania teachers plan from PA Core Standards for Mathematics, standard area CC.2.2 Algebraic Concepts, and, in PreK–Grade 2, Mathematical Thinking and Expression in the 2024 Pennsylvania Learning Standards for Early Childhood. 0012.7 is not Algebra I. It scores whether you can put true equations, open number sentences, and growing patterns in children's hands — and whether you can catch the child who thinks equals means compute.

PA Core files algebra as its own area (CC.2.2). Geometry is CC.2.3. Measurement sits with data in CC.2.4. Pearson still binds algebra to geometry and measurement on 0012.7. Name the content the stem asks for; do not fight the grouping.

Patterns that do algebraic work

0012.5 asked children to recognize, create, and expand patterns. 0012.7 algebra asks what the pattern is doing — a core, a rule, a prediction you can defend.

Exam trap: A pretty ABAB border with no "what comes next, and why?" is decoration, not algebra. A growing pattern taught as keep adding pretty things is still 0012.5 play.

Repeating-pattern core. In red-blue-red-blue, the core is AB (red-blue). Children who only chant colors often fail when you ask for the 9th cube. Children who can say it goes in twos can find it. That is algebraic structure, still with cubes.

Growing patterns. A PreK staircase of one cube, then two, then three is a growing pattern children can build. Grade 3–4 should name the change (each step is one more) and extend it. Do not require recursive formulas, function notation, or slope.

Equality as balance — the equals-sign trap

This is the highest-yield algebra misconception in elementary teaching — and a likely PECT distractor family.

Many children treat = as now write the answer — an operator that means compute — instead of a relation that means the two sides have the same value. That reading is fed by a diet of problems that always look like 3 + 7 = ? with the unknown parked on the right.

Instructional response at PreK–4 grain:

1. A pan balance or two sides of cubes. 3 and a bag on the left; 10 on the right. What is in the bag?
2. True/false equations as daily number talk: Is 6 = 6 true? Is 5 + 3 = 8 true? Is 5 + 3 = 7 + 1 true? Is 5 + 3 = 53 false?
3. Open number sentences with the unknown on either side, including ? = 4 + 5, so children must not wait until after the equals sign.
4. Read = as is the same as, not as a green light to calculate.

Worked classroom example. A Grade 2 child writes 8 + 4 = 12 + 5 = 17. Do not only mark it wrong and assign more a + b = ? facts. Put 8 + 4 on a balance against 12 + 5. The sides are not the same: 12 is not 17. Rebuild as 8 + 4 = 12 and, separately, 12 + 5 = 17, or as 8 + 4 = 7 + 5 if the goal is relational thinking (8 is one more than 7; 4 is one less than 5). The error is not they cannot add. The error is what equals means.

3 + ? = 10 vs 3 + 7 = ? Keep both. The first is an unknown addend — algebra as missing-part thinking. The second is a result unknown — needed arithmetic that, alone, rebuilds the compute myth. A class that never sees an unknown on the left will look fluent and still fail 0012.7.

Unknowns, boxes, and properties as algebraic thinking

An unknown in PreK–4 is a missing number, a box, a blank, or later a letter standing for one number in this sentence. It is not solve for x, check extraneous roots, or graph a line.

• PreK–K: I am hiding some bears. You see 3. There are 5 in all. How many are hiding?
• Grade 1–2: 3 + ? = 10 and 10 = 3 + ? and 7 + 5 = ? + 6
• Grade 3–4: n × 4 = 20; same number in both boxes in ☐ + ☐ = 10 when the boxes must match

Properties are algebraic structure, not posters to chant and forget.

If a stem offers FOIL or the quadratic formula, it is the wrong exam. If it offers children explaining why 8 + 4 = 7 + 5 without computing either side first, that is relational thinking — 0012.7 algebra.

Function tables at Grade 3–4 grain

A function table (in/out, what's my rule?) is the elementary picture of a function: each input has one output, and a rule connects them.

Worked example.

Children who only memorize pairs cannot fill 8. Children who say double, times 2, or add itself can. The algebraic object is the rule, not the completed cells. Extend: if in is 0, out is 0; if in is 10, out is 20. Do not require y = 2x notation in Grade 3, though a teacher may write a words-rule: out is two times in. A machine that sometimes doubles and sometimes adds 5 is not a function at this grain — call that a broken machine and repair the rule.

Developmental grain

Scenarios

PreK. Ms. Patel hides 2 of 5 bears. Children show with fingers what is in the cave. Equality as parts and whole, not a written equation yet.

Grade 2. Mr. Diaz posts 7 + 3 = 10, 7 + 3 = 8 + 2, and 7 + 3 = 73. Children vote true/false and prove with cubes. The third is the equals-means-mash-the-digits trap.

Grade 4. Ms. Cho's class uses an in/out machine, then justifies 6 × 8 = 6 × 5 + 6 × 3 with an array — distributive property as algebra, not as a trick.

English learners and children with IEPs still do this work. Balance pans, cubes, and same / not same cards keep the target. A calculator may compute a sum; it does not teach what = means. Skipping true/false equations because a child isn't ready for algebra is not an accommodation (0012.2).

Exam traps for 0012.7 algebra

• Treating = as compute.
• Only a + b = ? items, never a + ? = c or ? = a + b.
• Calling a decorative repeating border algebra with no core or prediction.
• Dumping Algebra I (slope, quadratic, isolate x as the whole program).
• Teaching properties as posters children cannot use.
• Confusing this section with 0012.5 prenumeracy (sorting and AB play with no structure talk) or 0012.6 (operations as calculation only).
• Function tables as memorize the pairs, not name the rule.

Plan: Children will use [balance, cubes, table] to show that [two quantities are the same / a pattern has this rule / this unknown makes both sides balance]. If the unknown always sits after equals, you are still teaching arithmetic, not 0012.7 algebra.